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On Electronic Representations in Molecular Reaction Dynamics

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PAGE 1

ON ELECTR ONIC REPRESENT A TIONS IN MOLECULAR REA CTION D YNAMICS By BENJAMIN J. KILLIAN A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2005

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Cop yrigh t 2005 b y Benjamin J. Killian

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This work is de dic ate d to Br e ana, A ntonia, and Blake Jones. May the love and wonder of scienc e always b e ne ar.

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A CKNO WLEDGMENTS I am v ery grateful for the en vironmen t of the Quan tum Theory Pro ject and its mem b ers. The QTP is a w onderful place of education where con tin ual discussion and strong science foster in tellect and curiosit y I wish to express tremendous gratitude to Dr. Remigio Cabrera-T rujillo for his hours of discussion, correction, and teac hing throughout m y graduate education, as w ell as the help and friendship oered b y the other END Researc h Group mem b ers with whom I collab orated, Dr. Mauricio Coutinho-Neto, Dr. Anatol Blass, Dr. Da vid Masiello, Dr. Denis Jacquemin, Dr. Sv etlana Malino vsky a, and Mr. Virg F ermo. I oer sp ecial ac kno wledgmen t to Prof. Yngv e Ohrn and Dr. Erik Deumens for their patien t and masterful da y-to-da y teac hing and men toring in the eld of quan tum dynamics. My grateful thanks are giv en to m y mother and father, Jane and James Killian, for alw a ys encouraging me to striv e in ev ery endea v or and for supp ort throughout the long education pro cess. Without a paren t's motiv ation and lo ving supp ort, v ery little can b e truly accomplished in a p erson's life. Finally and most imp ortan tly I wish to express m y utmost lo v e and gratitude to m y wife, Donna, for her understanding, compassion, and emotional supp ort throughout our life together. Without her dev oted lo v e and encouragemen t, h me j Ph : D : i = 0. iv

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T ABLE OF CONTENTS pageA CKNO WLEDGMENTS. . . . . . . . . . . . . . . ivLIST OF T ABLES. . . . . . . . . . . . . . . . . viiLIST OF FIGURES. . . . . . . . . . . . . . . . viiiABSTRA CT. . . . . . . . . . . . . . . . . . xiCHAPTERS 1 INTR ODUCTION. . . . . . . . . . . . . . 11.1 Pr ecis of Quan tum Dynamics. . . . . . . . . 31.2 Solving the Sc hr odinger Equation. . . . . . . . 41.3 Time-Dep enden t Hartree-F o c k (TDHF). . . . . . 61.4 Electron-Nuclear Dynamics (END). . . . . . . . 81.4.1 The END Equations of Motion. . . . . . . 91.4.2 Minimal Electron-Nuclear Dynamics. . . . . 102 THEOR Y OF COLLISIONS. . . . . . . . . . . 122.1 Scattering Theory. . . . . . . . . . . . 122.1.1 Derection F unctions and Scattering Angles. . . 132.1.2 Cross Sections. . . . . . . . . . . 152.1.3 Iden tical P articles. . . . . . . . . . 172.1.4 Reference F rame T ransformations. . . . . . 182.2 Quan tum Mec hanical T reatmen t of Scattering Phenomena. 222.2.1 The In tegral Equation and its Relation to the Scattering Amplitude. . . . . . . . . . 222.2.2 The Born Series. . . . . . . . . . . 242.3 Semi-Classical T reatmen t of Scattering Phenomena: The Sc hi Appro ximation. . . . . . . . . . . . 262.3.1 Sc hi Scattering Amplitude for Large Angles. . . 272.3.2 Sc hi Scattering Amplitude for Small Angles. . . 343 BASIS SETS F OR D YNAMICAL HAR TREE-F OCK CALCULATIONS. . . . . . . . . . . . . . . . 413.1 The Hartree-F o c k Appro ximation. . . . . . . . 423.1.1 P artitioning of the Molecular W a v e F unction. . . 42 v

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3.1.2 The Hartree-F o c k W a v e F unction . . . . . . 45 3.1.3 Solving the HF Equations: Basis Set Expansions . 49 3.2 General F orms and Prop erties of Basis Sets . . . . . 52 3.2.1 Slater-T yp e and Gaussian-T yp e Orbitals . . . . 52 3.2.2 The Structure of Basis Sets . . . . . . . 56 3.3 Metho d for Constructing Basis Sets Consisten t with Dynamical Calculations . . . . . . . . . . . . 63 3.3.1 Basis Set Prop erties for Dynamical Calculations . 64 3.3.2 Ph ysical Justication for the Basis Set Construction Metho d . . . . . . . . . . . . 66 3.3.3 Construction of the Basis Set . . . . . . . 71 3.4 Comparativ e Results . . . . . . . . . . . 82 3.4.1 A tomic Energetics . . . . . . . . . . 83 3.4.2 Charge T ransfer Results . . . . . . . . 87 3.4.3 Prop erties of Diatomic and T riatomic Molecules . 98 4 VECTOR HAR TREE-F OCK: A MUL TI-CONFIGURA TIONAL REPRESENT A TION IN ELECTR ON-NUCLEAR D YNAMICS 104 4.1 In tro duction of the Lagrangian and V erication of the Equations of Motion . . . . . . . . . . . . 104 4.2 P arameterization of the State V ector . . . . . . . 106 4.3 The VHF Equations of Motion . . . . . . . . 114 4.3.1 Deriv ation of the Equations of Motion . . . . 115 4.3.2 Ev aluating the Equations of Motion . . . . . 120 4.4 Implemen tation of the V ector Hartree-F o c k Metho d . . 140 5 CONCLUSION . . . . . . . . . . . . . . 147 APPENDIX BASIS SET LIBRAR Y . . . . . . . . . . . 149 REFERENCES . . . . . . . . . . . . . . . . . 168 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 173 vi

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LIST OF T ABLES T able page 3.1 Notation emplo y ed in this review. . . . . . . . . . . 42 3.2 Slater Exp onen ts and Co ecien ts for He, Li, and Be. . . . . . 79 3.3 Slater Exp onen ts and Co ecien ts for B, C, and N. . . . . . 80 3.4 Slater Exp onen ts and Co ecien ts for O, F, and Ne. . . . . . 81 3.5 A tomic energies and electronic excitations in Helium . . . . . 84 3.6 A tomic energies and electronic excitations in Lithium . . . . 85 3.7 Molecular prop erties of the nitrogen molecule. . . . . . . . 99 3.8 Molecular prop erties of the carb on mono xide molecule. . . . . 100 3.9 Molecular prop erties of the w ater molecule. . . . . . . . 102 vii

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LIST OF FIGURES Figure page 2.1 Diagram of a classical collision system. . . . . . . . . 13 2.2 Three p ossible derection angles. A demonstrates a p ositiv e derection angle. B demonstrates a negativ e derection angle. C demonstrates a derection angle with magnitude greater than 2 . . . . . 14 2.3 Collision of distinguishable particles. . . . . . . . . . 17 2.4 Collision of iden tical particles. . . . . . . . . . . . 17 2.5 Relation of collision pair and cen ter-of-mass to origin. . . . . 19 2.6 Relation of the scattering angles b et w een reference frames. . . . 19 3.1 Comparison of STO and GTO represen tations of the radial part of the h ydrogen 1 s orbital. T op: A single GTO function t to a single STO function. Bottom: A linear com bination of six GTO functions t to a single STO function. . . . . . . . . . 55 3.2 T op: Plot of the radial distribution function for the 1 s (|), 2 s (), 3 s ( ), and 4 s (-) orbitals of the h ydrogen atom. Bottom: Plot of the radial distribution function for the 2 p (|), 3 p (-), and 4 p ( ) orbitals of the h ydrogen atom. . . . . . . 69 3.3 T op: Plot of the radial distribution function for the 1 s (|), 2 s (), and 3 s ( ) orbitals of the argon atom. The 1 s -orbital is scaled b y a factor of t w o-thirds. Bottom: Plot of the radial distribution function for the 2 p (|) and 3 p (-) orbitals of the argon atom. 70 3.4 Plot of the 1 s orbital exp onen t for the atoms through Kr as a function of atomic n um b er. The data are from Clemen ti and Raimondi (+). . . . . . . . . . . . . . . . . . 73 3.5 Plot of the 2 s and 2 p orbital exp onen ts for the atoms through Kr as a function of atomic n um b er. T op: The 2 s orbital exp onen ts. Bottom: The 2 p orbital exp onen ts. The data are from Clemen ti and Raimondi (+) and from the presen t w ork ( ). . . . . . . 74 viii

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3.6 Plot of the 3 s and 3 p orbital exp onen ts for the atoms through Kr as a function of atomic n um b er. T op: The 3 s orbital exp onen ts. Bottom: The 3 p orbital exp onen ts. The data are from Clemen ti and Raimondi (+) and from the presen t w ork ( ). . . . . . . 75 3.7 Plot of the 4 s and 4 p orbital exp onen ts for the atoms through Kr as a function of atomic n um b er. T op: The 4 s orbital exp onen ts. Bottom: The 4 p orbital exp onen ts. The data are from Clemen ti and Raimondi (+) and from the presen t w ork ( ). . . . . . . 76 3.8 Comparison of the probabilit y for near-resonan t c harge transfer b et w een H + and Li at 10 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02 (blac k). The BJK01 basis set for H w as used for eac h run. . . 89 3.9 Comparison of the probabilit y for near-resonan t c harge transfer b et w een H + and Li at 1 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02 (blac k). The BJK01 basis set for H w as used for eac h run. . . 90 3.10 Comparison of the dieren tial cross section for near-resonan t c harge transfer b et w een H + and Li at 10 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02 (blac k). The BJK01 basis set for H w as used for eac h run. . . . . . . . . . . . . . . . . . 91 3.11 Comparison of the total cross section for near-resonan t c harge transfer b et w een H + and Li as a function of energy The exp erimen tal data are: V arghese et al. (+) and Auma yr et al. ( ). The theoretical data are: Allan et al (|) and F ritsc h and Lin (-). F or the END data the follo wing Li basis sets are used: 6-31G ( 2 ), 6-31B ( ), BJK01 ( 4 ), BJK02 ( 3 ). The BJK01 basis set for H w as used for eac h run. . . . . . . . . . . . . . . . 92 3.12 Comparison of the probabilit y for resonan t c harge transfer b et w een Li + and Li at 1 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), BJK02 (blue). . . . . . . . . 93 3.13 Comparison of the probabilit y for resonan t c harge transfer b et w een Li + and Li at 10 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), BJK02 (blue). . . . . . . . . 94 3.14 Comparison of the dieren tial cross section for resonan t c harge transfer b et w een Li + and Li at 1 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), BJK02 (blue). . . . . . 95 ix

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3.15 Comparison of the total cross section for resonan t c harge transfer b et w een Li + and Li at as a function of collision energy The exp erimen tal data are form Loren ts et al. ( ). The theoretical data are from Sak ab e and Iza w a (|). F or the END data the follo wing Li basis sets are used: 6-31G ( 2 ), 6-31B ( ), BJK02 ( 4 ). . . . 96 3.16 Comparison of the probabilit y for resonan t c harge transfer b et w een He + and He at 5 k eV collision energy The follo wing He basis sets are used: 6-31G** (red), 6-31B** (blue), and BJK01 (purple). . 97 3.17 Comparison of the dieren tial cross section for resonan t c harge transfer b et w een He + and He at 5 k eV collision energy The exp erimental data are from Gao et al. ( ). F or the END data the follo wing He basis sets are used: 6-31G** (red), 6-31B** (blue), and BJK01 (purple). . . . . . . . . . . . . . . . . 98 4.1 Structural ro w c hart for minimal END electronic calculations. . . 141 4.2 Structural ro w c hart for V ector Hartree-F o c k END electronic calculations. . . . . . . . . . . . . . . . . . 142 x

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y ON ELECTR ONIC REPRESENT A TIONS IN MOLECULAR REA CTION D YNAMICS By Benjamin J. Killian August 2005 Chair: Nils Y. Ohrn Ma jor Departmen t: Chemistry F or man y decades, the eld of c hemical reaction dynamics has utilized computational metho ds that rely on p oten tial energy surfaces that are constructed using stationary-state calculations. These metho ds are t ypically dev oid of dynamical couplings b et w een the electronic and n uclear degrees of freedom, a fact that can result in incorrect descriptions of dynamical pro cesses. Often, non-adiabatic coupling expressions are included in these metho dologies. The Electron-Nuclear Dynamics (END) formalism, in con trast, circum v en ts these deciencies b y calculating all in termolecular forces directly at eac h time step in the dynamics and b y explicitly main taining all electronic-n uclear couplings. The purp ose of this w ork is to oer t w o new framew orks for implemen ting electronic represen tations in dynamical calculations. Firstly a new sc hema is prop osed for dev eloping atomic basis sets that are consisten t with dynamical calculations. T raditionally basis sets ha v e b een designed for use in stationary-state xi

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calculations of the structures and prop erties of molecules in their ground states. As a consequence of common construction tec hniques that utilize energy optimization metho ds, the uno ccupied orbitals b ear little resem blance to ph ysical virtual atomic orbitals. W e dev elop and implemen t a metho d for basis set construction that relies up on ph ysical prop erties of atomic orbitals and that results in meaningful virtual orbitals. These basis sets are sho wn to pro vide a signican t impro v emen t in the accuracy of calculated dynamical prop erties suc h as c harge transfer probabilities. Secondly the theoretical framew ork of END is expanded to incorp orate a m ulti-congurational represen tation for electrons. This formalism, named V ector Hartree-F o c k, is based in the theory of v ector coheren t states and utilizes a complete activ e space electronic represen tation. The V ector Hartree-F o c k metho d is fully disclosed, with deriv ation of the equations of motion. The expressions for the equation of motion are deriv ed in full and a plan for implemen ting the V ector Hartree-F o c k formalism within the curren t end yne computer co de is giv en. xii

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CHAPTER 1 INTR ODUCTION Chemistry is the study of ho w atoms and molecules c hange with time, and fundamen tally requires a dynamical description to correctly describ e these c hanges. Exp erimen tal ph ysical c hemistry is concerned with measuremen t of c hemical c hanges and the en umeration of the prop erties asso ciated with these c hanges. Quan tum c hemistry on the other hand, seeks to describ e the mec hanism through whic h c hemical c hanges o ccur and to calculate prop erties asso ciated with these c hanges as a function of fundamen tal prop erties of the molecule itself. Virtually all of c hemistry is describ ed in the v o cabulary of dynamics. Chemists sp eak of electron transfers, transition states, equilibria. The most fundamen tal measuremen ts made in a ph ysical c hemistry lab oratory in v olv e transference of heat o v er time, the n um b er of vibrations a molecule undergo es in a unit of time, the in v ersion of electron p opulations as a function of time. Chemistry is dynamic. Despite all of this, non-dynamical approac hes are still most often emplo y ed when p erforming c hemical calculations. These appro ximations are often sucien t for obtaining a v erage prop erties, but fail to oer a deep er understanding of c hemical pro cesses that comes through dynamical metho ds. T raditionally c hemists ha v e b een in terested in answ ering t w o general classes of questions regarding c hemical reactions. The rst, \T o what exten t will a reaction o ccur?" is purely thermo dynamic in nature. Pro vided the dierence in the free energy b et w een the pro duct sp ecies and the reactan t sp ecies is negativ e, the reaction has the p oten tial to o ccur sp on taneously to some exten t. Calculations of this t yp e are w ell-suited to time-indep enden t quan tum mec hanics; stationary state 1

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2 energy v alues can b e calculated for v arious ground and excited states, whic h can yield (through the application of statistical thermo dynamical principles) the free energies of the pro ducts and the reactan ts. The second class of question that a c hemist migh t ask is, \Ho w fast will the reaction o ccur?" This t yp e of question is kinetic in nature and relies up on some (usually detailed) kno wledge of the dynamics of the reaction. This t yp e of question is, in generally only successfully answ ered using time-dep enden t quan tum mec hanical treatmen ts. T o correctly describ e the kinetics of a reaction, one m ust rst describ e the in teractions through and resp onses to in terand in tramolecular forces as a function of the time-scale of the reaction. The fo cus of this dissertation is to in v estigate metho ds of impro ving representations of electrons for use in time-dep enden t dynamical calculations. Sp ecically this w ork will in v estigate t w o sides of the v ery same coin. Firstly a discussion will b e made to w ard basis set expansion for use in construction of the electronic w a v e function. A new metho d will b e prop osed for the construction of dynamically consisten t basis sets. Tw o particular features of the prop osed construction metho d are the ph ysical basis that underlies the metho d and its ease of application. Secondly discussion will b e made to w ard the construction of a m ulti-congurational w a v e function for use with the Electron-Nuclear Dynamics formalism. The full set of equations of motion are deriv ed for the V ector Hartree-F o c k implemen tation of Electron-Nuclear Dynamics in terms of a general expansion of atomic basis functions. Additionally discussion is made as to a p ossible sc heme for implemen tation. This c hapter pro vides a basic review of quan tum molecular dynamics. Subsequen tly a brief discussion is made of the general structure of the Time-Dep enden t Hartree-F o c k metho d. Finally the Electron-Nuclear Dynamics formalism in its simplest form is in tro duced and discussed and compared to Time-Dep enden t Hartree-F o c k.

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3 1.1 Pr ecis of Quan tum Dynamics The ph ysical description of an y c hemical ob ject is predicated on the concepts of a quan tum mec hanical state and a quan tum mec hanical conguration [ 1 ]. A quan tum mec hanical conguration is the set of all descriptiv e v ariables in the dynamical phase space (e.g., momen tum and p osition) and/or in the electronic Hilb ert space (e.g., orbital and spin angular momen tum) whic h describ es (within the completeness of the set) a giv en c hemical ob ject at an y giv en instan t in time, t = [ 1 ]. The conguration is usually represen ted as a k et in Dirac notation, written as j ; t = i where is a represen tativ e of the required set of descriptiv e v ariables. If is sp ecically dened, then the conguration ma y b e more con v enien tly expressed as j i A quan tum mec hanical state, or w a v e-function, can also b e expressed in Dirac notation, j ; t i where t indicates the general dep endence on time. The state can b e dened as an y subset of the complete set of congurations that con tains a giv en conguration, j ; t = i as w ell as all congurations that result from the dynamical ev olution of j ; t = i and all congurations that ev olv e in to j ; t = i with the passage of time (assuming no p erturbation of the system b y outside inruences) [ 1 ]. Some descriptions m ust include the time dep endence explicitly in the w a v efunction, while others allo w for the time-dep endence to b e treated as separable factor, dep ending on the app earance of time in the Sc hr odinger Equation [ 2 ]. F or a state to b e completely describ ed, one m ust kno w the relation b et w een all v ariables that determine the state at a sp ecic time, usually denoted as time t = 0. In v estigation of c hemical pro cesses requires the kno wledge of ho w a giv en state ev olv es with time | ho w the state c hanges dynamically The principle to ol in the descriptions of quan tum c hemical dynamics is the Sc hr odinger W a v e Equation, a time-dep enden t dieren tial equation of second order spatially and rst order temp orally The Sc hr odinger Equation denes the equations of motion for a giv en

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4 quan tum mec hanical state; once the initial state is completely describ ed, then the time ev olution of the state as dened b y the state-sp ecic Sc hr odinger Equation allo ws one to correctly predict the future prop erties of the state based on its initial conditions. The state-sp ecic Sc hr odinger Equation can b e expressed in function notation as N X i =1 h 2 M i r 2i j ; t i + V ( ; t ) j ; t i = i h @ @ t j ; t i : (1.1) On the left hand side of Equation ( 1.1 ), the rst term is the kinetic energy of the state; the summation is o v er the N particles comprising the state, M i is the mass of the i th particle, and r 2i is the Laplacian of the i th particle. The second term on the left is the p oten tial energy of the state. The p oten tial energy of the state is generally a complicated function of sev eral or all of the dynamical v ariables and is of particular imp ortance to the description of quan tum dynamics. One can com bine these t w o terms and dene the action of these terms b y an op erator, ^ H called the Hamiltonian op erator. Th us, Equation ( 1.1 ) can also b e written in op erator form, as ^ H j ; t i = i h @ @ t j ; t i : (1.2) Equation ( 1.2 )is the set of equations of motion required to calculate the dynamics of a quan tum mec hanical state. 1.2 Solving the Sc hr odinger Equation The bulk of quan tum dynamical researc h is in v olv ed with devising and rening metho ds to solv e Equation ( 1.2 ). A molecular Hamiltonian op erator will tak e the form (in atomic units, where h = c = e = m = 1) ^ H = 1 2 K X =1 1 M r 2 1 2 N X i =1 r 2i + K X =1 K X = +1 Z Z r ; K X =1 N X i =1 Z r i;a + N X j =1 N X i = j +1 1 r i;j ; (1.3) where the indices and refer to elemen ts of the K n uclei and i and j lab el elemen ts of the N electrons. F urthermore, Z is the c harge of n ucleus and r ;

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5 r ;i and r i;j are the distance b et w een n uclei and the distance b et w een n ucleus and electron i and the distance b et w een electrons i and j resp ectiv ely The complexit y of the molecular Hamiltonian op erator renders the Sc hr odinger equation analytically in tractable for all but the simplest systems. F or this reason, certain appro ximations m ust b e made to simplify the system of equations. Man y metho ds treat the n uclear and electronic degree of freedom separately in what is usually referred to as the Born-Opp enheimer appro ximation. Born and Opp enheimer [ 4 ] rst in tro duced the concept of an adiabatic separation, in whic h the electronic and n uclear degrees of freedom are decoupled. In a simplistic description, the mass of the electrons, m is so small in comparison to the n uclei, M that the electronic motion is signican tly faster than the motion of the n uclei, allo wing one to p erform a separate electronic calculation at eac h n uclear conguration. The result of this appro ximation is that the electronic p oten tial energy can b e constructed at a large n um b er of n uclear congurations, resulting in a p oten tial energy surface that is a function of the n uclear co ordinates. A t this p oin t, the n uclear Hamiltonian is emplo y ed in whic h the p oten tial energy term is no w a sum of the Coulom b repulsion of the n uclei and the sp ecic electronic p oten tial energy that corresp ond to the giv en n uclear conguration. This appro ximation giv es rise to suc h metho ds as molecular dynamics (or quan tum molecular dynamics if the n uclei are quan tum). By emplo ying excited state surfaces, m ulti-surface metho ds arise, suc h as surface hopping metho ds. One immediate deciency of these Born-Opp enheimer-t yp e metho ds is the lac k of dynamical coupling b et w een the electrons and the n uclei. As a result, it is desirable to discuss metho ds in whic h no p oten tial energy surfaces are utilized. This requires direct calculation of the p oten tial energy b et w een n uclei and electrons at eac h time-step. Suc h metho ds will b e the fo cus of the remainder of this dissertation.

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6 Tw o sp ecic metho ds will b e discussed. First is the Time-Dep enden t HartreeF o c k metho d. Second is the Electron-Nuclear Dynamics formalism. Both metho ds are time-dep enden t metho ds that do not require the calculation of p oten tial energy surfaces. 1.3 Time-Dep enden t Hartree-F o c k (TDHF) The TDHF metho d w as rst prop osed b y Dirac in 1930 [ 5 ]. The principal assumption of the TDHF metho d is that the self-consisten t cen tral eld appro ximation holds. In this appro ximation (whic h is equiv alen t to the Hartree-F o c k appro ximation) the total electronic w a v e function is comp osed of a pro duct of w a v e functions for eac h electron. F urthermore, eac h electron mo v es in an a v erage p oten tial. The reference state for TDHF metho ds is the single determinan t Hartree-F o c k ground state, represen ted as a Slater determinan t (using second quan tization), j D i = a y1 :::a yN j v ac i : (1.4) The TDHF state v ector has the form j i = e i P r s r s ( t ) a yr a s j D i (1.5) whic h has a corresp onding Dirac densit y op erator of the form = e i P r s r s ( t ) a yr a s j i ih i j e i P r s r s ( t ) a yr a s : (1.6) The densit y op erator has a corresp onding matrix form that is idemp oten t ( 2 = ) and the trace is equal to the n um b er of electrons in the system [ 6 ]. The quan tum mec hanical Lagrangian tak es the form L = 1 2 h i h @ @ t j i + h j i h @ @ t i h j F j i ; (1.7)

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7 where F is the F o c k op erator, whic h has matrix elemen ts of the form F r s = h r s + 1 2 X pq h r s jj pq i q p (1.8) where h r s is an elemen t of the one-electron Hamiltonian matrix. The equations of motion for the TDHF metho d are deriv ed from the fact that the Lagrangian should b e stationary with resp ect to small v ariations in the TDHF state v ector, suc h that j i i X r s r s a yr a s j i ; (1.9) as w ell as a corresp onding v ariation in the densit y op erator. By requiring the Lagrangian to b e stationary under suc h an arbitrary v ariation, one obtains the TDHF equations of motion i = [ F ; ] ; (1.10) where the dot indicates the time deriv ativ e and the brac k ets denote the comm utation relation [ a; b ] = ab ba A t this p oin t, it is assumed that the time-ev olution will act as a small p erturbation to the F o c k op erator. The one-electron Hamiltonian will b e p erturb ed suc h that h r s h 0r s + h r s ; (1.11) where h 0r s is the one-electron Hamiltonian that corresp onds to the unp erturb ed HF stationary ground state. The second term con tains the time-dep endence. Secondly the t w o-electron comp onen t is p erturb ed through the densit y suc h that 0 + ; (1.12) where, again, the subscript indicates the HF stationary ground state solution and the time-dep endence is carried b y the p erturbativ e term. The ab o v e p erturbations are then substituted in to the TDHF equations of motion and, traditionally terms

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8 of order t w o or greater in the p erturbations are truncated. This implemen tation results in a metho d that is linearized and limits the electron-n uclear couplings within TDHF metho ds [ 7 ]. F urthermore, as b oth comp onen ts of the F o c k op erator dep end explicitly up on the electron densit y the equations of motion m ust b e solv ed iterativ ely at eac h time-step. Again, this is traditionally not done. In most TDHF metho ds, the reference state is determined as either a conguration in teraction (CI) expansion to the lev el of single excitations (resulting in the T amm-Dank o Appro ximation) [ 8 ] or as a p erturbation expansion to the lev el of single and some double excitations (resulting in the Random Phase Appro ximation) [ 9 ]. By allo wing the expansion co ecien ts to b e time-dep enden t, the time-ev olution of the individual orbitals is no w no longer required and the single reference is used throughout the dynamics [ 7 ]. This is also a limitation to the TDHF metho d, as the state is not p ermitted dynamically outside of the limit of the single conguration included in the HF reference state. 1.4 Electron-Nuclear Dynamics (END) The END theory is a non-adiabatic form ulation allo wing a complete dynamical treatmen t of electrons and n uclei that eliminates the need for p oten tial energy surfaces that is equiv alen t to a generalized TDHF appro ximation. ElectronNuclear Dynamics is deriv ed from application of the time-dep enden t v ariational principle (TD VP) on a family of appro ximate state v ectors parameterized in terms of Thouless co ecien ts. The END formalism and application of END to v aried ph ysical problems has b een explained in detail in previous w orks [ 10 11 12 13 ]. A brief accoun t of the deriv ation of the END equations of motion will b e pro vided in this section.

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9 1.4.1 The END Equations of Motion The deriv ation b egins with the quan tum mec hanical action functional A = Z t 2 t 1 L dt; (1.13) where L is the quan tum mec hanical Lagrangian, dened as L = h j i d dt ^ H j i h j i : (1.14) Here denes a complete set of time-dep enden t parameters that describ e the particular c hoice of state v ector and ^ H is the quan tum mec hanical Hamiltonian op erator for the system. By requiring that the action remain stationary under v ariations of the parameters one obtains the Euler-Lagrange equations d dt @ L @ i = @ L @ i ; (1.15) for the set of dynamical v ariables f i g [ 14 ]. Application of the principle of least action results in a coupled set of rst-order dieren tial equations of motion 0B@ 0 i C i C 0 1CA 0B@ _ 1CA = 0B@ @ E @ @ E @ 1CA ; (1.16) where E ( ; ) = h j ^ H j i h j i (1.17) is the energy of the system and acts as the generator of innitesimal time translations and where C is an in v ertible metric matrix with elemen ts dened as C = @ 2 ln S @ @ : (1.18) The metric matrices dene the couplings b et w een the dynamical v ariables. The argumen t of the logarithm in Equation ( 1.18 ) is the o v erlap, dened as S = h j i The equations of motion giv en in Equation ( 1.16 ) are exact; an y appro ximations

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10 to the END form ulation arise due to the c hoice of dynamical parameters and the completeness of the electronic basis set. 1.4.2 Minimal Electron-Nuclear Dynamics The minimal END theory is implemen ted in the end yne computer co de. In the curren t v ersion, n uclei are treated classically and electrons are represen ted using single Thouless determinan ts that are complex and single v alued [ 15 ]. All electron-n uclear couplings are retained. The dynamical parameters are c hosen as = f R k ; P k ; z ph ; z ph g where R k and P k are the a v erage p osition and momen tum of the k th n ucleus and z ph and z ph (spanned b y a set of atomic orbitals) are the Thouless co ecien t corresp onding to the p th atomic orbital of the h th spin orbital and its complex conjugate, resp ectiv ely Cho osing the state to b e represen ted as a pro duct coheren t state v ector j i = j z ; R ij R ; P i j z ij i (1.19) where the n uclear w a v e function is expressed as a pro duct of tra v eling Gaussians j i = Y k exp 1 2 X k R k w k 2 + iP k ( X k R k ) # (1.20) tak en in the narro w w a v e pac k et limit ( w k 0, for all w k ), and where the electronic state is represen ted with a single determinan t electronic w a v e function j z i = det f h ( x p ) g ; (1.21) allo ws for a consisten t description of the electron-n uclear dynamics. The molecular spin orbitals in Equation ( 1.21 ), h are spanned b y a set of basis functions, f u h ( x ) g that are Gaussian-t yp e atomic orbitals (GTOs) cen tered on the a v erage

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11 n uclear p ositions. By making these assumptions, Equation ( 1.16 ) tak es the form 0BBBBBBB@ i C 0 i C R i C P 0 i C i C R C P i C yR i C TR C RR I + C RP i C yP i C TP I + C PR C PP 1CCCCCCCA 0BBBBBBB@ z z R P 1CCCCCCCA = 0BBBBBBB@ @ E =@ z @ E =@ z @ E =@ R @ E =@ P 1CCCCCCCA ; (1.22) where the dynamic metric elemen ts of Equation ( 1.22 ) are ( C X Y ) ij ; k l = 2 Im @ 2 ln S @ X ik @ Y j l R 0 = R ; (1.23) ( C X ik ) ph = @ 2 ln S @ z ph @ X ik R 0 = R ; (1.24) and C ph ; q g = @ 2 ln S @ z ph @ z q g R 0 = R : (1.25) The coupling is explicit in the metric elemen ts giv en in Equations ( 1.23 ) ( 1.25 ), n uclear-n uclear coupling, non-adiabatic n uclear-electronic coupling, and pure electronic-electronic coupling, resp ectiv ely Minimal Electron-Nuclear Dynamics is a generalization of the TDHF metho d [ 10 ]. The electronic and n uclear in teractions are still regulated through the F o c k matrix, but there are some dierences in the dynamical ev olution of the state. Firstly the v ariation of the one-densit y in the END formalism is ac hiev ed through a general v ariation of the Thouless co ecien ts and is not truncated at an y particular order. As a result, END is equiv alen t to fully non-linearized TDHF. Secondly the reference state is p ermitted to c hange as w arran ted b y the dynamics of the system. This pro vides additional rexibilit y to the dynamical ev olution, as the reference state is not limited to a stationary state as calculated previous to the dynamical ev olution.

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CHAPTER 2 THEOR Y OF COLLISIONS Man y c hemical reaction principles and results can b e elucidated and obtained b y virtue of so-called \scattering exp erimen ts". The exp erimen ts most generally in v olv e a b eam of pro jectile sp ecies inciden t up on a reaction cell con taining target sp ecies or up on a second b eam of target sp ecies. The reactions and energy transferences o ccur in the v olume of the reaction cell or the union v olume of the crossed b eams. Theoretical descriptions of suc h scattering phenomena are con tained in the classical, quan tal, and semi-classical realms of collision theory 2.1 Scattering Theory While exp erimen tal results are obtained for bulk phase reactions in general, excellen t theoretical descriptions of these scattering pro cesses can b e made b y considering the in teraction of a single pro jectile particle with a single target particle. A t it's simplest, classical scattering theory in v olv es the reduction of a t w o-b o dy problem in to a reduced one-b o dy problem. This transformation results in a description of the motion in the cen ter-of-mass reference frame. This t yp e of analysis is similar to the analysis of un b ound Kepler motion, ho w ev er, the nature of the cen tral force is considerably dieren t (though the functional form ma y b e v ery similar) and all information ab out the orbits is lost (only the inciden t momen tum, the nal momen tum, and the angle b et w een the t w o is observ ed) [ 16 ]. In this section, w e will discuss the fundamen tals of classical scattering theory and metho ds to extend these fundamen tals in to the language of atomic and molecular collisions, quan tum scattering theory 12

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13 2.1.1 Derection F unctions and Scattering Angles One b egins b y considering a classical collision system consisting of t w o p oin t masses in the lab oratory frame with some undened p oten tial acting b et w een them that is a function only of the separation b et w een the particles. It will b e assumed that, initially one particle is in motion (the pro jectile) and one is a rest (the target). Figure 2.1 demonstrates suc h a collision system. The scattering axis is dened as the axis parallel to the inciden t motion of the pro jectile and con taining the lo cation of the target, as demonstrated b y A in Figure 2.1 The impact parameter, lab eled as b is dened as the distance at whic h the pro jectile is initially lo cated in a direction p erp endicular to the scattering axis. As one can assume a spherically symmetric p oten tial, V ( r ), this can b e in an y direction p erp endicular to the scattering axis. The other principle feature of Figure 2.1 is the derection angle, lab eled as The derection angle is the angle through whic h the pro jectile is derected b y the p oten tial, as measured in a coun ter-clo c kwise manner from the p ositiv e scattering axis. The derection angle can b e p ositiv e due to a repulsiv e p oten tial (as demonstrated in A in Figure 2.2 ), or it can b e negativ e due to an attractiv e p oten tial (see B in Figure 2.2 ). F urthermore, under certain conditions, the pro jectile can orbit b A Y Target Projectile Figure 2.1: Diagram of a classical collision system.

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14 Y > 0 Y < 0 B. A. C. Y > 2p Figure 2.2: Three p ossible derection angles. A demonstrates a p ositiv e derection angle. B demonstrates a negativ e derection angle. C demonstrates a derection angle with magnitude greater than 2 the target for one or more p erio ds, resulting in a derection with magnitude greater than 2 radians (see C in Figure 2.2 ). It should b e noted that, exp erimen tally it is imp ossible to distinguish b et w een the three derections sho wn in Figure 2.2 Exp erimen t can only determine the angle at at whic h the pro jectile is scattered, relativ e to the scattering axis, and cannot elucidate whether a particle is scattered through a p ositiv e derection, a negativ e derection, or through an orbiting derection with magnitude greater than 2 radians. Th us, w e m ust dene a parameter that corresp onds to what is ph ysically measured; a parameter called the scattering angle. The scattering angle, denoted

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15 is dened as = j mo d 2 j : (2.1) F or an y scattering system there exists a mapping that relates the impact parameter to a resulting scattering. This mapping is called the derection function, ( b ). It is ob vious from the previous paragraph that the derection function is not an injectiv e mapping, as more than one impact parameter ma y result in the same scattering angle. F urthermore, the derection function is not a surjectiv e mapping, as there is no guaran tee that the pro jectile will b e scattered in to ev ery angle b et w een 0 and 2.1.2 Cross Sections While reaction rates are usually the quan tities desired from \scattering exp erimen ts", the fundamen tal observ able of suc h exp erimen ts is the dieren tial cross section [ 17 ]. The dieren tial cross section ( d =d n) is dened as d d n = scattered curren t p er unit solid angle inciden t curren t p er unit area : (2.2) A t this p oin t it b ecomes necessary to assume some limited quan tum nature of the scattering system. Without further justication at this p oin t in the discussion, w e will in tro duce the general form of the scattering w a v e function. It is customary to c ho ose the ansatz w a v e function for the target b eam b y considering the asymptotic regions where the eect of the scattering p oten tial pro duced b y the target (assumed to b e nite) is negligible. In the p ost-scattering region, the scattered w a v e function will b e a linear com bination of inciden t plane w a v e and scattered spherical w a v es [ 18 ], e ik i r + f ( ; ) e ik f r r ; (2.3) where k i and k f are the magnitudes of the initial and nal momen ta of the projectile, resp ectiv ely Both the n umerator and the denominator in Equation ( 2.2 )

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16 require ev aluation of the probabilit y curren t densit y j for the w a v e function, dened as j = 1 2 mi ( r r ) ; (2.4) where m is the mass of the pro jectile. It should b e noted that v ectors are denoted using b oldfaced letters, while resp ectiv e magnitudes are represen ted using nonb oldfaced letters. The n umerator of Equation ( 2.2 ) is found from the curren t densit y of the outgoing spherical comp onen t b y the expression j out d A where d A is the unit dieren tial area normal to the solid angle subtended b y d n. F or this case, d A = r 2 d r It should b e noted that, due to the area, only the r comp onen t of j out is required, greatly simplifying the calculation. The denominator of Equation ( 2.2 ) is lik ewise found considering the curren t densit y of the incoming plane w a v e, j in As all inciden t particles are considered, the denominator is just the incoming curren t densit y Th us, the the dieren tial cross section can no w b e expressed as d d n = j out d A j in = k f k i j f ( ; ) j 2 : (2.5) Th us, the dieren tial cross section dep ends only on the square of the amplitude of the scattered spherical w a v e. It should b e furthermore noted that a state-to-state dieren tial cross section ma y b e obtained if a probabilit y amplitude for transition is included in the scattering amplitude The second quan tit y of imp ort in \scattering exp erimen ts" is the total cross section, dened as = Z 2 0 Z 0 d d n sin d d: (2.6) The total cross section dep ends only up on the relativ e kinetic energy of the colliding particles and is an eectiv e area of scattering or reaction. If the pro jectile strik es within this eectiv e area it will b e scattered or reacted, ho w ev er, b ecause

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17 equivalent to is not A A A A B B q q B B Figure 2.3: Collision of distinguishable particles. of the a v eraging, the sp ecic information con tained within the dieren tial cross section is lost. 2.1.3 Iden tical P articles Consider the collision of t w o particles in the cen ter of mass frame (Figure 2.3 ). T o simplify the explanation, it will b e assumed that axial symmetry exists, eliminating a dep endence on Because eac h particle is distinguishable throughout the en tire collision pro cess, eac h can b e determined to scatter through either and angle of or an angle of Ho w ev er, a in teresting problem arises when iden tical particles are considered. Ev en if the particles are distinguishable at some time of separation, once the particles en ter the in teraction region they are no longer distinguishable. The particle scattered through angle is indistinguishable from the particle scattered through angle (Figure 2.4 ). equivalent to A A A q q A is A A A A Figure 2.4: Collision of iden tical particles.

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18 When considering iden tical particles, the ansatz w a v e function for the pro jectile no w tak es the form [ 19 ] e ik i r + [ f ( ) f ( )] e ik f r r ; (2.7) where the sign is determined b y whether F ermi-Dirac or Bose-Einstein statistics are emplo y ed. As the particles are iden tical, momen tum is conserv ed in the collision. F rom the same argumen t pro vided in Section 2.1.2 the dieren tial cross section for iden tical particles in the cen ter-of-mass frame is d d n = j f ( ) f ( ) j 2 : (2.8) 2.1.4 Reference F rame T ransformations One b enet of the END formalism is that calculations need not b e p erformed in the cen ter-of-mass frame of reference. This is b enecial as exp erimen tal results are generally rep orted in the lab frame. Ho w ev er, it w as demonstrated in the previous section that the calculation of the dieren tial cross section for the reaction of iden tical particles is b est handled in the cen ter-of-mass frame. Th us, a sc hema for transformation b et w een lab and cen ter-of-mass frames m ust b e dev elop ed. Consider the collision of t w o particles, as depicted in Figure 2.5 The rst particle (pro jectile) has a mass, m 1 the second particle (target) has mass, m 2 In the lab frame, the particles ha v e p ositions, r 1 and r 2 and momen ta, k 1 = m 1 r 1 and k 2 = m 2 r 2 (where the dot indicates dieren tiation with resp ect to time). In the frame relativ e to the cen ter-of-mass, the particles are dened b y p ositions, s 1 and s 2 and momen ta, p 1 = m 1 s 1 and p 2 = m 2 s 2 The cen ter-of-mass for the collision system is dened as ha ving mass, M = m 1 + m 2 p osition, R and momen tum, P = M R relativ e to the origin. It is eviden t from the denition of the system that the p osition of a particle in the lab frame diers from the p osition in the cen ter-of-mass frame, at an y time

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19 Origin 1 r r 2 s 1 s 2 R Figure 2.5: Relation of collision pair and cen ter-of-mass to origin. during the dynamics, b y the p osition of the cen ter-of-mass, r 1 = s 1 + R : (2.9) Dieren tiation of the ab o v e equation and m ultiplication b y m 1 yields the relation amongst momen ta, k 1 = p 1 + m 1 M P : (2.10) If one considers the momen ta for the pro jectile in the t w o reference frames after the collision, one ma y obtain the relationship b et w een the scattering angles in the t w o reference frames. The pro jectile p ossesses a momen tum k f1 in the lab frame p q a k f1 P 1 f mM 1 Figure 2.6: Relation of the scattering angles b et w een reference frames.

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20 and p f1 in the cen ter-of-mass frame. The sup erscript f indicates a nal condition. The pro jectile is scattered through an angle of in the lab frame and an angle of in the cen ter-of-mass frame (Figure 2.6 ). It is eviden t that the transv erse comp onen ts of the momen ta are equal, p f1 sin( ) = k f 1 sin( ) ; (2.11) and that the longitudinal comp onen ts dier b y the momen tum of the cen ter-ofmass, p f1 cos( ) = k f 1 cos( ) m 1 M P : (2.12) T ransformations b et w een the scattering angles are obtained b y taking the appropriate ratios of Equations ( 2.11 ) and ( 2.12 ). The cen ter-of-mass scattering angle is obtained from the expression tan( ) = sin( ) cos( ) m 1 M P k f 1 : (2.13) This generalized equation can b e m uc h simplied under t w o assumptions. If the particles are iden tical then m 1 M = 1 2 If the target is initially stationary in the lab frame, then the magnitude of the cen ter-of-mass momen tum is equal to the magnitude of the initial momen tum of the pro jectile ( k i i ) at an y time during the collision. Th us, the simplied expression tak es the form tan( ) = sin( ) cos( ) 1 2 r ; (2.14) where r = k i 1 k f 1 Lik ewise, the rev erse relation can also b e obtained b y in v erting the ratio. The lab frame scattering angle can b e obtained from the general expression tan( ) = sin ( ) cos( ) + m 1 M P p f1 : (2.15)

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21 Again, the expression can b e simplied. If the particles are assumed to b e iden tical, then (as b efore) m 1 M = 1 2 Additionally the collision is elastic in the cen ter-of-mass frame, requiring that p 1 = 1 2 P at all times in the course of the dynamics. Th us, tan( ) = sin ( ) cos ( ) + 1 ; (2.16) whic h indicates that = 1 2 Once the scattering angle is con v erted to the cen ter-of-mass frame, the derection function and the dieren tial cross section can b e calculated. The cen terof-mass frame dieren tial cross section m ust then b e transformed bac k to the lab frame. This is accomplished b y realizing that fact that the n um b er of particles scattered in to a giv en solid angle m ust b e conserv ed b et w een the t w o frames [ 20 ], suc h that d d n l ab sin ( ) j d j = d d n C M sin ( ) j d j ; (2.17) or d d n l ab = d d n C M d [cos ( )] d [cos( )] : (2.18) The m ultiplicativ e factor in Equation ( 2.18 ) is ev aluated b y rst applying the La w of Cosines to Figure 2.6 and substitution in to Equation ( 2.12 ), yielding cos ( ) = cos( ) + p 1 + 2 + 2 cos( ) ; (2.19) where = 1 2 P p f1 No w the deriv ativ e can b e tak en with resp ect to cos ( ), d [cos( )] d [cos( )] = cos( ) + 1 (1 + 2 + 2 cos( )) 3 = 2 : (2.20) As the n umerator ab o v e nev er go es to zero in the domain = [ ; ], Equation ( 2.20 ) can b e in v erted to yield the required factor, d [cos( )] d [cos( )] = (1 + 2 + 2 cos( )) 3 = 2 cos( ) + 1 : (2.21)

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22 The expression in Equation ( 2.21 ) is completely general to a collision pair. As b efore, simplications can b e made b y considering the particles to b e iden tical. As seen earlier, this assumption requires that = 1. Under this assumption, the transformation from the cen ter-of-mass frame to the lab frame tak es the form d d n l ab = 4 cos 2 d d n C M : (2.22) 2.2 Quan tum Mec hanical T reatmen t of Scattering Phenomena T o this p oin t the description of scattering phenomena has b een tacitly quan tal. The scattering w a v e function ansatz w as in tro duced and a discussion of the iden tical particle problem w as made, but no reference to the Sc hr odinger Equation has y et b een made. In this section, the dieren tial form of the scattering Sc hr odinger Equation will b e in tro duced, a deriv ation of the in tegral form of the Sc hr odinger Equation will b e made, and this in tegral form will b e related to the scattering amplitude. The Born Series and the v arious Born Appro ximations, a set of self-consisten t solutions to the in tegral form of the Sc hr odinger Equation is then in tro duced. Finally the Sc hi Appro ximations for large and small scattering angles, whic h are deriv ed from the Born Series, will b e fully deriv ed and discussed. 2.2.1 The In tegral Equation and its Relation to the Scattering Amplitude The time-indep enden t Sc hr odinger Equation h 2 2 m r 2 ( r ) + V ( r ) ( r ) = E ( r ) (2.23) can b e rewritten to tak e the form r 2 + k 2 ( r ) = U ( r ) ( r ) ; (2.24) where k = p (2 mE = h 2 ) is the magnitude of the momen tum v ector for the scattered particle and where U ( r ) = 2 mV ( r ) = h 2 is the scattering p oten tial. Equation ( 2.24 )

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23 is an inhomogeneous dieren tial equation with a solution [ 17 18 ] ( r ) = 0 ( r ) + Z G ( r ; r 0 ) U ( r 0 ) ( r 0 ) d r 0 ; (2.25) where 0 ( r ) and G ( r ; r 0 ) are, resp ectiv ely a general solution and the Green's F unction that corresp onds to the homogeneous coun terpart of Equation ( 2.24 ), whic h is of the form r 2 + k 2 0 ( r ) = 0 : (2.26) The ab o v e homogeneous dieren tial equation is nothing more than the free particle Sc hr odinger Equation, and the solution is 0 ( r ) = e ik r Additionally the Equation ( 2.26 ) is in the form of the Helmholtz Equation and therefore the corresp onding Green's F unction that satises Equation ( 2.25 ) tak es the form [ 14 ] G ( r ; r 0 ) = e ik j r r 0 j 4 j r r 0 j : (2.27) Th us, one nds that Equation ( 2.25 ) no w b ecomes, up on substitution, ( r ) = e ik r 1 4 Z e ik j r r 0 j j r r 0 j U ( r 0 ) ( r 0 ) d r 0 ; (2.28) whic h is the in tegral form of the Sc hr odinger Equation with scattering p oten tial lo cated at p osition r 0 It is no w desirable to nd an expression that relates the scattering amplitude to the in tegral form of the Sc hr odinger Equation. T o accomplish this, one returns to the p ost-scattering conditions in whic h the ansatz of Equation ( 2.3 ) is assumed to hold, namely that r r 0 Under this requiremen t, the angle b et w een r and r 0 approac hes zero and j r r 0 j r r 0 ^ r This appro ximation allo ws one to express the in tegral form of the Sc hr odinger Equation as ( r ) = e ik r 1 4 e ik r r Z e i k r 0 U ( r 0 ) ( r 0 ) d r 0 : (2.29)

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24 One can then immediately compare the mo died in tegral form of the Sc hr odinger Equation as giv en ab o v e with Equation ( 2.3 ), whic h demonstrates clearly that f ( ) = f ( k ) = 1 4 Z e i k r 0 U ( r 0 ) ( r 0 ) d r 0 : (2.30) 2.2.2 The Born Series Though one has no w seen the solution to the scattering w a v e function in Equation ( 2.28 ), the solution is analytically in tractable, as the in tegrand itself dep ends on Solution of the scattering w a v e function requires n umerical in tegration to self-consistency This self-consisten t metho d yields the Born Series, with truncations yielding the Born Appro ximations [ 19 21 ] of v arious orders. The First Born Appro ximation assumes that the scattering p oten tial has a negligible eect on the incoming plane w a v e, th us the scattered w a v e function tak es the form = e i k i r 1 ; (2.31) The subscript on r is for indexing purp oses only This term can b e substituted in to Equation ( 2.30 ), and the scattering amplitude in the First Born Appro ximation then b ecomes f ( k i ; k f ) = 1 4 Z e i k f r 1 U ( r 1 )e i k i r 1 d r 1 : (2.32) In some cases conditions are suc h that the First Born Appro ximation is sucien t for analysis, most notably when the scattering p oten tial is not v ery strong and its eectiv e range is quite small [ 19 ]. Ho w ev er, the First Born Appro ximation rarely pro vides adequate accuracy The Second Born Appro ximation is b egun b y iterating on the w a v e function. The scattered w a v e function for the second order of appro ximation no w b ecomes = e i k i r 1 + Z G ( r 1 r 2 ) U ( r 2 ) ( r 2 ) d r 2 : (2.33)

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25 Substitution of Equation ( 2.31 ) in to the ab o v e equation yields the w a v e function in the Second Born Appro ximation = e i k i r 1 + Z G ( r 1 r 2 ) U ( r 2 )e i k i r 2 d r 2 (2.34) whic h generates the second order scattering amplitude f ( k i ; k f ) = 1 4 Z e i k f r 1 U ( r 1 )e i k i r 1 d r 1 + Z e i k f r 1 U ( r 1 ) G ( r 1 r 2 ) U ( r 2 )e i k i r 2 d r 2 d r 1 : (2.35) This pattern can then b e con tin ued, and one can mak e successiv e iteration to an arbitrary degree and obtain the n th order w a v e function = e i k f r 1 + Z G ( r 1 r 2 ) U ( r 2 )e i k i r 2 d r 2 + Z Z G ( r 1 r 2 ) U ( r 2 ) G ( r 2 r 3 ) U ( r 3 )e i k i r 3 d r 3 d r 2 + : : : + + Z Z G ( r 1 r 2 ) U ( r 2 ) G ( r 2 r 3 ) U ( r 3 ) : : : G ( r n 1 r n ) U ( r n )e i k i r n d r n : : : d r 2 : (2.36) The innite Born Series is obtained b y letting n 1 (with careful reordering of the indices). The innite Born Series leads to a scattering amplitude with the general form f ( k i ; k f ) = 1 4 1 X n =1 Z Z e i k f r n U ( r n ) G ( r n r n 1 ) U ( r n 1 ) G ( r 2 r 1 ) U ( r 1 )e i k i r 1 d r n d r 1 : (2.37) No w, while Equation ( 2.37 ) is exact, it is v ery un wieldy to solv e and still dep ends up on the scattering p oten tial function, a prop ert y not utilized within nor obtained from END. F urthermore, the Born Series is often aicted with slo w or no con v ergence [ 2 ]. So it is desirable for a n um b er of reasons to express the Born

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26 Series in a simplied form, in particular a form that do es not dep end explicitly up on the scattering p oten tial. 2.3 Semi-Classical T reatmen t of Scattering Phenomena: The Sc hi Appro ximation The w ork in this section w as rst in tro duced b y L. I. Sc hi in 1956 [ 22 ]. This discussion is based on that pap er. The scattering p oten tial and the pro jectile particle can b e c haracterized b y sev eral ph ysical prop erties. The scattering p oten tial can b e describ ed b y V and R whic h are rough indications of the strength of the p oten tial and its eectiv e range, resp ectiv ely Lik ewise, the pro jectile is c haracterized b y its kinetic energy T ; its w a v e n um b er (magnitude of the momen tum), k ; its sp eed, v ; and its scattering angle, These c haracteristics pro vide a qualitativ e means b y whic h to discuss the scattering pro cess. F or example, the First Born Appro ximation is v alid under conditions when the scattered w a v e is insignican tly p erturb ed b y the scattering p oten tial (c.f. Equation ( 2.31 )). This can b e expressed in a qualitativ e manner b y the magnitude of V b eing v ery small compared to the collision energy (w eak scattering p oten tial) or b y the magnitude of R b eing v ery small (short range scattering p oten tial). Sc hi pro vides a general condition under whic h the First Born Appro ximation is v alid as giv en b y the expression ( j V j R ) = ( h v ) << 1 [ 22 ]. The Sc hi Appro ximation pro vides an appro ximate scattering amplitude for large collision energy collisions. Sp ecically the assumptions imp osed are that j V j =T << 1, that b e v ery large or v ery small in comparison to p 1 = ( k R ), and that the scattering p oten tial b e slo wing v arying when compared to the incoming w a v elength. In con trast with the First Born Appro ximation, the Sc hi Appro ximation is v alid for an y magnitude of ( j V j R ) = ( hv ) [ 22 ].

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27 2.3.1 Sc hi Scattering Amplitude for Large Angles The Sc hi Appro ximation consists of appro ximately represen ting eac h of the terms in the Born series using the stationary state appro ximation [ 22 23 24 ]. One b egins the deriv ation with the innite Born series, Equation ( 2.37 ), rewritten sligh tly to the form f ( k i ; k f ) = 1 4 1 X n =1 Z Z e i ( k f r n k i r 1 ) U ( r n ) G ( r n r n 1 ) U ( r n 1 ) G ( r 2 r 1 ) U ( r 1 ) d r n d r 1 : (2.38) A c hange of v ariables is made, suc h that n 1 = r n r n 1 so that r n = n 1 + r n 1 The argumen t of the exp onen tial term can b e expanded iterativ ely as k f r n + k i r 1 = k f ( n 1 + r n 1 ) + k i ( r 2 1 ) = k f n 1 k f ( n 2 + r n 2 ) + k i ( r 3 2 ) k i 1 ... = k f n 1 k f n 2 k f n 3 ::: k f ( m + r m ) + k i ( r m m 1 ) ::: k i 2 k i 1 = k f n 1 k f n 2 k f n 3 ::: k f m + q r m k i m 1 ::: k i 2 k i 1 ; (2.39) where q = k i k f is the momen tum transfer for the collision pro cess. T o complete the transformation of v ariables in Equation ( 2.38 ), one m ust calculate the Jacobian of the transformation, sp ecically J = @ 1 @ r 1 @ 2 @ r 1 @ m 1 @ r 1 @ m +1 @ r 1 @ n 2 @ r 1 @ n 1 @ r 1 @ 1 @ r 2 @ 2 @ r 2 @ m 1 @ r 2 @ m +1 @ r 2 @ n 2 @ r 2 @ n 1 @ r 2 @ 1 @ r n @ 2 @ r n @ m 1 @ r n @ m +1 @ r n @ n 2 @ r n @ n 1 @ r n; ; (2.40)

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28 noting that the r m ro w and column are omitted. The deriv ativ es tak e the general form @ f @ r g = 8>>>><>>>>: 1 for f = g 1 for f = g 1 0 otherwise : (2.41) F rom this, it can b e justied that the Jacobian matrix has the v alues of 1 along the diagonal, the v alues of 1 for eac h elemen t immediately b elo w the diagonal, and the v alue 0 elsewhere. F rom this is can b e seen that the determinan t tak es the v alue J = 1. Therefore, d n 1 :::d 1 = d r n :::d r m +1 d r m 1 :::d r 1 ; (2.42) and the limits of in tegration do not c hange. The c hange of v ariables ma y no w b e accomplished, transforming Equation ( 2.38 ) in to the follo wing form f ( k i ; k f ) = 1 4 1 X n =1 n X m =1 Z Z e i [ k f n 1 ::: k f m +1 + q r m k i m 1 ::: k i 1 ] U ( r m + n 1 + n 2 + ::: + m ) G ( n 1 ) U ( r m + n 2 + n 3 + ::: + m ) G ( n 2 ) ::: U ( r m + m ) G ( m ) U ( r m ) G ( m 1 ) U ( r m m 1 ) G ( m 2 ) ::: U ( r m m 1 m 2 ::: 3 ) G ( 2 ) U ( r m m 1 m 2 ::: 2 ) G ( 1 ) U ( r m m 1 m 2 ::: 1 ) d r m d n 1 d n 1 :::d 2 d 1 : (2.43)

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29 where the fact that r f = 8><>: r m + f 1 + f 2 + ::: + m for f > m r m m 1 m 2 ::: 1 for f < m (2.44) w as used to indicate the dep endence of the p oten tial terms. The second summation (o v er m ) arises from the the stationary phase appro ximation. The ma jorit y of the in tegral will come from the regions where the phase is stationary sp ecically where the deriv ativ e of the phase with resp ect to the w a v e v ector is zero [ 24 ]. Th us, one m ust sum o v er all of these stationary phase p oin ts to completely en umerate the in tegral. No w, recognizing that the transformed Green's function for the scattering amplitude equation no w tak es the form G ( r n r n 1 ) G ( n 1 ) = e ik f 4 ; (2.45) the ab o v e equation can b e partitioned in to the form f ( k i ; k f ) = 1 4 1 X n =1 n X m =1 ( 1 4 n 1 Z e i q r m d r m Z 1 n 1 e i ( k f n 1 k f n 1 ) g ( n 1 ) d n 1 Z 1 n 2 e i ( k f n 2 k f n 2 ) g ( n 2 ) d n 2 ::: Z 1 2 e i ( k f 2 k i 2 ) g ( 2 ) d 2 Z 1 1 e i ( k f 1 k i 1 ) g ( 1 ) d 1 : (2.46) In Equation ( 2.46 ), the terms g ( j ) represen t a pro duct of all U terms that dep end explicitly up on j Th us, for an elastic scattering pro cess, the scattering amplitude has b een reduced to a pro duct of in tegrals of the form I = Z 1 g ( )e i ( k k ) d : (2.47)

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30 T o ev aluate this in tegral, one needs to transform to spherical p olar co ordinates ( ; ; ) where measures the angle of k with resp ect to I = Z 2 0 d Z 0 sin d Z 1 0 g ( ; ; )e i ( k k cos ) d; (2.48) where the scalar pro duct is geometrically ev aluated. One can no w mak e the familiar c hange of v ariable = cos where d = sin d Substitution (with appropriate c hange of limits) yields I = Z 2 0 d Z 1 1 d Z 1 0 g ( ; ; )e ik (1 ) d: (2.49) One can no w in tegrate b y parts with resp ect to resulting in the expression I = Z 2 0 d Z 1 0 d ( i k g ( ; ; )e ik (1 ) 1 1 Z 1 1 i k @ g ( ; ; ) @ e ik (1 ) d ) : (2.50) In tegration of the second term in the braces will bring do wn another factor of 1 = ( k ). By applying the limits on the rst term, one can write I = Z 2 0 d Z 1 0 d i k g ( ; 1 ; ) g ( ; 1 ; )e ik + O ( k 2 ) : (2.51) As the collision energy is assumed to b e large ( k >> 1), one can reasonably exp ect the magnitude of k to b e somewhat greater than 1. Th usly the exp onen tial will b e highly oscillatory and subsequen t in tegration o v er will result in a negligible con tribution to I As a result, one can omit this term, as w ell as the terms of order k 2 and higher. F urthermore, if one assumes a spherically symmetric scattering p oten tial, the v alue of b ecomes arbitrary and in tegrates out to a factor of 2 Th us, I 2 i k Z 1 0 g ( ^ k ) d: (2.52) The ^ k arises due to the facts that = 1 ( = 0) and that is arbitrary and so the p oten tial term no w dep ends only on the magnitude of in the direction of the

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31 scattered momen tum. So, the in tegration o v er the angles serv es to transform the dep endence on the v ector in the scattering p oten tial in to a dep endence up on the magnitude of in the direction of the momen tum. No w, returning to Equation ( 2.46 ) one nds that in tegration of the angular dep endence results in the equation f ( k i ; k f ) = 1 4 1 X n =1 n X m =1 ( i 2 k n 1 Z e i q r m d r m Z 1 0 g ( ^ k f n 1 ) d n 1 Z 1 0 g ( ^ k f n 2 ) d n 2 ::: Z 1 0 g ( ^ k i 2 ) d 2 Z 1 0 g ( ^ k i 1 ) d 1 : (2.53) The individual g terms can b e factored bac k to their resp ectiv e U terms, with the dep endences replaced b y ^ k dep endences. Sp ecically one nds that f ( k i ; k f ) = 1 4 1 X n =1 n X m =1 ( i 2 k n 1 Z d r m Z 1 0 d 1 Z 1 0 d n 1 e i q r m U r m + ^ k f ( n 1 + n 2 + ::: + m ) ::: U r m + ^ k f m U ( r m ) U r m ^ k i m 1 ::: U r m ^ k i ( m 1 + m 2 + ::: + 1 ) o (2.54) A t this p oin t, a second c hange of v ariables can b e made, suc h that s j = 8>>>><>>>>: j + j +1 + ::: + m 2 + m 1 for j < m m + m +1 + ::: + j 2 + j 1 for j > m m for j = m (2.55) There are t w o recursion relations that arise, sp ecically s j = j + s j +1 for j < m and s j = s j 1 + j for j > m It is clear from follo wing the previous analysis that the Jacobian is unit y but the lo w er limits of in tegration m ust b e transformed. When in tegrating o v er j for j < m one nds that the lo w er limit of j = 0 transforms to s j +1 Lik ewise, when one in tegrates o v er j for j > m one nds that the lo w er limit transforms to s j 1 Therefore, the c hange of v ariables ma y b e

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32 imp osed, transforming Equation ( 2.54 ) in to the form f ( k i ; k f ) = 1 4 1 X n =1 n X m =1 ( i 2 k n 1 Z d r m e i q r m Z 1 s 2 ds 1 U r m ^ k i s 1 Z 1 s 3 ds 2 U r m ^ k i s 2 ::: Z 1 s m 1 ds m 2 U r m ^ k i s m 2 Z 1 0 ds m 1 U r m ^ k i s m 1 Z 1 0 ds m U r m ^ k f s m Z 1 s m ds m +1 U r m + ^ k f s m +1 ::: Z 1 s n 3 ds n 2 U r m + ^ k f s n 2 Z 1 s n 2 ds n 1 U r m + ^ k f s n 1 : (2.56) A t this p oin t, the ab o v e equation can b e partitioned in to t w o parts, the part explicitly dep ending up on the scattered direction ( ^ k f ) and the part dep ending explicitly up on the inciden t direction ( ^ k i ). If one considers the pro duct dep ending only on the inciden t direction, then y et another substitution can b e made, suc h that W j = Z 1 s j U r m ^ k i s j 1 ds j 1 ; (2.57) and W 0 = Z 1 0 U r m ^ k i s m 1 ds m 1 : (2.58) Th us, calling the pro duct in question K one nds that K = Z 1 s 2 ds 1 U r m ^ k i s 1 Z 1 s 3 ds 2 U r m ^ k i s 2 ::: Z 1 s m 1 ds m 2 U r m ^ k i s m 2 Z 1 0 ds m 1 U r m ^ k i s m 1 = Z W 0 0 dW m 1 Z W m 1 0 dW m 2 ::: Z W 3 0 dW 2 Z W 2 0 dW 1 : (2.59)

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33 The expression for K can no w b e in tegrated, with the follo wing results: K = Z W 0 0 dW m 1 Z W m 1 0 dW m 2 ::: Z W 3 0 dW 2 Z W 2 0 dW 1 = Z W 0 0 dW m 1 Z W m 1 0 dW m 2 ::: Z W 3 0 dW 2 W 2 = 1 2 Z W 0 0 dW m 1 Z W m 1 0 dW m 2 ::: Z W 4 0 dW 3 W 2 3 = 1 2 3 Z W 0 0 dW m 1 Z W m 1 0 dW m 2 ::: Z W 5 0 dW 4 W 3 4 ... = 1 2 3 ::: ( m 3) Z W 0 0 dW m 1 Z W m 1 0 dW m 2 W m 3 m 2 = 1 2 3 ::: ( m 3) ( m 2) Z W 0 0 dW m 1 W m 2 m 1 = 1 2 3 ::: ( m 3) ( m 2) ( n 1) W m 1 0 = [( m 1)!] 1 Z 1 0 U r m ^ k i s ds m 1 : (2.60) A similar substitution can b e made for the pro duct in v olving the direction of scattered momen tum, allo wing for the ev aluation of these n m in tegrals, Z 1 0 ds m U r m ^ k f s m Z 1 s m ds m +1 U r m + ^ k f s m +1 ::: Z 1 s n 3 ds n 2 U r m + ^ k f s n 2 Z 1 s n 2 ds n 1 U r m + ^ k f s n 1 = [( n m )!] 1 Z 1 0 U r m + ^ k f s ds n m : (2.61) The scattering amplitude can then b e expressed using these terms, f ( k i ; k f ) = 1 4 1 X n =1 n X m =1 ( i 2 k n 1 Z d r m e i q r m U ( r m ) [( m 1)!] 1 Z 1 0 U r m ^ k i s ds m 1 [( n m )!] 1 Z 1 0 U r m + ^ k f s ds n m ) (2.62)

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34 A t this p oin t, the ab o v e equation can b e further simplied, but cannot b e made dev oid of its dep endence up on the scattering p oten tial, whic h w as the in ten tion. Ho w ev er, a second appro ximation can b e made whic h will allo w a general expression that do es not dep end explicitly up on the scattering p oten tial. 2.3.2 Sc hi Scattering Amplitude for Small Angles The ab o v e expression for the scattering amplitude is v alid under conditions that k is large, so that the second term in Equation ( 2.51 ) is negligible and that the scattering angle is large, requiring that the n dieren t stationary phase p oin ts are distinct. F or this discussion, one b egins with the assumption that the scattering angle is v ery small, suc h that << ( k R ) 1 = 2 This results in t w o sp ecic requiremen ts. First, the n distinct stationary phase p oin ts no w con v erge to a single p oin t, th us eliminating the requiremen t for summation o v er m Secondly the scalar pro duct in the exp onen tial of Equation ( 2.62 ) can b e written in its Cartesian comp onen ts, e i q r m = e i ( q x x m + q y y m + q z z m ) : (2.63) Since the scattering angle is assumed to b e so small, the angle b et w een k i and k f is nearly zero, ensuring that q z is negligible. This can b e explicitly demonstrated if one requires that ^ k i coincide with the z-axis. Th us, q z = q ^ k i = ( k i k f ) ^ k i = k i ^ k i k f ^ k i = k i k f cos = k f (1 cos ) k f 1 1 2 2 k f 2 2 : (2.64)

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35 In the ab o v e demonstration, the fth line w as obtained b y virtue of the fact that k i = k f for elastic scattering. F urthermore, the sixth line w as obtained b y emplo ying the Maclaurin series represen tation of cos with truncation at the second term. It b ecomes clear that, b ecause is so small, the longitudinal comp onen t of the momen tum transfer is negligible, as w ould b e exp ected ph ysically This is not true of the transv erse comp onen ts of the momen tum transfer. Again, one m ust consider the momen tum transfer, q q = q 2 x + q 2 y ; (2.65) where the longitudinal comp onen t has b een omitted. Using the denition of the momen tum transfer, one nds that q 2 x + q 2 y = ( k i k f ) 2 = k 2 i + k 2 f 2 k i k f cos = 2 k 2 f 2 k 2 f cos 2 k 2 f (1 cos ) 2 k 2 f 2 2 k 2 f 2 : (2.66) Th us, it is clear that the transv erse comp onen ts of the momen tum transfer are b oth of the order of k and cannot b e neglected. Con v erting to Cartesian comp onen ts, the scattering amplitude no w b ecomes f ( k i ; k f ) = 1 4 1 X n =1 ( i 2 k n 1 Z 1 1 dx m Z 1 1 dy m e i ( q x x m + q y y m ) Z 1 1 dz m U ( x m ; y m ; z m ) [( m 1)!] 1 Z z m 1 U ( x m ; y m ; z ) dz m 1 [( n m )!] 1 Z 1 z m U ( x m ; y m ; z ) dz n m ) : (2.67)

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36 T o obtain the ab o v e form, it m ust b e recognized that the rst in tegral o v er s in Equation ( 2.62 ) represen ts the pre-scattering tra jectory while the second in tegration o v er s represen ts the p ost-scattering tra jectory As the scattering angle is so small, these to w a v e-v ectors are eectiv ely the same. Therefore, instead of in tegrating o v er t w o dieren t tra jectories one can in tegrate a single tra jectory for b oth regions ( 1 to z m and then z m to 1 ). The in tegration o v er z m is of principle in terest, so one can dene it as M suc h that M = Z 1 1 dz m U ( x m ; y m ; z m ) [( m 1)!] 1 Z z m 1 U ( x m ; y m ; z ) dz m 1 [( n m )!] 1 Z 1 z m U ( x m ; y m ; z ) dz n m : (2.68) One can then in tro duce the notation that w = Z z m 1 U ( x m ; y m ; z ) dz (2.69) and a = Z 1 1 U ( x m ; y m ; z ) dz : (2.70) In order to b e able to in tegrate o v er w one m ust ev aluate the deriv ativ e of w with resp ect to z m dw dz m = d dz m Z z m 1 U ( x m ; y m ; z ) = U ( x m ; y m ; z m ) : (2.71) Therefore, it is clear that dw = dz m U ( x m ; y m ; z m ). Finally the new limits of in tegration m ust b e ev aluated, leading one to nd that w = 8><>: R 1 1 U ( x m ; y m ; z ) = 0 for z m = 1 R 1 1 U ( x m ; y m ; z ) = a for z m = 1 (2.72)

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37 Th us, the in tegral o v er z m no w can b e rewritten as M = [ ( m 1) ( n m )!] 1 Z a 0 w m 1 ( a w ) n m dw : (2.73) This equation m ust no w b e in tegrated b y parts through m 1 iterations. F or the rst in tegration, one can let u = w m 1 and dv = ( a m ) n m dw Th us, M = 1 ( m 1) ( n m )! Z a 0 w m 1 ( a w ) n m dw = 1 ( m 1) ( n m )! 1 n ( m 1) w m 1 ( a w ) n ( m 1) a0 + m 1 m ( m 1) Z a 0 w m 2 ( a w ) n ( m 1) dw = 1 ( m 2)!( n ( m 1))! Z a 0 w m 2 ( a w ) n ( m 1) dw : (2.74) F or the second in tegration, one can let u = w m 2 and let dv = ( a w ) n ( m 1) dw suc h that M = 1 ( m 2)!( n ( m 1))! Z a 0 w m 2 ( a w ) n ( m 1) dw = 1 ( m 2)!( n ( m 1))! 1 n ( m 2) w m 2 ( a w ) n ( m 2) a0 + m 2 n ( m 2) Z a 0 ( w m 3 ( a w ) n ( m 2) ) dw = 1 ( m 3)!( n ( m 2))! Z a 0 w m 3 ( a w ) n ( m 2) dw : (2.75) F ollo wing this pattern, one nds that after m 1 in tegrations, M tak es the form M = 1 ( n 1)! Z a 0 ( a w ) n 1 dw : (2.76) The nal in tegration can no w b e p erformed, yielding the follo wing expression, M = 1 ( n 1)! Z a 0 ( a w ) n 1 dw = 1 ( n 1)! 1 n ( a w ) n a0 = a n n : (2.77)

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38 The scattering amplitude is no longer dep enden t up on m as should b e exp ected. Th us, f ( k i ; k f ) = 1 4 1 X n =1 i 2 k n 1 Z 1 1 dx Z 1 1 dy e i ( q x x + q y y ) a n n = 1 4 Z 1 1 dx Z 1 1 dy e i ( q x x + q y y ) 1 X n =1 i 2 k n 1 a n n = k 2 i Z 1 1 dx Z 1 1 dy e i ( q x x + q y y ) 1 X n =1 ia 2 k n 1 n = k 2 i Z 1 1 dx Z 1 1 dy e i ( q x x + q y y ) e ia 2 k 1 = ik 2 Z 1 1 dx Z 1 1 dy e i ( q x x + q y y ) 1 exp i 2 k Z 1 1 U ( x; y ; z ) dz : (2.78) A t this p oin t, it should b e noted that the in tegration o v er x and y is tak en o v er the plane p erp endicular to the scattering axis. In the limit of an axially symmetric scattering p oten tial, x and y can b e represen ted b y the impact parameter, b = p x 2 + y 2 and the azim uthal angle, suc h that x = b cos and y = b sin Th us, the transformation to cylindrical co ordinates renders the scattering amplitude in to the form f ( k i ; k f ) = ik 2 Z 2 0 d Z 1 0 e i ( q x b cos + q y b sin ) bdb 1 exp i 2 k Z 1 1 U ( b; z ) dz : (2.79) No w, as the scattering p oten tial is spherically symmetric, the momen tum transfer will alw a ys b e parallel with b As a result, the expression q x b cos + q y b sin will ha v e the same v alue regardless of the v alue of Therefore, one can c ho ose an y v alue of and then sum all of the con tributions from 0 to 2 Sp ecically if one allo ws = 0 = 0, then q x = q and q y = 0. Consequen tly q x b cos + q y b sin = q b cos 0 : (2.80)

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39 The in tegral o v er no w tak es the form of the zeroth-order Bessel function [ 14 ], Z 2 0 e q b cos d = 2 J 0 ( q b ) : (2.81) The small angle Sc hi Appro ximation to the scattering amplitude can no w b e expressed as f ( k i ; k f ) = ik Z 1 0 J 0 ( q b ) 1 exp i 2 k Z 1 1 U ( b; z ) dz bdb: (2.82) Unfortunately the scattering amplitude is still an explicit function of the scattering p oten tial, a term that is not obtained during an END calculation. This expression can, ho w ev er, b e transformed in to one that do es not dep end on the scattering p oten tial [ 23 ]. If one assumes that r is the p osition of the pro jectile at an y p oin t during the tra jectory then the fact that the scattering angle is v ery small requires that r r = ( z + b ) ( z + b ) ; (2.83) where z is the p osition of the pro jectile along the scattering axis. As the impact parameter is p erp endicular to the scattering axis, it is clear that r 2 = z 2 + b 2 F rom this, one nds that dz = dr 1 b r 1 = 2 : (2.84) Therefore, the argumen t of the exp onen tial no w b ecomes 1 2 k Z 1 1 U ( b; z ) = 1 2 k Z 1 1 U ( r ) 1 b r 1 = 2 dr = 1 k Z 1 0 U ( r ) 1 b r 1 = 2 dr ; (2.85) where the fact that the limits of in tegration remain the same when the transformation from z to r is made. F urthermore, the symmetry of the scattering p oten tial w as utilized in the second step. The ab o v e expression is the Massey-Mohr appro ximation for the semi-classical phase shift, ( b ), in the small scattering angle

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40 limit [ 21 25 26 ] whic h is the semi-classical analog of the Kennard appro ximation for classical scattering pro cesses [ 23 27 ]. Sp ecically the Massey-Mohr appro ximation is giv en the form [ 23 ] ( b ) k 2 Z 1 b U k 2 1 b r 2 # 1 = 2 dr : (2.86) In the asymptotic limit, r >> b therefore, 1 2 k Z 1 1 U ( b; z ) 2 ( b ) : (2.87) Finally this means that the Sc hi Appro ximation scattering amplitude for small angle scattering tak es the form [ 23 28 ] f ( k i ; k f ) ik Z 1 0 J 0 ( q b ) (1 exp [2 i ( b )] ) bdb: (2.88) While Equation ( 2.88 ) no longer dep ends explicitly up on the p oten tial, it has in tro duced the semi-classical phase shift, whic h itself is not a v alue provided b y END calculations. This is easily remedied, though, b y the fact that the semi-classical phase shift is directly related to the derection function b y the expression [ 19 21 23 ] ( b ) = 2 k d ( b ) db : (2.89) The Sc hi Appro ximation pro vides an eectiv e metho d for calculating the scattering amplitude without an y prior kno wledge of the scattering p oten tial. The principle p o w er of the Sc hi Appro ximation is the fact that it explicitly con tains all of the terms in the innite Born series. F urthermore, the Sc hi Appro ximation pro vides go o d dieren tial cross sections in the region of small scattering angle, whic h the region most often rep orted b y exp erimen tal studies.

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CHAPTER 3 BASIS SETS F OR D YNAMICAL HAR TREE-F OCK CALCULA TIONS Quan tum mec hanical calculations on molecular systems are traditionally divided in to t w o general categories, the lo calized v alence b ond-t yp e (VB) calculations and the delo calized molecular orbital-t yp e (MO) calculations. In man y w a ys, these t w o descriptions serv e as complemen ts to eac h other. Where VB metho ds, in general, pro vide more visual ph ysical descriptions, suc h as molecular geometries, they generally fail with imp ortan t prop erties, suc h as paramagnetism and b ond strengths. The MO metho d tends to pro vide a more rigorous quan tum mec hanical description of molecules, suc h as electron delo calization o v er the en tiret y of a molecule and v ariations in b ond strengths, but generally oers little in the w a y of ph ysically in tuitiv e descriptions of the molecular system. Both classes of metho ds are form ulated from rather coarse appro ximations, and, as a consequence, eac h p ossesses deciencies, in particular with relation to electron correlation. The VB metho ds tend to o v er-estimate electron correlation, while MO metho ds tend to under-estimate correlation eects [ 29 ]. While eac h has its strengths and w eaknesses, and while eac h con v erges to a common molecular w a v e function represen tation in the absence of their resp ectiv e generalized appro ximations, a large comm unit y of quan tum c hemists has c hosen to emplo y MO metho ds for quan tum c hemical calculations. The MO metho d, and its application in the Hartree-F o c k appro ximation, will b e the fo cus of this c hapter. Some address m ust b e made to w ard notation. The literature of ab initio quan tum c hemistry as with most other elds, is not completely consisten t. While it is not crucial to describ e a set of canonical v ariables, clarication of sym b olic 41

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42 T able 3.1: Notation emplo y ed in this review. T erm Meaning Exact electronic w a v e function Generally appro ximated electronic w a v e function Electronic spin-orbital Spatial electronic factor Spin electronic factor General electronic basis function dV Elemen t of spatial v olume (no spin included) d Elemen t of v olume (spin included) Upp ercase Roman indices Nuclear indices Lo w ercase Roman indices Molecular orbital expansion co ecien ts Lo w ercase Greek indices A tomic orbital expansion co ecien ts notation up fron t is the most lucid manner of handling the issue. The notation of Szab o [ 30 ] will b e most closely follo w ed, with minor adjustmen ts as needed. T able 3.1 pro vides an exhaustiv e list of notation used in this c hapter. 3.1 The Hartree-F o c k Appro ximation The MO metho d in quan tum c hemical calculations of molecular systems has b ecome synon ymous with the Hartree-F o c k (HF) appro ximation. In this section, the HF appro ximation and the HF w a v e functions will b e in tro duced. Additionally the concept of electron correlation and the correlation eects will b e addressed. Finally an in depth discussion of HF basis sets will b e made. 3.1.1 P artitioning of the Molecular W a v e F unction In Section 1.2 the non-relativistic molecular Hamiltonian op erator w as in tro duced, with the explicit form ^ H = 1 2 K XA =1 1 M A r 2A 1 2 N X i =1 r 2i + K XA =1 K X B = A +1 Z A Z B r A;B K XA =1 N X i =1 Z A r i;a + N X j =1 N X i = j +1 1 r i;j ; (3.1) where the indices A and B refer to elemen ts of the K n uclei and i and j lab el elemen ts of the N electrons. F urthermore, Z A is the c harge of n ucleus A and r A;B r A;i and r i;j are the distance b et w een n uclei A and B the distance b et w een n ucleus

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43 A and electron i and the distance b et w een electrons i and j resp ectiv ely The complicated coupling b et w een the n uclear and the electronic degrees of freedom cause the Sc hr odinger equation to b e practically insoluble b y direct metho ds. T o simplify the problem, the Born-Opp enheimer (BO) appro ximation w as in tro duced, in whic h the n uclear degrees of freedom are assumed to v ary on a m uc h larger timescale than the electronic degrees of freedom. In essence, the BO appro ximation allo ws for a \clamp ed n ucleus" mo del, in whic h the n uclei are xed in space and the electronic degrees of freedom are allo w ed to v ary on this xed n uclear framew ork. As the n uclei are motionless, the n uclear kinetic energy terms (the rst summation in Equation ( 3.1 ), ab o v e) b ecome iden tically zero. This assumption decouples to n uclear degrees of freedom from the n uclear degrees of freedom. The resulting form of the Sc hr odinger equation is no w called the electronic Sc hr odinger equation and tak es the form ^ H el = 1 2 N X i =1 r 2i + K XA =1 K X B = A +1 Z A Z B r A;B K X A =1 N X i =1 Z A r i;a + N X j =1 N X i = j +1 1 r i;j : (3.2) It is the solutions to this equation with whic h the ma jorit y of quan tum c hemistry is fo cused. F ollo wing the example of L owdin [ 31 ], one can partition the electronic Hamiltonian op erator in to three parts, ^ H el = h 0 + N X i =1 h 1i + N X i =1 N X j = i +1 h 2i;j : (3.3) In the ab o v e equation, the term h 0 is the n uclear repulsion (or zero-electron Hamiltonian) term, whic h dep ends only up on the xed n uclear degrees of freedom and is dened as h 0 = 1 2 K X A =1 K X B = A +1 Z A Z B r A;B : (3.4)

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44 The second term in Equation 3.3 h 1i is the one-electron Hamiltonian. The one-electron Hamiltonian includes the additiv e kinetic energies of the individual electrons as w ell as instan taneous attractiv e p oten tials b et w een the N electrons and the K n uclei and tak es the form h 1i = 1 2 r 2i K XA =1 Z A r i;a : (3.5) The nal term, h 2i;j is the t w o-electron Hamiltonian, whic h generates the instan taneous repulsiv e forces b et w een eac h individual electron and the remaining molecular electrons. The t w o-electron Hamiltonian tak es the form h 2i;j = 1 r i;j : (3.6) The terms denoted b y h 0 result in a constan t for a giv en n uclear conguration, and therefore will not ha v e an y b earing on our c hoice of molecular w a v e function. Rather, the total energy will only b e increased b y the xed p oten tial energy v alues asso ciated with h 0 The ab o v e partitioning of the molecular Hamiltonian op erator in Equation ( 3.3 ) allo ws one to write the molecular w a v e function as a pro duct w a v e function of the form el ( f r i g ; f r A g ) = 1 ( f r i g ; f r A g ) 2 ( f r i g ; f r A g ) ; (3.7) where the factor w a v e functions are eigenfunctions of the op erators in Equation 3.3 sp ecically ^ H el 1 = E 1 1 ; (3.8) ^ H el 2 = E 2 2 ; (3.9) where E 1 and E 2 are the energy eigen v alues. The total energy for the giv en n uclear conguration, E b ecomes the sum of the energy eigen v alues and the n uclear

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45 p oten tial energy dened b y h 0 E = E 1 + E 2 + V nuc : (3.10) The ma jor fo cus of quan tum c hemistry in v olv es the solution of Equations ( 3.3 ), ( 3.7 ), and ( 3.10 ), and w e will use these equations as the starting p oin t for the discussion of the Hartree-F o c k appro ximation. 3.1.2 The Hartree-F o c k W a v e F unction The HF appro ximation b egins with the assumption that the total electronic w a v e function can b e appro ximated b y a pro duct of one-electron w a v e functions. F urthermore, one m ust assume that the p oten tial exp erienced b y a giv en electron is an a v erage of the p oten tials pro duced b y the remaining electrons [ 29 ]. In this appro ximation, the term h 0 and the N dieren t h 1 terms are main tained, but the N 2 ( N 1) t w o-electron terms are replaced b y N additional one electron terms. By replacing the t w o-electron terms with one-electron terms, the HF appro ximation do es not explicitly treat the instan taneous in teraction of individual electrons. Rather, eac h electron is treated as if it w ere inruenced b y an a v erage eld pro duced b y the other electrons in the molecule. This appro ximation allo ws for relativ ely accurate quan tum c hemical calculations despite the gross appro ximations imp osed, though sev eral imp ortan t ph ysical descriptions are omitted. These correlation eects will not b e discussed in this c hapter. The new electronic Hamiltonian can b e rewritten in the form ^ H el = N X i =1 h 1i + V H F i ; (3.11) where h 1i is the one-electron Hamiltonian (as previously dened) and V H F is the new HF one-electron p oten tial energy term (to b e dened later in this section). Because the zero-electron Hamiltonian only adds a constan t factor to the electronic

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46 energy it is omitted. The most striking b enet of this reform ulation of the electronic Hamiltonian is the fact that, b ecause the Hamiltonian op erator is no w a sum of one-electron terms, one can appro ximate a single N -electron w a v e function as a pro duct of N one-electron w a v e functions (called a Hartree pro duct), H P ( f x i g ; f r A g ) = N Y j =1 j ( f x i g ; f r A g ) ; (3.12) whic h is an eigenfunction of the one-electron Hamiltonian, ^ H el H P = E H P : (3.13) As b efore, the eigen v alue is the energy of the system. The w a v e function dened in Equation ( 3.12 ) is a pro duct of a set of N spinorbitals, j whic h are themselv es eigenfunctions of the resp ectiv e one-electron Hamiltonians, h 1j + V H F j j = j j ; (3.14) where j is the energy eigen v alue of the j th orbital. The total energy E is a sum of the orbital energies, E = N X j =1 j : (3.15) The electronic Hamiltonian in Equation ( 3.11 ) do es not ha v e a spin dep endence, therefore a transformation from spatial electronic co ordinates, f r i g to spatial-spin co ordinates, f x i g b y emplo ying a pro duct form of the spin-orbitals do es not c hange the energy eigen v alues of Equation ( 3.13 ). Th us, in the absence of relativistic eects, the N spin-orbitals can b e represen ted b y a pro duct of a spatial-dep enden t factor, j ( f r i g ), and a spin-dep enden t factor, either \spin up", ( ), or \spin do wn", ( ), suc h that j = 8><>: j ( f r i g ) ( ) j ( f r i g ) ( ) : (3.16)

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47 The v ariable represen ts the general spin v ariable. It should b e noted that the pro duct w a v e functions indicated in the ab o v e equation ha v e the same spatial factors with v arying spin factors. This is consisten t with restricted HF metho d. Allo wing eac h spin-orbital to ha v e a unique spin AND spatial factor results in the unrestricted HF metho d. [ 30 ]. The partitioning of the spin-orbitals giv en b y Equation ( 3.16 ) satises the P auli exclusion principle b y allo wing t w o spin-orbitals to ha v e the exact same spatial factor, but opp osite spins (that is, one molecular orbital with a paired set of electrons). Ho w ev er, a further requiremen t of the molecular electronic w a v e function is that it m ust ob ey the F ermi-Dirac statistics, in particular that the electronic w a v e function m ust b e an ti-symmetric with resp ect to exc hange of electron indices. The Hartree pro duct giv en b y Equation ( 3.12 ) do es not satisfy the F ermi-Dirac statistics, b eing that the sign of H P remains unc hanged if t w o indices are exc hanged. Adherence to the F ermi-Dirac statistics traditionally required detailed group algebra. This w as greatly simplied b y Slater, who circum v en ted group theoretical descriptions b y in tro duction of the spin-orbital function directly in to a determinan t that w ould later b ear his name [ 32 ]. Slater exploited the prop ert y of matrix algebra that, giv en a matrix, in terc hanging an y t w o columns of the matrix will c hange the sign of the determinan t of the matrix. Th us, Slater constructed a matrix in whic h the spin-orbitals are placed as the columns and the o ccup ying electrons are placed as the ro ws of the matrix. The determinan t of this matrix is the most general an ti-symmetrized pro duct w a v e function. Mathematically one nds that, S l ater = ( N !) 1 1 ( x 1 ) 2 ( x 1 ) ::: N ( x 1 ) 1 ( x 2 ) 2 ( x 2 ) ::: N ( x 2 ) ::: ::: ::: ::: 1 ( x N ) 2 ( x N ) ::: N ( x N ) = ( N !) 1 det f j ( x i ) g : (3.17)

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48 A t this p oin t some discussion of the appro ximated one-electron p oten tial in Equation ( 3.11 ) can b e made. It is b ey ond the scop e of this review to explicitly deriv e the form of the term V H F i so it m ust suce to sa y that the p oten tial form can b e obtained b y emplo ying Lagrange's metho d of undetermined m ultipliers to the energy eigen v alue equation [ 30 ] ^ H el S l ater = E S l ater : (3.18) The resulting HF Hamiltonian (v ariously referred to as the F o c kian and denoted F ) is a sum of N F o c k op erators, whic h satisfy the eigen v alue equations of the form h 1i + N X j 6 = i J j N X j 6 = i K j # i = i i : (3.19) The term in brac k ets in the ab o v e equation is the F o c k op erator for orbital i It can b e seen that the appro ximated one-electron p oten tial has b een split in to t w o comp onen ts. The rst is the coulom b op erator, whic h denes the in teraction of electrons with an a v erage p oten tial. The action of the coulom b op erator is to pro vide an a v erage repulsiv e p oten tial felt b y an electron at the p osition x 1 that arises from an electron in a second orbital. The coulom b op erator has the in v erse r form of a coulom b in teraction, w eigh ted b y the probabilit y densit y of the orbital to b e a v eraged, sp ecically J i ( x 1 ) j ( x 1 ) = Z d x 2 j i ( x 2 ) j 2 r 1 ij j ( x 1 ) : (3.20) The coulom b op erator arises as a consequence of the assumption that the electronic Hamiltonian is a sum of one-electron op erators only The second op erator results from the an ti-symmetrization of the w a v e function through the use of a Slater determinan t. This op erator, the exc hange op erator, results in the exc hange of t w o electrons and pro duces a one-electron p oten tial that is dep enden t up on the v alue of the orbital in question throughout all space [ 30 ]. The form of the exc hange

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49 op erator is K i ( x 1 ) j ( x 1 ) = Z d x 2 j ( x 2 ) r 1 ij i ( x 2 ) j ( x 1 ) : (3.21) The eect of the coulom b op erator is lo calized and has a classically in tuitiv e in terpretation, ho w ev er, the action of the exc hange op erator is non-lo cal and dep ends up on the lo cation of these t w o orbitals in the spin-orbital space. 3.1.3 Solving the HF Equations: Basis Set Expansions In Section 3.1.2 w e in tro duced the Hartree-F o c k in tegral-dieren tial equations (no w rewritten using Dirac notation), F S l ater = E S l ater ; (3.22) b y using Equations ( 3.18 ) ( 3.21 ). T raditionally one mak es consisten t use of Slater determinan ts for the molecular w a v e function, and sup erscripts on will consequen tly b e dropp ed for the remainder of the c hapter. The solution of these equations is still a non-trivial task. The rst metho ds of solving these equations, sp ecically for small atomic systems, w as through n umerical in tegration [ 29 ]. A considerable breakthrough w as in tro duced b y Ro othaan [ 33 ] in 1951. The computation routine of Ro othaan in v olv ed expanding the molecular w a v e function in a basis of atomic spin-orbitals with the general form of a linear com bination of spatial atomic orbital basis functions m ultiplied b y the appropriate spin function. This expansion allo w ed the Hartree-F o c k dieren tial equations to b e written as a set of algebraic matrix equations, whic h could b e solv ed using a v ailable linear algebraic tec hniques. The general form of the basis set expansion of the ith spatial comp onen t of the w a v e function tak es the form i = K X =1 c i ; (3.23)

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50 where the basis set is comp osed of K functions, f j = 1 ; 2 ; :::; K g The term c i represen ts the expansion co ecien t for the th basis function to form the ith spatial comp onen t of the w a v e function. The size of the expansion ( K ) is in general not limited to a sp ecic n um b er. In fact, an innite expansion w ould b e desirable, as this w ould corresp ond to the full Hartree-F o c k w a v e function. Ho w ev er, this is practically imp ossible due to computational limitations. The size of K should ideally b e large enough to oer the b est descriptions of the molecular orbitals without b ecoming computationally inecien t. This, along with other sp ecic criteria that m ust b e considered will b e addressed in the next section. A t this p oin t it will suce to assume that some general function form and expansion size has b een decided up on. The Hartree-F o c k equation for a giv en spatial comp onen t of the w a v e function can no w b e written as F j i i = i j i i ; (3.24) where i is the orbital energy of the ith spatial orbital. The basis set expansion for i can no w b e in tro duced. The result b ecomes F K X =1 j i c i = i K X =1 j i c i : (3.25) A t this p oin t, one can then m ultiply through on the left of the equation b y an arbitrary basis comp onen t h j and in tegrate. The ith orbital energy is a n um b er, and th us can b e extracted from the in tegration, as can the expansion co ecien ts. Th us, the equation no w b ecomes K X =1 h j F j i c i = K X =1 h j i c i i : (3.26) A t this p oin t, one m ust in tro duce t w o matrices with elemen ts that are related to the terms in the ab o v e equation. The rst is the F o c k matrix, F The F o c k matrix

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51 has elemen ts dened as F = h j F j i : (3.27) The second matrix is the o v erlap matrix, S The elemen ts of the o v erlap matrix are giv en as S = h j i (3.28) and arises from the fact that the basis functions are not necessarily orthogonal. The o v erlap matrix pro vides a measure of the linear dep endence of the set of basis functions. Due to assumed normalization, the diagonal elemen ts of the o v erlap matrix all ha v e a magnitude of one. The o-diagonal elemen ts will range in magnitude b et w een zero and one. Elemen ts approac hing one will demonstrate a strong linear dep endence b et w een t w o basis functions, while a v alue approac hing zero indicates a strong linear indep endence. The HF equation can th usly b e expressed using the newly dened matrix elemen ts, K X =1 F c i = K X =1 S c i i : (3.29) It is clear at this p oin t that the ab o v e equation is an elemen t of one single matrix equation of the form F C = SC ; (3.30) where F and S are as dened ab o v e, C is the K K matrix of expansion co ecien ts, and is a K K diagonal matrix with the orbital energies as the diagonal elemen ts. Equation ( 3.30 ) is commonly referred to as the Ro othaan-Hall equation [ 34 ]. A companion equation exists for unrestricted determinan ts, called the P ople-Nesb et equation [ 30 35 ]. The P ople-Nesb et equations ha v e individual matrix equations for eac h set of spin-orbitals, as eac h pair of spin-orbitals has a dieren t spatial comp onen t in the unrestricted formalism.

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52 As the o v erlap matrix and the F o c k matrix are b oth Hermitian (and in man y cases real and symmetric), relativ ely simple solution tec hniques are a v ailable for solving the Ro othaan-Hall equations [ 36 37 ]. The most common metho d in v olv es diagonalization of the matrices b y use of a unitary transformation. This metho d presen ts the eigen v alues (elemen ts of ) and eigenfunctions (elemen ts of C ) for the matrix equation. The rudimen ts of solving the Ro othaan-Hall or P ople-Nesb et equations will not b e discussed an y further in this dissertation. Rather, in the next t w o sections sp ecic in terest will b e placed on the general form and construction of basis sets for use with these matrix equations. 3.2 General F orms and Prop erties of Basis Sets T o this p oin t, no men tion has b een made as to the functional form that the basis set should tak e. In general, an y functional form is p ossible, but certain prop erties are desirable. Sp ecically the w a v e function for the system m ust b e single-v alued, nite, con tin uous, and square-in tegrable [ 38 ]. It is th us desirable that one c ho ose basis functions that p ossess these c haracteristics. A set of atomic orbitals (A Os) is an immediate c hoice for a giv en basis set, as A Os satisfy the ab o v e criteria as w ell as oer a c hemical in tuitiv eness lac king in other c hoices. 3.2.1 Slater-T yp e and Gaussian-T yp e Orbitals A rst c hoice of basis sets w as a set of spatial orbitals with the same functional form as the h ydrogenic orbitals, S T O = (2 ) 2 n +1 (2 n )! 1 = 2 r n 1 e r Y m l ( ; ) : (3.31) Equation ( 3.31 ) is called a Slater-t yp e orbital (STO) [ 29 34 ]. The equation in v olv es the terms whic h is the orbital exp onen t; n whic h is the principle quan tum n um b er; l whic h is the azim uthal quan tum n um b er; and m whic h is

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53 the magnetic quan tum n um b er. The term Y m l ( ; ) is the spherical harmonic. As the functional form of h ydrogenic orbitals are deriv ed from sp ecic linear com binations of STO basis functions, one ma y reasonably exp ect that they will b e go o d appro ximations to orbitals in m ulti-electron atoms. In fact, STO functions ha v e correct functional forms at small r v alues and also at v ery large r v alues. As can b e seen from Equation ( 3.31 ), STO functions do not ha v e an y no dal structure, and therefore linear com binations m ust b e constructed to prop erly mimic the no dal structure of atomic orbitals. Ho w ev er, basis sets built from relativ ely small linear com binations of STO basis functions ha v e b een quite successfully emplo y ed in quan tum c hemical calculations [ 29 ]. Despite the desirable functional form of the STOs, the principle dra wbac k is n umerical [ 34 ]. The large p ortion of computational time in the HF metho d is the calculation of the man y-cen ter t w o-electron in tegrals. Slater-t yp e orbital basis functions do not admit simple analytical expression for suc h t w o-cen ter in tegrals, and m ust therefore b e n umerically in tegrated [ 39 ]. This is a time consuming pro cess and, as a result, for an y system with more than a few atoms the accuracy obtained b y using STO bases is out w eighed b y the sev ere decrease in computational eciency This limitation w as circum v en ted b y Bo ys, who suggested using Gaussiant yp e functions instead of exp onen tial functions [ 40 ]. The functional form of a Gaussian-t yp e orbital (GTO) is [ 41 ] GT O = (2 = ) 1 = 2 (4 ) 2 n +1 2 (2 n 1)!! # 1 = 2 r n 1 e r 2 Y m l ( ; ) ; (3.32) where n; l and m are the same quan tum n um b ers as giv en in Equation ( 3.31 ). The parameter is a dieren t orbital exp onen t sp ecic to the GTO basis. Note that angularly STO and GTO basis functions ha v e the same functional form, the only dierence lies in the radial factors.

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54 The primary deciency of the GTO basis is immediately clear. A GTO function do es not pro vide the correct functional form as r 0 or at r 1 In particular, an STO function will ha v e d S T O =dr > 0 at r = 0, whereas a GTO function will alw a ys ha v e d GT O =dr = 0 at r = 0. F urthermore, the GTO basis function has a m uc h faster drop-o in the tail of the orbital as r 1 than do es the STO function. This is a fairly sev ere limitation to the accuracy of GTO bases in computations, as the functional form of the STO basis sets leads to sup erior accuracy o v er GTO bases. The top panel in Figure 3.1 demonstrates the considerable dierence b et w een the radial part of the h ydrogen 1 s orbital represen ted as a single STO function and a single GTO function. This red curv e is the h ydrogenic 1 s STO function and the blue line is a single GTO function in whic h the exp onen t has b een optimized to pro vide the b est least-squares t to the STO [ 41 ]. One can impro v e the structure b y building a linear com bination of GTO basis functions (with optimized exp onen ts) for eac h STO function emplo y ed. Both the set of exp onen ts for eac h primitiv e Gaussian orbital and the set of con traction co ecien ts m ust b e optimized to t the Slater in question. The b ottom panel in Figure 3.1 sho ws that a linear com bination of six GTO functions (blue) pro vides a m uc h b etter t for the h ydrogen 1 s orbital as represen ted b y a single STO function (red), but that it is still not particularly go o d at the cusp of the orbital. In general, a v ery large n um b er of GTO functions with large exp onen ts w ould b e needed to correctly mimic the cusp of an STO function, but no matter ho w man y terms where included in the linear com bination, the deriv ativ e of the GTO function w ould still b e zero at r = 0. Despite this cusp deciency quite go o d accuracy can b e obtained from a go o d sized linear com bination of GTO basis functions p er Slater. Y et, one m ust ask wh y a linear com bination of six or more GTO functions is fa v orable o v er a single STO

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55 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 Radial Amplitude (a.u.)Radial Distance (a.u.) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 Radial Amplitude (a.u.)Radial Distance (a.u.) Figure 3.1: Comparison of STO and GTO represen tations of the radial part of the h ydrogen 1 s orbital. T op: A single GTO function t to a single STO function. Bottom: A linear com bination of six GTO functions t to a single STO function. function, particularly when the accuracy of the STO basis is sup erior to the larger GTO basis. The answ er lies in the fact that when a pro duct of t w o GTOs is tak en, the result is a third GTO [ 30 ]. This reduces a m ulti-cen ter t w o-electron in tegral to a considerably simpler analytic form [ 39 ]. As a consequence, a calculation utilizing a larger GTO basis is m uc h more ecien t that a calculation using a smaller STO basis. This substan tially greater computational sp eed has lead to the fact that most calculations of p oly atomic systems ha v e traditionally emplo y ed GTO basis sets [ 42 ].

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56 Other basis sets ha v e b een emplo y ed in v arious studies, suc h plane w a v e basis sets. Ho w ev er, these are still not as widely used as STO bases and, particularly GTO bases. The remainder of this c hapter will sp ecically fo cus on Slater and Gaussian basis sets. 3.2.2 The Structure of Basis Sets No w that the general form of an orbital basis function has b een c hosen, either STO or GTO, the full basis set for an atom m ust b e constructed. While there are (theoretically) no limitations on the construction of basis sets, practicalit y has restricted quan tum c hemists to certain accepted forms of con traction. Minimal basis sets An y atom m ust ha v e at least enough orbitals a v ailable to completely con tain the required n um b er of electrons. An y basis set designed to completely represen t only the ground state orbital structure of an atom is referred to a minimal basis set [ 30 ]. F or example, a minimal basis set for a Mg atom w ould con tain no less than 6 orbitals, the 1 s 2 s 2 p x 2 p y 2 p z and the 3 s orbital. It w ould therefore b e sucien t to build a minimal basis of 6 STO basis functions, three with n = 0 and three dieren t v alues of as w ell as three others with n = 1 and a fourth v alue of F urthermore, it w ould b e p ossible to build a minimal basis set of six single, uncon tracted GTO basis functions in the same manner. Ho w ev er, as the previous discussion indicated, the accuracy w ould b e considerably less than the for the comparable STO basis. So, to remedy this, a sp ecic n um b er of Gaussian primitiv e functions (single GTO basis functions) are generally t to eac h STO function in the minimal set. This describ es the formalism to construct a minimal STON G basis set [ 43 ]. The title of this basis set indicates that N Gaussian primitiv es are optimized to t a single STO function. While an y n um b er of primitiv es ma y b e t to a giv en STO function, in practice the STO-6G is the largest Gaussian minimal

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57 basis that is emplo y ed. Without exception, the STO-3G is the most widely used minimal basis set. Minimal basis sets are notoriously inaccurate basis sets. In general STON G basis sets oer qualitativ ely correct descriptions of fundamen tal c hemical prop erties suc h as b onding and can b e emplo y ed for initial guesses and for calculations in v olving v ery large molecules where more complete basis sets w ould b e computationally inecien t. Ho w ev er, the small size of the STON G basis set is prohibitiv e to the use of minimal bases for calculations in whic h ev en mo derate accuracy is required [ 30 44 ]. As a nal p oin t concerning STON G minimal bases it should b e noted that for certain atoms these bases are not truly minimal. Man y times c hemical b onding is not correctly mimic k ed using a truly minimal basis set. F or this reason, the Group 1A and Group 2A metals, as w ell as the rst t w o ro ws of transition metals, will include the lo w-lying p orbitals ev en though they are uno ccupied in the un b ound atom [ 34 ]. The s and p exp onen ts for a giv en principal energy lev el are iden tical. This is computationally more ecien t than allo wing for separate s and p exp onen ts as eac h set of s and p orbitals will then ha v e the same radial b eha vior and can, consequen tly b e in tegrated together [ 30 ]. Double-zeta and split-v alence basis sets One of the main limitations of the minimal basis set is the fact that there is no rexibilit y for the generated orbitals to c hange size under the inruence of in tramolecular surroundings. Eac h orbital has a single set of exp onen ts that con trol the size and shap e of the orbital, and while the amplitude of the orbital can b e adjusted through the HF co ecien ts, the spatial size cannot b e c hanged. T o remedy this, one w ould desire to include more than a single linear com bination of basis functions for a giv en orbital. This idea giv es rise to the next lev el of basis sets, the double-zeta and the split v alence basis sets. Muc h of this discussion

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58 will b e made in terms of STO bases. It should b e remem b ered that these ideas can b e translated directly to GTO bases b y requiring that eac h STO function b e constructed as a linear com bination of GTO functions. Sp ecically the double-zeta functions allo w for eac h orbital to b e a comp osed of a linear com bination of t w o STO basis functions, eac h with a dieren t exp onen t. The larger exp onen t (a tigh ter function) is, in general, sligh tly larger than the optimal exp onen t for the single zeta function, while the smaller exp onen t (a more diuse function) is sligh tly smaller [ 30 ]. F urthermore, as the double-zeta basis is more complete than a comparable single-zeta basis, a correctly optimized linear com bination of double-zeta functions will b e a b etter represen tation of the ph ysical orbital than a single-zeta function. This results in an impro v emen t in the ground state energy as demonstrated b y Clemen ti and Ro etti in their seminal pap er on double-zeta functions for atoms [ 45 ]. A simplication of the double-zeta basis can b e made b y realizing that, during a c hemical pro cess, the size and shap e of core atomic orbitals will not c hange signican tly Therefore, under most an y conditions, a single w ell-optimized STO function will pro vide a sucien t represen tation of a core orbital. One can then allo w for the v alence orbitals to b e represen ted using a com bination of t w o STO basis functions, as in the double-zeta basis. This is the form ula for constructing split-v alence basis sets [ 34 ]. Split-v alence basis sets pro vide ground state energies that sho w impro v emen ts o v er minimal basis sets, but that are not as go o d as double-zeta bases. Again, this is due to the fact that a more complete description of core orbitals is obtained with the double-zeta functions. Ho w ev er, this energy dierence is small when compared to the computational eciency gained through using split-v alence bases [ 30 ]. The most common split-v alence basis sets that are emplo y ed in computational c hemistry are the l : mn G basis sets. These are com binations of GTO functions suc h

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59 that the core orbitals for the atom are represen ted as a con traction of l GTO basis functions. A giv en v alence orbital is represen ted as a linear com bination of t w o basis functions, one comp osed of m GTO primitiv es and the second comp osed of n GTO primitiv es [ 34 ]. The only exceptions o ccur with H and He, in whic h there are no core shells and only the v alence shell structure is used (and as a consequence the 4:31G bases for H and He are iden tical to the 6:31G bases). The most common examples of split-v alence bases are the 3:21G [ 46 ], 4:31G [ 47 ], and 6:31G [ 48 ] bases. This pattern can b e extended to larger basis sets suc h as the 6:311G basis [ 44 ]. Double-zeta and split-v alence basis sets impro v e the electronic represen tation in sev eral w a ys. First, as men tioned ab o v e, the double-zeta bases allo w for a b etter description of atomic orbitals b y virtue of the increased completeness of the basis set. Also, the v alence orbitals are no w rexible enough to c hange size during a c hemical pro cess. In particular, this allo ws for b etter descriptions of b onding and anisotropic c hemical pro cesses, suc h as the anisotrop y of the b onding p -orbitals when forming and b onds in systems with b ond orders greater than unit y [ 34 ]. One last feature of split-v alance bases is that, with prop er optimization, the more diuse v alence functions can b eha v e as virtual orbitals, a prop ert y not a v ailable in minimal basis sets. This feature will b ecome increasingly imp ortan t when one desires to in v estigate dynamical pro cesses. P olarization basis sets Man y ph ysical pro cesses require not only a c hange in the size of atomic orbitals o v er time, but also a c hange in the shap e of the orbital. Examples include the b eha vior of the orbitals for an atom sub ject to an external electric eld or the orbitals of an atom whic h has some non-zero momen tum. Both of these pro cesses result in a p olarization of the atomic orbitals. This p olarization causes a net increase of electron densit y in one area o-cen ter from the n uclear cen ter and a

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60 corresp onding net decrease in electron densit y in the region of space immediately opp osite the n uclear cen ter. While split-v alence basis sets allo w for the size of the orbitals to ructuate, they do not p ermit the shap e of the orbital c hange. This cannot b e accomplished b y merely c hanging the size of the orbital exp onen ts. In order to c hange the shap e of atomic orbitals, the basis m ust rexible enough to allo w com binations of basis functions that represen t o ccupied atomic orbitals with higher angular momen tum basis functions[ 30 34 44 ]. The most common metho ds of p olarization in v olv e the addition of basis functions that mimic a d -orbital to the elemen ts from Li to Ar. This lev el of p olarization is denoted using a single asterisk. The most common example is the 6:31G* basis [ 49 ], whic h is the basic split-v alence 6:31G basis describ ed ab o v e with the addition of a single d -symmetry basis function [ 34 ] (or f -symmetry basis function to transition metals) [ 44 ]. The second form includes the d -( f -)orbitals for hea vy atoms as w ell as an basis function with p -symmetry to H and He. This lev el is denoted with t w o asterisks (suc h as 6:31G**)[ 49 ]. Again, this pattern can b e emplo y ed using larger split-v alence bases, resulting in suc h com binations as 6:311G**, a basis that is commonly emplo y ed for correlated calculations [ 50 ]. The eect of including p olarization functions has traditionally b een observ ed in structural prop erties, particularly in constrained systems where the electron densit y is shifted a w a y from the n uclear cen ters [ 34 ], and in systems sub jected to external electric elds [ 30 ]. In particularly atoms whic h can b e m ultiply b onded will, in particular, require a greater degree of p olarization. This is eviden t from the v alence-b ond description of c hemical b onding, in whic h the formation of \h ybrid" atomic orbitals (whic h are nothing more than p olarized atomic orbitals) is the underlying principle of c hemical b ond formation [ 51 ]. F or this reason, it has long b een accepted that it is more imp ortan t to include p olarization basis sets on main

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61 group elemen ts rather than singly b onding sp ecies, suc h as Group 1 elemen ts, as indicated b y the strong statemen t of Szab o and Ostlund [ 30 ] that it \has b een empirically determined that adding p olarization functions to the hea vy atoms is more imp ortan t than adding p olarization functions to h ydrogen." While this is generally true for structure calculations, it cannot b e accepted when dynamical calculations are b eing p erformed. In particular, an y atom that p ossesses non-zero momen tum will exp erience a p olarization of its electronic orbitals due to the motion of the atom. This eect will b e presen t in all atoms, including H and He. F or this reason, it is of crucial imp ortance to include p olarizing p -functions on H atoms for dynamical calculations. Diuse basis sets The previously men tioned basis set structures do a go o d job of describing v arious c hemical pro cesses, ho w ev er, all of them lo cate the electronic densit y relativ ely close to the n uclear cen ters. The split-v alence structure allo ws for increasing the size of orbitals, but the exp onen ts are alw a ys close in magnitude to the exp onen t in a comparable single-v alence basis function. This limits the abilit y of the orbital constructed from a split-v alence basis to expand b ey ond small ructuations around the size of the orbital constructed from a single-v alence basis. Additionally the p olarization functions allo w for shifting of the electron densit y a w a y from the n uclear cen ter, sa y to a c hemical b ond. Again, this shift in the densit y is not large. As a consequence, systems with large electron densities that are lo cated a signican t distance from the n uclear cen ter (suc h as anions and systems in v olving Rydb erg states) are not prop erly mo deled using minimal, split-v alence, or p olarization basis sets. T o prop erly describ e suc h systems, diuse basis sets m ust b e emplo y ed [ 34 44 ]. Diuse basis sets are structured in a manner v ery similar to split-v alence basis sets. A minimal (or split-v alence or p olarization) basis set in constructed and

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62 additional basis functions are included to pro vide for the diuse atomic orbitals. Ho w ev er, the exp onen ts of these diuse basis functions are m uc h smaller than for the v alence basis functions, resulting in an electron densit y that is lo cated m uc h further a w a y from the n uclear cen ter. In general, the diuse functions are of the same angular momen tum as the v alence basis functions. This means that a carb on atom w ould incorp orate one additional s and one additional p -function. This diuse structure is denoted using a + sym b ol [ 34 ]. If a single diuse s -function is added to a h ydrogen or helium atom, then this is denoted b y t w o plus signs. Th us, one can no w b egin to emplo y a virtual alphab et soup of suc h com binations as 3:21+G*, 6:31++G*, or 6:311+G**. Ev en-temp ered and univ ersal ev en-temp ered basis sets F urther adv ances in the building of basis sets w ere made when it w as realized that, as a basis set got larger, the orbital exp onen ts within a giv en angular momentum con v erged to a geometric sequence. This geometric sequence tak es the form i = 0 i ; (3.33) where i is the ith exp onen t in the sequence, 0 is the largest exp onen t, and is a constan t that is sp ecic to the angular momen tum. A basis set that is constructed using this t yp e of metho d is called an ev en-temp ered basis set [ 39 42 ]. The general feature of an ev en-temp ered basis set is that it limits the n um b er of parameters that m ust b e optimized. F urthermore, ev en-temp ering ensures that a GTO expansion of an STO is w ell-spanned, with no regions in whic h the represen tation is particularly p o or. There is in general a small energy price that m ust b e paid, but this usually is on the order of sev eral h undredths of a Hartree [ 42 ]. F urthermore, it has b een p ostulated that, if enough ev en-temp ered exp onen ts are include in a con traction, then this set of exp onen ts w ould ev en tually b ecome

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63 iden tical o v er an en tire ro w of the p erio dic table. This leads to a univ ersal ev entemp ered basis set [ 39 42 ]. Other basis set structures In this section, a brief discussion has b een made of the general structure of basis sets, with details giv en ab out those basis sets most commonly emplo y ed in quan tum c hemical calculations. This is only a v ery small sampling of the basis sets a v ailable for computation, ho w ev er, most of these basis sets include the basic principles listed ab o v e. It is b ey ond the scop e of this w ork to pro vide an in depth discussion of the dieren t t yp es of basis sets used in calculations. The comparisons made in the forthcoming sections will, in general, b e related to the t yp es of basis sets review ed in this section. 3.3 Metho d for Constructing Basis Sets Consisten t with Dynamical Calculations The structure of a basis set is hea vily dep enden t up on the t yp es of ph ysical prop erties that one desires to calculate. In some cases diuse functions are required, in others p olarization functions are required. In most cases, some balanced com bination of all of the prop erties is needed. Because of this, man y ha v e view ed basis set construction as an art (or blac k magic in some cases). Y et, no matter what form the basis set tak es, most ha v e one trait in common: with v ery few exceptions, basis sets m ust b e optimized. Usually basis sets are optimized with resp ect to the energy of the ground state of the system b y means of the v ariational principle. Ho w ev er, the HF equations are non-linear, and therefore an y optimization pro cess b ecomes computationally costly An additional feature that is common to most basis sets is that they are built for use in stationary state calculations of the ground state of a giv en system. While most of these basis sets do pro vide represen tations of uno ccupied orbitals either due

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64 to constructing a split-v alence or diuse basis set, these uno ccupied orbitals b ear little or no resem blance to the virtual orbitals in the system. In this section a new metho d for the construction of basis sets will b e in troduced. This metho d has a simple ph ysical underlying justication. The metho d do es not require exp ensiv e and complex energy optimizations (and do es not, in fact, require an y optimizations at all). Finally this metho d allo ws for the construction of ph ysically meaningful virtual orbitals, a necessit y for the computation of a wide v ariet y of dynamical prop erties. 3.3.1 Basis Set Prop erties for Dynamical Calculations In stationary state calculations minimal basis sets are rarely if ev er, sucien t for the description of the c hemical sp ecies in question. As outlined in the previous section, a v ariet y of extra basis functions m ust b e included to impro v e the description. In this case an y set of functions added to the minimal basis set generally demonstrates no ph ysical resem blance to atomic orbitals in the system. Rather, they just act to pro vide a more complete spanning of the electronic Hilb ert space, increasing the rexibilit y of the basis set. These extra basis functions serv e to lo w er the ground state energy (due to an increase in the accuracy of the represen tation of the o ccupied atomic orbitals and in some cases b y partially accoun ting for the correlation energy in the system) but they do little else. F or dynamical calculations, particularly c harge transfer pro cesses, the basis set m ust b e rexible enough to allo w for correct description of electronic transitions b et w een atomic orbitals, either within a single atom or molecule or b et w een the collision pair. One sp ecic asp ect of this requiremen t is the fact that virtual orbitals (atomic or molecular) m ust b e a v ailable for o ccupation throughout the dynamical pro cesses. T o this p oin t, little eort has b een made b y the computational comm unit y to construct atomic basis sets that prop erly describ e virtual atomic orbitals, mainly due the fact that most basis sets are optimized with resp ect to the

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65 total ground state energy of the system rather than optimized with resp ect to the individual atomic orbital energies. F or this reason, and with v ery few exceptions, sto c k basis sets that are most commonly emplo y ed in computational c hemistry are inadequate for use in dynamical calculations. Metho ds for impro ving sto c k basis sets traditionally follo w along the lines of increasing the size of the basis sets b y the inclusion of more and more uncon tracted diuse primitiv es. This metho d will, in the limit of innite expansion sizes, lead to correct virtual energy lev els b y virtue of the Hylleraas-Uns old separation theorem [ 52 ]. Ho w ev er, this brute force metho d is extremely inecien t for basis set construction. As the n um b er of basis function increases, so do es computation time. While this is extremely limiting in the area of structure theory it is virtually imp ossible in dynamical metho ds suc h and END, where a large n um b er of calculations m ust b e made p er tra jectory (with man y tra jectories required for a single collision energy). F or this reason, a new metho d m ust b e devised for building basis sets that include correct represen tations of virtual orbitals. The construction of dynamically meaningful virtual orbitals is dep enden t most strongly up on t w o prop erties of the atomic orbitals, the energetics of the orbitals and the shap es of the orbitals. The energetics of the orbitals are the most ob vious concern. If the orbital energies are not correct, then the energy required for electronic transitions within the basis set will not prop erly mo del the energy dierences in nature. T o o small of an energy gap will result in increased transfer probabilit y while to o large of a energy dierence will ha v e the opp osite eect. In a single determinan tal treatmen t of the electrons, the orbital energies should mimic the energetics of the system. Therefore, the correct energetic of the atomic orbitals should b e a measure of the abilit y to correctly sim ulate dynamical transitions. The shap e of the orbital w a v e function is also imp ortan t for dynamical calculations. The no dal structure of the w a v e function determines regions in whic h

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66 electron densit y is zero or where it is non-zero. Again, a basis set m ust correctly mo del the electron densit y in an atom. More discussion ab out these prop erties will b e made in the next sections. Ho w ev er, at this p oin t it will suce to sa y that b oth of these prop erties can b e addressed quite eectiv ely through the use of STO basis sets. Sp ecically the energetics of an orbital is largely dep enden t up on the structure of the tail of the orbital w a v e function. As w as men tioned in the previous section, STO bases correctly describ e the tail of h ydrogenic orbitals and, lik ewise, repro duce the orbital tails in man y-electron atoms quite w ell. Additionally while single STO functions do not con tain an y no dal structure, linear com binations of STO functions can if carefully built. F or this reason, it seems most reasonable to construct basis sets using STO functions, at least initially 3.3.2 Ph ysical Justication for the Basis Set Construction Metho d As a part of the ph ysical justication of the prop osed metho d for basis set construction, one m ust rst return to the radial factor of a Slater-t yp e orbital basis function, giv en b y the form R S T O = (2 ) 2 n +1 (2 n )! 1 = 2 r n 1 e r : (3.34) When one compares this form to the h ydrogenic orbital functions, one nds that the orbital exp onen t is related directly to the n uclear c harge of the h ydrogenic atom in question, sp ecically = Z n ; (3.35) where Z is the n uclear c harge and n is the principal quan tum n um b er. In the case of a h ydrogenic system, only one electron is asso ciated with the system and therefore the electron will alw a ys feel the full n uclear c harge (that is, there is no n uclear shielding due to the presence of other electrons).

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67 This idea can b e extended to the construction of a w a v e function for an orbital in an y c hosen atom. It should b e noted, ho w ev er, that for a giv en electron in an arbitrary orbital in a man y-electron atom the n uclear c harge felt b y that electron will not b e the full n uclear c harge. Rather, the full n uclear c harge will b e shielded b y electron densit y lo cated b et w een the orbital in question and the n ucleus. This giv es rise to the concept of an eectiv e n uclear c harge, Z ef f The orbital exp onen t no w tak es the form = Z ef f n : (3.36) The eectiv e n uclear c harge will v ary as a function of the principal quan tum n um b er, in general an orbital with a smaller principal quan tum n um b er will ha v e a larger eectiv e c harge. Zener pro vided v alues for these eectiv e c harges based on v ariational calculations [ 53 ]. Slater [ 54 ] determined an empirical metho d for calculating the eectiv e n uclear c harge for an y arbitrary orbital and presen ted the equation Z ef f = Z s; (3.37) where Z in the full n uclear c harge and s is a screening constan t that is a function of the orbital and the n um b er of electrons. F or s and p -orbitals, the screening constan t w as dened to b e s = 0 : 35 N n + 0 : 85 N n 1 + 1 : 00 N n 2 ; (3.38) where N n is the n um b er of additional electrons in the same principal lev el, N n 1 is the n um b er of electrons in the principal lev el immediately lo w er, and N n 2 is the n um b er of all remaining electrons in lo w er principal lev els [ 54 ]. Ha ving dened the eectiv e c harge, Slater then suggests construction of atomic orbitals as single STO functions of the same form as Equation ( 3.31 ), with the exception that the orbital exp onen t tak es the form of Equation ( 3.36 ) and the principal quan tum n um b er n is replaced b y an eectiv e quan tum n um b er, n The eectiv e quan tum

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68 n um b er deviates from the true quan tum n um b er only for n > 3 [ 54 ]. This, the form of the w a v e function consists of a single STO for eac h set of n and l quan tum n um b ers, with the orbital exp onen t equal to the eectiv e n uclear c harge felt b y the corresp onding atomic orbital divided b y the eectiv e principal quan tum n um b er asso ciated with the orbital. Ha ving discussed a sp ecic metho d for the construction of basis sets for man yelectron atoms, it is no w time to consider some ph ysical asp ects of atomic orbitals in a bit more detail. One prop ert y that a basis set should prop erly mo del is the radial distribution of the electronic orbital, dened as [ 55 ] D nl ( r ) = r 2 [ R nl ( r )] 2 : (3.39) In the ab o v e equation, r is the radial distance from the n ucleus, n is the principal quan tum n um b er, l is the azim uthal quan tum n um b er, and R nl is the radial factor of the w a v e function for a giv en n and l The top panel of Figure 3.2 demonstrates the radial distribution functions for the rst four s -orbitals in the h ydrogen atom. The h umps in the radial densit y function for a giv en orbital indicate regions in whic h the probabilit y for the electron to exist is greatest. This feature demonstrates that the electron densit y corresp onding to an orbital tak es the form of concen tric shells of electron densit y [ 29 55 ]. F rom this it b ecomes clear that an atomic orbital with principal quan tum n um b er n will p osses n regions of electron densit y eac h b ecoming increasingly closer to the n ucleus but with smaller probabilit y This fact is mirrored in the n uclear screening expression adv anced b y Slater and giv en in Equation ( 3.38 ). The shielding due to electrons in the next lo w est principal lev el do not completely shield the n ucleus, rather they ha v e an only an 85% eectiv e shielding due to the p enetration of the higher principal lev els. This ph ysical feature of atomic orbitals is not limited to s -orbitals, nor is it limited to the description of the h ydrogen atomic orbitals. In Figure 3.2 the

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69 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 25 30 Dnl(r) (a.u.)Radial Distance (a.u.) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 5 10 15 20 25 30 Dnl(r) (a.u.)Radial Distance (a.u.) Figure 3.2: T op: Plot of the radial distribution function for the 1 s (|), 2 s (-), 3 s ( ), and 4 s (-) orbitals of the h ydrogen atom. Bottom: Plot of the radial distribution function for the 2 p (|), 3 p (-), and 4 p ( ) orbitals of the h ydrogen atom. b ottom panel demonstrates the radial densit y functions for the 2 p -, 3 p -, and 4 p -orbitals of the h ydrogen atom, where a similar shell structure is observ ed. F urthermore, Figure 3.3 sho ws the radial distribution functions for the o ccupied s and p -orbitals in the Ar atom, in the top and b ottom panel, resp ectiv ely In b oth plots, the orbitals w a v e functions that are plotted are the double-zeta w a v e functions of Clemen ti and Ro etti [ 45 ]. Again, the shell structure is clearly eviden t. A t this p oin t, the most signican t ph ysical attribute of these shells of electron densit y b ecomes apparen t. In the case of s and p -orbitals in the atoms of the rst

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70 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 Dnl(r) (a.u.)Radial Distance (a.u.) 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 Dnl(r) (a.u.)Radial Distance (a.u.) Figure 3.3: T op: Plot of the radial distribution function for the 1 s (|), 2 s (-), and 3 s ( ) orbitals of the argon atom. The 1 s -orbital is scaled b y a factor of t w o-thirds. Bottom: Plot of the radial distribution function for the 2 p (|) and 3 p (-) orbitals of the argon atom. few ro ws of a p erio dic table, the radial lo cation of the shells is largely indep enden t of the principal quan tum n um b er asso ciated with an orbital. In other w ords, all s -orbitals ha v e a shell of electron densit y that has roughly the same radial lo cation as the shell of electron densit y due to the 1 s orbital. Lik ewise, all s -orbitals with principle quan tum n um b er n ha v e n -1 shells that ha v e roughly the same radial lo cation as the n -1 lo w er energy s -orbitals. This is also true for p -orbitals. The consequence is that an y orbital of s symmetry p ossesses a partial c haracter of all of the lo w er energy s -orbitals. F urther, eac h of these c haracteristic shells exists within

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71 the same region of eectiv e n uclear c harge that is sp ecic to that shell of electron densit y No w, relating the regions of eectiv e n uclear c harge bac k to the idea of the Slater orbital exp onen t, this no w means that a giv en orbital can b e constructed as a linear com bination of all of the previous orbital basis functions, eac h with a sp ecic orbital exp onen t that relates to the eectiv e c harge region exp erienced b y the corresp onding shell of electron densit y Sp ecically the orbital w a v e function can b e written as n;l = n X i =1 c i N n;l e i r ; (3.40) where N n;l is the appropriate normalization factor for the STO in question, the c i 's are the expansion co ecien ts, and the i is the orbital exp onen t (eectiv e n uclear c harge) for the i th shell. On the surface, this is nothing new. Relating the orbital exp onen t to an eectiv e c harge w as prop osed b y Slater and the linear com bination is nothing more than a restatemen t of the sup erp osition principle [ 1 ] through whic h the HF metho d determines the HF eigenstates as a linear com bination of the basis v ectors [ 29 ]. Ho w ev er, this ph ysical insigh t do es serv e as an imp ortan t under-tone for the basis set construction prop osed in this w ork. 3.3.3 Construction of the Basis Set One b egins construction of the basis set b y dening a linear com bination of STO functions, eac h with orbital exp onen ts deriv ed from the eectiv e c harge exp erienced b y the orbital in question. The form of the eectiv e c harge ma y b e determined in an y n um b er of w a ys, the simplest of whic h is to emplo y Slater's form ulation for the screening constan t [ 54 ]. Ho w ev er, Slater's screening constan ts are empirically mo deled and ma y not b e as accurate as those determined b y other metho ds. Instead, one ma y consider the w ork of Clemen ti and Raimondi [ 56 ] as an extension of the earlier w ork of Slater and Zener. Clemen ti and Raimondi made

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72 a study of the elemen ts through Kr, represen ting eac h atomic orbital as a single STO function (that is a minimal basis set) with v ariable orbital exp onen ts. The exp onen ts where then optimized (with resp ect to the ground state energy) using an SCF pro cedure. A t this p oin t, the prop osed metho d has not deviated from older metho ds of basis set construction. By emplo ying this metho d, one can construct a minimal basis set for the ground state of the atom in question, ho w ev er, no recourse is a v ailable for construction of virtual orbitals. The optimization pro cess could b e extended to determine the orbital co ecien ts for the virtual states, but this w ould require calculations b ey ond the HF lev el to do so, as the transition energies w ould b e dep enden t up on electron correlation. The minim um lev el of theory that could b e emplo y ed w ould b e conguration in teraction. This w ould increase the computational eort required to optimize the virtual orbital exp onen ts. T o remedy this, a new approac h to determining these virtual orbital exp onen ts is prop osed in this w ork; a metho d that is extremely simplistic in its application, y et has pro v en to b e v ery p o w erful. The metho d b egins through the in v estigation of the b eha vior of Clemen ti's shielded orbital exp onen ts as a function of the atomic n um b er. Figure 3.4 demonstrates the functional b eha vior for the 1 s orbital exp onen ts through Kr. As can b e seen from the gure, the orbital exp onen ts exhibit a v ery linear dep endence on the atomic n um b er. The top panel of Figure 3.5 sho ws the dep endence of the 2 s orbital exp onen ts on the atomic n um b er. There are t w o data sets in this gure, the data of Clemen ti and Raimondi [ 56 ], whic h are denoted using the plus sym b ols, and the data from the presen t w ork, whic h are denoted using the op en circles. Clemen ti's data still demonstrates a linear dep endence of the exp onen t on the atomic n um b er, ho w ev er, it b ecomes clear at this p oin t that more than one linear region is observ ed. As an example, the orbital exp onen ts for the elemen ts from Li through Ne ha v e a

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73 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Orbital Exponent (a.u.)Atomic Number Figure 3.4: Plot of the 1 s orbital exp onen t for the atoms through Kr as a function of atomic n um b er. The data are from Clemen ti and Raimondi (+). sligh tly dieren t slop e and in tercept than for the remaining elemen ts. Eac h blo c k of elemen ts will ha v e a sligh tly dieren t slop e. The data p oin ts curren t to this w ork will b e discussed at length later. The data in the b ottom panel of Figure 3.5 sho w the same relation for the 2 p orbital exp onen ts. Lik ewise, the plots in Figure 3.6 demonstrate the dep endence for the 3 s (top) and 3 p (b ottom) orbital exp onen ts and those in Figure 3.7 presen t the data for the 4 s (top) and 4 p (b ottom) orbitals. The most striking feature when comparing all of the previous plots is that, while the individual linear regions b ecome more distinct from one another as the principal and azim uthal quan tum n um b ers increase, the (lo cal) linear dep endence of the orbital exp onen t on the atomic n um b er is still quite strong. It is this feature that denes the prop osed metho d for virtual orbital construction. Discussion m ust no w b e made with regards to the remaining sets of data p oin ts, those denoted with the op en circles. These data p oin ts are new to this w ork and are deriv ed from the data of Clemen ti and Raimondi. Sp ecically these p oin ts are orbital exp onen ts corresp onding to virtual orbitals for the atoms in question.

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74 0 2 4 6 8 10 12 14 0 5 10 15 20 25 30 35 40 Orbtial Exponent (a.u.)Atomic Number 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 30 35 40 Orbtial Exponent (a.u.)Atomic Number Figure 3.5: Plot of the 2 s and 2 p orbital exp onen ts for the atoms through Kr as a function of atomic n um b er. T op: The 2 s orbital exp onen ts. Bottom: The 2 p orbital exp onen ts. The data are from Clemen ti and Raimondi (+) and from the presen t w ork ( ). These virtual orbital exp onen ts are determined b y rst considering the h ydrogen atom. As the h ydrogen atom has only a single electron, there will nev er b e an y n uclear shielding for that atom. This the electron will alw a ys exp erience the same eectiv e n uclear c harge (of unit magnitude) no matter in whic h orbital the electron has probabilit y for existing. This means that Z ef f = 1 alw a ys, and the orbital exp onen t for an y orbital in the H atom is just equal to the recipro cal of the principal quan tum n um b er. A t this p oin t, one mak es reference to the linear b eha vior of the o ccupied orbital exp onen ts. The virtual orbital exp onen ts are then

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75 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 35 40 Orbital Exponent (a.u.)Atomic Number 0 1 2 3 4 5 6 7 0 5 10 15 20 25 30 35 40 Orbital Exponent (a.u.)Atomic Number Figure 3.6: Plot of the 3 s and 3 p orbital exp onen ts for the atoms through Kr as a function of atomic n um b er. T op: The 3 s orbital exp onen ts. Bottom: The 3 p orbital exp onen ts. The data are from Clemen ti and Raimondi (+) and from the presen t w ork ( ). determined b y making a linear in terp olation b et w een the H atom virtual orbital exp onen t and the exp onen t corresp onding to the rst a v ailable o ccupied orbital in that symmetry F or example, in the case of the 3 p orbitals, the in terp olation is made b et w een the exp onen t corresp onding to the H atom 3 p orbital and the exp onen t that corresp onds to the 3 p orbital of Al (atomic n um b er = 13). This metho d for determining the exp onen ts that corresp ond to virtual atomic orbitals relies on w ell-do cumen ted trends exhibited b y a parameter that is related directly to a ph ysical prop ert y namely the trend in the regions of eectiv e c harge

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76 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 35 40 Orbital Exponent (a.u.)Atomic Number 0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 30 35 40 Orbtial Exponent (a.u.)Atomic Number Figure 3.7: Plot of the 4 s and 4 p orbital exp onen ts for the atoms through Kr as a function of atomic n um b er. T op: The 4 s orbital exp onen ts. Bottom: The 4 p orbital exp onen ts. The data are from Clemen ti and Raimondi (+) and from the presen t w ork ( ). as demonstrated earlier in this section. Ho w ev er, it is imp ortan t to note that a n um b er of sev ere assumptions ha v e b een made. P erhaps the t w o strongest assumptions are that the virtual orbital exp onen ts demonstrate the same linear b eha vior as do the o ccupied orbital exp onen ts and the assumption that there is not a large c hange in the magnitude of the orbital exp onen t as one transitions from the o ccupied orbital exp onen ts to the uno ccupied orbital exp onen ts. While neither of these assumptions can b e tested without the construction of energy-optimized virtual orbital w a v e functions, the sev erit y of the assumptions is tolerated in lieu

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77 of the ease of application. And, in spite of these assumptions, the calculations using basis sets constructed from this starting p oin t ha v e yielded surprisingly go o d results for b oth stationary state and dynamical calculations, as will b e demonstrated in the next section. Once the in terp olations ha v e b een accomplished, the next step is to construct the atomic orbital w a v e functions. This is b egun with the w a v e function for the 1 s orbital, whic h is represen ted b y a single Slater-t yp e orbital, 1 s = N 1 s ( 1 s )e 1 s r : (3.41) The expansion co ecien t in just the normalization co ecien t for an STO basis function with n = 1 and with orbital exp onen t 1 s F rom this, the w a v e function of the 2 s orbital can then b e constructed as a linear com bination of the 1 s w a v e function (pro viding for the cusp) and a single STO basis function with n = 2 that is used to represen t the tail p ortion of the orbital. The orbital w a v e function tak es the form 2 s = c 1 s N 1 s ( 1 s )e 1 s r + c 2 s N 2 s ( 2 s ) r e 2 s r : (3.42) Again, the terms N 1 s and N 2 s are the normalization co ecien ts for the sp ecied STOs. In Equation ( 3.42 ), t w o expansion co ecien ts m ust b e determined. This requires the sim ultaneous solution of a set of t w o equations. In this case the t w o equations are the normalization condition for the 2 s orbital ( h 2 s j 2 s i = 1) and the orthogonalit y of the 1 s orbital with the 2 s orbital ( h 1 s j 2 s i = 0). It b ecomes clear at this p oin t that the ab o v e pro cess can b e iterated o v er as man y s -orbitals as are required for an atomic basis set, b e they o ccupied or virtual. Sp ecically the ns orbital w a v e function is dened as ns = n X i =1 c is N is ( is ) r i 1 e is r : (3.43)

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78 As b efore, the w a v e function has n undetermined co ecien ts and therefore requires the solution of a set of n sim ultaneous equations. These equations tak e the form the normalization of the ns w a v e function ( h ns j ns i = 1) and the orthogonalit y of the ns w a v e function with the other w a v e functions ( h 1 s j ns i = h 2 s j ns i = ::: = h ( n 1) s j ns i = 0). This forms a set of n sim ultaneous equations that can b e used to determine a set of expansion co ecien ts. The construction of the p -orbital w a v e functions follo ws the same sc hema. This prop osed construction metho d is emplo y ed through a template designed for an y commercial computational pac k age, suc h as Maple, in to whic h the orbital exp onen ts are input. The program then calculates the expansion co ecien ts b y solving the orthonormalit y conditions for eac h set of orbitals. The p o w er of this metho d is that it do es not require an y exp ensiv e non-linear energy optimizations. F urthermore, it allo ws for a general construction of an y atomic orbital, either o ccupied or virtual, pro vided that the orbital exp onen t is kno wn or can b e in terp olated using the ab o v e men tioned metho d. W a v e functions ha v e b een built for the atoms from He through Ne. T able 3.2 presen ts the STO exp onen ts and co ecien ts for He, Li, and Be. These exp onen ts and co ecien ts w ere determined using the previously describ ed metho d. T able 3.3 lists the w a v e function parameters for B, C, and N. Lastly T able 3.4 con tains the exp onen ts and co ecien ts for O, F, and Ne. As a nal step, the STO w a v e functions m ust then b e expanded in a basis GTO functions to allo w for computations to b e p erformed. There are t w o main metho ds b y whic h this is accomplished. The rst metho d is through use of a linear least-squares tting program whic h will determine the b est set of GTO exp onen ts and expansion co ecien ts. The program that has b een emplo y ed for some of the results rep orted in this w ork used a v ariation of the Amo eba program from Numerical Recip es [ 37 ]. In this metho d the GTO orbital exp onen ts w ere

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79 T able 3.2: Slater Exp onen ts and Co ecien ts for He, Li, and Be. A tom: Helium Conguration: H e : 1 s 2 2 s 0 2 p 0 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 1.6875 0.5698 0.3836 0.2847 0.6777 0.4185 0.2935 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.29929 -1.04383 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.22486 -0.97897 1.37212 0.00000 3 p 0.75715 -1.25430 0.00000 4 s 0.17763 -0.85950 1.84082 -1.72886 4 p 0.58192 -1.40551 1.52534 A tom: Lithium Conguration: Li : 1 s 2 2 s 1 2 p 0 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 2.6906 0.6396 0.4338 0.3193 0.8554 0.5036 0.3370 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.16487 -1.01350 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.13471 -0.90398 1.36057 0.00000 3 p 0.69166 -1.21589 0.00000 4 s 0.04890 -0.38438 1.11657 -1.56430 4 p 0.48801 -1.20388 1.42094 A tom: Beryllium Conguration: B e : 1 s 2 2 s 2 2 p 0 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 3.6848 0.9560 0.4841 0.3540 1.0330 0.5888 0.3805 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.18228 -1.01832 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.09188 -0.54041 1.13214 0.00000 3 p 0.65076 -1.19310 0.00000 4 s 0.03755 -0.23713 0.87282 -1.46603 4 p 0.43111 -1.08063 1.35846

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80 T able 3.3: Slater Exp onen ts and Co ecien ts for B, C, and N. A tom: Boron Conguration: B : 1 s 2 2 s 2 2 p 1 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 4.6795 1.2881 0.5343 0.3886 1.2107 0.6740 0.4241 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.21294 -1.02242 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.07096 -0.37100 1.06382 0.00000 3 p 0.62253 -1.17794 0.00000 4 s 0.03208 -0.17493 0.78864 -1.42357 4 p 0.39304 -0.99801 1.31754 A tom: Carb on Conguration: C : 1 s 2 2 s 2 2 p 2 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 5.6727 1.6083 0.5846 0.4233 1.5679 0.7591 0.4676 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.22393 -1.02477 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.05798 -0.28600 1.03828 0.00000 3 p 0.47856 -1.10861 0.00000 4 s 0.02788 -0.14169 0.75459 -1.40332 4 p 0.29066 -0.86509 1.27113 A tom: Nitrogen Conguration: N : 1 s 2 2 s 2 2 p 3 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 6.6651 1.9237 0.6348 0.4580 1.9170 0.8443 0.5111 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.23080 -1.02629 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.04925 -0.23459 1.02580 0.00000 3 p 0.39933 -1.07678 0.00000 4 s 0.02866 -0.14023 0.90396 -1.52899 4 p 0.23578 -0.79242 1.24296

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81 T able 3.4: Slater Exp onen ts and Co ecien ts for O, F, and Ne. A tom: Oxygen Conguration: O : 1 s 2 2 s 2 2 p 4 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 7.6579 2.2458 0.6851 0.4926 2.2266 0.9294 0.5546 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.23710 -1.02772 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.04298 -0.19877 1.01854 0.00000 3 p 0.35976 -1.06274 0.00000 4 s 0.01173 -0.05606 0.46221 -1.27638 4 p 0.20750 -0.75025 1.22490 A tom: Fluorine Conguration: F : 1 s 2 2 s 2 2 p 5 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 8.6501 2.5638 0.7353 0.5273 2.5500 1.0146 0.5981 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.24136 -1.02871 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.03835 -0.17388 1.01419 0.00000 3 p 0.32735 -1.05222 0.00000 4 s 0.02018 -0.09298 0.71660 -1.37637 4 p 0.18516 -0.71747 1.21094 A tom: Neon Conguration: N e : 1 s 2 2 s 2 2 p 6 3 s 0 3 p 0 4 s 0 4 p 0 s-F unctions p-F unctions Exp onen ts Exp onen ts 1 s 2 s 3 s 4 s 2 p 3 p 4 p 9.6421 2.8792 0.7856 0.5619 2.8792 1.0997 0.6416 Co ecien ts Co ecien ts 1 s 1.00000 0.00000 0.00000 0.00000 2 s 0.24439 -1.02943 0.00000 0.00000 2 p 1.00000 0.00000 0.00000 3 s 0.03483 -0.15570 1.01138 0.00000 3 p 0.30169 -1.04452 0.00000 4 s 0.01858 -0.08429 0.71010 -1.37115 4 p 0.16788 -0.69204 1.20000

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82 constrained to b e ev en-temp ered. Dep ending on the use of the basis set, an y n um b er of GTO functions (up to ten) w ere used for eac h STO t. While the rst tting metho d emplo ys a fairly simple linear optimization to determine exp onen ts and co ecien ts for GTO expansions of STO orbitals, the second metho d do es not require an y suc h optimizations. This metho d uses the results of Stew art [ 41 ]. Stew art p erformed least squares calculations to determine the b est set of parameters that allo ws for the construction of an expansion of up to six GTO functions for a sp ecied single STO function. As previously men tioned, this metho d do es not require tedious optimizations, as they ha v e already b een p erformed. This allo ws for a simple template to b e constructed in a n umerical spreadsheet program. The orbital exp onen ts are input in to this spreadsheet template, along with the required parameters from Stew art's pap er, and the GTO orbital exp onen ts and expansion co ecien ts are the resulting output. While the metho d for basis set construction outlined in this section is v ery simplistic in its application, the ph ysical underpinning of the construction is quite strong and main tained throughout the metho d. F urthermore, this formalism is easy to emplo y with no costly or time-consuming optimizations required if engineered correctly In the follo wing section, results that ha v e b een obtained with basis sets constructed using this metho d are presen ted for comparison with sev eral common sto c k basis sets that w ere built using energy optimization metho ds. 3.4 Comparativ e Results This section will pro vide comparisons b et w een basis sets constructed using the metho d prop osed in the previous section and with some of the more commonly used sto c k basis sets, suc h as the 3-21G, the 6-31G, and the 6-31G* basis sets. The comparisons will b e made suc h that a computation p erformed with, for example, a 6-31G basis set will b e compared with a newly constructed basis set with the same size parameters (same n um b er of expansions p er orbital). Comparisons will b e

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83 made for sev eral t yp es of ph ysical prop erties, sp ecically for the excitation energies within the sp ecic atom, for c harge transfer probabilities and cross sections, and for the vibrational and electronic prop erties of sev eral diatomic and triatomic molecules. These rst t w o comparisons are for prop erties that are imp ortan t to dynamical pro cesses. The third comparison relates prop erties that are traditionally calculated based on energy (suc h as diagonalization of the Hessian to obtain vibrational frequencies). This will oer comparison b et w een energy optimized basis sets with the basis sets constructed from the curren t metho d (with no energy optimizations).3.4.1 A tomic Energetics In this section, the relativ e accuracy of electronic excitations within atoms will b e compared. Six t yp es of basis sets will b e used in this comparison, including three sto c k basis sets (the 3-21G, 6-31G, and 6-31G** basis sets) and three comparable basis sets built using the previously prop osed metho d (here called the 3-21B, 6-31B, and 6-31B** basis sets). The new basis sets ha v e b een constructed suc h that the same n um b er of primitiv e Gaussians are used for eac h orbital con traction. This will help to ensure that meaningful comparisons can b e made b et w een the energy optimized sto c k basis sets and the newly constructed basis sets. The structure of the new, dynamically consisten t basis sets can b e found in the basis set library lo cated in the App endix. The data in this section are comprised of excitation energies as calculated using the Gaussian 98 computational suite [ 57 ]. The excitation energies are calculated as the absolute energy dierence b et w een the t w o states in question. The absolute energies are calculated using the m ulti-congurational capabilities of Gaussian 98, as found within the CASSCF routine [ 44 ]. The complete activ e space is dened to b e the set of v alence shell electrons and the v alence shell and all

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84 T able 3.5: A tomic energies and electronic excitations in Helium Sto c k Basis Sets A tomic HF Energy (Exp erimen tal: -2.9031840 a.u.) 3-21G 6-31G 6-31G** (a.u.) (a.u.) (a.u.) -2.8505767 -2.8701621 -2.8873650 Excitation Energies Expt. 3-21G % 6-31G % 6-31G** % T ransition Excitation Excitation Error Excitation Error Excitation Error (cm 1 ) (cm 1 ) (cm 1 ) (cm 1 ) 1 s 2 ( 1 S ) 1 s 2 s ( 3 S ) 159843.3 442932.5 177 322814.3 102 326589.9 104 1 s 2 ( 1 S ) 1 s 2 s ( 1 S ) 166264.7 557807.8 235 421708.2 154 425452.4 156 1 s 2 ( 1 S ) 1 s 2 p ( 3 P ) 169074.1 N/A N/A 483411.3 183 1 s 2 ( 1 S ) 1 s 2 p ( 1 P ) 171122.2 N/A N/A 558949.8 227 Dynamically Consisten t Basis Sets A tomic HF Energy (Exp erimen tal: -2.9031840 a.u.) 3-21B 6-31B 6-31B** (a.u.) (a.u.) (a.u.) -2.7123567 -2.8118598 -2.8119307 Excitation Energies Expt. 3-21B % 6-31B % 6-31B** % T ransition Excitation Excitation Error Excitation Error Excitation Error (cm 1 ) (cm 1 ) (cm 1 ) (cm 1 ) 1 s 2 ( 1 S ) 1 s 2 s ( 3 S ) 159843.3 146507.5 -8.34 155196.7 -2.91 155212.3 -2.90 1 s 2 ( 1 S ) 1 s 2 s ( 1 S ) 166264.7 155066.0 -6.74 162821.9 2.07 162778.0 -2.10 1 s 2 ( 1 S ) 1 s 2 p ( 3 P ) 169074.1 N/A N/A 169004.0 -0.04 1 s 2 ( 1 S ) 1 s 2 p ( 1 P ) 171122.2 N/A N/A 171592.9 0.28

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85 T able 3.6: A tomic energies and electronic excitations in Lithium Sto c k Basis Sets A tomic HF Energy (Exp erimen tal: -7.8848995 a.u.) 3-21G 6-31G 6-31G** (a.u.) (a.u.) (a.u.) -7.3815132 -7.4312358 -7.4313723 Excitation Energies Expt. 3-21G % 6-31G % 6-31G** % T ransition Excitation Excitation Error Excitation Error Excitation Error (cm 1 ) (cm 1 ) (cm 1 ) (cm 1 ) 2 s 1 ( 2 S ) 2 p 1 ( 2 P ) 14903.8 14620.3 -1.90 15626.9 4.85 15647.1 4.99 2 s 1 ( 2 S ) 3 s 1 ( 2 S ) 27205.8 43415.8 59.6 46379.9 70.5 48098.6 76.8 2 s 1 ( 2 S ) 3 p 1 ( 2 P ) 30925.9 43662.6 41.2 49549.6 60.2 46408.9 50.1 2 s 1 ( 2 S ) 3 d 1 ( 2 D ) 31283.2 N/A N/A 112671.0 260 Dynamically Consisten t Basis Sets A tomic HF Energy (Exp erimen tal: -7.8848995 a.u.) 3-21B 6-31B 6-31B** (a.u.) (a.u.) (a.u.) -7.3252176 -7.4161167 -7.4161167 Excitation Energies Expt. 3-21B % 6-31B % 6-31B** % T ransition Excitation Excitation Error Excitation Error Excitation Error (cm 1 ) (cm 1 ) (cm 1 ) (cm 1 ) 2 s 1 ( 2 S ) 2 p 1 ( 2 P ) 14903.8 14810.7 -0.62 15678.5 5.20 15678.5 5.20 2 s 1 ( 2 S ) 3 s 1 ( 2 S ) 27205.8 26096.7 -4.08 26795.2 -1.51 26795.2 -1.51 2 s 1 ( 2 S ) 3 p 1 ( 2 P ) 30925.9 43823.9 41.7 44431.2 43.7 44431.2 43.7 2 s 1 ( 2 S ) 3 d 1 ( 2 D ) 31283.2 N/A N/A 35503.5 13.5

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86 virtual orbitals in the giv en basis set. By using the CASSCF metho dology essen tial correlation can b e accoun ted for in the basis set comparisons. The data in T able 3.5 demonstrates the excitations for the helium atom. The top part of the table relates the exp erimen tal ground state atomic energy and the excitation energies using the sto c k basis sets. The b ottom half of the table sho ws the same v alues calculated in the newly constructed and dynamically consisten t basis sets. It b ecomes clear immediately up on insp ection of the data that the sto c k basis sets pro vide a m uc h more accurate ground state atomic energy Ho w ev er, this is to b e exp ected, as the sto c k basis sets w ere constructed in suc h a w a y as to minimize the Hartree-F o c k atomic energy and the dynamically consisten t basis sets w ere not. It should b e noted that, ev en in the w orst case, the dynamically consisten t basis sets are no more than 6.57 p ercen t in error with the exp erimen tal energy tak en from Mo ore [ 58 ]. The most striking feature of the table is the excitation energy comparisons. The exp erimen tal excitation energies are tak en from Bac her and Goudsmit [ 59 ]. The sto c k basis sets do not represen t the orbital-to-orbital excitation energies w ell. One's atten tion can rst b e directed to the p ercen t errors of the calculated excitation energies (using sto c k basis sets) with resp ect to the exp erimen tal v alues. The smallest error is ab out 100 p ercen t. Ev en more imp ortan t is the fact that all of the excitations corresp ond to virtual orbitals that are b ound states in the He atom. The excitation energies are all greater in magnitude than the ionization energy of the atom (ab out 197000 cm 1 ) [ 59 ]. This fact is denoted b y the red coloration of the excitation energy v alues. By con trast, the dynamically consisten t basis sets constructed in this w ork sho w a remark able impro v emen t in the excitation energetics. The p ercen t errors are reduced from a minim um of 102 p ercen t in the sto c k basis sets to a maxim um

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87 of 8.34 p ercen t in the newly constructed basis sets. Moreo v er, all of the newly constructed orbitals represen t b ound states in the atom. T able 3.6 relates the same data, though for the lithium atom. The exp erimental atomic energy v alue is calculated in this case b y considering the correlation energy obtained b y Eggarter and Eggarter using second-order p erturbation metho ds [ 60 ], and the excitation energies are from Bac her and Goudsmit [ 59 ]. The atomic energies demonstrate the same pattern as the helium atom: the sto c k basis sets oer go o d represen tations of the atomic energy while the newly constructed basis sets are not as go o d (though no w the largest p ercen t error is only 7.1 p ercen t). The excitation energy data oers a considerably dieren t comparison than in the case of helium. P articularly one nds that in this case the sto c k basis sets do allo w for a go o d represen tation of the 2 p virtual orbital excitation whic h are, in fact, b etter than the orbitals arising from the new basis sets. Ho w ev er, the 3 s 3 p and 3 d orbitals are all un b ound. The dynamically consisten t basis sets all pro vide go o d 2 p 3 s and 3 d virtual orbitals, y et p erform v ery p o orly in the description of the 3 p orbitals. This can b e attributed to either of the principal assumptions made in the in terp olation metho d (c.f. Section 3.3.3 ). 3.4.2 Charge T ransfer Results F rom the preceding data, it b ecomes clear that a basis set can b e constructed that has a relativ ely small size and ph ysically meaningful excitations in to virtual orbitals. While this is a ph ysical c haracteristic that is v ery imp ortan t in man y dynamical prop erties, one m ust sp ecically in v estigate the b eha vior of the basis sets when calculating suc h dynamical prop erties. In this section, c harge transfer probabilities are calculated using the 6-31G sto c k basis set and sev eral basis sets constructed using the newly prop osed metho dology The con traction sc hemes for eac h of the basis sets are giv en in the App endix.

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88 Three c harge transfer systems will b e in v estigated. The rst is the nearresonan t c harge transfer in the H + /Li collision system. In this pro cess, the n = 2 orbitals are in near-resonance with the 2 p orbitals in Li (with orbital energies of -0.125 a.u. and -0.130 a.u., resp ectiv ely) [ 61 ]. Lik ewise, the n = 3 orbitals of H are in near-resonance with the 3 p orbitals in Li (with orbital energies of -0.055 a.u. and 0.057 a.u., resp ectiv ely) [ 61 ]. These near-resonances should result in a few regions of large probabilit y for transfer, as the electron can b e excited in to one of the virtual orbitals in Li and then transfer o v er to the H atom. The second system is the resonan t c harge transfer in the Li + /Li collision system. In this case, the orbitals in b oth collision sp ecies ha v e the same energies, promoting strong resonances in the c harge transfer pro cess. Finally the resonan t transfer b et w een He + and He is in v estigated. The minimal END formalism has b een applied successfully to in v estigations of resonan t c harge transfer pro cesses, particularly the collision of H + /H [ 62 ]. The same metho ds are emplo y ed in this in v estigation. Sp ecically the probabilit y for electron transfer is calculated to b e the dierence in Mullik en p opulation b et w een the inciden t pro jectile and the fastest particle after collision, the dieren tial cross section is calculated using the distinguishable or iden tical particle scattering amplitudes (as the case w arran ts), the scattering amplitude is calculated from the small angle Sc hi Appro ximation, and the total cross section for resonan t transfer is calculated using the semi-classical form ula RT = Z 1 0 bP ( b )d b; (3.44) where b is the impact parameter and P ( b ) is the transfer probabilit y All of the calculations in this section w ere p erformed using end yne, v ersion 5 [ 63 ]. The rst comparison to b e made is the transfer probabilit y Figure 3.8 demonstrates the near-resonan t c harge transfer probabilit y in the H + /Li collision

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89 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 Transfer ProbabilityImpact Parameter (au) Figure 3.8: Comparison of the probabilit y for near-resonan t c harge transfer b et w een H + and Li at 10 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02 (blac k). The BJK01 basis set for H w as used for eac h run. system at 10 k eV collision energy The red line represen ts the transfer probabilit y as calculated using the 6-31G basis set. The blue line sho ws the probabilit y as calculated using the comparable 6-31B basis set. It b ecomes immediately ob vious that the transfer probabilit y is signican tly dep enden t up on the basis set emplo y ed. As sho wn in the previous section, the 6-31B basis set for Li pro vides m uc h more accurate excitations in to the 2 p and 3 s orbitals than do es the 6-31G basis set. A t small impact parameter, more collisional energy is utilized for electronic excitation, therefore, the increase in transfer probabilit y at small impact parameter that is accrued b y c hanging from the 6-31G to the 6-31B basis set is due to the impro v ed description of the virtual orbitals on the Li atom. F urther impro v emen t is made b y altering the orbital exp onen t in the 3 p STO for the Li atom. This results in an impro v ed represen tation of the 3 p virtual orbital. F urthermore, a larger n um b er of con tractions can also b e utilized. The BJK01 (purple line) and BJK02 (blac k line) basis sets are a result of these mo dications to the 6-31B basis set. The

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90 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 Transfer ProbabilityImpact Parameter (au) Figure 3.9: Comparison of the probabilit y for near-resonan t c harge transfer b et w een H + and Li at 1 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02 (blac k). The BJK01 basis set for H w as used for eac h run. BJK02 basis set pro vides the most accurate excitation energies (all less than 5 cm 1 ). Due to this fact and the rapid con v ergence seen b et w een the dynamically consisten t basis sets, it can b e inferred that the BJK02 basis set pro vides an excellen t description of the near-resonan t c harge transfer b et w een H + and Li. Figure 3.9 represen ts the transfer probabilit y for the same reaction, though at a collision energy of 1000 eV. A t this energy the higher virtual orbitals are less accessible energetically Therefore, it can b e exp ected that the sto c k basis sets will con v erge more strongly with the dynamically consisten t basis set. This is what can b e inferred from the gure. A t v ery small impact parameter, the energy dep osition from the pro jectile to the target is enough to allo w for excitation to the lo w er lying states, but b ey ond ab out 3 atomic units of impact parameter the probabilities con v erge. Ho w ev er, as the description of excitation energies impro v es, the probabilit y in the small impact parameter region is substan tially impro v ed.

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91 0.001 0.01 0.1 1 10 100 1000 10000 100000 1e+06 1e+07 1e+08 0.01 0.1 1 10 Differential Cross Section ( 2/ster.)Lab Frame Scattering Angle (deg.) Figure 3.10: Comparison of the dieren tial cross section for near-resonan t c harge transfer b et w een H + and Li at 10 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02 (blac k). The BJK01 basis set for H w as used for eac h run. Once the transfer probabilit y distribution is determined, the dieren tial cross section for transfer can b e calculated. Figure 3.10 pro vides the dieren tial cross sections for electron transfer at 10 k eV collision energy While all four basis sets pro vide for the same structure for the dieren tial cross section, the newly constructed basis sets all sho w an increase in the magnitude o v er the sto c k basis set. F urthermore, the three newly constructed basis sets, all of v arying con traction sizes, sho w a rapid con v ergence. Unfortunately no exp erimen tally measured dieren tial cross sections are a v ailable for this system, whic h prohibits an y comparison with exp erimen t and only allo ws for discussion of con v ergence. Finally the total cross section for transfer can b e ev aluated. This prop ert y is calculated using Equation ( 3.44 ). Figure 3.11 sho ws the total cross section as calculated using the END formalism and compared to other theoretical as w ell as exp erimen tal v alues. The exp erimen tal data come from the w ork of V arghese et al. [ 64 ] and Auma yr et al. [ 65 ]. The theoretical data are from the w ork of Allan

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92 10 100 100 1000 10000 Total Cross Section (1016cm2)Collision Energy (eV) Figure 3.11: Comparison of the total cross section for near-resonan t c harge transfer b et w een H + and Li as a function of energy The exp erimen tal data are: V arghese et al. (+) and Auma yr et al. ( ). The theoretical data are: Allan et al (|) and F ritsc h and Lin (-). F or the END data the follo wing Li basis sets are used: 631G ( 2 ), 6-31B ( ), BJK01 ( 4 ), BJK02 ( 3 ). The BJK01 basis set for H w as used for eac h run. et al. [ 66 ] and F ritsc h and Lin [ 67 ]. The most striking feature of this gure is the fact that all but one of the basis sets pro vide data that are quite go o d in comparison with the exp erimen tal data. F urthermore, those basis sets that oer go o d comparison to exp erimen t also sho w go o d precision amongst the cross section v alues. The 6-31B basis set app ears to oer the b est comparison with exp erimen t. A p oin t of consternation arises from the fact that data set that deviates most from the exp erimen tal data corresp onds to the BJK02 basis set, in whic h the excitation energies are most accurate in comparison with the exp erimen tal v alues. Also of in terest is the comparison of the BJK02 data with the data of Allan et al. [ 66 ] whic h is represen ted b y the t w o solid lines. Allan emplo y ed a molecular basis set of Slater orbital functions that had to b e cen tered on either the pro jectile or the target. The lo w er line represen ts the cross section calculated when the molecular basis set w as cen tered on the Li, while the upp er line is the cross section when

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93 the basis set w as cen tered on the proton. The data corresp onding to the BJK02 basis set matc hes the data with the molecular basis cen tered on the Li atom. While this feature is y et to b e completely understo o d, it do es suggest that the newly constructed atomic basis set can b e emplo y ed in the construction of molecular basis sets with comparable results as Slater functions. F urthermore, the data reiterates the p oin t that basis set construction is only one part of the panoply of factors con tributing to dynamical calculations, and no particular dynamically consisten t basis set will ev er act as a silv er bullet to correct for all of these features. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 Transfer ProbabilityImpact Parameter (a.u.) Figure 3.12: Comparison of the probabilit y for resonan t c harge transfer b et w een Li + and Li at 1 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), BJK02 (blue). The next set of data demonstrate the c harge transfer probabilit y as a function of impact parameter for the resonan t c harge transfer in the Li + Li collision system. Figure 3.12 demonstrates a similar trend to the previous collision system. The transfer probabilit y for the 6-31G basis set con v erges with the transfer probabilit y for the BJK02 basis set at larger impact parameters, but the BJK02 basis set sho ws an increase in transfer probabilit y at smaller impact parameter, where the higher excited states will b e energetically accessible. The same probabilities

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94 are plotted in Figure 3.13 except that the collision energy is 10 k eV. It can b e seen that there is a region of impact parameters in whic h the transfer probabilit y is greater for the sto c k basis set than for the dynamically consisten t basis set. This ma y b e a result of excitation energy in to the 2 p orbital in the 6-31G b eing larger in exp erimen tally determined. This will result in an a v ailable energy lev el that can b e accessed in the 6-31G basis that is not presen t in the BJK02 basis set. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 Transfer ProbabilityImpact Parameter (a.u.) Figure 3.13: Comparison of the probabilit y for resonan t c harge transfer b et w een Li + and Li at 10 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), BJK02 (blue). Lik ewise, Figure 3.14 sho ws the features of the dieren tial cross section at 1.0 k eV collision energy As b efore, the c hange in dieren tial cross section is subtle, but the dynamically consisten t basis set sho ws an increase in magnitude o v er the sto c k basis set. Ho w ev er, without exp erimen tal data to compare with, one can only sp eculate as to an y p ossible impro v emen t in the dieren tial cross section. Finally Figure 3.15 pro vides a comparison of the calculated total cross sections for transfer with exp erimen tal and theoretical data. The exp erimen tal data are tak en from Loren ts et al. [ 68 ] and the theoretical data are from Sak ab e and Iza w a [ 69 ]. Immediately one can see that the total cross sections calculated using

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95 1 10 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09 1e+10 0.01 0.1 1 10 Differential Cross Section ( 2/ster.)Lab Frame Scattering Angle (deg.) Figure 3.14: Comparison of the dieren tial cross section for resonan t c harge transfer b et w een Li + and Li at 1 k eV collision energy The follo wing Li basis sets are used: 6-31G (red), BJK02 (blue). the 6-31G basis set sho w the p o orest comparison with exp erimen t. The 6-31B demonstrates b etter agreemen t, but the BJK02 basis set sho ws the b est agreemen t with exp erimen t. It should b e noted that Sak ab e and Iza w a calculated there cross section using a parameterized form ula based on ionization p oten tials rather than through dynamical calculations. F urthermore, the BJK02 basis set sho ws the predicted oscillatory structure at higher collision energy The Li systems where c hosen for comparisons b ecause they are b oth pseudoone-electron systems, in whic h the single v alence electron is transfered and correlation eects are minimal. Ho w ev er, as demonstrated in the previous section, the 2 p excitation energy for the sto c k basis set is relativ ely close to the exp erimen tally observ ed v alue. As one w old reasonably exp ect the transition to the 2 p orbital to predominate at most collision energies, it is not unreasonable to exp ect only a mo derate c hange in the measured c harge transfer in these systems. The nal collision system, He + on He, will allo w for a m uc h clearer demonstration of the dierence

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96 10 100 1000 100 1000 10000 Total Cross Section ( 2)Lab Frame Collision Energy (eV) Figure 3.15: Comparison of the total cross section for resonan t c harge transfer b et w een Li + and Li at as a function of collision energy The exp erimen tal data are form Loren ts et al. ( ). The theoretical data are from Sak ab e and Iza w a (|). F or the END data the follo wing Li basis sets are used: 6-31G ( 2 ), 6-31B ( ), BJK02 ( 4 ). that can o ccur in c harge transfer when energetically correct excitations are mo deled in the basis set. Figure 3.16 sho ws the resonan t c harge transfer probabilit y as a function of impact parameter for the He + He collision system at 5 k eV. One immediately can see the highly oscillatory nature that is indicativ e of resonan t c harge transfer pro cesses b et w een atoms. It b ecomes clear that for impact parameters greater than ab out 1 a.u., the data for the three basis sets con v erge. This is the region in whic h the transfer o ccurs directly from the 1 s orbital of the target in to the 1 s orbital of the pro jectile. A t impact parameters less than ab out 1remem b ered a.u., the 2 s and 2 p orbitals b egin to tak e part, and the transfer probabilit y c hanges signican tly for the dynamically consisten t basis sets. This c hange is noted ev en more clearly Figure 3.17 whic h compares the dieren tial cross section for resonan t c harge transfer as calculated using END to

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97 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 Transfer ProbabilityImpact Parameter (a.u.) Figure 3.16: Comparison of the probabilit y for resonan t c harge transfer b et w een He + and He at 5 k eV collision energy The follo wing He basis sets are used: 631G** (red), 6-31B** (blue), and BJK01 (purple). the exp erimen tal v alues rep orted b y Gao et al. [ 70 ]. While the o v erall structure of the dieren tial cross section calculated using the 6-31G** basis is reasonably go o d in comparison with exp erimen t, one can see impro v emen t in the data obtained when the dynamically consisten t basis set are emplo y ed. There is a substan tial phase shift that o ccurs b et w een 0.1 and 1 degrees. F urthermore, the BJK01 basis set demonstrates a decrease in magnitude b et w een the angle of 0.06 and 0.1 degrees that b egins to con v erge with the exp erimen tal data. It m ust b e remem b ered that the END calculations utilize a single determinan tal represen tations of the electrons, and will nev er exactly repro duce the exp erimen tal results, ho w ev er, it is eviden t that the energetically accessible virtual orbitals pro vided through the dynamically consisten t bases are imp ortan t to the correct description of the dieren tial cross section. In this section, it has b een demonstrated that the dynamically consisten t basis sets do result in increased c harge transfer probabilities in regions of impact parameter and collision energy that allo w virtual orbitals to b e accessible to

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98 100 1000 10000 100000 1e+06 1e+07 0.01 0.1 1 Differential Cross Section ( 2/ster.)Lab Frame Scattering Angle (deg.) Figure 3.17: Comparison of the dieren tial cross section for resonan t c harge transfer b et w een He + and He at 5 k eV collision energy The exp erimen tal data are from Gao et al. ( ). F or the END data the follo wing He basis sets are used: 6-31G** (red), 6-31B** (blue), and BJK01 (purple). excitations of the v alence electrons. This eect, ho w ev er, is most noticeable in systems where the sto c k basis sets pro vide a p o or description of these virtual orbitals. While these dynamically consisten t basis sets are not the only factor in impro ving dynamical calculations, they do pla y a role in impro ving result from suc h calculations.3.4.3 Prop erties of Diatomic and T riatomic Molecules One nal collection of prop erties will b e in v estigated. Man y structure prop erties of molecules are based on ground state energy of the molecule in question. The most common of these prop erties is the calculation of vibrational normal mo des of the molecule. This is accomplished b y diagonalization of the Hessian matrix, the matrix of second deriv ativ es of the energy with resp ect to the n uclear co ordinates [ 44 ]. It is reasonable to exp ect that energy optimized basis sets w ould oer b etter vibrational analyses for molecules than basis sets that ha v e not b een sub ject

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99 T able 3.7: Molecular prop erties of the nitrogen molecule. N 2 Bond Length, r (Exp erimen tal: 1.0940 A) Basis r ( A) % Err. 3-21G 1.0829 -1.01 6-31G 1.0892 -0.44 6-31G** 1.0784 -1.43 Basis r ( A) % Err. 3-21B 1.1877 8.57 6-31B 1.1674 3.86 6-31B** 1.1787 7.74 Vibrational F requency (Exp erimen tal: 2359.61 cm 1 ) Basis (cm 1 ) % Err. 3-21G 2611.84 10.7 6-31G 2660.99 12.8 6-31G** 2758.00 16.9 Basis (cm 1 ) % Err. 3-21B 2325.75 -1.43 6-31B 2377.98 0.78 6-31B** 2329.08 -1.29 to optimizations. Ho w ev er, if these non-optimized basis set are to b e used in c hemical calculations, one w ould desire that the basis sets pro vide reasonable results for non-dynamical ph ysical prop erties as w ell. In this section, vibrational mo des calculated for sev eral molecules using sto c k basis sets and the newly constructed basis sets are compared, along with other molecular prop erties, suc h as b ond lengths, b ond angles, and dip ole momen ts. The calculations w ere all p erformed using Gaussian 98. The rst collection of data, giv en in T able 3.7 relates the b ond length and normal mo de frequency of the N 2 molecule as calculated using the v arious basis sets. The exp erimen tal v alues are tak en from Herzb erg [ 71 ]. Up on consideration of the b ond lengths as calculated using the v arious basis sets, it b ecomes clear that the sto c k basis sets pro vide more accurate v alues (all in error b y less than 1.5 p ercen t). Ho w ev er, the v alues obtained using the dynamically consisten t basis sets do not demonstrate signican tly increased inaccuracy All of the v alues pro vided sho w less than 9 p ercen t error. F urthermore, comparison of the vibrational frequency sho ws v ery go o d results. The normal mo de frequency calculated using the energy optimized basis sets demonstrates an consisten t discrepancy of greater than ten p ercen t, while the non-optimized basis sets are consisten tly in error b y less

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100 T able 3.8: Molecular prop erties of the carb on mono xide molecule. CO Bond Length, r (Exp erimen tal: 1.1282 A) Basis r ( A) % Err. 3-21G 1.1285 0.03 6-31G 1.1307 0.22 6-31G** 1.1139 -1.27 Basis r ( A) % Err. 3-21B 1.2271 8.77 6-31B 1.2132 7.53 6-31B** 1.2149 7.68 Vibrational F requency (Exp erimen tal: 2170.21 cm 1 ) Basis (cm 1 ) % Err. 3-21G 2315.30 6.69 6-31G 2286.00 5.34 6-31G** 2438.37 12.4 Basis (cm 1 ) % Err. 3-21B 2055.77 -5.27 6-31B 2023.84 -6.74 6-31B** 1986.84 -8.45 Dip ole Momen t, (Exp erimen tal: 0.117 D) Basis (D) % Err. 3-21G 0.3972 240 6-31G 0.5729 390 6-31G** 0.2642 126 Basis (D) % Err. 3-21B 0.5606 379 6-31B 0.4967 325 6-31B** 0.4646 297 than 1.5 p ercen t. Ho w ev er, it m ust b e noted that a systematic error is in tro duced in to vibrational calculations b y virtue of correlation eects. This error is generally treated using w ell-accepted empirical m ultiplicativ e factors. The rep orted v alues do not accoun t for these correlation eects. Ho w ev er, if one assumes that the newly constructed basis sets demonstrate comparable correlation eects, then the v alues computed using the dynamically consisten t basis sets will b e roughly 10% lo w er than the exp erimen tal frequency This is reasonable, in ligh t of the fact that the b ond lengths are to o long in comparison the exp erimen tal b ond length. But, this 10% error is quite go o d in ligh t of the simplicit y of the prop osed basis set construction metho d. The second diatomic molecule submitted for in v estigation is carb on mono xide. The exp erimen tal v alues of the b ond length and the normal mo de frequency are tak en from Herzb erg [ 71 ], while the dip ole momen t is tak en from A tkins [ 72 ]. T able 3.8 pro vides the b ond lengths, normal mo de frequencies, and dip ole momen ts as calculated using the v arious basis sets. A t this p oin t, the trends are b ecoming

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101 clear. The b ond lengths are more accurate using the sto c k basis sets, though the dynamically consisten t basis sets are not in error b y more than 9%. Again, the frequencies calculated using the newly constructed basis sets are, up on initial in v estigation, more accurate than the v alues obtained using sto c k basis sets. Ho w ev er, when correlation eects are considered, the dynamically consisten t basis sets lose some accuracy The vibrational frequencies obtained b y using the dynamically consisten t basis sets can b e reasonably exp ected to incur ab out 10% error. The nal quan tit y to compare is the magnitude of the dip ole momen t. In the case of the dip ole momen t of CO, neither the sto c k basis sets nor the dynamically consisten t basis sets p erform w ell. The error in the sto c k basis sets is ab out 200% on a v erage, while the error in the newly constructed basis sets is ab out 330%. F urthermore, all of the basis sets pro vided dip ole momen ts that are an ti-parallel to the exp erimen tally determined dip ole momen t. This is a w ell-kno wn failure of the Hartree-F o c k analysis of CO [ 34 ], but it demonstrates that, at this lev el of theory the newly constructed basis sets p erform comparably to the sto c k basis sets. Finally the calculated v alues for w ater are compared; these are giv en in T able 3.9 The exp erimen tal data are tak en from the CR C Handb o ok [ 73 ]. Again, the trends are conrmed. The b ond lengths from the newly constructed basis sets are in error b y ab out 7% and the vibrational frequencies sho w a comparable error of less than ab out 10%. The b ond angle in w ater sho ws ab out a 5% error when the sto c k basis sets are emplo y ed. This decreases to ab out 3% when the dynamically consisten t basis sets are used. This can p ossibly b e attributed to the impro v ed virtual orbital structure in the newly constructed basis sets and p olarization eects. Finally the dip ole momen t can b e considered. Both the sto c k basis sets and the newly constructed basis sets demonstrate go o d comparison with exp erimen t in the case of w ater. Of particular in terest is the fact that the v alues calculated using the dynamically consisten t basis sets are in error b y ab out half in comparison

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102 T able 3.9: Molecular prop erties of the w ater molecule. O-H Bond Length, r (Exp erimen tal: 0.9584 A) Basis r ( A) % Err. 3-21G 0.9668 0.88 6-31G 0.9497 -0.91 6-31G** 0.9428 -1.63 Basis r ( A) % Err. 3-21B 1.0328 7.76 6-31B 1.0200 6.43 6-31B** 1.0185 6.27 H-O-H Angle, (Exp erimen tal: 104.45 ) Basis ( ) % Err. 3-21G 107.68 3.09 6-31G 111.55 6.80 6-31G** 105.95 1.44 Basis ( ) % Err. 3-21B 104.35 -0.10 6-31B 108.13 3.52 6-31B** 106.80 2.25 Dip ole Momen t, (Exp erimen tal: 1.85 D) Basis (D) % Err. 3-21G 2.387 29.0 6-31G 2.501 35.2 6-31G** 2.148 16.1 Basis (D) % Err. 3-21B 2.169 17.2 6-31B 2.172 17.4 6-31B** 1.930 4.32 1st Normal Mo de, 1 (Exp erimen tal: 1595 cm 1 ) Basis 1 cm 1 % Err. 3-21G 1799.61 12.8 6-31G 1737.26 8.92 6-31G** 1769.02 10.9 Basis 1 cm 1 % Err. 3-21B 1666.66 4.49 6-31B 1606.26 0.71 6-31B** 1743.98 9.34 2nd Normal Mo de, 2 (Exp erimen tal: 3657 cm 1 ) Basis 2 cm 1 % Err. 3-21G 3810.29 4.33 6-31G 3987.03 9.17 6-31G** 4151.50 13.7 Basis 2 cm 1 % Err. 3-21B 3703.32 1.41 6-31B 3668.15 0.44 6-31B** 3742.36 2.47 3rd Normal Mo de, 3 (Exp erimen tal: 3756 cm 1 ) Basis 3 cm 1 % Err. 3-21G 3943.74 5.00 6-31G 4143.95 10.3 6-31G** 4268.43 13.6 Basis 3 cm 1 % Err. 3-21B 3998.78 6.46 6-31B 4034.29 7.41 6-31B** 4072.82 8.44

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103 the v alues from the sto c k basis sets. The 6-31B** basis, in particular, pro vides a remark ably go o d dip ole momen t magnitude, whic h is lik ely a consequence of the impro v ed p olarization orbitals included in that basis set. This c hapter has demonstrated that a simple and computationally inexp ensiv e metho d for construction of atomic basis sets can b e ac hiev ed. This metho d fo cuses on the ph ysical prop erties of o ccupied and virtual atomic orbitals, and as a consequence do es not require exp ensiv e and tedious energy optimizations. These basis sets pro vide for ph ysically meaningful electronic excitations within the atoms, mark ed impro v emen t in c harge transfer descriptions in comparison to sto c k basis sets, and are still capable of returning reasonable structural prop erties of ground state molecules. F urthermore, these prop erties can all b e impro v ed through minor mo dications that ma y b e made in the future.

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CHAPTER 4 VECTOR HAR TREE-F OCK: A MUL TI-CONFIGURA TIONAL REPRESENT A TION IN ELECTR ON-NUCLEAR D YNAMICS In this c hapter, the equations of motion for a m ulti-congurational formalism of Electron-Nuclear Dynamics are deriv ed. The formalism is dubb ed V ector Hartree-F o c k (VHF). First, a Lagrangian for the system is in tro duced, with v erication that the Sc hr odinger Equation is returned. Secondly the new Lagrangian is parameterized in terms of the dynamical v ariables that describ e the system. F rom this parameterized Lagrangian, the VHF equations of motion are deriv ed. 4.1 In tro duction of the Lagrangian and V erication of the Equations of Motion In dynamics, a Lagrangian function in tro duced for a dynamical system is not unique. In fact, pro vided a giv en Lagrangian function returns the correct equations of motion, then that Lagrangian is a v alid Lagrangian for the system in question [ 74 ]. With this in mind, the VHF equations of motion are more con v enien tly deriv ed if one alters the form of the quan tum mec hanical Lagrangian as in tro duced in the rst c hapter. Consider a state j i dened b y a set of dynamical v ariables = f ; g with quan tum mec hanical Hamiltonian op erator H A quan tum mec hanical Lagrangian can b e constructed of the form L ( ; ) = h j i @ @ t H j i : (4.1) One can exploit the hermiticit y of the time deriv ativ e to rewrite the Lagrangian as L ( ; ) = i 2 h j @ @ t ~ @ @ t j i h j H j i (4.2) 104

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105 where the op erators @ @ t and ~ @ @ t act exclusiv ely on the k et and the bra, resp ectiv ely Hamilton's Principle requires that v ariations in the action of the system v anish [ 75 ], S = Z t 2 t 1 L dt = 0 : (4.3) The b oundary conditions will b e imp osed suc h that h ( t 1 ) j ( t 1 ) i = h ( t 2 ) j ( t 2 ) i If one assumes that the dynamical terms and are the only v ariable parameters, then b y v arying one obtains h ( t 2 ) j ( t 2 ) i h ( t 1 ) j ( t 1 ) i = 0 and b y v ariation of one obtains h ( t 2 ) j ( t 2 ) i h ( t 1 ) j ( t 1 ) i = 0. F urthermore, v ariation of the Lagrangian results in an expression L = i 2 h h j i + h j i h j i h j i i h j H j i h j H j i : (4.4) The ab o v e expression can b e simplied b y in tro ducing the total time deriv ativ es of h j i and h j i L = i 2 d dt h j i d dt h j i + i h h j i h j i i h j H j i h j H j i : (4.5) Substitution of the ab o v e in to Equation ( 4.3 ) and subsequen t in tegration results in the expression 0 = i 2 h j ij t 2 t 1 h j ij t 2 t 1 + Z t 2 t i i h h j i h j i i h j H j i h j H j i dt: (4.6) The rst term on the righ t-hand side v anishes due to the constrain ts imp osed b y the b oundary conditions, lea ving 0 = Z t 2 t i i h j i h j H j i + complex conjugate dt: (4.7) By the fact that the in tegral ev aluates to zero and the fact that the parameters and are p ermitted to v ary indep enden tly of eac h other, b oth terms in the

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106 in tegrand m ust b e iden tically zero. Considering only the rst term, one obtains 0 = h j n i j i H j i o : (4.8) As no constrain ts on the v ariation of or w ere imp osed, h j is completely arbitrary requiring that the brac k eted term b e iden tically zero. Th us, H j i = i @ @ t j i : (4.9) F urthermore, if the same analysis is p erformed on the complex conjugate term, the resulting equation tak es the form ( h j H ) = i h j ~ @ @ t : (4.10) Equations ( 4.9 ) and ( 4.10 ) are the Sc hr odinger Equations for the system j i v erifying that the c hosen Lagrangian do es indeed return the correct dynamical equations. 4.2 P arameterization of the State V ector The state v ector will b e assumed to tak e the form of a pro duct w a v e function, j i = j R ; P ij z ; R ; P i = j ij z i : (4.11) The rst factor is the n uclear comp onen t of the w a v e function, j i = Y j;k s 1 b k p exp 1 2 X j k R j k b k 2 + iP j k ( X j k R j k ) # ; (4.12) where X j k is the k th Cartesian comp onen t of the j th n ucleus, R j k and P j k are the corresp onding a v erage n uclear p ositions and momen ta, and b is a non-dynamical width parameter. The n uclear w a v e pac k et will b e treated in a \zero width" limit (i.e. classically), in the manner that b k 0 (for all k ) and h 0, suc h that h b 2k 1. The electronic factor j z i is constructed from a collection of single

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107 N -electron Thouless determinan ts that can b e written as D = det f i ( x j ) g (4.13) and are expressed in terms of a set of dynamical spin-orbitals of the form h = u h + K X p = N +1 u p z ph ; (4.14) where f u l g is a set of basis functions of rank K constructed from a set of Gaussian atomic orbitals that are cen tered on the a v erage atomic p ositions. F urthermore, the complex-v alued parameter z ph is the p th Thouless co ecien t corresp onding to the h th atomic orbital in the sp ecied conguration. F rom this, it can b e seen that the m ulti-congurational w a v e function no w tak es the form j z i = X d j D i ; (4.15) where pro vides an index o v er included congurations. The expansion co ecien ts are p ermitted to b e complex-v alued. It is assumed in the curren t discussion that the spin orbitals within a giv en conguration are orthogonal, ho w ev er, this restriction is not imp osed on t w o spin orbitals from diering congurations. Therefore, the normalization of the w a v e-function m ust b e addressed, and is giv en the form S = X X d d D : (4.16) In the ab o v e equation, the term D is the o v erlap in tegral b et w een t w o dieren t congurations, giv en the form D = h D j D i = det f g ; (4.17)

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108 where the matrix is giv en as = J y I z y 0B@ I z 1CA J : (4.18) The previous equation utilizes a v ector represen tation that corresp onds to Equation ( 4.14 ) as w ell as the atomic o v erlap matrix, The electronic represen tation is constructed using a complete activ e space (CAS) formalism [ 76 ] whic h allo ws one to partition the electronic space. There is a total of N electrons in a basis of K spin orbitals. Of these, there are K 1 \core orbitals", those spin orbitals that are o ccupied in ev ery conguration. Secondly there is an activ e space, whic h is dened as the set of spin orbitals that ha v e some probabilit y of b eing o ccupied that is b et w een zero and unit y The rank of the activ e space is denoted b y K 2 This requires that the virtual space (those orbitals that are nev er o ccupied) has a rank of K K 2 All full matrices ha v e dimension K K and can b e partitioned using the follo wing construction M = 264 M M > M M 375 : (4.19) The sup erscripts pro vide a mnemonic for remem b ering the structure of the matrices: the bullet ( ) represen ts the activ e blo c k with dimensions K 2 K 2 the op en circle ( ) represen ts the virtual blo c k with dimensions ( K K 2 ) ( K K 2 ), the > sup erscript p oin ts to the righ t and sym b olizes the upp er righ t blo c k of the matrix with dimensions K ( K K 2 ), and the sup erscript p oin ts do wn and sym b olizes the ( K K 2 ) K lo w er left blo c k of the matrix. F urthermore, the sym b ols I and I represen t unit matrices with dimensions K K and ( K K 2 ) ( K K 2 ), resp ectiv ely

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109 A t this p oin t, one can p erform the prescrib ed matrix m ultiplication indicated in Equation ( 4.18 ), resulting the expanded form = J y + z y + > z + z y z J : (4.20) A t this p oin t, the J matrices m ust b e discussed. The matrix denoted as J is a matrix of ones an zeros with dimensions of K 2 N The elemen ts are dened suc h as to select the Thouless co ecien ts that corresp ond to a giv en conguration. The sp ecic v alue of the elemen t ( J ) hj is a Kronec k er delta ( J ) hj = hh j ; (4.21) where 1 h K 2 and where h j is a spin orbital in D In this parameterization, the dynamical v ariables are f R ; P ; z ; z ; d; d g Before the parameterized Lagrangian can b e in tro duced, sev eral commen ts concerning the prop erties of n uclear w a v e pac k ets m ust b e made. As the n uclear w a v e function is normalized at eac h time-step, the exp ectation v alue of an observ able O is dened as h O i = h j O j i F rom this, the a v erage width of the normalized n uclear w a v e pac k et can b e found, h ( X j k R j k ) i = 1 b p Z 1 1 ( X j k R j k )e X j k R j k b 2 dX = 0 ; (4.22) due to the symmetry of the in tegrand. Secondly the exp ectation v alue of the square of the a v erage width of the w a v e pac k et can b e found, h ( X j k R j k ) 2 i = 1 b p Z 1 1 ( X j k R j k ) 2 e X j k R j k b 2 dX = 1 b p b 3 p 2 = b 2 2 : (4.23)

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110 F rom this, the width (the ro ot-mean-square v alue of the a v erage width of the n uclear w a v e function, x = p h ( X j k R j k ) 2 i ) is b p 2 No w one m ust consider exp ectation v alues in v olving the momen tum op erator, ^ P = i h @ @ X Th us, h ^ P i = i h b p Z 1 1 e 1 2 X j k R j k b 2 iP j k ( X j k R j k ) @ @ X j k e 1 2 X j k R j k b 2 + iP j k ( X j k R j k ) dX = i h b 3 p h j X j k R j k + iP j k j i = i h b 3 p ( h X j k R j k i + iP j k h j i ) = P j k hb p b 3 p h j i = P j k h b 2 h j i = P j k (4.24) in our limit b 0 ; h 0 ; h =b 2 1 and with normalized n uclear w a v e functions. F urthermore, h ^ P 2 i = 1 b p Z 1 1 e 1 2 X j k R j k b 2 iP j k ( X j k R j k ) @ 2 @ X 2 j k e 1 2 X j k R j k b 2 + iP j k ( X j k R j k ) dX ) = P 2 j k + 1 b 2 h j i + 1 b 4 h ( X j k R j k ) 2 i = P 2 j k + 1 2 b 2 (4.25) again with the limits men tioned ab o v e. Lik ewise, it is v ery simple to sho w that h P j k i = P j k h j i Finally realizing that the n umerical v alue of h j i is 1, then h ^ P 2 P 2 j k i = h ^ P 2 i h P 2 j k i = 1 2 b 2 ; (4.26)

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111 from whic h it immediately follo ws that P = q h ^ P 2 P 2 j k i = 1 p 2 b The result that can no w b e concluded is that the minim um uncertain t y in the n uclear Gaussian w a v e pac k et is X P = 1 2 in atomic units. Additionally t w o sp ecic exp ectation v alues of deriv ativ es m ust b e ev aluated. Firstly in atomic units, @ @ R j k = @ @ X j k = i h ^ P j k i = iP j k h j i : (4.27) Secondly @ @ P j k = h ( X j k R j k ) i = 0 ; (4.28) as previously demonstrated. One nal commen t m ust b e made concerning the exp ectation v alue of the Hamiltonian op erator, h j H j i The total energy of the system can b e expressed as E = X j k p 2j k 2 M k h j H el j i h j i ; (4.29) where H el is the separable electronic comp onen t of the Hamiltonian op erator. Using the previously deriv ed prop erties of Gaussian w a v e pac k ets, it is easy to sho w that the kinetic energy term in the Equation ( 4.29 ) can b e rewritten as X j k p 2j k 2 M k = h j T j i h j i ; (4.30) where T is the a v erage kinetic energy op erator, T = P j k ^ P 2 j k 2 M k No w, substitution of Equation ( 4.30 ) in to Equation ( 4.29 ) and recognizing that H = T + H el then

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112 it b ecomes eviden t that E = h j H j i h j i : (4.31) Therefore, it can easily b e seen that h j H j i = h H i = E S; (4.32) whic h is the energy of the system w eigh ted b y the electronic normalization. F urthermore, from the earlier explored prop erties of the n uclear w a v e function ( h ^ P 2 i = P 2j k in the narro w w a v e pac k et limit) and the normalization condition, it can b e seen that h H i = h z j H j z i : (4.33) One imp ortan t feature of the electronic norm is that if the electronic congurations are normalized at time t = 0, then they will remain normalized throughout the dynamics. Therefore, if sp ecial care is tak en that the electronic congurations are normalized initially then the exp ectation v alue of the Hamiltonian is equal to the energy of the system at all time-steps. As a result, an y deriv ativ e of the normalized exp ectation v alue of the Hamiltonian with resp ect to a giv en dynamical v ariable is a force expression. This is of sp ecial imp ortance when the matrix form of the equations of motion is in v estigated later. No w, from the previous section, one can separate the Lagrangian in the form L = i 2 h j @ @ t j i h j ~ @ @ t j i # h z j z i + i 2 h z j @ @ t j z i h z j ~ @ @ t j z i # h j i h H i : (4.34)

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113 Expansion of the time deriv ativ e with resp ect to the dynamical v ariables allo ws the Lagrangian to b e expressed as L = i 2 X j k ( h j @ @ R j k j i h j ~ @ @ R 0 j k j i R j k h z j z i + h j @ @ P j k j i h j ~ @ @ P 0 j k j i P j k h z j z i + h z j @ @ R j k j z i h z j ~ @ @ R 0 j k j z i R j k h j i + h z j @ @ P j k j z i h z j ~ @ @ P 0 j k j z i P j k h j i ) + i 2 X X ph h z j @ @ z ph j z i z ph h z j ~ @ @ z ph j z i z ph + h z j @ @ d j z i d h z j ~ @ @ d j z i d # h j i h H i ; (4.35) where it is understo o d that the bra is a function of R 0 ; P 0 ; z ; d and the k et is a function of R ; P ; z ; d in the limit where R 0 ; P 0 R ; P F rom this p oin t forw ard, the dep endence of z and z on will not b e explicitly indicated, suc h that z ph = z ph and z ph = z ph No w, in tro duction of Equations ( 4.27 ) and ( 4.28 ) in to the ab o v e form of the Lagrangian allo ws one to write L = X j k (" P j k h z j z i + 1 2 h z j @ @ R j k j z i h z j ~ @ @ R 0 j k j z i !# R j k + i 2 h z j @ @ P j k j z i h z j ~ @ @ P 0 j k j z i # P j k ) h j i + i 2 X ( X ph h z j @ @ z ph j z i z ph h z j @ @ z ph j z i z ph # + h z j @ @ d j z i d h z j @ @ d j z i d h j i h H i : (4.36) Before the nal form of the parameterized Lagrangian can b e in tro duced, the deriv ativ e expressions m ust b e rewritten to prop erly rerect the dep endence on the

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114 c hosen dynamical v ariables. Recalling the fact that h z j = h R 0 ; P 0 ; z ; d j and that j z i = j R ; P ; z ; d i then it is trivial to sho w that @ @ R h z j z i = h z j @ @ R j z i @ @ R 0 h z j z i = h z j ~ @ @ R 0 j z i @ @ P h z j z i = h z j @ @ P j z i @ @ P 0 h z j z i = h z j ~ @ @ P 0 j z i @ @ z h z j z i = h z j @ @ z j z i @ @ z h z j z i = h z j ~ @ @ z j z i @ @ d h z j z i = h z j @ @ d j z i @ @ d h z j z i = h z j ~ @ @ d j z i : No w, b y recalling that the m ulti-congurational electronic o v erlap is S = h z j z i and in tro ducing the ab o v e iden tities, the parameterized Lagrangian no w tak es the form L = ( X j k S P j k R j k + i 2 @ S @ R j k @ S @ R 0 j k R j k + i 2 @ S @ P j k @ S @ P 0 j k P j k # + X X ph @ S @ z ph z ph @ S @ z ph z ph + @ S @ d d @ S @ d d #) h j i h H i : (4.37) 4.3 The VHF Equations of Motion It has b een demonstrated that the prop er ph ysical description of the system can b e ac hiev ed b y imp osing Hamilton's Principle up on the quan tum mec hanical Lagrangian. The deriv ation of the equations of motion can b e facilitated b y emplo ying a mathematical equiv alen t the Hamilton's Principle, the Euler-Lagrange equations [ 75 ], d dt @ L @ q = @ L @ q ; (4.38) where q is an elemen t of the set of dynamical v ariables. By solving the ab o v e equation for eac h dynamical v ariable, a set of coupled dieren tial equations of motion will b e obtained. The dynamics of the system can then b e calculated as solutions to this coupled set of equations of motion.

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115 4.3.1 Deriv ation of the Equations of Motion As the deriv ation of the equations of motion for eac h dynamical v ariable is a length y and tedious undertaking, a detailed deriv ation with resp ect to only one v ariable will b e used to demonstrate the pro cess. Let q = R indicating that @ L @ R il = S P il + i 2 @ S @ R il @ S @ R 0 il : (4.39) Again, it is necessary to in tro duce the full expansion of the time deriv ativ e with resp ect to the dynamical v ariables, suc h that d dt @ L @ R il = X j k P il @ S @ R j k + @ S @ R 0 j k R j k + i 2 @ 2 S @ R j k @ R il + @ 2 S @ R 0 j k @ R il @ 2 S @ R j k @ R 0 il @ 2 S @ R 0 j k @ R 0 il R j k + S ij;l k P j k + P il @ S @ P j k + @ S @ P 0 j k P j k + i 2 @ 2 S @ P j k @ R il + @ 2 S @ P 0 j k @ R il @ 2 S @ P j k @ R 0 il @ 2 S @ P 0 j k @ R 0 il P j k # + i 2 X @ 2 S @ d @ R il @ 2 S @ d @ R 0 il d + @ 2 S @ d @ R il @ 2 S @ d @ R 0 il d + X ph @ 2 S @ z ph @ R il @ 2 S @ z ph @ R 0 il z ph + @ 2 S @ z ph @ R il @ 2 S @ z ph @ R 0 il # z ph !# : (4.40)

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116 Lik ewise, one nds that @ L @ R il = X j k @ S @ R il + @ S @ R 0 il P j k R j k + i 2 @ 2 S @ R il @ R j k + @ 2 S @ R 0 il @ R j k @ 2 S @ R il @ R 0 j k @ 2 S @ R 0 il @ R 0 j k R j k + i 2 @ 2 S @ R il @ P j k + @ 2 S @ R 0 il @ P j k @ 2 S @ R il @ P 0 j k @ 2 S @ R 0 il @ P 0 j k P j k ) + i 2 X @ 2 S @ R il @ d + @ 2 S @ R 0 il @ d d @ 2 S @ R il @ d + @ 2 S @ R 0 il @ d d + X ph @ 2 S @ R il @ z ph + @ 2 S @ R 0 il @ z ph z ph @ 2 S @ R il @ z ph + @ 2 S @ R 0 il @ z ph z ph # ) @ h H i @ R il : (4.41) When the previous t w o equations are substituted in to the Euler-Lagrange equations, one nds that @ h H i @ R il = X j k (" @ S @ R il + @ S @ R 0 il P j k P il @ S @ R j k + @ S @ R 0 j k !# R j k + i @ 2 S @ R 0 il @ R j k @ 2 S @ R il @ R 0 j k R j k + S ij;l k P il @ S @ P j k + @ S @ P 0 j k !# P j k + i @ 2 S @ R 0 il @ P j k @ 2 S @ R j k @ P 0 j k P j k ) + X P il @ S @ d + i @ 2 S @ R 0 il @ d d + P il @ S @ d i @ 2 S @ R il @ d d + X ph P il @ S @ z ph + i @ 2 S @ R 0 il @ z ph z ph + P il @ S @ z ph i @ 2 S @ R il @ z ph z ph #) : (4.42)

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117 The t w o terms con taining the second deriv ativ es of the o v erlap found under the summation o v er j and k eac h represen t the dierence of a complex n um b er and its complex conjugate. In general, for a complex n um b er c = a + ib one nds that c c = ( a + ib ) ( a ib ) = 2 ib = 2 i Im ( c ). Lik ewise, the single deriv ativ e terms under the j and k summation represen t the sum of a complex n um b er and its complex conjugate. F ollo wing the previous example, c + c = ( a + ib ) + ( a ib ) = 2 a = 2Re( c ). These iden tities can b e used to further simplify the ab o v e equation, yielding @ h H i @ R il = X j k 2Re @ S @ R il P j k P il 2Re @ S @ R j k R j k 2Im @ 2 S @ R 0 il @ R j k R j k + S ij;l k P il 2Re @ S @ P j k 2Im @ 2 S @ R 0 il @ P j k P j k + X P il @ S @ d + i @ 2 S @ R 0 il @ d d + P il @ S @ d i @ 2 S @ R il @ d d + X ph P il @ S @ z ph + i @ 2 S @ R 0 il @ z ph z ph + P il @ S @ z ph i @ 2 S @ R il @ z ph z ph #) ; (4.43) whic h is the nal form of the equation of motion for q = R The ab o v e equation of motion can b e rewritten as a v ector equation b y collecting the n uclear Cartesian comp onen ts (summation of j ) and b y com bining the z parameters and expansion co ecien ts d in to a separate rectangular matrices, z and d By in tro ducing the coupling matrices ( C z z ) = @ 2 S @ z @ z ; ( C z X ) = @ 2 S @ z @ X ; ( C dd ) = @ 2 S @ d @ d ; ( C dX ) = @ 2 S @ d @ X ; ( C z d ) = @ 2 S @ z @ d ; ( C z ) = @ S @ z ; C X = 2Re @ S @ X ; C X Y = 2Im @ 2 S @ X 0 @ Y X 0 X Y 0 Y ;

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118 where X ; Y 2 f R ; P g Equation ( 4.43 ) can b e written in the v ector form r R l h H i = X k n [ C R l P k P l C R k + C R l R k ] R k + [ S 1 P l C P k + C R l P k ] P k o + h P l C z + i C yz R l i z + P l C z i C Tz R l z + h P l C d + i C ydR l i d + P l C d i C TdR l d : (4.44) The term 1 in the ab o v e v ector equation is the unit matrix of appropriate dimension. F ollo wing the same deriv ation sc hema as just sho wn the remaining v e equations of motion can b e deriv ed. When q = P one obtains, r P l h H i = X k n [ S 1 + C P l P k + C P l R k ] R k + C P l P k P k o + i C yz P l z i C Tz P l z + i C ydP l d i C TdP l d : (4.45) When q = z one obtains @ h H i @ z = X k n C z P k i C z R k R k i C z P k P k o i C z z z i C z d d : (4.46) When q = z one obtains @ h H i @ z = X k n [ C z P k + i C z R k ] R k + i C z P k P k o + i C z z z + i C z d d : (4.47) When q = d one obtains @ h H i @ d = X k n C d i C dR k R k i C dP k P k o i C Tz d z i C dd d : (4.48) Finally when q = d one obtains @ h H i @ d = X k n [ C d P k + i C dR k ] R k + i C dP k P k o + i C yz d z + i C dd d : (4.49) By com bining Equations ( 4.44 ) ( 4.49 ) the VHF equations of motion can no w b e represen ted as one grand matrix equation of the form giv en in Equation ( 4.50 ), found on the follo wing page.

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1192666666666666664 i Cz z0 CzP + i Cz Ri Cz Pi Cz d0 0 i Cz zCzP i Cz R i Cz P0 i Cz dPCz+ i Cyz RPCz i CTz RCRP PCR+ CRRS 1 PCP+ CRPPCd+ i CydRPCd i CTdRi Cyz P i CTz PS 1 + CPP + CP RCP Pi CydP i CTdPi Cyz d0 CdP + i CdRi CdPi Cdd0 0 i CTz dCd i CdR i CdP0 i Cdd3777777777777775 2666666666666664 z z_ R P d d3777777777777775 = 2666666666666664 @ h H i =@ z@ h H i =@ z @ h H i =@ R @ h H i =@ P @ h H i =@ d@ h H i =@ d 3777777777777775 (4.50)

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120 Some v ery striking similarities b et w een the END equations of motion and Hamilton's equations of motion are immediately obtained. First, if one represen ts the matrix in Equation ( 4.50 ) as C and represen ts the leftand righ t-hand side v ectors as and @ E @ resp ectiv ely and if C 1 is assumed to exist, then Equation ( 4.50 ) can b e in v erted to the form = C 1 @ E @ (4.51) whic h is analogous to the symplectic form of Hamilton's equations [ 74 ]. F urthermore, a generalized P oisson brac k et for an y t w o dieren tiable functions of the dynamical v ariables can b e in tro duced in this symplectic form, f f ( ) ; g ( ) g = @ f @ T C 1 @ g @ : (4.52) F rom this and the general symplectic form of the equations of motion, it is easy to v erify that the time-ev olution of the dynamical v ariable can b e expressed using the generalized P oisson brac k et, = f ; E g ; (4.53) indicating that the dynamics are go v erned b y Hamilton-lik e equations. 4.3.2 Ev aluating the Equations of Motion The END equations of motion giv en in Equation ( 4.50 ) con tain 34 dieren t sub-blo c ks that eac h dep end directly up on v arious deriv ativ es of the electronic o v erlap and the exp ectation v alue of the Hamiltonian op erator. In this section, these deriv ativ es will b e explicitly ev aluated. First, one m ust recall that the electronic state v ector tak es the form j z i = X d j D i : (4.54)

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121 Th us, the electronic o v erlap has the form S = X d d D ; (4.55) where D = det 8><>: J y I z y 0B@ I z 1CA J 9>=>; = det J y + z y + > z + z y z J = det f g ; (4.56) where is the o v erlap of the atomic basis. It will b e helpful to in tro duce some auxiliary expressions that will allo w for simplication of the deriv ativ e expressions. First, the one-electron densit y matrix (the one-densit y) can b e in tro duced. The one-densit y can b e partitioned according to congurations. The building blo c ks of the one-densit y are the matrices of the form = 0B@ I z 1CA J 1 J y I z y ; (4.57) from whic h the one-densit y can b e constructed. The one-densit y has the form = 1 S X d d D : (4.58) The deriv ativ es of the electronic o v erlap with resp ect to the n uclear co ordinates and momen ta will con tain terms that ha v e the general form A ;k = T r r jiX k (4.59) B ;k l = T r r jiX k r jiX l (4.60) F ;k l = T r r jiX k r jiX l (4.61)

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122 The deriv ativ es of the o v erlap with resp ect to electronic degrees of freedom will generate expressions that utilize the follo wing expressions Q = [ + z ] J (4.62) Q > = Q _y = J y > + z y (4.63) P = Q 1 (4.64) P > = 1 Q > (4.65) R = Q 1 Q > (4.66) V = 0 I Q 1 J y I z y (4.67) W = 0B@ I z 1CA J 1 (4.68) U = V r jiX k W (4.69) As with the application of the Euler-Lagrange equations, the deriv ations of the individual terms for the symplectic form are tedious. F or this reason, an y rep etition will b e a v oided. One can b egin with the rst deriv ativ es of the electronic o v erlap. Sp ecically if one tak es the deriv ativ e with resp ect to z ph one nds that ( C z ) ph = @ S @ z ph = X d d @ det f g @ z ph (4.70)

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123 No w, the determinan t of an y non-singular N N matrix, M can b e expressed using the Laplace expansion [ 77 ], det f M g = 1 N X i;j M ij ad j i ( M ) ; (4.71) where ad j i ( M ) is the j; i th elemen t of the adjoin t matrix. Th us, ( C z ) ph = X d d 1 K 2 X f ;g @ ( ) f g @ z ph ad g f ( ) : (4.72) In the ab o v e equation, the deriv ativ e of the adjoin t is not required. This is b ecause the adjoin t is the transp ose of the minors of the matrix in question. Since the minor of a giv en matrix is the determinan t of the sub-matrix generated when the i th column and the j th ro w are remo v ed, ad j i ( ) do es not dep end on ( ) ij A t this p oin t, one can consider only the deriv ativ e, @ ( ) f g @ z ph = @ @ z ph J y + z y + > z + z y z J f g = @ @ z ph J y J f g + X r ( J z ) r f ( J ) r g + X s J y > f s ( zJ ) sg + X r ( J z ) r f ( zJ ) r g # : (4.73) A t this p oin t, it m ust b e recognized that the deriv ativ e of the rst and third term inside the brac k ets will b e zero. F urthermore, as the J y selects the Thouless co ecien ts for the conguration, it is only only meaningful to dieren tiate with resp ect to that conguration. Therefore, the conguration is implied and the J y matrix is no w redundan t. Th us, @ ( ) f g @ z ph = X r ( J ) r g r p f h + X r ( zJ ) r g r p f h # = ([ + z ] J ) pg f h : (4.74)

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124 Th us, ( C z ) ph = X d d 1 K 2 X f ;g ([ + z ] J ) pg ad g f ( ) f h : (4.75) No w, emplo ying the in v erse of Equation ( 4.71 ), one nds that ad ij ( ) = K 2 D 1 ij ; (4.76) and therefore, ( C z ) ph = X d d D X f ;g ([ + z ] J ) pg 1 g f f h = X d d D ( P ) ph : (4.77) Rep eating the pro cess with z ph results in the expression ( C ) ph = X d d D ( P > ) hp ; (4.78) whic h is the Hermitian conjugate of the deriv ativ e with resp ect to z as w ould b e exp ected from the symplectic structure. As the determinan t do es not dep end up on d or d the deriv ativ es of the o v erlap with resp ect to the conguration expansion co ecien ts are trivial: ( C d ) = @ @ d X d d D = X d D (4.79) and ( C d ) = @ @ d X d d D = X d D : (4.80) The dep endence of the determinan t up on the n uclear degrees of freedom arises in the o v erlap of the atomic basis only Ho w ev er, it is imp ortan t that one b e sp ecic

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125 as to whether the deriv ativ e is b eing tak en with resp ect to the primed (bra) or unprimed (k et) n uclear degrees of freedom, as the deriv ativ es are tak en b efore the limits of R 0 R and P 0 P are imp osed. T o sp ecify whic h deriv ativ e is b eing tak en, the n uclear sym b ol will b e sup erscripted with either a bra or a k et. The deriv ativ e with resp ect to the n uclear degrees of freedom (sym b olized as X or Y ) in v olv es application of the c hain rule to the determinan t r jiX D = X f ;g @ det f g @ ( ) f g @ ( ) f g @ X k = X f ;g 1 K 2 X h;i @ ( ) hi @ ( ) f g ad ih ( ) @ ( ) f g @ X k = X f ;g 1 K 2 ad g f ( ) @ ( ) f g @ X k = D X f ;g 1 g f 264 J y I z y r jiX k 0B@ I z 1CA J 375 f g = D T r 264 1 J y I z y r jiX k 0B@ I z 1CA J 375 = D A ;k : (4.81) Th us, it b ecomes clear that ( C X ) k = r jiX k X d d D = X d d D A ;k ; (4.82) and ( C X ) k = r hjX k X d d D = X d d D A ;k ; (4.83) where prop er atten tion is giv en to whic h deriv ativ e is b eing tak en.

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126 A t this juncture, the second deriv ativ es of the electronic o v erlap can b e ev aluated b y direct dieren tiation of the rst deriv ativ e terms. The pro cess is v ery similar to that for the rst deriv ativ e ev aluation, ho w ev er, in most cases deriv ativ es of the in v erse of are required. Again, the c hain rule can b e applied, resulting in the general form @ 1 f g @ = X ij 1 f i @ ( ) ij @ 1 j g ; (4.84) for the selected dynamical degrees of freedom. The ev aluation of the deriv ativ es of 1 will not b e rep eated here, as they dep end up on expression that ha v e previously b een deriv ed. The deriv ativ e of 1 ha v e the form @ 1 f g @ z ph = 1 f h [ + z ] J 1 pg = 1 f h ( P ) pg (4.85) and @ 1 f g @ z ph = 1 J y > + z y g p 1 hf = ( P > ) g p 1 hf (4.86) for the electronic gradien ts and r jiX k 1 f g = 0B@ 1 J y I z y r jiX k 0B@ I z 1CA J 1 1CA f g = W y r jiX k W f g (4.87) and r hjX k 1 f g = 0B@ 1 J y I z y r hjX k 0B@ I z 1CA J 1 1CA f g = W y r hjX k W f g (4.88)

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127 for the n uclear gradien ts. The deriv ativ es of 1 with resp ect to the expansion co ecien ts are all zero. Using these terms, the second deriv ativ es can no w b e ev aluated. The deriv ativ es with resp ect to the electronic parameters will b e considered rst: ( C z z ) q g ; ph = @ @ z q g @ S @ z ph = @ @ z q g X d d D 1 J y > + z y hp = X d d D n ( + z ) J 1 q g 1 J y > + z y hp +[ 1 ] hg ( + z ) J 1 J y > + z y q p o = X d d D ( P ) q g ( P > ) hp + ( 1 ) hg ( R ) q p (4.89) and therefore, ( C z z ) q g ; ph = X d d D ( P > ) g q ( P ) ph + ( R ) pq ( 1 ) g h ; (4.90) also ( C dd ) = @ @ d @ S @ d = D (4.91) and therefore, ( C dd ) = @ @ d @ S @ d = D : (4.92) The deriv ativ es that couple the electronic parameters with the expansion co ecien ts are p erformed in the same manner, but the transp ose and the Hermitian

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128 conjugate m ust also b e ev aluated, suc h that ( C z d ) q g ; = @ @ z q g @ S @ d = @ @ z q g X d D = X d D ( P ) q g (4.93) and ( C z d ) q g ; = @ @ z q g @ S @ d = @ @ z q g X d D = X d D ( P > ) g q ; (4.94) and C T z d ;ph = @ @ d @ S @ z ph = @ @ d X d d D ( P ) ph = X d D P ph (4.95) and nally C yz d ;ph = @ @ d @ S @ z ph = @ @ d X d d D ( P > ) ph = X d D P > ph : (4.96) It should b e noted that the transp ose and Hermitian conjugate terms are iden tical to the t w o previous deriv ativ es. Ho w ev er, the transp ose and Hermitian conjugate terms are main tained in order to emphasize the symplectic structure of the equations of motion.

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129 Using the same pro cedure, the deriv ativ es that couple the electronic and n uclear degrees of freedom can b e ev aluated. Firstly ( C z X ) q g ; k = @ @ z q g r jiX k S = X d d 8><>: @ D @ z q g T r 264 1 J y I z y r jiX k 0B@ I z 1CA J 375 + D @ @ z q g X f ;g 1 g f 264 J y I z y r jiX k 0B@ I z 1CA J 375 f g 9>=>; = X d d D n ( + z ) J 1 q g T r 264 1 J y I z y r jiX k 0B@ I z 1CA J 375 + 0 I ( + z ) J 1 J y I z y r jiX k 0B@ I z 1CA J 1 375 q g 9>=>; = X d d D f ( P ) q g A ;k + ( U ) q g g : (4.97) And, b y appropriately transp osing and conjugating, one nds ( C z X ) q g ; k = @ @ z q g r hjX k S = X d d D n ( P > ) g q A ;k + ( U > ) g q o (4.98) and C Tz X l ; ph = r jiX l @ S @ z ph = X d d D n A ;l ( P ) ph + ( U ) ph o (4.99)

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130 and, lastly C yz X l ; ph = r hjX l @ S @ z ph = X d d D n A ;k ( P > ) hp + ( U > ) hp o : (4.100) No w the second deriv ativ e terms in v olving the n uclear degrees of freedom and the expansion co ecien ts can b e ev aluated. The symmetries of the deriv ativ es b ecome more ob vious, but eac h term will b e k ept explicit to preserv e the symplectic structure. Sp ecically ( C dX ) ; k = @ @ d r jiX k S = @ @ d X d d D T r 264 1 J y I z y r jiX k 0B@ I z 1CA J 375 = X d D A ;k ; (4.101) and ( C dX ) ; k = @ @ d r hjX k S = X d D A ;k ; (4.102) and ( C dX ) ; l = r jiX l @ S @ d = X d D A ;l ; (4.103) and nally C ydX ; l = r hjX l @ S @ d = X d D A ;l : (4.104)

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131 The nal comp onen ts of the symplectic form are the second deriv ativ es with resp ect to the n uclear parameters only These four expressions can b e generalized b y a single term, ( C X Y ) l ;k = r hjX l r jiY k S = X d d D 8><>: T r 264 1 J y I z y r hjX l 0B@ I z 1CA J 375 T r 264 1 J y I z y r jiX k 0B@ I z 1CA J 375 + T r 264 1 J y 8><>: I z y r hjX l r jiY k 0B@ I z 1CA I z y r hjX l 0B@ I z 1CA J 1 J y I z y r jiY k 0B@ I z 1CA 9>=>; J 375 9>=>; = X d d D f A ;l A ;k + B ;l k F ;l k g : (4.105) Ha ving completed the ev aluation of the left-hand side of the VHF equation of motion, one can no w in v estigate the deriv ativ es found on the righ t-hand side. These terms are the deriv ativ es of the exp ectation v alue of the Hamiltonian op erator with resp ect to the dynamical v ariables, and are analogous to the dynamical forces exp erienced b y the system. In an orthonormal basis of spin orbitals, f i g the full ab initio molecular Hamiltonian can b e written in second quan tization in the form [ 6 ] H = N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) + K X i;j =1 h ij b yi b j + 1 4 K X i;j;k ;l =1 V ij ; k l b yi b yj b l b k : (4.106)

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132 In the ab o v e expression, Z k is the atomic n um b er of the k th n ucleus, h is the Hermitian one-electron in tegral matrix with elemen ts h ij = Z i ( r ) 1 2 r 2r N at X k =1 Z k j r R k j # j ( r ) d 3 r ; (4.107) and V represen ts the an ti-symmetrized t w o-electron in tegrals V ij ; k l = h ij j k l i h ij j l k i ; (4.108) where h ij j k l i = Z i ( r 1 ) j ( r 2 ) k ( r 1 ) l ( r 2 ) j r 1 r 2 j : (4.109) F rom this, it b ecomes clear that the molecular Hamiltonian can b e partitioned in to a zero-electron comp onen t, H (0) = N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) ; (4.110) a one-electron comp onen t, H (1) = K X i;j =1 h ij b yi b j ; (4.111) and a t w o-electron comp onen t, H (2) = 1 4 K X i;j;k ;l =1 V ij ; k l b yi b yj b l b k : (4.112) Th us, the exp ectation v alue of the Hamiltonian b ecomes h H i = X d d h D j H j D i = X d d h H (0) + H (1) + H (2) i ; (4.113) where H (0) = h D j H (0) j D i and so forth. F rom this, it can b e seen that the deriv ativ es of the exp ectation v alue of the Hamiltonian op erator with resp ect to the

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133 expansion co ecien ts are trivial to ev aluate. P articularly @ h H i @ d = X d h H (0) + H (1) + H (2) i (4.114) and @ h H i @ d = X d H (0) + H (1) + H (2) : (4.115) T o ev aluate the remaining deriv ativ es it is con v enien t to rewrite the oneand t w o-electron comp onen ts of the exp ectation v alue in terms of the blo c ks of the oneand t w o-densit y matrices. The blo c ks of the one-densit y w ere in tro duced previously and the blo c ks of the t w o-densit y can b e constructed as [ 6 ] (2) ij;k l = ( ) k i ( ) l j ( ) k j ( ) l i : (4.116) The oneand t w o-electron comp onen ts of the exp ectation v alue no w b ecome [ 3 ] H (1) = T r [ h ] (4.117) and H (1) = 1 4 T r h V (2) i : (4.118) F urthermore, as the zero-electron comp onen t do es not op erate on the elemen ts of the electronic space, one can write H (0) = H (0) D : (4.119) A t this p oin t, it b ecomes clear that the dep endence of the exp ectation v alue of the Hamiltonian on the Thouless co ecien ts and the n uclear parameters o ccurs ultimately in the blo c ks of the one-densit y It b ecomes necessary then, to ev aluate dieren tiate the one-densit y with resp ect to the electronic parameters and the n uclear parameters. In v estigation of the one-densit y blo c k giv en in Equation ( 4.57 ) clearly indicates that there is no dep endence up on the expansion co ecien ts and, as a consequence,

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134 the deriv ativ es with resp ect to d and d are b oth zero. The deriv ativ es of the one-densit y blo c k can b e tak en with resp ect to the Thouless co ecien ts, with the results that @ ( ) ij @ z ph = X r 264 0B@ I z 1CA J 375 ir @ @ z ph 1 J y I z y r j = 264 0B@ I z 1CA J 1 375 ih 0 I ( + z ) J 1 J y I z y pj = [ W ] ih [ V ] pj (4.120) and @ ( ) ij @ z ph = X r @ @ z ph 264 0B@ I z 1CA J 1 375 ir J y I z y r j = h V y i j p h W y i hi : (4.121) The deriv ativ es with resp ect to the n uclear co ordinates can b e similarly ev aluated with the result that r X k ( ) ij = X r s 264 0B@ I z 1CA J 375 ir r X k 1 r s J y I z y sj = 264 0B@ I z 1CA J 1 J y I z y r X k 0B@ I z 1CA J 1 J y I z y 375 ij = [ r X k ] ij : (4.122)

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135 The previous expressions for the deriv ativ es of the one-densit y blo c k can no w b e emplo y ed in the ev aluation of the exp ectation v alue of the Hamiltonian. Starting with the electronic degrees of freedom, one nds that @ H (0) @ z ph = H (0) @ D @ z ph = H (0) [ + z ] J 1 ph = H (0) ( P ) ph (4.123) and @ H (1) @ z ph = X ij h ij @ ( ) j i @ z ph = X ij h ij [ W ] j h [ V ] pi = X ij [ V ] pi h ij [ W ] j h = [ V h W ] ph : (4.124) T o ev aluate the deriv ativ e of the t w o-electron comp onen t, one m ust recall that (assuming that ETFs are not included) real-v alued basis functions are emplo y ed in END. Therefore, the t w o-electron in tegral term ( V ) will b e hermitian with resp ect to the exc hange of an y sets of electronic indices. Ho w ev er, an y terms in v olving the one-densit y or its deriv ativ es are still an ti-symmetric, requiring b o okk eeping for the

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136 signs. Th us, @ H (2) @ z ph = 1 4 X ij k l V ik ; j l @ @ z ph h ( ) j i ( ) l k ( ) j k ( ) l i i = 1 4 X ij k l V ik ; j l h [ W ] j h [ V ] pi ( ) l k + ( ) j i [ W ] l h [ V ] pk [ W ] j h [ V ] pk ( ) l i ( ) j k [ W ] l h [ V ] pi i = 1 2 X ij k l V ik ; j l h [ W ] j h [ V ] pi ( ) l k ( ) j k [ W ] l h [ V ] pi i = X ij k l V ik ; j l [ W ] j h [ V ] pi ( ) l k (4.125) where the i and k indices w ere exc hanged in the second and third terms in the second step and the l and j indices w ere exc hanged in the third step. No w, @ H (2) @ z ph = X ij [ V ] pi X l k V ik ; j l ( ) l k [ W ] j h = X ij [ V ] pi [T r ( V ) b ] ij [ W ] j h = [ V T r ( V ) b W ] ph ; (4.126) where the partial trace has has b een utilized, T r ( V ) b = X l k V ik ; j l ( ) l k : (4.127) No w, the individual con tributions can b e summed to yield the deriv ativ e of the exp ectation v alue of the Hamiltonian with resp ect to the Thouless parameters, @ h H i @ z ph = X d d ( @ H (0) @ z ph + @ H (1) @ z ph + @ H (2) @ z ph ) = X d d n H (0) ( P ) ph + [ V h W ] ph + [ V T r ( V ) b W ] ph o = X d d n H (0) P + V ( h + T r ( V ) b ) W o ph : (4.128)

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137 It should b e noted that the term h + T r ( V ) b has the same form as the F o c k matrix except that it is ev aluated o v er t w o dieren t congurations. In the case of a single determinan tal reference, this expression reduces to the F o c k matrix. A t this p oin t, the deriv ativ es with resp ect to the n uclear parameters m ust b e ev aluated. The zero-electron comp onen t of the Hamiltonian con tains explicit dep endence up on b oth the n uclear p ositions and momen ta. Therefore, the deriv ativ e with resp ect to eac h part m ust b e ev aluated separately Both deriv ativ es are simple, and dieren tiation results in the forms r R k H (0) = r R k D N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) + D r R k N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) = D A ;k N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) + D X l ( 6 = k )=1 Z k Z l e 2 ( R k R l ) j R k R l j 3 = H (0) A ;k + D X l ( 6 = k )=1 Z k Z l e 2 ( R k R l ) j R k R l j 3 (4.129) and lik ewise, r P k H (0) = r P k D N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) + D r R k N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) = D A ;k N at X k =1 ( P 2k 2 M k + N at X l = k +1 Z k Z l j R k R l j ) + D P k M k = H (0) A ;k + D P k M k : (4.130)

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138 The deriv ativ es of the oneand t w o-electron comp onen ts will ha v e the same general structure. F or the one-electron comp onen t, r X k H (1) = X ij h r X k h ij ( ) j i + h ij r X k ( ) j i i = X ij h r X k h ij ( ) j i h ij [ r X k ] j i i = T r ( r X k h ij ) T r ( h ij r X k ) = T r ( r X k h ij ) T r ( r X k h ij ) : (4.131) In the same manner, dieren tiation of the t w o-electron comp onen t (with careful reordering of the electronic indices) results in the form r X k H (2) = 1 4 X ij k l n r X k V ik ; j l h ( ) j i ( ) l k ( ) j k ( ) l i i + V ik ; j l r X k h ( ) j i ( ) l k ( ) j k ( ) l i io = 1 2 X ij k l n r X k V ik ; j l h ( ) j i ( ) l k i + V ik ; j l r X k h ( ) j i ( ) l k io = 1 2 T r T r ( r X k V ) a b + 1 2 X ij k l V ik ; j l r X k h ( ) j i ( ) l k i : (4.132)

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139 No w the deriv ativ e in the second term m ust b e ev aluated, r X k H (2) = 1 2 T r T r ( r X k V ) a b + 1 2 X ij k l V ik ; j l h r X k ( ) j i ( ) l k + ( ) j i r X k ( ) l k i = 1 2 T r T r ( r X k V ) a b 1 2 X ij k l V ik ; j l [ r X k ] j i ( ) l k + ( ) j i [ r X k ] l k = 1 2 T r T r ( r X k V ) a b X ij k l V ik ; j l [ r X k ] j i ( ) l k = 1 2 T r T r ( r X k V ) a b T r T r ( V r X k ) a b = 1 2 T r T r ( r X k V ) a b T r T r ( r X k V ) a b (4.133) In the ab o v e deriv ation, the double partial trace w as used, in whic h T r (T r ( V ) a ) b = X l k [T r ( V ) a ] l k ( ) l k = X ij l k V ik ; j l ( ) j i ( ) l k : (4.134) No w the deriv ativ es of the exp ectation v alue of the Hamiltonian can b e written in full form, rst for the n uclear p ositions r R k h H i = X d d n r R k H (0) + r R k H (1) + r R k H (2) o = X d d D 8<: 24 H (0) A ;k X l ( 6 = k )=1 Z k Z l e 2 ( R k R l ) j R k R l j 3 35 + T r ( r R k h ) T r ( r R k h ) + 1 2 T r T r ( r R k V ) a b T r T r ( r R k V ) a b (4.135)

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140 and secondly for the n uclear momen ta r P k h H i = X d d n r P k H (0) + r P k H (1) + r P k H (2) o = X d d D H (0) A ;k P k M k + T r ( r P k h ) T r ( r P k h ) + 1 2 T r T r ( r P k V ) a b T r T r ( r P k V ) a b : (4.136) 4.4 Implemen tation of the V ector Hartree-F o c k Metho d As the equations of motion ha v e b een fully disclosed, a metho d for the implemen tation of the V ector Hartree-F o c k metho d m ust b e prop osed. Some w ork has already b een made to w ard this implemen tation, through a com bination of altering the curren t structure of the co de and inclusion of new calculation subroutines as needed. It is desirable to implemen t the VHF co de in suc h a w a y as to parallel the curren t co de as m uc h as p ossible, th us eliminating the need for massiv e structural reorganization of the existing framew ork. The curren t implemen tation of the minimal END formalism is through the program end yne, v ersion 5 [ 63 ]. As w as previously men tioned, the electronic w a v e function is treated as a single determinan t in this v ersion. Figure 4.1 pro vides a general sc hematic of the computer co de structure that is emplo y ed to p erform calculations in end yne, v ersion 5. The top lev el indicates that the k ernel of end yne, v ersion 5 is called using the mo dule endkerndrv.f90 Inside the k ernel, once the conditions are suc h that ev olution of the dynamics is c hosen, the subroutine dynevo is called. Inside of dynevo the subroutine runkern is called. The subroutine runkern con tains the v arious subroutines that will compute the symplectic form, the forces, and other prop erties that are desired b y the user. Of the subroutines called b y runkern t w o are of particular in terest for the dynamical ev olution of the state. The rst, runforce allo ws for the calculation

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141 enkerndrv.f90 runspeed runkern dynevo runforce endthouless.f90 endkernwrk.f90 endmolecule.f90 Minimal END Structure Figure 4.1: Structural ro w c hart for minimal END electronic calculations. of the force v ector and the individual terms in the symplectic form. In particular, runforce will call subroutines from the mo dule endthouless.f90 in order to calculate expressions that dep end explicitly up on the Thouless parameters. T o calculate the elemen ts of the molecular Hamiltonian, runforce calls subroutines found in the mo dule endmolecule.f90 Finally subroutines from the mo dule endkernwork are called whic h results in the calculation of the n uclear symplectic form and the force v ector. Once the force v ector and the symplectic form are ev aluated and stored, the subroutine runspeed is called b y runkern Inside runspeed the symplectic form is in v erted using standard LAP A CK [ 78 ] routines. F rom the in v erted symplectic and the force v ector, runspeed calculates the v elo cit y v ector whic h is passed bac k to the dieren tial equation solv er to determine the dynamics of the system for the giv en time-step.

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142 It is imp ortan t to realize, that this is a v ery generalized sc hematic of only a small part of the curren t end yne computer co de, sp ecically the part that relates to the dynamical ev olution of the electronic state. The details of these calculations (that is to sa y the explicit form of the individual expressions found in the symplectic form) can b e found in the review b y Deumens et al. [ 10 ]. VHF Structure enkerndrv.f90 endthouless.f90 endkernwrk.f90 endmolecule.f90 runconfigpair runconfigloop runwork runspeed runkern dynevo runforce Figure 4.2: Structural ro w c hart for V ector Hartree-F o c k END electronic calculations. Figure 4.2 pro vides a prop osed sc hematic ro w c hart for dynamical ev olution in the VHF metho d. It is clear that the o v erall structure of the dynamical calculations is v ery similar to that for minimal END. While some structural c hanges ma y b e required within the v arious subroutines, the most substan tial alteration

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143 o ccurs when the runwork subroutine is called b y runforce As w as demonstrated in the deriv ations of the previous sections, the equations of motion are con v enien tly calculated b y considering only t w o congurations at a time. T o accomplish this, runkern will call the subroutine runconfigloop if a m ulti-congurational reference is pro vided in the input dec k. Once inside runconfigloop the program will cycle through all p ossible com binations of congurations. F or eac h set of congurations, and runconfigloop will call the subroutine runconfigpair The subroutine runconfigpair has a similar structure to runkern in the curren t v ersion of end yne, but the structure of man y of the subroutines and the order in whic h they are called m ust b e altered. This is principally due to the fact that a n um b er of the terms that are used in the curren t implemen tation (suc h as the END phase) are meaningless in the VHF implemen tation and the fact that the summation o v er all congurations m ust b e done through runconfigloop rather than inside runconfigpair In the curren t implemen tation, the summations o v er the single conguration can b e tallied as eac h expression is ev aluated, ho w ev er, in the VHF metho d, the expressions are ev aluated in blo c ks that corresp ond to a giv en pair of congurations. It is only after these ev aluations are made for a giv en pair of congurations that the sum can b e incremen ted. The same trio of mo dules will b e accessed from within runconfigpair as are accessed from runkern in the curren t v ersion, with sev eral alterations made to eac h. The most substan tial alterations (as could b e exp ected) will o ccur in endthouless.f90 where those expressions that are explicitly dep enden t up on the Thouless parameters are ev aluated. In particular, the curren t metho d for construction of the deriv ativ es with resp ect to the dynamical parameters in v olv es transformation of the Thouless co ecien ts from the o ccupied subspace to the virtual subspace, as outlined in the review b y Deumens et al. [ 10 ]. This transformation is not necessary and is imp osed for aesthetic reasons, as it simplies the

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144 equations in to a more compact form. The virtual space in the single determinan tal reference is the set of spin orbitals that are not o ccupied at a giv en time. Ho w ev er, in a CAS reference the set of uno ccupied spin orbitals will b e a union of the K K 2 spin orbitals that are nev er o ccupied in an y of the congurations and the spin orbitals in the activ e space that are not o ccupied at a giv en time. As the uno ccupied orbitals in eac h conguration are alw a ys dieren t from the other congurations, there is no meaningful w a y to mak e this transformation from the o ccupied subspace to the uno ccupied subspace. This necessitates a restructuring of the subroutines con tained in endthouless.f90 While this is only one of the c hanges that m ust o ccur, it is b y far one of the most signican t. One nal asp ect to the new implemen tation is that it m ust b e consisten t with the curren t implemen tation when a single determinan t reference is c hosen in the input dec k. T o accomplish this with minimal co de c hanges, a n um b er of the curren t expressions will b e ev aluated only in case of the single determinan t. The follo wing pseudo-co de pro vides a prop osed structure for the runconfigpair subroutine. While the details of the subroutines are not included, the result of eac h is indicated. The subroutines will b e lo cated in the appropriate mo dules, either endthouless.f90 endmolecule.f90 or endkernwork.f90

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145 subroutine runconfigpair !! Description of routine !! Compute oneand two-electron integrals and build the one-density blocks and auxiliary blocks from the configurations kappa and lambda as well as contributions to the energies, gradients, symplectic form. !Declare variables !! Build quantities using the overlap matrix Delta and its derivatives wrt P and R. !! Calculates the blocks of the one-density call densth if single determinant Compute a set of virtual orbitals, v, that orthonormal with the occupied orbitals, z. call coefvth Calculate the intermediate expressions needed for the construction of the symplectic form for a single configuration. call auxvth else (if multi-determinant) Calculate the intermediate expressions needed for the construction of the symplectic form for multiple configurations. call auxth Calculate the contribution to the MC electronic overlap for the current configurations. end if Calculate the electronic components of the symplectic form. call elespformth Calculate the intermediate expressions needed for the construction of the nuclear symplectic form. call auxdr Calculate the nuclear components of the symplectic form. call nucspform Calculates the coupled components of the symplectic form.

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146 call cplspformth Increment the total electronic overlap by adding the contribution of the current pair of configurations. !! Build quantities using the 2-electron integrals and the derivative integrals. !! Loop through electronic indices for the two given configurations to calculate the two-electron integrals. call teloop !! Build quantities using the one-electron integrals, the derivative integrals and the auxiliary quantities from before. !! Calculate the one-electron energy. call oeegy Calculate the sum of the electronic kinetic and potential energies (in the form of the Fock matrix). call oefock Calculate the gradient of the energy with respect to nuclear parameters (done in two parts). call oegrad call gradfin Calculate the gradient of the energy with respect to the Thouless coefficients. call gradth Return the needed expressions returnend

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CHAPTER 5 CONCLUSION F or sev eral y ears, it has b een clear that the electronic represen tations in END m ust b e impro v ed. The sto c k basis sets that had b een emplo y ed did not pro vide for prop er dynamical descriptions and the single determinan t w a v e function w as inadequate in describing more complex spin states. In this w ork, impro v emen ts to b oth the construction of basis sets and the form of the w a v e function are prop osed. The metho d of constructing dynamically consisten t basis sets has a n um b er of adv an tages o v er traditional metho ds of basis set construction. Sp ecically the new dynamically consisten t metho d emplo ys simple, ph ysically in tuitiv e principles in the construction. The prop osed metho d is computationally inexp ensiv e, as no non-linear energy optimizations are required. The basis sets constructed from the prop osed metho d are v ery general, as they are not constructed with particular prop erties in mind. Finally the basis sets deriv ed from the prop osed construction metho d pro vide go o d descriptions of electronic excitations in atoms and demonstrate impro v emen t in c harge transfer cross sections while still allo wing for go o d descriptions of commonly ev aluated structural prop erties suc h as magnitudes of dip ole momen ts and vibrational frequencies. The V ector Hartree-F o c k metho d has b een adv anced as a route of implemen ting m ulti-congurational represen tations of electrons in the END formalism. The V ector Hartree-F o c k metho d utilizes a non-orthonormal complete activ e space represen tation of the electrons, whic h is consisten t with the generalized theory of v ector coheren t states. The V ector Hartree-F o c k equations of motion arise from the 147

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148 application of the Time-Dep enden t V ariational Principle on the quan tum mec hanical action and ha v e b een completely deriv ed. The structure of the equations of motion p ermit a symplectic form, guaran teeing that the ev olution of the n uclei can b e calculated using Hamilton's equations and that the ev olution of the electronic degrees of freedom can b e calculated using Hamilton-lik e equations. Finally a suggested implemen tation is giv en that preserv es as m uc h of the curren t end yne computer co de as p ossible. The implemen tation of these new electronic represen tations, either individually or in congress, will allo w for increased rexibilit y from the end yne computational suite. Most imp ortan tly the prop osed impro v emen ts to the electronic representations will p ermit b etter descriptions of energy barriers, allo wing for increased accuracy in c hemical reactions that o ccur at lo w er energies. This will pro vide a complemen t to the curren t implemen tation of end yne that has demonstrated remark able accuracy at higher reaction energies.

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APPENDIX BASIS SET LIBRAR Y This App endix pro vides a library of the dynamically consisten t basis sets discussed in this dissertation. The library structure emplo y ed in this w ork is the A CESI I basis sets library structure. The sto c k basis sets are a v ailable in an y quan tum c hemistry basis set library and will not b e rep eated here. H:3-21BBuilt to same standards as 3-21G 1 023 1.784754 0.267713 0.025304 0.8717415 0.0000000 2.8264375 0.0000000 0.0000000 1.0000000 H:6-31BBuilt to same standards as 6-31G 1 024 2.227661 0.405771 0.109818 0.014150 0.5470819 0.0000000 1.8976888 0.0000000 1.5761883 0.0000000 0.0000000 1.0000000 149

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150 H:BJK01Constructed from scratch 3 0 1 2 3 2 1 7 4 1 25.298480 3.7947720 1.1067370 0.32132200 0.10085400 0.0252135 0.0063034 0.0115056 0.0000000 0.0000000 0.0515950 0.0000000 0.0000000 0.1960018 0.0000000 0.0000000 0.5082867 0.0000000 0.0000000 0.3753349 0.1425000 0.0000000 0.0000000 -1.0000000 0.1000000 0.0000000 0.0000000 -1.0000000 0.5623620 0.1023930 0.0276610 0.00691525 0.0509264 0.0000000 0.4163883 0.0000000 0.6755409 0.1000000 0.0000000 1.0000000 0.0150000 1.0000000

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151 H:6-31B**Built to same standards as 6-31G** 2 0 1 2 1 4 1 2.227661 0.405771 0.109818 0.014150 0.5470819 0.0000000 1.8976888 0.0000000 1.5761883 0.0000000 0.0000000 1.0000000 0.015482 1.0000000 HE:3-21BBuilt to same standards as 3-21G 1 023 2.425687 0.431771 0.032862 1.5247658 0.0000000 2.4066857 0.0000000 0.0000000 1.0000000

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152 HE:6-31BBuilt to same standards as 6-31G 1 024 6.343612 1.155497 0.312723 0.032862 0.5470819 0.0000000 1.8976888 0.0000000 1.5761883 0.0000000 0.0000000 1.0000000 HE:6-31B**Built to same standards as 6-31G** 2 0 1 2 1 4 1 6.343612 1.155497 0.312723 0.032862 0.5470819 0.0000000 1.8976888 0.0000000 1.5761883 0.0000000 0.0000000 1.0000000 0.080818 1.0000000

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153 HE:633BBuilt from scratch 2 0 1 2 1 9 3 65.789492 12.062431 3.374634 1.159278 0.450181 0.185410 0.838166 0.050896 0.019540 0.0324841 0.0000000 0.1749819 0.0000000 0.5974527 0.0000000 1.3134109 0.0000000 1.4764240 0.0000000 0.4620223 0.0000000 0.0000000 0.2124986 0.0000000 -2.1129015 0.0000000 -1.6242009 0.422185 0.108352 0.036787 0.66473192.31751101.7286327

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154 LI:3-21BBuilt to same standards as 3-21G 2 0 1 3 2 6 3 16.126766 2.937511 0.795005 0.052866 0.020080 0.009968 0.5470819 0.0000000 0.0000000 1.8976888 0.0000000 0.0000000 1.5761883 0.0000000 0.0000000 0.0000000 2.6483533 0.0000000 0.0000000 1.0124187 0.0000000 0.0000000 0.0000000 1.0000000 0.316384 0.078252 0.023133 1.8512504 0.0000000 2.7478869 0.0000000 0.0000000 1.0000000

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155 LI:6-31BBuilt to same standards as 6-31G 2 0 1 3 2 10 4 56.034549 10.366392 2.827920 1.079960 0.518147 0.271519 2.518400 0.615480 0.050150 0.009502 0.1437761 0.0000000 0.0000000 0.6068918 0.0000000 0.0000000 1.5266054 0.0000000 0.0000000 1.3194583 0.0000000 0.0000000 0.4352521 0.0000000 0.0000000 0.0399570 0.0000000 0.0000000 0.0000000 0.5263900 0.0000000 0.0000000 0.3533601 0.0000000 0.0000000 -3.6409892 0.0000000 0.0000000 0.0000000 1.0000000 1.597154 0.295474 0.080313 0.016313 0.1085890 0.0000000 0.8613239 0.0000000 1.3740381 0.0000000 0.0000000 1.0000000

PAGE 168

156 LI:6-31B**Built to same standards as 6-31G** 2 0 1 2 3 2 1 10 4 1 56.034549 10.366392 2.827920 1.079960 0.518147 0.271519 2.518400 0.615480 0.050150 0.009502 0.1437761 0.0000000 0.0000000 0.6068918 0.0000000 0.0000000 1.5266054 0.0000000 0.0000000 1.3194583 0.0000000 0.0000000 0.4352521 0.0000000 0.0000000 0.0399570 0.0000000 0.0000000 0.0000000 0.5263900 0.0000000 0.0000000 0.3533601 0.0000000 0.0000000 -3.6409892 0.0000000 0.0000000 0.0000000 1.0000000 1.597154 0.295474 0.080313 0.016313 0.1085890 0.0000000 0.8613239 0.0000000 1.3740381 0.0000000 0.0000000 1.0000000 0.037595 1.0000000

PAGE 169

157 LI:BJK01Modified from 6-31B 2 0 1 3 2 9 4 921.300000 138.700000 31.940000 9.353000 3.158000 1.157000 0.444600 0.076660 0.010250 0.0013670 0.0000000 0.0000000 0.0104250 0.0000000 0.0000000 0.0498590 0.0000000 0.0000000 0.1607010 0.0000000 0.0000000 0.3446040 0.0000000 0.0000000 0.4251970 0.0000000 0.0000000 0.1694680 -0.2223110 0.0000000 0.0000000 1.1164770 0.0000000 0.0000000 0.0000000 1.0000000 1.488000 0.266700 0.0738501 0.0117250 0.0387700 0.0000000 0.2362570 0.0000000 0.8304480 0.0000000 0.0000000 1.0000000

PAGE 170

158 LI:BJK02Modified from BJK01 2 0 1 3 2 14 7 196.918997 29.537856 8.245056 4.135330 1.838264 0.789296 0.371748 0.083903 0.624800 0.097215 0.037250 0.061045 0.015458 0.008525 0.0399726 0.0000000 0.0000000 0.2193090 0.0000000 0.0000000 0.5469290 0.0000000 0.0000000 0.6273337 0.0000000 0.0000000 1.3313663 0.0000000 0.0000000 1.0528811 0.0000000 0.0000000 0.3070866 0.0000000 0.0000000 0.0088767 0.0000000 0.0000000 0.0000000 0.5490562 0.0000000 0.0000000 -0.9872475 0.0000000 0.0000000 -2.8545279 0.0000000 0.0000000 0.0000000 -3.0560960 0.0000000 0.0000000 1.6002684 0.0000000 0.0000000 2.8407566 1.615800 0.401248 0.121326 0.045016 0.020500 0.009850 0.005102 0.0327336 0.0000000 0.2021071 0.0000000 0.8964778 0.0000000 0.9642687 0.0000000 0.2252693 -0.8998926 0.0000000 -1.3680179 0.0000000 -0.6532625

PAGE 171

159 C:3-21BBuilt to same standards as 3-21G 2 0 1 3 2 6 7 71.685060 13.057523 3.533875 0.334265 0.126967 0.018102 0.5470819 0.0000000 0.0000000 1.8976888 0.0000000 0.0000000 1.5761883 0.0000000 0.0000000 0.0000000 2.6483533 0.0000000 0.0000000 1.0124187 0.0000000 0.0000000 0.0000000 1.0000000 1.062951 0.262901 0.052516 1.8512501 0.0000000 2.7478869 0.0000000 0.0000000 1.0000000

PAGE 172

160 C:6-31BBuilt to same standards as 6-31G 2 0 1 3 2 10 4 743.444586 136.309751 38.134556 13.100249 5.087210 2.095194 6.677585 0.405486 0.155672 0.018102 0.0324841 0.0000000 0.0000000 0.1749819 0.0000000 0.0000000 0.5974527 0.0000000 0.0000000 1.3136109 0.0000000 0.0000000 1.4764240 0.0000000 0.0000000 0.4620223 0.0000000 0.0000000 0.0000000 0.2124986 0.0000000 0.0000000 -2.1129015 0.0000000 0.0000000 -1.6242009 0.0000000 0.0000000 0.0000000 1.0000000 2.259772 0.579963 0.196906 0.052516 0.6647319 0.0000000 2.3175110 0.0000000 1.7286327 0.0000000 0.0000000 1.0000000

PAGE 173

161 C:6-31B**Built to same standards as 6-31G** 2 0 1 2 3 2 1 10 4 1 743.444586 136.309751 38.134556 13.100249 5.087210 2.095194 6.677585 0.405486 0.155672 0.018102 0.0324841 0.0000000 0.0000000 0.1749819 0.0000000 0.0000000 0.5974527 0.0000000 0.0000000 1.3136109 0.0000000 0.0000000 1.4764240 0.0000000 0.0000000 0.4620223 0.0000000 0.0000000 0.0000000 0.2124986 0.0000000 0.0000000 -2.1129015 0.0000000 0.0000000 -1.6242009 0.0000000 0.0000000 0.0000000 1.0000000 2.259772 0.579963 0.196906 0.052516 0.6647319 0.0000000 2.3175110 0.0000000 1.7286327 0.0000000 0.0000000 1.0000000 0.096212 1.0000000

PAGE 174

162 N:3-21BBuilt to same standards as 3-21G 2 0 1 3 2 6 7 98.960609 18.025798 4.878485 0.478223 0.181648 0.021345 0.5470819 0.0000000 0.0000000 1.8976888 0.0000000 0.0000000 1.5761883 0.0000000 0.0000000 0.0000000 2.6483533 0.0000000 0.0000000 1.0124187 0.0000000 0.0000000 0.0000000 1.0000000 1.588988 0.393007 0.064966 1.8512501 0.0000000 2.7478869 0.0000000 0.0000000 1.0000000

PAGE 175

163 N:6-31BBuilt to same standards as 6-31G 2 0 1 3 2 10 4 1026.318860 188.174439 52.644427 18.084782 7.022850 2.892397 9.553445 0.580118 0.222716 0.021345 0.0324841 0.0000000 0.0000000 0.1749819 0.0000000 0.0000000 0.5974527 0.0000000 0.0000000 1.3136109 0.0000000 0.0000000 1.4764240 0.0000000 0.0000000 0.4620223 0.0000000 0.0000000 0.0000000 0.2124986 0.0000000 0.0000000 -2.1129015 0.0000000 0.0000000 -1.6242009 0.0000000 0.0000000 0.0000000 1.0000000 3.378097 0.866978 0.294351 0.294351 0.6647319 0.0000000 2.3175110 0.0000000 1.7286327 0.0000000 0.0000000 1.0000000

PAGE 176

164 N:6-31B**Built to same standards as 6-31G** 2 0 1 2 3 2 1 10 4 1 1026.318860 188.174439 52.644427 18.084782 7.022850 2.892397 9.553445 0.580118 0.222716 0.021345 0.0324841 0.0000000 0.0000000 0.1749819 0.0000000 0.0000000 0.5974527 0.0000000 0.0000000 1.3136109 0.0000000 0.0000000 1.4764240 0.0000000 0.0000000 0.4620223 0.0000000 0.0000000 0.0000000 0.2124986 0.0000000 0.0000000 -2.1129015 0.0000000 0.0000000 -1.6242009 0.0000000 0.0000000 0.0000000 1.0000000 3.378097 0.866978 0.294351 0.294351 0.6647319 0.0000000 2.3175110 0.0000000 1.7286327 0.0000000 0.0000000 1.0000000 0.116370 1.0000000

PAGE 177

165 O:3-21BBuilt to same standards as 3-21G 2 0 1 3 2 6 7 130.637663 23.795813 6.440076 0.651776 0.247570 0.024862 0.5470819 0.0000000 0.0000000 1.8976888 0.0000000 0.0000000 1.5761883 0.0000000 0.0000000 0.0000000 2.6483533 0.0000000 0.0000000 1.0124187 0.0000000 0.0000000 0.0000000 1.0000000 2.143685 0.530201 0.078722 1.8512501 0.0000000 2.7478869 0.0000000 0.0000000 1.0000000

PAGE 178

166 O:6-31BBuilt to same standards as 6-31G 2 0 1 3 2 10 4 1354.841066 248.408626 69.495782 23.873677 9.270847 3.818247 13.020494 0.790649 0.303542 0.024862 0.0324841 0.0000000 0.0000000 0.1749819 0.0000000 0.0000000 0.5974527 0.0000000 0.0000000 1.3136109 0.0000000 0.0000000 1.4764240 0.0000000 0.0000000 0.4620223 0.0000000 0.0000000 0.0000000 0.2124986 0.0000000 0.0000000 -2.1129015 0.0000000 0.0000000 -1.6242009 0.0000000 0.0000000 0.0000000 1.0000000 4.557349 1.169629 0.397106 0.078722 0.6647319 0.0000000 2.3175110 0.0000000 1.7286327 0.0000000 0.0000000 1.0000000

PAGE 179

167 O:6-31B**Built to same standards as 6-31G** 2 0 1 2 3 2 1 10 4 1 1354.841066 248.408626 69.495782 23.873677 9.270847 3.818247 13.020494 0.790649 0.303542 0.024862 0.0324841 0.0000000 0.0000000 0.1749819 0.0000000 0.0000000 0.5974527 0.0000000 0.0000000 1.3136109 0.0000000 0.0000000 1.4764240 0.0000000 0.0000000 0.4620223 0.0000000 0.0000000 0.0000000 0.2124986 0.0000000 0.0000000 -2.1129015 0.0000000 0.0000000 -1.6242009 0.0000000 0.0000000 0.0000000 1.0000000 4.557349 1.169629 0.397106 0.078722 0.6647319 0.0000000 2.3175110 0.0000000 1.7286327 0.0000000 0.0000000 1.0000000 0.142838 1.0000000

PAGE 180

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PAGE 185

BIOGRAPHICAL SKETCH Benjamin J. (Ben) Killian w as b orn in Lak e Cit y FL, on Ma y 12, 1976, to Jane C. and James W. Killian. His rst scien tic in terest w as for dinosaurs and w as fostered b y a visit to the Smithsonian Natural History Museum in W ashington, D.C., at the age of v e. F rom that p oin t on w ard he has b een in terested in ph ysical sciences. Ben graduated from Colum bia High Sc ho ol, lo cated in Lak e Cit y in 1994. He then attended Lak e Cit y Comm unit y College for t w o y ears, where he w as named the Math and Science Studen t of Y ear in 1996. He earned his A.A. degree with a ph ysical science emphasis in 1996. That same y ear he transfered to the Univ ersit y of Florida, where he ma jored in c hemistry In 1998 he graduated with his Bac helor of Science degree in c hemistry with American Chemical So ciet y certication. In 1999, Ben w as accepted as a graduate studen t in the Chemistry Departmen t at UF. Tw o y ears later, he w as accepted to candidacy and b egan his do ctoral w ork in Electron-Nuclear Dynamics with Yngv e Ohrn and Erik Deumens. As a graduate studen t, Ben sp en t a large p ortion of his time as a teac hing assistan t, for whic h he w as t wice recognized with a UF Chemistry Departmen t T eac hing Aw ard. In 2005, Ben w as a w arded a second place Stasc h Aw ard for Publication Excellence. In Ma y of 2001, Ben married his lo ving wife Donna, who has since b een an increasingly strong supp orter of his w ork. 173


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ON ELECTRONIC REPRESENTATIONS IN MOLECULAR REACTION
DYNAMICS















By

BENJAMIN J. KILLIAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Benjamin J. Killian















This work is dedicated to Breana, Antonia, and Blake Jones.

May the love and wonder of science always be near.















ACKNOWLEDGMENTS


I am very grateful for the environment of the Quantum Theory Project and its

members. The QTP is a wonderful place of education where continual discussion

and strong science foster intellect and curiosity.

I wish to express tremendous gratitude to Dr. Remigio Cabrera-Trujillo for his

hours of discussion, correction, and teaching throughout my graduate education, as

well as the help and friendship offered by the other END Research Group members

with whom I collaborated, Dr. Mauricio Coutinho-Neto, Dr. Anatol Blass, Dr.

David Masiello, Dr. Denis Jacquemin, Dr. Svetlana Malinovskya, and Mr. Virg

Fermo.

I offer special acknowledgment to Prof. Yngve Ohrn and Dr. Erik Deumens

for their patient and masterful day-to-day teaching and mentoring in the field of

quantum dynamics.

My grateful thanks are given to my mother and father, Jane and James

Killian, for always encouraging me to strive in every endeavor and for support

throughout the long education process. Without a parent's motivation and loving

support, very little can be truly accomplished in a person's life.

Finally, and most importantly, I wish to express my utmost love and gratitude

to my wife, Donna, for her iii.1'i -i.,iiiir. compassion, and emotional support

throughout our life together. Without her devoted love and encouragement,

(melPh.D.) 0.















TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ...................... ......... vii

LIST OF FIGURES ................... ......... viii

ABSTRACT ....................... ........... xi

CHAPTERS

1 INTRODUCTION ........................... 1

1.1 Pricis of Quantum Dynamics ...... ......... 3
1.2 Solving the Schridinger Equation ...... ...... ... 4
1.3 Time-Dependent Hartree-Fock (TDHF) ............ 6
1.4 Electron-Nuclear Dynamics (END) ....... ........ 8
1.4.1 The END Equations of Motion ............. 9
1.4.2 Minimal Electron-Nuclear Dynamics .......... 10

2 THEORY OF COLLISIONS .......... ............. 12

2.1 Scattering Theory ........... ..... ....... 12
2.1.1 Deflection Functions and Scattering Angles . 13
2.1.2 Cross Sections ................. . .. 15
2.1.3 Identical Particles .............. .. .. 17
2.1.4 Reference Frame Transformations . .... 18
2.2 Quantum Mechanical Treatment of Scattering Phenomena 22
2.2.1 The Integral Equation and its Relation to the Scat-
tering Amplitude .................. .. 22
2.2.2 The Born Series ............ .. .. .. 24
2.3 Semi-Classical Treatment of Scattering Phenomena: The Schiff
Approximation .. . . ... . 26
2.3.1 Schiff Scattering Amplitude for Large Angles . 27
2.3.2 Schiff Scattering Amplitude for Small Angles . 34

3 BASIS SETS FOR DYNAMICAL HARTREE-FOCK CALCULA-
TIONS .................. ....... ..... 41

3.1 The Hartree-Fock Approximation . . ..... 42
3.1.1 Partitioning of the Molecular Wave Function . 42









3.1.2 The Hartree-Fock Wave Function . . ... 45
3.1.3 Solving the HF Equations: Basis Set Expansions 49
3.2 General Forms and Properties of Basis Sets . ... 52
3.2.1 Slater-Type and Gaussian-Type Orbitals . ... 52
3.2.2 The Structure of Basis Sets . . .... 56
3.3 Method for Constructing Basis Sets Consistent with Dynam-
ical Calculations . . ..... ..... 63
3.3.1 Basis Set Properties for Dynamical Calculations 64
3.3.2 Physical Justification for the Basis Set Construction
Method .......... .... .. ....... 66
3.3.3 Construction of the Basis Set . . 71
3.4 Comparative Results ................... .. 82
3.4.1 Atomic Energetics .............. .. .. 83
3.4.2 Charge Transfer Results . . . 87
3.4.3 Properties of Diatomic and Triatomic Molecules 98

4 VECTOR HARTREE-FOCK: A MULTI-CONFIGURATIONAL
REPRESENTATION IN ELECTRON-NUCLEAR DYNAMICS 104

4.1 Introduction of the Lagrangian and Verification of the Equa-
tions of Motion . . .......... ... 104
4.2 Parameterization of the State Vector . . 106
4.3 The VHF Equations of Motion . . 114
4.3.1 Derivation of the Equations of Motion . ... 115
4.3.2 Evaluating the Equations of Motion . ... 120
4.4 Implementation of the Vector Hartree-Fock Method . 140

5 CONCLUSION.... ....... ........ .......... .. 147

APPENDIX BASIS SET LIBRARY ................. .. .. 149

REFERENCES ................... ............. 168

BIOGRAPHICAL SKETCH .................. ......... .. 173















LIST OF TABLES

Table page

3.1 Notation employed in this review. ............. . 42

3.2 Slater Exponents and Coefficients for He, Li, and Be. . ... 79

3.3 Slater Exponents and Coefficients for B, C, and N. . .... 80

3.4 Slater Exponents and Coefficients for O, F, and Ne. . ... 81

3.5 Atomic energies and electronic excitations in Helium . ... 84

3.6 Atomic energies and electronic excitations in Lithium . ... 85

3.7 Molecular properties of the nitrogen molecule. .. . ..... 99

3.8 Molecular properties of the carbon monoxide molecule. . ... 100

3.9 Molecular properties of the water molecule. . . 102















LIST OF FIGURES

Figure page

2.1 Diagram of a classical collision system. .... . ... 13

2.2 Three possible deflection angles. A demonstrates a positive deflection
angle. B demonstrates a negative deflection angle. C demonstrates
a deflection angle with magnitude greater than 27. . ... 14

2.3 Collision of distinguishable particles. ................ .. 17

2.4 Collision of identical particles. ................ ..... 17

2.5 Relation of collision pair and center-of-mass to origin. . ... 19

2.6 Relation of the scattering angles between reference frames. . 19

3.1 Comparison of STO and GTO representations of the radial part of
the hydrogen Is orbital. Top: A single GTO function fit to a sin-
gle STO function. Bottom: A linear combination of six GTO func-
tions fit to a single STO function. ................. .. 55

3.2 Top: Plot of the radial distribution function for the Is (-), 2s (- -
), 3s (. .), and 4s (- --) orbitals of the hydrogen atom. Bottom:
Plot of the radial distribution function for the 2p (-), 3p (- -),
and 4p (. .) orbitals of the hydrogen atom. . . 69

3.3 Top: Plot of the radial distribution function for the Is (-), 2s (- -
), and 3s (. .) orbitals of the argon atom. The Is-orbital is scaled
by a factor of two-thirds. Bottom: Plot of the radial distribution
function for the 2p (-) and 3p (- -) orbitals of the argon atom. .70

3.4 Plot of the Is orbital exponent for the atoms through Kr as a func-
tion of atomic number. The data are from Clementi and Raimondi
(+). ...... .. .. .. .. ... .. .. .... ... ..... 73

3.5 Plot of the 2s and 2p orbital exponents for the atoms through Kr as
a function of atomic number. Top: The 2s orbital exponents. Bot-
tom: The 2p orbital exponents. The data are from Clementi and
Raimondi (+) and from the present work (o). . . ... 74









3.6 Plot of the 3s and 3p orbital exponents for the atoms through Kr as
a function of atomic number. Top: The 3s orbital exponents. Bot-
tom: The 3p orbital exponents. The data are from Clementi and
Raimondi (+) and from the present work (o). . . ... 75

3.7 Plot of the 4s and 4p orbital exponents for the atoms through Kr as
a function of atomic number. Top: The 4s orbital exponents. Bot-
tom: The 4p orbital exponents. The data are from Clementi and
Raimondi (+) and from the present work (o). . . ... 76

3.8 Comparison of the probability for near-resonant charge transfer be-
tween H+ and Li at 10 keV collision energy. The following Li basis
sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02
(black). The BJK01 basis set for H was used for each run. . 89

3.9 Comparison of the probability for near-resonant charge transfer be-
tween H+ and Li at 1 keV collision energy. The following Li basis
sets are used: 6-31G (red), 6-31B (blue), BJK01 (purple), BJK02
(black). The BJKO1 basis set for H was used for each run. . 90

3.10 Comparison of the differential cross section for near-resonant charge
transfer between H+ and Li at 10 keV collision energy. The follow-
ing Li basis sets are used: 6-31G (red), 6-31B (blue), BJKO1 (pur-
ple), BJK02 (black). The BJKO1 basis set for H was used for each
run .............. ................ .. 91

3.11 Comparison of the total cross section for near-resonant charge trans-
fer between H+ and Li as a function of energy. The experimental
data are: Varghese et al. (+) and Aumayr et al. (x). The theoret-
ical data are: Allan et al (-) and Fritsch and Lin (- -). For the
END data the following Li basis sets are used: 6-31G (0), 6-31B
(o), BJK01 (A), BJK02 (0). The BJK01 basis set for H was used
for each run. . . . . .. ... . 92

3.12 Comparison of the probability for resonant charge transfer between
Li+ and Li at 1 keV collision energy. The following Li basis sets
are used: 6-31G (red), BJK02 (blue). ............... .. 93

3.13 Comparison of the probability for resonant charge transfer between
Li+ and Li at 10 keV collision energy. The following Li basis sets
are used: 6-31G (red), BJK02 (blue). ............... .. 94

3.14 Comparison of the differential cross section for resonant charge trans-
fer between Li+ and Li at 1 keV collision energy. The following Li
basis sets are used: 6-31G (red), BJK02 (blue). . . ... 95









3.15 Comparison of the total cross section for resonant charge transfer be-
tween Li+ and Li at as a function of collision energy. The experi-
mental data are form Lorents et al. (x). The theoretical data are
from Sakabe and Izawa (-). For the END data the following Li
basis sets are used: 6-31G (0), 6-31B (o), BJK02 (A). . 96

3.16 Comparison of the probability for resonant charge transfer between
He+ and He at 5 keV collision energy. The following He basis sets
are used: 6-31G** (red), 6-31B** (blue), and BJKO1 (purple). 97

3.17 Comparison of the differential cross section for resonant charge trans-
fer between He+ and He at 5 keV collision energy. The experimen-
tal data are from Gao et al. (*). For the END data the following
He basis sets are used: 6-31G** (red), 6-31B** (blue), and BJKO1
(purple). . . . . . .. . 98

4.1 Structural flow chart for minimal END electronic calculations. . 141

4.2 Structural flow chart for Vector Hartree-Fock END electronic calcula-
tions. ........ .. .. .. .. ... .. .. .. .. ... 142















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



ON ELECTRONIC REPRESENTATIONS IN MOLECULAR REACTION
DYNAMICS



By

Benjamin J. Killian

August 2005

Chair: Nils Y. Ohrn
M.,. ir Department: Chemistry

For many decades, the field of chemical reaction dynamics has utilized compu-

tational methods that rely on potential energy surfaces that are constructed using

stationary-state calculations. These methods are typically devoid of dynamical cou-

plings between the electronic and nuclear degrees of freedom, a fact that can result

in incorrect descriptions of dynamical processes. Often, non-adiabatic coupling

expressions are included in these methodologies. The Electron-Nuclear Dynamics

(END) formalism, in contrast, circumvents these deficiencies by calculating all

intermolecular forces directly at each time step in the dynamics and by explicitly

maintaining all electronic-nuclear couplings.

The purpose of this work is to offer two new frameworks for implementing

electronic representations in dynamical calculations. Firstly, a new schema is

proposed for developing atomic basis sets that are consistent with dynamical

calculations. Traditionally, basis sets have been designed for use in stationary-state









calculations of the structures and properties of molecules in their ground states. As

a consequence of common construction techniques that utilize energy optimization

methods, the unoccupied orbitals bear little resemblance to 1.r, -i .,l virtual atomic

orbitals. We develop and implement a method for basis set construction that relies

upon 1ir, -i, .,l properties of atomic orbitals and that results in meaningful virtual

orbitals. These basis sets are shown to provide a significant improvement in the

accuracy of calculated dynamical properties such as charge transfer probabilities.

Secondly, the theoretical framework of END is expanded to incorporate a

multi-configurational representation for electrons. This formalism, named Vector

Hartree-Fock, is based in the theory of vector coherent states and utilizes a

complete active space electronic representation. The Vector Hartree-Fock method

is fully disclosed, with derivation of the equations of motion. The expressions for

the equation of motion are derived in full and a plan for implementing the Vector

Hartree-Fock formalism within the current ENDyne computer code is given.















CHAPTER 1
INTRODUCTION


Chemistry is the study of how atoms and molecules change with time, and

fundamentally requires a dynamical description to correctly describe these changes.

Experimental ]li-, -i .,1 chemistry is concerned with measurement of chemical

changes and the enumeration of the properties associated with these changes.

Quantum chemistry, on the other hand, seeks to describe the mechanism through

which chemical changes occur and to calculate properties associated with these

changes as a function of fundamental properties of the molecule itself.

Virtually all of chemistry is described in the vocabulary of dynamics. Chemists

speak of electron transfers, transition states, equilibria. The most fundamental

measurements made in a 1ir, -i .,1 chemistry laboratory involve transference of

heat over time, the number of vibrations a molecule undergoes in a unit of time,

the inversion of electron populations as a function of time. Chemistry is dynamic.

Despite all of this, non-dynamical approaches are still most often employed when

performing chemical calculations. These approximations are often sufficient for

obtaining average properties, but fail to offer a deeper understanding of chemical

processes that comes through dynamical methods.

Traditionally, chemists have been interested in answering two general classes of

questions regarding chemical reactions. The first, "To what extent will a reaction

occur?" is purely thermodynamic in nature. Provided the difference in the free

energy between the product species and the reactant species is negative, the

reaction has the potential to occur spontaneously to some extent. Calculations of

this type are well-suited to time-independent quantum mechanics; stationary state









energy values can be calculated for various ground and excited states, which can

yield (through the application of statistical thermodynamical principles) the free

energies of the products and the reactants.

The second class of question that a chemist might ask is, "How fast will

the reaction occur?" This type of question is kinetic in nature and relies upon

some (usually detailed) knowledge of the dynamics of the reaction. This type of

question is, in generally, only successfully answered using time-dependent quantum

mechanical treatments. To correctly describe the kinetics of a reaction, one must

first describe the interactions through and responses to inter- and intramolecular

forces as a function of the time-scale of the reaction.

The focus of this dissertation is to investigate methods of improving represen-

tations of electrons for use in time-dependent dynamical calculations. Specifically,

this work will investigate two sides of the very same coin. Firstly, a discussion will

be made toward basis set expansion for use in construction of the electronic wave

function. A new method will be proposed for the construction of dynamically con-

sistent basis sets. Two particular features of the proposed construction method are

the pl-i, -.1l basis that underlies the method and its ease of application. Secondly,

discussion will be made toward the construction of a multi-configurational wave

function for use with the Electron-Nuclear Dynamics formalism. The full set of

equations of motion are derived for the Vector Hartree-Fock implementation of

Electron-Nuclear Dynamics in terms of a general expansion of atomic basis func-

tions. Additionally, discussion is made as to a possible scheme for implementation.

This chapter provides a basic review of quantum molecular dynamics. Subse-

quently, a brief discussion is made of the general structure of the Time-Dependent

Hartree-Fock method. Finally the Electron-Nuclear Dynamics formalism in its

simplest form is introduced and discussed and compared to Time-Dependent

Hartree-Fock.









1.1 Pr6cis of Quantum Dynamics

The lr, l-i, .,1 description of any chemical object is predicated on the concepts

of a quantum mechanical state and a quantum mechanical configuration [1]. A

quantum mechanical configuration is the set of all descriptive variables in the

dynamical phase space (e.g., momentum and position) and/or in the electronic

Hilbert space (e.g., orbital and spin angular momentum) which describes (within

the completeness of the set) a given chemical object at any given instant in time,

t = r [1]. The configuration is usually represented as a ket in Dirac notation,

written as 1|, t = r), where 9 is a representative of the required set of descrip-

tive variables. If r is specifically defined, then the configuration may be more

conveniently expressed as 1|).

A quantum mechanical state, or wave-function, can also be expressed in Dirac

notation, 1|, t), where t indicates the general dependence on time. The state can

be defined as any subset of the complete set of configurations that contains a given

configuration, 90, t = r), as well as all configurations that result from the dynamical

evolution of |1, t = r) and all configurations that evolve into 91, t = r) with the

passage of time (assuming no perturbation of the system by outside influences) [1].

Some descriptions must include the time dependence explicitly in the wave-

function, while others allow for the time-dependence to be treated as separable

factor, depending on the appearance of time in the Schridinger Equation [2]. For a

state to be completely described, one must know the relation between all variables

that determine the state at a specific time, usually denoted as time t = 0.

Investigation of chemical processes requires the knowledge of how a given state

evolves with time -how the state changes dynamically. The principle tool in the

descriptions of quantum chemical dynamics is the Schridinger Wave Equation,

a time-dependent differential equation of second order spatially and first order

temporally. The Schridinger Equation defines the equations of motion for a given









quantum mechanical state; once the initial state is completely described, then the

time evolution of the state as defined by the state-specific Schrodinger Equation

allows one to correctly predict the future properties of the state based on its initial

conditions. The state-specific Schrodinger Equation can be expressed in function

notation as
N
V 21tt. + t) 1 (1.1)
i= 1
On the left hand side of Equation (1.1), the first term is the kinetic energy of the

state; the summation is over the N particles comprising the state, M, is the mass

of the ith particle, and V2 is the Laplacian of the ith particle. The second term

on the left is the potential energy of the state. The potential energy of the state is

generally a complicated function of several or all of the dynamical variables and is

of particular importance to the description of quantum dynamics. One can combine

these two terms and define the action of these terms by an operator, 4r, called the

Hamiltonian operator. Thus, Equation (1.1) can also be written in operator form,

as

\0 t} = h t). (1.2)

Equation (1.2)is the set of equations of motion required to calculate the dynamics

of a quantum mechanical state.


1.2 Solving the Schr6dinger Equation

The bulk of quantum dynamical research is involved with devising and refining

methods to solve Equation (1.2). A molecular Hamiltonian operator will take the

form (in atomic units, where h = c m 1)

K N K K K N N N
4 IV'772 7v72 Y y Za yyZa z z' ,(1.3)
a=1 i=1 a=1 3 % +1 = a,13 I 1 i ,a j1 i=j+1 i'

where the indices a and 3 refer to elements of the K nuclei and i and j label

elements of the N electrons. Furthermore, Za is the charge of nucleus a and r,/p,









rF,i, and rij are the distance between nuclei a and 3, the distance between nucleus

a and electron i, and the distance between electrons i and j, respectively.

The complexity of the molecular Hamiltonian operator renders the Schr6dinger

equation analytically intractable for all but the simplest systems. For this reason,

certain approximations must be made to simplify the system of equations. Many

methods treat the nuclear and electronic degree of freedom separately, in what is

usually referred to as the Born-Oppenheimer approximation. Born and Oppen-

heimer [4] first introduced the concept of an adiabatic separation, in which the

electronic and nuclear degrees of freedom are decoupled. In a simplistic description,

the mass of the electrons, m, is so small in comparison to the nuclei, M, that the

electronic motion is significantly faster than the motion of the nuclei, allowing one

to perform a separate electronic calculation at each nuclear configuration. The re-

sult of this approximation is that the electronic potential energy can be constructed

at a large number of nuclear configurations, resulting in a potential energy surface

that is a function of the nuclear coordinates. At this point, the nuclear Hamilto-

nian is cinl'-'v,-d in which the potential energy term is now a sum of the Coulomb

repulsion of the nuclei and the specific electronic potential energy that correspond

to the given nuclear configuration. This approximation gives rise to such methods

as molecular dynamics (or quantum molecular dynamics if the nuclei are quantum).

By epinll-,ving excited state surfaces, multi-surface methods arise, such as surface

hopping methods.

One immediate deficiency of these Born-Oppenheimer-type methods is the

lack of dynamical coupling between the electrons and the nuclei. As a result, it

is desirable to discuss methods in which no potential energy surfaces are utilized.

This requires direct calculation of the potential energy between nuclei and elec-

trons at each time-step. Such methods will be the focus of the remainder of this

dissertation.









Two specific methods will be discussed. First is the Time-Dependent Hartree-

Fock method. Second is the Electron-Nuclear Dynamics formalism. Both methods

are time-dependent methods that do not require the calculation of potential energy

surfaces.

1.3 Time-Dependent Hartree-Fock (TDHF)

The TDHF method was first proposed by Dirac in 1930 [5]. The principal

assumption of the TDHF method is that the self-consistent central field approx-

imation holds. In this approximation (which is equivalent to the Hartree-Fock

approximation) the total electronic wave function is composed of a product of

wave functions for each electron. Furthermore, each electron moves in an average

potential.

The reference state for TDHF methods is the single determinant Hartree-Fock

ground state, represented as a Slater determinant (using second quantization),


ID) =a...a>vac). (1.4)

The TDHF state vector has the form


I|) = eiF,," A(t a lD) (1.5)

which has a corresponding Dirac density operator of the form

P e x (rs t)a ', 'i I, .- ty r,, (t)a~ (1.6)


The density operator has a corresponding matrix form that is idempotent (F2 F)

and the trace is equal to the number of electrons in the system [6]. The quantum

mechanical Lagrangian takes the form

= [(ih- W ) + (lih l) -] ('WF P '), (1.7)
2 Ot Ot









where # is the Fock operator, which has matrix elements of the form


rs = hrs Y (rs |pq)F,p (1.8)
pq

where hr, is an element of the one-electron Hamiltonian matrix.

The equations of motion for the TDHF method are derived from the fact that

the Lagrangian should be stationary with respect to small variations in the TDHF

state vector, such that

"') -> t 6A,,at aI), (1.9)
r'S
as well as a corresponding variation in the density operator. By requiring the

Lagrangian to be stationary under such an arbitrary variation, one obtains the

TDHF equations of motion

i = [F, r], (1.10)

where the dot indicates the time derivative and the brackets denote the commuta-

tion relation [a, b] = ab ba.

At this point, it is assumed that the time-evolution will act as a small pertur-

bation to the Fock operator. The one-electron Hamiltonian will be perturbed such

that

h,s h, + 6hs, (1.11)

where ho, is the one-electron Hamiltonian that corresponds to the unperturbed HF

stationary ground state. The second term contains the time-dependence. Secondly,

the two-electron component is perturbed through the density, such that


p F + 6F, (1.12)


where, again, the subscript indicates the HF stationary ground state solution and

the time-dependence is carried by the perturbative term. The above perturbations

are then substituted into the TDHF equations of motion and, traditionally, terms









of order two or greater in the perturbations are truncated. This implementation

results in a method that is linearized and limits the electron-nuclear couplings

within TDHF methods [7].

Furthermore, as both components of the Fock operator depend explicitly upon

the electron density, the equations of motion must be solved iteratively at each

time-step. Again, this is traditionally not done. In most TDHF methods, the

reference state is determined as either a configuration interaction (CI) expansion to

the level of single excitations (resulting in the Tamm-Dankoff Approximation) [8]

or as a perturbation expansion to the level of single and some double excitations

(resulting in the Random Phase Approximation) [9]. By allowing the expansion

coefficients to be time-dependent, the time-evolution of the individual orbitals is

now no longer required and the single reference is used throughout the dynam-

ics [7]. This is also a limitation to the TDHF method, as the state is not permitted

dynamically outside of the limit of the single configuration included in the HF

reference state.


1.4 Electron-Nuclear Dynamics (END)

The END theory is a non-adiabatic formulation allowing a complete dynamical

treatment of electrons and nuclei that eliminates the need for potential energy

surfaces that is equivalent to a generalized TDHF approximation. Electron-

Nuclear Dynamics is derived from application of the time-dependent variational

principle (TDVP) on a family of approximate state vectors parameterized in terms

of Thouless coefficients. The END formalism and application of END to varied

pllv-i, .,l problems has been explained in detail in previous works [10, 11, 12, 13]. A

brief account of the derivation of the END equations of motion will be provided in

this section.









1.4.1 The END Equations of Motion

The derivation begins with the quantum mechanical action functional

A dt, (1.13)

where Y is the quantum mechanical Lagrangian, defined as

(O (1.14)

Here ( defines a complete set of time-dependent parameters that describe the

particular choice of state vector and A is the quantum mechanical Hamiltonian
operator for the system. By requiring that the action remain stationary under

variations of the parameters (, one obtains the Euler-Lagrange equations

d OY OY
(1.15)
dtA aQ1'

for the set of dynamical variables {(Q} [14]. Application of the principle of least
action results in a coupled set of first-order differential equations of motion

o -iC* (
0 ( ) (1.16)
iC 0 *

where

E((*, () (1.17)

is the energy of the system and acts as the generator of infinitesimal time transla-
tions and where C is an invertible metric matrix with elements defined as

C 2 In S (
C 3 0(* (1.18)

The metric matrices define the couplings between the dynamical variables. The

argument of the logarithm in Equation (1.18) is the overlap, defined as S = ((I).

The equations of motion given in Equation (1.16) are exact; any approximations









to the END formulation arise due to the choice of dynamical parameters ( and the
completeness of the electronic basis set.

1.4.2 Minimal Electron-Nuclear Dynamics
The minimal END theory is implemented in the ENDyne computer code. In
the current version, nuclei are treated classically and electrons are represented
using single Thouless determinants that are complex and single valued [15]. All
electron-nuclear couplings are retained. The dynamical parameters are chosen as

(={Rk, Pk, Zph, Zph}, where Rk and Pk are the average position and momentum
of the kth nucleus and Zph and z*h (spanned by a set of atomic orbitals) are the
Thouless coefficient corresponding to the pth atomic orbital of the hth spin orbital
and its complex conjugate, respectively. Choosing the state to be represented as a
product coherent state vector


I|) = z; R IR, P) } \z)\ (1.19)

where the nuclear wave function is expressed as a product of traveling Gaussians


|)J exp ( R- )2 k(Xk Rk)j (1.20)
k
taken in the narrow wave packet limit (-,, 0, for all -,, ), and where the
electronic state is represented with a single determinant electronic wave function

4z = det{xh(Xp)}, (1.21)

allows for a consistent description of the electron-nuclear dynamics. The molecular
spin orbitals in Equation (1.21), Xh, are spanned by a set of basis functions,

{uh(x)}, that are Gaussian-type atomic orbitals (GTOs) centered on the average









nuclear positions. By


iC

0



iCp


0

-iC*

iC

-iC
-


where the dynamic m


making these assumptions, Equation (1.16) takes

iCR iCp z E/Oz

-iCh -Cp z* oE/Oz*

CRR -I +CRpP E/OR

I+ CpR Cpp P \ E/OP

etric elements of Equation (1.22) are

a2 In S
(CxY)ij;kl -2 Im 0--
OXikOiYj R'=R


the form




(1.22)







(1.23)


(1.24)


82 In S
(Cxk)ph n S
Ozd, Xik
1 R' =R


and

Chg 02 nS (1.25)
O-Z iOZqg

The coupling is explicit in the metric elements given in Equations (1.23) -

(1.25), nuclear-nuclear coupling, non-adiabatic nuclear-electronic coupling, and pure

electronic-electronic ('liii, respectively.

Minimal Electron-Nuclear Dynamics is a generalization of the TDHF

method [10]. The electronic and nuclear interactions are still regulated through

the Fock matrix, but there are some differences in the dynamical evolution of the

state. Firstly, the variation of the one-density in the END formalism is achieved

through a general variation of the Thouless coefficients and is not truncated at

any particular order. As a result, END is equivalent to fully non-linearized TDHF.

Secondly, the reference state is permitted to change as warranted by the dynamics

of the system. This provides additional flexibility to the dynamical evolution, as

the reference state is not limited to a stationary state as calculated previous to the

dynamical evolution.















CHAPTER 2
THEORY OF COLLISIONS


Many chemical reaction principles and results can be elucidated and obtained

by virtue of so-called "- II iiii, exp. i iiil I.-". The experiments most generally

involve a beam of projectile species incident upon a reaction cell containing

target species or upon a second beam of target species. The reactions and energy

transferences occur in the volume of the reaction cell or the union volume of

the crossed beams. Theoretical descriptions of such scattering phenomena are

contained in the classical, quantal, and semi-classical realms of collision theory.


2.1 Scattering Theory

While experimental results are obtained for bulk phase reactions in general,

excellent theoretical descriptions of these scattering processes can be made by

considering the interaction of a single projectile particle with a single target

particle. At it's simplest, classical scattering theory involves the reduction of a

two-body problem into a reduced one-body problem. This transformation results

in a description of the motion in the center-of-mass reference frame. This type of

analysis is similar to the analysis of unbound Kepler motion, however, the nature of

the central force is considerably different (though the functional form may be very

similar) and all information about the orbits is lost (only the incident momentum,

the final momentum, and the angle between the two is observed) [16]. In this

section, we will discuss the fundamentals of classical scattering theory and methods

to extend these fundamentals into the language of atomic and molecular collisions,

quantum scattering theory.










2.1.1 Deflection Functions and Scattering Angles

One begins by considering a classical collision system consisting of two point

masses in the laboratory frame with some undefined potential acting between

them that is a function only of the separation between the particles. It will be

assumed that, initially, one particle is in motion (the projectile) and one is a rest

(the target). Figure 2.1 demonstrates such a collision system. The scattering

axis is defined as the axis parallel to the incident motion of the projectile and

containing the location of the target, as demonstrated by A in Figure 2.1. The

impact parameter, labeled as b, is defined as the distance at which the projectile

is initially located in a direction perpendicular to the scattering axis. As one

can assume a spherically symmetric potential, V(r), this can be in any direction

perpendicular to the scattering axis.

The other principle feature of Figure 2.1 is the deflection angle, labeled as T.

The deflection angle is the angle through which the projectile is deflected by the

potential, as measured in a counter-clockwise manner from the positive scattering

axis. The deflection angle can be positive due to a repulsive potential (as demon-

strated in A in Figure 2.2), or it can be negative due to an attractive potential (see

B in Figure 2.2). Furthermore, under certain conditions, the projectile can orbit









Projectile

b
/A
Target
Figure 2.1: Diagram of a classical collision system.











---.....------"^


A. > ""








B.














Figure 2.2: Three possible deflection angles. A demonstrates a positive deflection
angle. B demonstrates a negative deflection angle. C demonstrates a deflection
angle with magnitude greater than 2w.


the target for one or more periods, resulting in a deflection with magnitude greater

than 2r radians (see C in Figure 2.2).

It should be noted that, experimentally, it is impossible to distinguish between

the three deflections shown in Figure 2.2. Experiment can only determine the angle

at at which the projectile is scattered, relative to the scattering axis, and cannot

elucidate whether a particle is scattered through a positive deflection, a negative

deflection, or through an orbiting deflection with magnitude greater than 2r

radians. Thus, we must define a parameter that corresponds to what is l1ier-i .'ily

measured; a parameter called the scattering angle. The scattering angle, denoted 0,









is defined as

0 = I mod 2I1. (2.1)

For any scattering system there exists a mapping that relates the impact

parameter to a resulting scattering. This mapping is called the deflection function,

O(b). It is obvious from the previous paragraph that the deflection function is

not an injective Im.,i iiinJ.. as more than one impact parameter may result in the

same scattering angle. Furthermore, the deflection function is not a surjective

iii..''iir-:. as there is no guarantee that the projectile will be scattered into every

angle between 0 and r.


2.1.2 Cross Sections

While reaction rates are usually the quantities desired from -, .ii I ig

exp. iinii. ii. the fundamental observable of such experiments is the differential

cross section [17]. The differential cross section (da/dQ) is defined as

da scattered current per unit solid angle (
(2.2)
df incident current per unit area

At this point it becomes necessary to assume some limited quantum nature of

the scattering system. Without further justification at this point in the discussion,

we will introduce the general form of the scattering wave function. It is customary

to choose the ansatz wave function for the target beam by considering the ..i -'mp-

totic regions where the effect of the scattering potential produced by the target

(assumed to be finite) is negligible. In the post-scattering region, the scattered

wave function will be a linear combination of incident plane wave and scattered

spherical waves [18],
eikfr
e e""ik i) f(, ikr (2.3)

where ki and kf are the magnitudes of the initial and final moment of the pro-

jectile, respectively. Both the numerator and the denominator in Equation (2.2)









require evaluation of the probability current density, j, for the wave function,

defined as
1
j (y*V V *) (2.4)
2mi

where m is the mass of the projectile. It should be noted that vectors are denoted

using boldfaced letters, while respective magnitudes are represented using non-

boldfaced letters. The numerator of Equation (2.2) is found from the current

density of the outgoing spherical component by the expression jot dA, where dA

is the unit differential area normal to the solid angle subtended by dQ. For this

case, dA = r2 dr. It should be noted that, due to the area, only the r component

of jot is required, greatly simplifying the calculation. The denominator of Equation

(2.2) is likewise found considering the current density of the incoming plane wave,

ji,. As all incident particles are considered, the denominator is just the incoming
current density. Thus, the the differential cross section can now be expressed as

da jot dA
dQ Jii (2.5)
kIf(0, 0) 2
ki
Thus, the differential cross section depends only on the square of the amplitude of

the scattered spherical wave. It should be furthermore noted that a state-to-state

differential cross section may be obtained if a probability amplitude for transition is

included in the scattering amplitude

The second quantity of import in '-, ., I iiin exp. I iii, iii is the total cross

section, a, defined as

a= sin Ododo. (2.6)

The total cross section depends only upon the relative kinetic energy of the

colliding particles and is an effective area of scattering or reaction. If the projectile

strikes within this effective area it will be scattered or reacted, however, because










A B



e e
is not
equivalent to
B B



B A

Figure 2.3: Collision of distinguishable particles.


of the avn.,.ii,-:. the specific information contained within the differential cross

section is lost.


2.1.3 Identical Particles

Consider the collision of two particles in the center of mass frame (Figure 2.3).

To simplify the explanation, it will be assumed that axial symmetry exists,

eliminating a dependence on 9. Because each particle is distinguishable throughout

the entire collision process, each can be determined to scatter through either and

angle of 0 or an angle of 7 0. However, a interesting problem arises when identical

particles are considered. Even if the particles are distinguishable at some time

of separation, once the particles enter the interaction region they are no longer

distinguishable. The particle scattered through angle 0 is indistinguishable from

the particle scattered through angle r 0 (Figure 2.4).

A A




A A

equivalent to




A
A


Figure 2.4: Collision of identical particles.









When considering identical particles, the ansatz wave function for the projec-

tile now takes the form [19]

eikfr
Se &ir + [f(0) f( -0)] eik (2.7)


where the sign is determined by whether Fermi-Dirac or Bose-Einstein statistics are

employed. As the particles are identical, momentum is conserved in the collision.

From the same argument provided in Section 2.1.2, the differential cross section for

identical particles in the center-of-mass frame is


= If(0)f( ) 2. (2.8)


2.1.4 Reference Frame Transformations

One benefit of the END formalism is that calculations need not be performed

in the center-of-mass frame of reference. This is beneficial as experimental results

are generally reported in the lab frame. However, it was demonstrated in the

previous section that the calculation of the differential cross section for the reaction

of identical particles is best handled in the center-of-mass frame. Thus, a schema

for transformation between lab and center-of-mass frames must be developed.

Consider the collision of two particles, as depicted in Figure 2.5. The first

particle (projectile) has a mass, mi, the second particle (target) has mass, m2. In

the lab frame, the particles have positions, ri and r2, and moment, kl = mn l and

k2 = r22 (where the dot indicates differentiation with respect to time). In the

frame relative to the center-of-mass, the particles are defined by positions, sl and

S2, and moment, pi = ms1 and p2 = m22. The center-of-mass for the collision

system is defined as having mass, M = mi + m2, position, R, and momentum,

P = MR, relative to the origin.

It is evident from the definition of the system that the position of a particle

in the lab frame differs from the position in the center-of-mass frame, at any time










S


r
2/










Origin

Figure 2.5: Relation of collision pair and center-of-mass to origin.


during the dynamics, by the position of the center-of-mass,


ri s1 + R.


(2.9)


Differentiation of the above equation and multiplication by mi yields the relation

amongst moment,


k P mi
kl = pl + -P.
M


(2.10)


If one considers the moment for the projectile in the two reference frames

after the collision, one may obtain the relationship between the scattering angles in

the two reference frames. The projectile possesses a momentum k{ in the lab frame

P
___i^_p
/, ^^


Figure 2.6: Relation of the scattering angles between reference frames.









and pl in the center-of-mass frame. The superscript f indicates a final condition.

The projectile is scattered through an angle of a in the lab frame and an angle

of 0 in the center-of-mass frame (Figure 2.6). It is evident that the transverse

components of the moment are equal,


p{ sin(0) kf sin(a), (2.11)

and that the longitudinal components differ by the momentum of the center-of-

mass,

p{ cos(0) -f cos(a) P. (2.12)
M
Transformations between the scattering angles are obtained by taking the

appropriate ratios of Equations (2.11) and (2.12). The center-of-mass scattering
angle is obtained from the expression

sin(a)
tan(0) si (a) (2.13)
cos(a) J

This generalized equation can be much simplified under two assumptions. If the

particles are identical then r- = If the target is initially stationary in the

lab frame, then the magnitude of the center-of-mass momentum is equal to the

magnitude of the initial momentum of the projectile (k ) at any time during the

collision. Thus, the simplified expression takes the form

sin(a)
tan(0) os(a) (2.14)
cos(a) 7

where 7 Likewise, the reverse relation can also be obtained by inverting the

ratio. The lab frame scattering angle can be obtained from the general expression

sin(0)
tan(a) sin (2.15)
cos(0) + 7
p17









Again, the expression can be simplified. If the particles are assumed to be identical,
then (as before) 4- = 1 Additionally, the collision is elastic in the center-of-mass
frame, requiring that pl = P at all times in the course of the dynamics. Thus,

sin(0)
tan(a) cos() (2.16)

which indicates that a = 10.
Once the scattering angle is converted to the center-of-mass frame, the
deflection function and the differential cross section can be calculated. The center-
of-mass frame differential cross section must then be transformed back to the lab
frame. This is accomplished by realizing that fact that the number of particles

scattered into a given solid angle must be conserved between the two frames [20],
such that
d- sin(a) da =(- sin(0)|| do (2.17)
"- lab dQ CM
or
(da dc \ d[cos(a)]
)dlab dJOM d[cos(a)] ( 21)
The multiplicative factor in Equation (2.18) is evaluated by first applying the
Law of Cosines to Figure 2.6 and substitution into Equation (2.12), yielding

cos(O) +
cos(a) cos( + (2.19)
V1 + e2 +2 cos(0)'

where = ~. Now the derivative can be taken with respect to cos(0),

d[cos(a)] cos() + 1
(2.20)
d[cos(O)] (l+ 2 2 cos(0))3 2'

As the numerator above never goes to zero in the domain 0 = [-7, r], Equation
(2.20) can be inverted to yield the required factor,

d[cos(O)] (1+ 2 + 2 cos(0))3/2
d[cos(a cos() (2.21)
d[cos(a)] ( cos(0) + 1









The expression in Equation (2.21) is completely general to a collision pair. As

before, simplifications can be made by considering the particles to be identical.

As seen earlier, this assumption requires that =- 1. Under this assumption, the

transformation from the center-of-mass frame to the lab frame takes the form

d 4 cos (2.22)
lab CM

2.2 Quantum Mechanical Treatment of Scattering Phenomena

To this point the description of scattering phenomena has been tacitly

quantal. The scattering wave function ansatz was introduced and a discussion

of the identical particle problem was made, but no reference to the Schrodinger

Equation has yet been made. In this section, the differential form of the scattering

Schr6dinger Equation will be introduced, a derivation of the integral form of the

Schr6dinger Equation will be made, and this integral form will be related to the

scattering amplitude. The Born Series and the various Born Approximations, a

set of self-consistent solutions to the integral form of the Schr6dinger Equation is

then introduced. Finally, the Schiff Approximations for large and small scattering

angles, which are derived from the Born Series, will be fully derived and discussed.


2.2.1 The Integral Equation and its Relation to the Scattering Ampli-
tude

The time-independent Schrodinger Equation

hV2(r)+ V(r)2(r) E (r) (2.23)
2m

can be rewritten to take the form


[V2 + k2] () (r)(r), (2.24)

where k = /(2mE/h2) is the magnitude of the momentum vector for the scattered

particle and where U(r) = 2mV(r)/h2 is the scattering potential. Equation (2.24)









is an inhomogeneous differential equation with a solution [17, 18]


y(r) o- o(r) + G(r, ro)U(ro)(ro)dro, (2.25)

where yo(r) and G(r, ro) are, respectively, a general solution and the Green's

Function that corresponds to the homogeneous counterpart of Equation (2.24),

which is of the form

[V2 + k2] o(r) 0. (2.26)

The above homogeneous differential equation is nothing more than the free particle

Schrodinger Equation, and the solution is yo(r) = eik. Additionally, the Equation

(2.26) is in the form of the Helmholtz Equation and therefore the corresponding

Green's Function that satisfies Equation (2.25) takes the form [14]

-ik|r-rol
G(r, ro) >- -ro (2.27)
47|r ro

Thus, one finds that Equation (2.25) now becomes, upon substitution,

1 eik|r-rol
y(r) ek I -U(ro)(ro)dro, (2.28)

which is the integral form of the Schrodinger Equation with scattering potential

located at position ro.

It is now desirable to find an expression that relates the scattering amplitude

to the integral form of the Schrodinger Equation. To accomplish this, one returns

to the post-scattering conditions in which the ansatz of Equation (2.3) is assumed

to hold, namely that r > ro. Under this requirement, the angle between r and ro

approaches zero and Ir rol r ro r. This approximation allows one to express

the integral form of the Schrodinger Equation as

1 eikr. (2.29)
(r) eikr e ikro U(ro) (ro)dro. (2.29)
47 r









One can then immediately compare the modified integral form of the Schr6dinger

Equation as given above with Equation (2.3), which demonstrates clearly that


f(0) = f(k) eikroU(ro)(ro)dro. (2.30)
47

2.2.2 The Born Series

Though one has now seen the solution to the scattering wave function in

Equation (2.28), the solution is analytically intractable, as the integrand itself

depends on y. Solution of the scattering wave function requires numerical integra-

tion to self-consistency. This self-consistent method yields the Born Series, with

truncations yielding the Born Approximations [19, 21] of various orders.

The First Born Approximation assumes that the scattering potential has a

negligible effect on the incoming plane wave, thus the scattered wave function takes

the form

= eki'r, (2.31)

The subscript on r is for indexing purposes only. This term can be substituted into

Equation (2.30), and the scattering amplitude in the First Born Approximation

then becomes

f(k, kf) -= / e ikfrU(rl)eiki.rldri. (2.32)
47
In some cases conditions are such that the First Born Approximation is sufficient

for analysis, most notably when the scattering potential is not very strong and its

effective range is quite small [19]. However, the First Born Approximation rarely

provides adequate accuracy.

The Second Born Approximation is begun by iterating on the wave function.

The scattered wave function for the second order of approximation now becomes

Seiki'1 + f G(ri r2)U(r2)2(r2)dr2. (2.33)








Substitution of Equation (2.31) into the above equation yields the wave function in
the Second Born Approximation

Seikir1 + G(ri r2)U(r2)eikr2dr2 (2.34)

which generates the second order scattering amplitude

f(k, kf) = [- eik-r l(rl)eikildr
(2.35)
+ e-k*rlU )G(r) r U r2(r2, k r2dr, rl

This pattern can then be continued, and one can make successive iteration to
an arbitrary degree and obtain the nth order wave function

Seikfr G(ri r2)U(r2)eikir2 dr2

+ f I G(r r2)U(r2)G(r2- r3)U(r3)e ik 3drdr

+...+ (2.36)

.+ G(r, r2)U(r2)G(r2 r3)U(r3) X ...

x G(r,_1 rn)U(rn)eiki.rndrn ... dr2.

The infinite Born Series is obtained by letting n oo (with careful reordering
of the indices). The infinite Born Series leads to a scattering amplitude with the
general form

f (k, kf) =- e-ik-f.U(rn)G(rn r,_l)U(r,_l)
rn=1 (2.37)
x .. x G(r2 rl)U(rl)eiki' dr... drl.

Now, while Equation (2.37) is exact, it is very unwieldy to solve and still
depends upon the scattering potential function, a property not utilized within nor
obtained from END. Furthermore, the Born Series is often afflicted with slow or
no convergence [2]. So it is desirable for a number of reasons to express the Born









Series in a simplified form, in particular a form that does not depend explicitly

upon the scattering potential.


2.3 Semi-Classical Treatment of Scattering Phenomena: The Schiff
Approximation

The work in this section was first introduced by L. I. Schiff in 1956 [22]. This

discussion is based on that paper.

The scattering potential and the projectile particle can be characterized by

several pir, -i. .,1 properties. The scattering potential can be described by V and R,

which are rough indications of the strength of the potential and its effective range,

respectively. Likewise, the projectile is characterized by its kinetic energy, T; its

wave number (magnitude of the momentum), k; its speed, v; and its scattering

angle, 0.

These characteristics provide a qualitative means by which to discuss the

scattering process. For example, the First Born Approximation is valid under

conditions when the scattered wave is insignificantly perturbed by the scattering

potential (c.f. Equation (2.31)). This can be expressed in a qualitative manner

by the magnitude of V being very small compared to the collision energy (weak

scattering potential) or by the magnitude of R being very small (short range

scattering potential). Schiff provides a general condition under which the First

Born Approximation is valid as given by the expression (IVIR)/(hv) << 1 [22].

The Schiff Approximation provides an approximate scattering amplitude

for large collision energy collisions. Specifically, the assumptions imposed are

that IV/T << 1, that 0 be very large or very small in comparison to 1/(R),

and that the scattering potential be slowing varying when compared to the

incoming wavelength. In contrast with the First Born Approximation, the Schiff

Approximation is valid for any magnitude of (IVIR)/(hv) [22].









2.3.1 Schiff Scattering Amplitude for Large Angles

The Schiff Approximation consists of approximately representing each of the

terms in the Born series using the stationary state approximation [22, 23, 24].

One begins the derivation with the infinite Born series, Equation (2.37), rewritten

slightly to the form


f (k, kf) ..-. e-i(kf -r~-krl)U(r)G(r r,_i)U(r,_i)
n=1 (2.38)
x x x G(r2 rl)U(rl)dr, dri.

A change of variables is made, such that p,- = r -r,_l, so that r, p,, _+r,_1.

The argument of the exponential term can be expanded iteratively as

-kf r, + ki ri -kf- (p,_- + r,_) + k (r2 pi)

-kf p,_1 kf (p~-2 + r_2) + ki (r3 P2) ki pi



S-kf p,_ kf p_2 kf p,_3 ...- kf *(p, + rm) (2.39)

+ ki (r, Pm-) ... ki p2 k *i pi

= -kf P_- kf Pn-2 kf p_ -... kf p,

+ q r, ki pi- ... ki p2 ki pi,

where q = ki kf is the momentum transfer for the collision process. To complete

the transformation of variables in Equation (2.38), one must calculate the Jacobian

of the transformation, specifically

pi '_ '' 'P 'P 2 L

Opl 'P- '' 'P I 'P 2 L
S_ Or2 Or2 Or2 Or2 r2 Or2 (2.40)


Opi 'p_ 'P L 'P L 'P 2 'P L
Dr.n Drn Drn Orn Orrn rrn,









noting that the r, row and column are omitted. The derivatives take the general

form
-1 forf = g
apf 1 for f = g (2.41)

0 otherwise

From this, it can be justified that the Jacobian matrix has the values of -1 along

the diagonal, the values of 1 for each element immediately below the diagonal, and

the value 0 elsewhere. From this is can be seen that the determinant takes the

value J = 1. Therefore,


dp,_l...dp = dr,...drm+ldrm _...dri, (2.42)


and the limits of integration do not change.

The change of variables may now be accomplished, transforming Equation

(2.38) into the following form


f (ki,k) = -- / / e' i-k-p 1-...-k*Pm+1+qrm-k'Pm- 1-...-kiP
.-n lrn 1

x U(rm Pn+ p- + Pn-2 + -. + Pm)G(Pni)

x U(rm + Pn-2 -+ P-3 P+m)G(pn-2)

x ...X

x U(rm + pm)G(pm)U(rm)G(pm -)U(rm Pm-1)G(pm-2) (2.43)

X ...X

x U(rm Pm-i Pm-2- ..- P3)G(P2)

x U(rm Pmi Pm-2 ..- P2)G(Pl)

X U(rm Pmi- Pm-2 Pi)

x drmdpn1 dp,- ...dp2dPl.









where the fact that

r im+pf_l+pf_i+...+P, for f >m
rf rm 1Pf2 f > (2.44)
rm Pm-1 Pm--2 -P1 for f < 7

was used to indicate the p dependence of the potential terms. The second summa-

tion (over m) arises from the the stationary phase approximation. The majority of

the integral will come from the regions where the phase is stationary, specifically

where the derivative of the phase with respect to the wave vector is zero [24].

Thus, one must sum over all of these stationary phase points to completely enumer-

ate the integral.

Now, recognizing that the transformed Green's function for the scattering

amplitude equation now takes the form

eikfP
G(r, r,_l) -- G(p,_-) e p (2.45)
47p

the above equation can be partitioned into the form
Sn n-l



x p ei(kfPn1-kf'Pn -k g(p n-)d 1
J (2.46)
X Pn_2ei(kP 2-k p 2)g(pn_2)dp_2 X ...

SP21ei(kfP2-ki'P2)g(P2)dP2 x f Pei(kp-ki pl)g(Pi)dl

In Equation (2.46), the terms g(pj) represent a product of all U terms that depend

explicitly upon pj. Thus, for an elastic scattering process, the scattering amplitude

has been reduced to a product of integrals of the form

I pg(p)ei(k-k)dp (2.47)
*2j7









To evaluate this integral, one needs to transform to spherical polar coordinates

(p, 0, q) where 0 measures the angle of k with respect to p,

I= j d sin Od pg(p, 0, )e(k-kcos)p, (2.48)

where the scalar product is geometrically evaluated. One can now make the
familiar change of variable p = cos 0, where dp = -sin OdO. Substitution (with
appropriate change of limits) yields

I= d0 dp pg(p, 0)eikp(-p)dp. (2.49)

One can now integrate by parts with respect to pt, resulting in the expression

I= f r pd i p ( )eikp(-) ikp(1-p)d}

(2.50)
Integration of the second term in the braces will bring down another factor of

1/(kp). By applying the limits on the first term, one can write

I = du pdp -i [g(p, 1,)- g(p, -1, )eikp] + (k-2). (2.51)
o J kp

As the collision energy is assumed to be large (k >> 1), one can reasonably expect
the magnitude of kp to be somewhat greater than 1. Thusly, the exponential will
be highly oscillatory and subsequent integration over p will result in a negligible
contribution to I. As a result, one can omit this term, as well as the terms of order
k-2 and higher. Furthermore, if one assumes a spherically symmetric scattering
potential, the value of 9 becomes arbitrary and integrates out to a factor of 27.
Thus,
I i- j g(kp)dp. (2.52)
k Jo
The kp arises due to the facts that p = 1 (0 = 0) and that 9 is arbitrary, and so
the potential term now depends only on the magnitude of p in the direction of the








scattered momentum. So, the integration over the angles serves to transform the
dependence on the vector p in the scattering potential into a dependence upon the
magnitude of p in the direction of the momentum.
Now, returning to Equation (2.46) one finds that integration of the angular
dependence results in the equation
i oo n ( / n-1 coo
f(k, kf) 4= --k- eiq'r- drm/ g(kfpn-1)dpn-1
n=lI m=/ k J (2.53)
x g(kfpn-2)dp.-2 x ... x g(kiP2)dP2 g(kip1)dpi}.

The individual g terms can be factored back to their respective U terms, with the p
dependence replaced by kp dependence. Specifically, one finds that

f.k<,-f +) {( -i) n-1j drn dP... dn-er-
oo n n-1n--2


x U (r + kf(pn_1 P+ n-2 ** Pmn)) X (2.54)

x U rm + kf Pm U (rm) U (rm kipm-i) x ...

xU (rn ki(Pm-l + Pm-2 +... + P)

At this point, a second change of variables can be made, such that

Pj + Pj+ + ... + pPm-2 Pm,-1 for j < m
sj P= m + Pm+i + --- + Pj-2 + j-i1 for j > m (2.55)

Pm for j = m

There are two recursion relations that arise, specifically sj = p + sj+l for j < m
and sj = sj-1 + pj for j > m. It is clear from following the previous analysis
that the Jacobian is unity, but the lower limits of integration must be transformed.
When integrating over pj for j < m, one finds that the lower limit of pj = 0
transforms to sj+l. Likewise, when one integrates over pj for j > m, one finds
that the lower limit transforms to sj-1. Therefore, the change of variables may be









imposed, transforming Equation (2.54) into the form
i n ( n-1 o
f(k kf) = { ( i /drmei- r ds1U (rm- m i1s)
47- 1 mn-l I 1 2

SdS2U (rm iS) X ... X j dSm r m -iSm-2)

X j dSm-1U (rm kism-,l) dsmU (rm kfSm)
0 0 0
X dS+iU (rm+ kfSm+l) x ... x dSn-2 (rm+ kfSn-2)
JS "Sn-3
X ds,-lU rm+ fSn-1) s }
Sn-2
(2.56)

At this point, the above equation can be partitioned into two parts, the part

explicitly depending upon the scattered direction (kf) and the part depending

explicitly upon the incident direction (k ). If one considers the product depending

only on the incident direction, then yet another substitution can be made, such
that

W1j U (rrn- kiSj -1_ S (2.57)

and

Wo U (rm- kism-i) dsm-1. (2.58)

Thus, calling the product in question K, one finds that

K j dsiU (ri sis j dS2U (m iS2) ...
K S2 J S300
X dSm-2U rm kim-2) dSm-lU (mr kiSm-1) (2.59)
SWm-l_ 1 0 d m2 ..

W/ dH0W- _2 X ... X W dW2 / d-i.
0 0 1 1








The expression for K can now be integrated, with the following results:
rWo rWm- fW3 rW2
Jo Jo Jo Jo
rWo rWm- 1W?
K dW, dW,- x ... x dWW2

2WO d,_1 id. X.. 2W
J O Jo Jo
2 x 3 d1WMl dWm_2 x ... x I dW4W4
2 x 3./n .n .O


(2.60)


S2x3x ( -- 3) dW m-
2x .." m )J Jo


dWm2W,7rn-3


1 fWO
2x3x... x (m- 3) x (m 2) O -l l
1
2 x 3 x ...x (m 3) x (m 2) x (n 1)
r/IW m-1 -
[(m -l)!]- [ U (rm- kis ds

A similar substitution can be made for the product involving the direction of
scattered momentum, allowing for the evaluation of these n m integrals,

j dSmU (rm kfSm) j dSm+U (rmn + kfSm+l) ..

X dSn-2U (rm +kfSn-2) X X dSn-U (rm + fS n-1
Sn-3 n-2
-[(n -m)!]-1 [ U (r + kfs) ds]

The scattering amplitude can then be expressed using these terms,
.
f(ki, kf) --- drme 1 e U (rm)
n 1 rn 1

x [(m -1)!] [ U (rm is) ds]

S[(n -m)!]1 U (rm+ k) ds] n-m}


(2.61)








(2.62)









At this point, the above equation can be further simplified, but cannot be

made devoid of its dependence upon the scattering potential, which was the

intention. However, a second approximation can be made which will allow a general

expression that does not depend explicitly upon the scattering potential.


2.3.2 Schiff Scattering Amplitude for Small Angles

The above expression for the scattering amplitude is valid under conditions

that k is large, so that the second term in Equation (2.51) is negligible and

that the scattering angle is large, requiring that the n different stationary phase

points are distinct. For this discussion, one begins with the assumption that the

scattering angle is very small, such that 0 << (kR)-1/2. This results in two specific

requirements. First, the n distinct stationary phase points now converge to a single

point, thus eliminating the requirement for summation over m. Secondly, the

scalar product in the exponential of Equation (2.62) can be written in its Cartesian

components,

eiq'rm ei(q9m+qyym+qzzm) (2.63)

Since the scattering angle is assumed to be so small, the angle between ki and kf

is nearly zero, ensuring that q, is negligible. This can be explicitly demonstrated if

one requires that ki coincide with the z-axis. Thus,

q q ki

S(k kf) k,

k- ki-kf '

= ki- kf cos (2.64)

= kf (1 cos 6)



kf 02
2









In the above demonstration, the fifth line was obtained by virtue of the fact

that ki = kf for elastic scattering. Furthermore, the sixth line was obtained

by nipl,-ving the Maclaurin series representation of cos 0 with truncation at

the second term. It becomes clear that, because 0 is so small, the longitudinal

component of the momentum transfer is negligible, as would be expected ]li, -i, .illy.

This is not true of the transverse components of the momentum transfer.

Again, one must consider the momentum transfer,


qAq q = + q, (2.65)

where the longitudinal component has been omitted. Using the definition of the

momentum transfer, one finds that

2 2 )2
q1 + qY (ki kf)2

= +k k+ 2kikf cos0

= 2k 22k cos0
(2.66)
2k (1 cos0)

2k2 (02)
kf 2



Thus, it is clear that the transverse components of the momentum transfer are both

of the order of ~ kO and cannot be neglected. Converting to Cartesian components,

the scattering amplitude now becomes

f(k kf) { f /" x do 1
,I I
n=l x o i(qx+q~y)
foo dzm
x dzmU (xm, [m, zm) [(m 1)!]1 U (xrm, ym, z) dz (2.67)
JOO -[J d

x [(- )!]1 U (r, yz) dz
,zm -









To obtain the above form, it must be recognized that the first integral over s

in Equation (2.62) represents the pre-scattering trajectory while the second

integration over s represents the post-scattering trajectory. As the scattering angle

is so small, these to wave-vectors are effectively the same. Therefore, instead of

integrating over two different trajectories one can integrate a single trajectory for

both regions (-oo to Zm and then Zm to oo).

The integration over Zm is of principle interest, so one can define it as M, such

that

M f dZmU (xm,, Ym, Zm) [(m 1)!]1 U (xm, Y, z) dz
-m (2.68)
x [(n m)J!]}~ U (xm", m, z) dz

One can then introduce the notation that


w U (xm, Ym, z) dz (2.69)
-00

and
r00
a U (xm, m,, z) dz. (2.70)
--oO
In order to be able to integrate over w, one must evaluate the derivative of w with

respect to m,,

dw d fzm
dz dz / U (Xm, ym, z)
dm dZm -oo (2.71)

SU (Xm, yn, Zm) .

Therefore, it is clear that dw = dzmU (xm, y,, Zm). Finally, the new limits of

integration must be evaluated, leading one to find that


j U (Xm, ym z,) = 0 for zm -= 0
w U = (2.72)
f J U (,m' Y z) -) a for zm = oc









Thus, the integral over Zm now can be rewritten as

M [(m 1)! (n m)!] 11 a -) -mdw. (2.73)

This equation must now be integrated by parts through m 1 iterations. For the
first integration, one can let u = w"1 and dv = (a m)"-"dw. Thus,

M j W- -(a w)ndw
S__m-__ wn-m _

(m )! (n m)! n (m )(2.

1 1 )Wjm(a ( w) w )--) dw"
M I 1-2


(m 2)!(n (m 1))! o

For the second integration, one can let u = w"-2 and let dv = (a w)"-(-l)1dw,

such that

M = ja W 2(a w)n-(m 1)dw
(m 2)!(n- (m 1))! o
1 -1 a
m-2(a W)n-(m-2)
(m 2)!(n- (m- 1))! n (m-2) 2
(2.75)
n (2 (%) -( W)n-(m-))d
n (m 2) 1 3

(m-3)!(n- (m 2))!(a 2

Following this pattern, one finds that after m 1 integration, M takes the form


M (a w)-ldw. (2.76)
(n 1)! f

The final integration can now be performed, yielding the following expression,

M = (a w)n-dw
(n 1)!
S- (a w) (2.77)
(n 1)! n

n!









The scattering

Thus,


amplitude is no longer dependent upon m, as should be expected.


f (k, k-) =- J ax 1!) a
47 t 2k k n!



24rf J x 2k) n!
k m "I
2 17 00 2k n!
2 1
2- J dx dyei(q +qy) (e i)
ik ( r" -
Ix .hpi,. +qyy) 1 exp U (x, U, z) dz .
27x 1- 2k )
(2.78)

At this point, it should be noted that the integration over x and y is taken

over the plane perpendicular to the scattering axis. In the limit of an axially sym-

metric scattering potential, x and y can be represented by the impact parameter,

b = /x2 2, and the azimuthal angle, 0, such that x = b cos 0 and y b sin 0.

Thus, the transformation to cylindrical coordinates renders the scattering ampli-

tude into the form


f(ki, kf) = d(1 e(qb cos+qb sin)bdb I exp U (b, z) dz]
27 Jo 2k-oo
(2.79)

Now, as the scattering potential is spherically symmetric, the momentum transfer

will always be parallel with b. As a result, the expression qb cos 0 + qyb sin 0 will

have the same value regardless of the value of 0. Therefore, one can choose any

value of 5, and then sum all of the contributions from 0 to 27. Specifically, if one

allows = 0' = 0, then q, = q and qy 0. Consequently,


qb cos 0 + qb sin = qb cos 0'.


(2.80)









The integral over 0 now takes the form of the zeroth-order Bessel function [14],

2 eqbcosdo 27Jo(qb). (2.81)

The small angle Schiff Approximation to the scattering amplitude can now be
expressed as

f(k, kf ) ik Jo(qb) exp U (b, z) dz bdb. (2.82)

Unfortunately, the scattering amplitude is still an explicit function of the
scattering potential, a term that is not obtained during an END calculation. This
expression can, however, be transformed into one that does not depend on the
scattering potential [23]. If one assumes that r is the position of the projectile at
any point during the trajectory, then the fact that the scattering angle is very small
requires that
r r (z + b) (z + b), (2.83)

where z is the position of the projectile along the scattering axis. As the impact
parameter is perpendicular to the scattering axis, it is clear that r2 = 2 + b2. From
this, one finds that
dr
dz = (2.84)
[1 (b)]1/2
Therefore, the argument of the exponential now becomes
I 1 f U(r)
i U (b, z) f U(r) ( )d
2k o 1 2 (b)]1/2
o [(2.85)
1 p 1U(r)
k [1 (b)]1/2

where the fact that the limits of integration remain the same when the transforma-
tion from z to r is made. Furthermore, the symmetry of the scattering potential
was utilized in the second step. The above expression is the Massey-Mohr ap-
proximation for the semi-classical phase shift, 6(b), in the small scattering angle








limit [21, 25, 26] which is the semi-classical analog of the Kennard approximation
for classical scattering processes [23, 27]. Specifically, the M.. -- -Ml 1ir approxima-
tion is given the form [23]


6(b) k 1 2 (2 12 dr. (2.86)

In the .i-'. mptotic limit, r >> b, therefore,

U (b, z) -26(b). (2.87)
2k

Finally, this means that the Schiff Approximation scattering amplitude for small
angle scattering takes the form [23, 28]

f (ki, kf) ik Jo(qb) (1 exp [2i(b)]) bdb. (2.88)

While Equation (2.88) no longer depends explicitly upon the potential, it
has introduced the semi-classical phase shift, which itself is not a value pro-
vided by END calculations. This is easily remedied, though, by the fact that
the semi-classical phase shift is directly related to the deflection function by the
expression [19, 21, 23]

(b) (b) (2.89)
k db
The Schiff Approximation provides an effective method for calculating the
scattering amplitude without any prior knowledge of the scattering potential. The
principle power of the Schiff Approximation is the fact that it explicitly contains
all of the terms in the infinite Born series. Furthermore, the Schiff Approximation
provides good differential cross sections in the region of small scattering angle,
which the region most often reported by experimental studies.















CHAPTER 3
BASIS SETS FOR DYNAMICAL HARTREE-FOCK CALCULATIONS


Quantum mechanical calculations on molecular systems are traditionally

divided into two general categories, the localized valence bond-type (VB) calcula-

tions and the delocalized molecular orbital-type (\10) calculations. In many ways,

these two descriptions serve as complements to each other. Where VB methods,

in general, provide more visual 1l., 1-i .,1 descriptions, such as molecular geome-

tries, they generally fail with important properties, such as paramagnetism and

bond strengths. The MO method tends to provide a more rigorous quantum me-

chanical description of molecules, such as electron delocalization over the entirety

of a molecule and variations in bond strengths, but generally offers little in the

way of 1ir, -i' .illy intuitive descriptions of the molecular system. Both classes of

methods are formulated from rather coarse approximations, and, as a consequence,

each possesses deficiencies, in particular with relation to electron correlation.

The VB methods tend to over-estimate electron correlation, while MO methods

tend to under-estimate correlation effects [29]. While each has its strengths and

weaknesses, and while each converges to a common molecular wave function rep-

resentation in the absence of their respective generalized approximations, a large

community of quantum chemists has chosen to employ MO methods for quantum

chemical calculations. The MO method, and its application in the Hartree-Fock

approximation, will be the focus of this chapter.

Some address must be made toward notation. The literature of ab initio

quantum chemistry, as with most other fields, is not completely consistent. While

it is not crucial to describe a set of canonical variables, clarification of symbolic











Term


Table 3.1: Notation employed in this review.

I Meaning


x

a,o3

dV
dr
Uppercase Roman indices
Lowercase Roman indices
Lowercase Greek indices


Exact electronic wave function
Generally approximated electronic wave function
Electronic spin-orbital
Spatial electronic factor
Spin electronic factor
General electronic basis function
Element of spatial volume (no spin included)
Element of volume (spin included)
Nuclear indices
Molecular orbital expansion coefficients
Atomic orbital expansion coefficients


notation up front is the most lucid manner of handling the issue. The notation

of Szabo [30] will be most closely followed, with minor adjustments as needed.

Table 3.1 provides an exhaustive list of notation used in this chapter.


3.1 The Hartree-Fock Approximation

The MO method in quantum chemical calculations of molecular systems has

become synonymous with the Hartree-Fock (HF) approximation. In this section,

the HF approximation and the HF wave functions will be introduced. Additionally,

the concept of electron correlation and the correlation effects will be addressed.

Finally, an in depth discussion of HF basis sets will be made.


3.1.1 Partitioning of the Molecular Wave Function

In Section 1.2, the non-relativistic molecular Hamiltonian operator was

introduced, with the explicit form
K 1 72 K NK K K N N
Y KV72 K ZAZB >1>1 ZA K ,(3.1)
A=l i= A=1B i= A+1a j i=j+l i,

where the indices A and B refer to elements of the K nuclei and i and j label

elements of the N electrons. Furthermore, ZA is the charge of nucleus A and TA,B,

TA,i, and rij are the distance between nuclei A and B, the distance between nucleus









A and electron i, and the distance between electrons i and j, respectively. The

complicated coupling between the nuclear and the electronic degrees of freedom

cause the Schrodinger equation to be practically insoluble by direct methods. To

simplify the problem, the Born-Oppenheimer (BO) approximation was introduced,

in which the nuclear degrees of freedom are assumed to vary on a much larger time-

scale than the electronic degrees of freedom. In essence, the BO approximation

allows for a "clamped nucleus" model, in which the nuclei are fixed in space

and the electronic degrees of freedom are allowed to vary on this fixed nuclear

framework. As the nuclei are motionless, the nuclear kinetic energy terms (the first

summation in Equation (3.1), above) become identically zero. This assumption

decouples to nuclear degrees of freedom from the nuclear degrees of freedom. The

resulting form of the Schr6dinger equation is now called the electronic Schr6dinger

equation and takes the form

N K K K NN N
ZC za ZAZ Z f ZA YEY (3.2)
i=1 A=l B=A+1 rAB Al i=1 r j= i=j+1 i

It is the solutions to this equation with which the majority of quantum chemistry is

focused.

Following the example of L6wdin [31], one can partition the electronic Hamil-

tonian operator into three parts,

N N N
ea = ho + h' j ih. (3.3)
i=1 i=1 j=i+1

In the above equation, the term ho is the nuclear repulsion (or zero-electron

Hamiltonian) term, which depends only upon the fixed nuclear degrees of freedom

and is defined as
K K
h A ZAZB (3.4)
A=1 B=A+1









The second term in Equation 3.3, h}, is the one-electron Hamiltonian. The

one-electron Hamiltonian includes the additive kinetic energies of the individual

electrons as well as instantaneous attractive potentials between the N electrons and

the K nuclei and takes the form


h' v.- Z (3.5)
A=1 'ia

The final term, hi, is the two-electron Hamiltonian, which generates the in-

stantaneous repulsive forces between each individual electron and the remaining

molecular electrons. The two-electron Hamiltonian takes the form


h = -(3.6)

The terms denoted by ho result in a constant for a given nuclear configuration,

and therefore will not have any bearing on our choice of molecular wave function.

Rather, the total energy will only be increased by the fixed potential energy values

associated with h.

The above partitioning of the molecular Hamiltonian operator in Equation

(3.3) allows one to write the molecular wave function as a product wave function of

the form

ed ({fi}; {(A}) I ({fi; {rA})I)2({ri}; { A}), (3.7)

where the factor wave functions are eigenfunctions of the operators in Equation

3.3, specifically,


4K1b = 1i, (3.8)

,42 eA2, (3.9)

where e1, and e2 are the energy eigenvalues. The total energy for the given nuclear

configuration, 8, becomes the sum of the energy eigenvalues and the nuclear









potential energy defined by h,


S= Si + e2 + Vn. (3.10)


The major focus of quantum chemistry involves the solution of Equations

(3.3), (3.7), and (3.10), and we will use these equations as the starting point for the

discussion of the Hartree-Fock approximation.


3.1.2 The Hartree-Fock Wave Function

The HF approximation begins with the assumption that the total electronic

wave function can be approximated by a product of one-electron wave functions.

Furthermore, one must assume that the potential experienced by a given electron

is an average of the potentials produced by the remaining electrons [29]. In this

approximation, the term ho and the N different h' terms are maintained, but the
-(N 1) two-electron terms are replaced by N additional one electron terms. By

replacing the two-electron terms with one-electron terms, the HF approximation

does not explicitly treat the instantaneous interaction of individual electrons.

Rather, each electron is treated as if it were influenced by an average field produced

by the other electrons in the molecule. This approximation allows for relatively

accurate quantum chemical calculations despite the gross approximations imposed,

though several important 1.r, -i, .,l descriptions are omitted. These correlation

effects will not be discussed in this chapter.

The new electronic Hamiltonian can be rewritten in the form
N
i [ =v[h HF] (3.11)
i= 1

where h1 is the one-electron Hamiltonian (as previously defined) and VHF is the

new HF one-electron potential energy term (to be defined later in this section).

Because the zero-electron Hamiltonian only adds a constant factor to the electronic









energy, it is omitted. The most striking benefit of this reformulation of the

electronic Hamiltonian is the fact that, because the Hamiltonian operator is now a

sum of one-electron terms, one can approximate a single N-electron wave function

as a product of N one-electron wave functions (called a Hartree product),

N
HP ({xi; {A}) j( ; FA}), (3.12)
j=1

which is an eigenfunction of the one-electron Hamiltonian,


j7lHP= EHP. (3.13)


As before, the eigenvalue is the energy of the system.

The wave function defined in Equation (3.12) is a product of a set of N spin-

orbitals, Xy, which are themselves eigenfunctions of the respective one-electron

Hamiltonians,

[h + VHF] X = X, (3.14)

where ce is the energy eigenvalue of the jth orbital. The total energy, E, is a sum

of the orbital energies,
N
E = e, (3.15)
j=1
The electronic Hamiltonian in Equation (3.11) does not have a spin dependence,

therefore a transformation from spatial electronic coordinates, {ri}, to spatial-spin

coordinates, {xi} by employing a product form of the spin-orbitals does not change

the energy eigenvalues of Equation (3.13). Thus, in the absence of relativistic

effects, the N spin-orbitals can be represented by a product of a spatial-dependent

factor, y)({ri}), and a spin-dependent factor, either "spin up", a(wc), or "spin

down", P(cw), such that


Xji = ( }) ) (3.16)
Y j({rt})X(1c)









The variable ac represents the general spin variable. It should be noted that the

product wave functions indicated in the above equation have the same spatial

factors with varying spin factors. This is consistent with restricted HF method.

Allowing each spin-orbital to have a unique spin AND spatial factor results in the

unrestricted HF method. [30].

The partitioning of the spin-orbitals given by Equation (3.16) satisfies the

Pauli exclusion principle by allowing two spin-orbitals to have the exact same

spatial factor, but opposite spins (that is, one molecular orbital with a paired set

of electrons). However, a further requirement of the molecular electronic wave

function is that it must obey the Fermi-Dirac statistics, in particular that the

electronic wave function must be anti-symmetric with respect to exchange of

electron indices. The Hartree product given by Equation (3.12) does not satisfy the

Fermi-Dirac statistics, being that the sign of IJHP remains unchanged if two indices

are exchanged. Adherence to the Fermi-Dirac statistics traditionally required

detailed group algebra. This was greatly simplified by Slater, who circumvented

group theoretical descriptions by introduction of the spin-orbital function directly

into a determinant that would later bear his name [32].

Slater exploited the property of matrix algebra that, given a matrix, inter-

changing any two columns of the matrix will change the sign of the determinant of

the matrix. Thus, Slater constructed a matrix in which the spin-orbitals are placed

as the columns and the occupying electrons are placed as the rows of the matrix.

The determinant of this matrix is the most general anti-symmetrized product wave

function. Mathematically, one finds that,

Xi(xi) X2(X) .. XN (X)

sater (N!)-1 (N!) det {Xj(xi)}. (3.17)


X1(xN) X2(XN) ... XN(XN)









At this point some discussion of the approximated one-electron potential in

Equation (3.11) can be made. It is beyond the scope of this review to explicitly

derive the form of the term HF, so it must suffice to say that the potential form

can be obtained by employing Lagrange's method of undetermined multipliers to

the energy eigenvalue equation [30]

esl3tSlater = E TSlater. (3.18)


The resulting HF Hamiltonian (variously referred to as the Fockian and denoted

F) is a sum of N Fock operators, which satisfy the eigenvalue equations of the

form
N N

hiJ+ A- Xi = iXi (3.19)
ji ji
The term in brackets in the above equation is the Fock operator for orbital i. It

can be seen that the approximated one-electron potential has been split into two

components. The first is the coulomb operator, which defines the interaction of

electrons with an average potential. The action of the coulomb operator is to

provide an average repulsive potential felt by an electron at the position xl that

arises from an electron in a second orbital. The coulomb operator has the inverse r

form of a coulomb interaction, weighted by the probability density of the orbital to

be averaged, specifically,


J(xi)x(xi) [/dx2,X(x2) 2rxi, j(X1). (3.20)

The coulomb operator arises as a consequence of the assumption that the electronic

Hamiltonian is a sum of one-electron operators only. The second operator results

from the anti-symmetrization of the wave function through the use of a Slater

determinant. This operator, the exchange operator, results in the exchange of two

electrons and produces a one-electron potential that is dependent upon the value

of the orbital in question throughout all space [30]. The form of the exchange









operator is

J(xi)xj(xi) dx2Xj(x2)r 1x(x2) Xj(X). (3.21)

The effect of the coulomb operator is localized and has a classically intuitive

interpretation, however, the action of the exchange operator is non-local and

depends upon the location of these two orbitals in the spin-orbital space.


3.1.3 Solving the HF Equations: Basis Set Expansions

In Section 3.1.2, we introduced the Hartree-Fock integral-differential equations

(now rewritten using Dirac notation),

j QSlater) E IISlater (3.22)


by using Equations (3.18) (3.21). Traditionally, one makes consistent use of

Slater determinants for the molecular wave function, and superscripts on T will

consequently be dropped for the remainder of the chapter. The solution of these

equations is still a non-trivial task. The first methods of solving these equations,

specifically for small atomic systems, was through numerical integration [29].

A considerable breakthrough was introduced by Roothaan [33] in 1951. The

computation routine of Roothaan involved expanding the molecular wave function

in a basis of atomic spin-orbitals with the general form of a linear combination of

spatial atomic orbital basis functions multiplied by the appropriate spin function.

This expansion allowed the Hartree-Fock differential equations to be written as

a set of algebraic matrix equations, which could be solved using available linear

algebraic techniques.

The general form of the basis set expansion of the ith spatial component of the

wave function takes the form
K
1 = (3.23)
P-11









where the basis set is composed of K functions, {q,|L = 1, 2,..., K}. The term

c1i represents the expansion coefficient for the ptth basis function to form the ith

spatial component of the wave function.

The size of the expansion (K) is in general not limited to a specific number.

In fact, an infinite expansion would be desirable, as this would correspond to the

full Hartree-Fock wave function. However, this is practically impossible due to

computational limitations. The size of K should ideally be large enough to offer

the best descriptions of the molecular orbitals without becoming computationally

inefficient. This, along with other specific criteria that must be considered will

be addressed in the next section. At this point it will suffice to assume that some

general function form and expansion size has been decided upon.

The Hartree-Fock equation for a given spatial component of the wave function

can now be written as

.) = e .), (3.24)

where ce is the orbital energy of the ith spatial orbital. The basis set expansion for

'. can now be introduced. The result becomes

K K
Y,#E ^ K (3.25)
p=1 p=1

At this point, one can then multiply through on the left of the equation by an

arbitrary basis component ({w1 and integrate. The ith orbital energy is a number,

and thus can be extracted from the integration, as can the expansion coefficients.

Thus, the equation now becomes
K K
> (9w|ci?|qV)cIO |)Cpw. (3.26)
p~ 1 p~ 1

At this point, one must introduce two matrices with elements that are related to

the terms in the above equation. The first is the Fock matrix, F. The Fock matrix









has elements defined as

F = ( | | ). (3.27)

The second matrix is the overlap matrix, S. The elements of the overlap matrix are

given as

S1 = (< ,|1) (3.28)

and arises from the fact that the basis functions are not necessarily orthogonal.

The overlap matrix provides a measure of the linear dependence of the set of

basis functions. Due to assumed normalization, the diagonal elements of the

overlap matrix all have a magnitude of one. The off-diagonal elements will range

in magnitude between zero and one. Elements approaching one will demonstrate

a strong linear dependence between two basis functions, while a value approaching

zero indicates a strong linear independence.

The HF equation can thusly be expressed using the newly defined matrix

elements,
K K
F- F.,pci = Scici. (3.29)
/p=1 p1=1
It is clear at this point that the above equation is an element of one single matrix

equation of the form

FC = SCe, (3.30)

where F and S are as defined above, C is the K x K matrix of expansion coeffi-

cients, and c is a K x K diagonal matrix with the orbital energies as the diagonal

elements. Equation (3.30) is commonly referred to as the Roothaan-Hall equa-

tion [34]. A companion equation exists for unrestricted determinants, called the

Pople-Nesbet equation [30, 35]. The Pople-Nesbet equations have individual matrix

equations for each set of spin-orbitals, as each pair of spin-orbitals has a different

spatial component in the unrestricted formalism.









As the overlap matrix and the Fock matrix are both Hermitian (and in many

cases real and symmetric), relatively simple solution techniques are available for

solving the Roothaan-Hall equations [36, 37]. The most common method involves

diagonalization of the matrices by use of a unitary transformation. This method

presents the eigenvalues (elements of c) and eigenfunctions (elements of C) for the

matrix equation.

The rudiments of solving the Roothaan-Hall or Pople-Nesbet equations will

not be discussed any further in this dissertation. Rather, in the next two sections

specific interest will be placed on the general form and construction of basis sets for

use with these matrix equations.


3.2 General Forms and Properties of Basis Sets

To this point, no mention has been made as to the functional form that the

basis set should take. In general, any functional form is possible, but certain

properties are desirable. Specifically, the wave function for the system must be

single-valued, finite, continuous, and square-integrable [38]. It is thus desirable

that one choose basis functions that possess these characteristics. A set of atomic

orbitals (AOs) is an immediate choice for a given basis set, as AOs satisfy the

above criteria as well as offer a chemical intuitiveness lacking in other choices.


3.2.1 Slater-Type and Gaussian-Type Orbitals

A first choice of basis sets was a set of spatial orbitals with the same functional

form as the hydrogenic orbitals,

(2( ))2n+'1 1/2
)STO [ (2) I e2- +l ). (3.31)

Equation (3.31) is called a Slater-type orbital (STO) [29, 34]. The equation

involves the terms (, which is the orbital exponent; n which is the principle

quantum number; 1, which is the azimuthal quantum number; and m, which is









the magnetic quantum number. The term YQ(0, 9) is the spherical harmonic.

As the functional form of hydrogenic orbitals are derived from specific linear

combinations of STO basis functions, one may reasonably expect that they will be

good approximations to orbitals in multi-electron atoms. In fact, STO functions

have correct functional forms at small r values and also at very large r values. As

can be seen from Equation (3.31), STO functions do not have any nodal structure,

and therefore linear combinations must be constructed to properly mimic the nodal

structure of atomic orbitals. However, basis sets built from relatively small linear

combinations of STO basis functions have been quite successfully employed in

quantum chemical calculations [29].

Despite the desirable functional form of the STOs, the principle drawback

is numerical [34]. The large portion of computational time in the HF method is

the calculation of the many-center two-electron integrals. Slater-type orbital basis

functions do not admit simple analytical expression for such two-center integrals,

and must therefore be numerically integrated [39]. This is a time consuming

process and, as a result, for any system with more than a few atoms the accuracy

obtained by using STO bases is outweighed by the severe decrease in computational

efficiency.

This limitation was circumvented by Boys, who -ii.'-., -ed using Gaussian-

type functions instead of exponential functions [40]. The functional form of a

Gaussian-type orbital (GTO) is [41]

[G1(2/)1/2 () 1/2
(GTO ) 2 n- 1 e-2C (,2(0)
YrO [ (2n 1)!! r (3.32)

where n, 1, and m are the same quantum numbers as given in Equation (3.31). The

parameter a is a different orbital exponent specific to the GTO basis. Note that

angularly, STO and GTO basis functions have the same functional form, the only

difference lies in the radial factors.









The primary deficiency of the GTO basis is immediately clear. A GTO

function does not provide the correct functional form as r 0 or at r oo.

In particular, an STO function will have dysTo/dr > 0 at r = 0, whereas a

GTO function will always have dGTO/dr = 0 at r = 0. Furthermore, the GTO

basis function has a much faster drop-off in the tail of the orbital as r oc

than does the STO function. This is a fairly severe limitation to the accuracy of

GTO bases in computations, as the functional form of the STO basis sets leads

to superior accuracy over GTO bases. The top panel in Figure 3.1 demonstrates

the considerable difference between the radial part of the hydrogen Is orbital

represented as a single STO function and a single GTO function. This red curve

is the hydrogenic Is STO function and the blue line is a single GTO function in

which the exponent has been optimized to provide the best least-squares fit to the

STO [41].

One can improve the structure by building a linear combination of GTO basis

functions (with optimized exponents) for each STO function cupll-v-d. Both the

set of exponents for each primitive Gaussian orbital and the set of contraction

coefficients must be optimized to fit the Slater in question. The bottom panel in

Figure 3.1 shows that a linear combination of six GTO functions (blue) provides a

much better fit for the hydrogen Is orbital as represented by a single STO function

(red), but that it is still not particularly good at the cusp of the orbital. In general,

a very large number of GTO functions with large exponents would be needed to

correctly mimic the cusp of an STO function, but no matter how many terms

where included in the linear combination, the derivative of the GTO function would

still be zero at r = 0.

Despite this cusp deficiency, quite good accuracy can be obtained from a good

sized linear combination of GTO basis functions per Slater. Yet, one must ask why

a linear combination of six or more GTO functions is favorable over a single STO















18

16

14

12
12 -



08 -

06

04

02
0 --------...........-------------- --
0 1 2 3 4 5
Radial Distance (a u)
2

18

16

14

12

E
S08 -

06 \

04

02 -

0 -
0 1 2 3 4 5
Radial Distance (a u )

Figure 3.1: Comparison of STO and GTO representations of the radial part of the

hydrogen Is orbital. Top: A single GTO function fit to a single STO function.

Bottom: A linear combination of six GTO functions fit to a single STO function.



function, particularly when the accuracy of the STO basis is superior to the larger


GTO basis. The answer lies in the fact that when a product of two GTOs is taken,


the result is a third GTO [30]. This reduces a multi-center two-electron integral to


a considerably simpler analytic form [39]. As a consequence, a calculation utilizing


a larger GTO basis is much more efficient that a calculation using a smaller STO


basis. This substantially greater computational speed has lead to the fact that


most calculations of polyatomic systems have traditionally cmiq-l..d GTO basis


sets [42].









Other basis sets have been i'pl,-v-id in various studies, such plane wave basis

sets. However, these are still not as widely used as STO bases and, particularly,

GTO bases. The remainder of this chapter will specifically focus on Slater and

Gaussian basis sets.


3.2.2 The Structure of Basis Sets

Now that the general form of an orbital basis function has been chosen, either

STO or GTO, the full basis set for an atom must be constructed. While there are

(theoretically) no limitations on the construction of basis sets, practicality has

restricted quantum chemists to certain accepted forms of contraction.

Minimal basis sets

Any atom must have at least enough orbitals available to completely contain

the required number of electrons. Any basis set designed to completely represent

only the ground state orbital structure of an atom is referred to a minimal basis

set [30]. For example, a minimal basis set for a Mg atom would contain no less

than 6 orbitals, the Is, 2s, 2p,, 2py, 2p,, and the 3s orbital. It would therefore be

sufficient to build a minimal basis of 6 STO basis functions, three with n = 0 and

three different values of ( as well as three others with n = 1 and a fourth value of



Furthermore, it would be possible to build a minimal basis set of six single,

uncontracted GTO basis functions in the same manner. However, as the previous

discussion indicated, the accuracy would be considerably less than the for the

comparable STO basis. So, to remedy this, a specific number of Gaussian primitive

functions (single GTO basis functions) are generally fit to each STO function in

the minimal set. This describes the formalism to construct a minimal STO-NG

basis set [43]. The title of this basis set indicates that N Gaussian primitives are

optimized to fit a single STO function. While any number of primitives may be fit

to a given STO function, in practice the STO-6G is the largest Gaussian minimal









basis that is eipl-v1d-. Without exception, the STO-3G is the most widely used

minimal basis set.

Minimal basis sets are notoriously inaccurate basis sets. In general STO-NG

basis sets offer qualitatively correct descriptions of fundamental chemical proper-

ties such as bonding and can be "ipl1-,vd for initial guesses and for calculations

involving very large molecules where more complete basis sets would be computa-

tionally inefficient. However, the small size of the STO-NG basis set is prohibitive

to the use of minimal bases for calculations in which even moderate accuracy is

required [30, 44].

As a final point concerning STO-NG minimal bases it should be noted that for

certain atoms these bases are not truly minimal. Many times chemical bonding is

not correctly mimicked using a truly minimal basis set. For this reason, the Group

1A and Group 2A metals, as well as the first two rows of transition metals, will

include the low-lying p orbitals even though they are unoccupied in the unbound

atom [34]. The s and p exponents for a given principal energy level are identical.

This is computationally more efficient than allowing for separate s and p exponents

as each set of s and p orbitals will then have the same radial behavior and can,

consequently, be integrated together [30].

Double-zeta and split-valence basis sets

One of the main limitations of the minimal basis set is the fact that there

is no flexibility for the generated orbitals to change size under the influence of

intramolecular surroundings. Each orbital has a single set of exponents that control

the size and shape of the orbital, and while the amplitude of the orbital can be

adjusted through the HF coefficients, the spatial size cannot be changed. To

remedy this, one would desire to include more than a single linear combination of

basis functions for a given orbital. This idea gives rise to the next level of basis

sets, the double-zeta and the split valence basis sets. Much of this discussion









will be made in terms of STO bases. It should be remembered that these ideas

can be translated directly to GTO bases by requiring that each STO function be

constructed as a linear combination of GTO functions.

Specifically, the double-zeta functions allow for each orbital to be a composed

of a linear combination of two STO basis functions, each with a different exponent.

The larger exponent (a tighter function) is, in general, slightly larger than the

optimal exponent for the single zeta function, while the smaller exponent (a more

diffuse function) is slightly smaller [30]. Furthermore, as the double-zeta basis is

more complete than a comparable single-zeta basis, a correctly optimized linear

combination of double-zeta functions will be a better representation of the ]li,, -i. .1

orbital than a single-zeta function. This results in an improvement in the ground

state energy, as demonstrated by Clementi and Roetti in their seminal paper on

double-zeta functions for atoms [45].

A simplification of the double-zeta basis can be made by realizing that, during

a chemical process, the size and shape of core atomic orbitals will not change

significantly. Therefore, under most any conditions, a single well-optimized STO

function will provide a sufficient representation of a core orbital. One can then

allow for the valence orbitals to be represented using a combination of two STO

basis functions, as in the double-zeta basis. This is the formula for constructing

split-valence basis sets [34].

Split-valence basis sets provide ground state energies that show improvements

over minimal basis sets, but that are not as good as double-zeta bases. Again, this

is due to the fact that a more complete description of core orbitals is obtained with

the double-zeta functions. However, this energy difference is small when compared

to the computational efficiency gained through using split-valence bases [30].

The most common split-valence basis sets that are employed in computational

chemistry are the l:mnG basis sets. These are combinations of GTO functions such









that the core orbitals for the atom are represented as a contraction of I GTO basis

functions. A given valence orbital is represented as a linear combination of two

basis functions, one composed of m GTO primitives and the second composed of

n GTO primitives [34]. The only exceptions occur with H and He, in which there

are no core shells and only the valence shell structure is used (and as a consequence

the 4:31G bases for H and He are identical to the 6:31G bases). The most common

examples of split-valence bases are the 3:21G [46], 4:31G [47], and 6:31G [48] bases.

This pattern can be extended to larger basis sets such as the 6:311G basis [44].

Double-zeta and split-valence basis sets improve the electronic representation

in several ways. First, as mentioned above, the double-zeta bases allow for a better

description of atomic orbitals by virtue of the increased completeness of the basis

set. Also, the valence orbitals are now flexible enough to change size during a

chemical process. In particular, this allows for better descriptions of bonding and

anisotropic chemical processes, such as the anisotropy of the bonding p-orbitals

when forming a and 7 bonds in systems with bond orders greater than unity [34].

One last feature of split-valance bases is that, with proper optimization, the more

diffuse valence functions can behave as virtual orbitals, a property not available

in minimal basis sets. This feature will become increasingly important when one

desires to investigate dynamical processes.

Polarization basis sets

Many ]li, -i. .,l processes require not only a change in the size of atomic

orbitals over time, but also a change in the shape of the orbital. Examples include

the behavior of the orbitals for an atom subject to an external electric field or the

orbitals of an atom which has some non-zero momentum. Both of these processes

result in a polarization of the atomic orbitals. This polarization causes a net

increase of electron density in one area off-center from the nuclear center and a









corresponding net decrease in electron density in the region of space immediately

opposite the nuclear center.

While split-valence basis sets allow for the size of the orbitals to fluctuate,

they do not permit the shape of the orbital change. This cannot be accomplished

by merely changing the size of the orbital exponents. In order to change the shape

of atomic orbitals, the basis must flexible enough to allow combinations of basis

functions that represent occupied atomic orbitals with higher angular momentum

basis functions[30, 34, 44].

The most common methods of polarization involve the addition of basis

functions that mimic a d-orbital to the elements from Li to Ar. This level of

polarization is denoted using a single asterisk. The most common example is the

6:31G* basis [49], which is the basic split-valence 6:31G basis described above

with the addition of a single d-symmetry basis function [34] (or f-symmetry basis

function to transition metals) [44]. The second form includes the d-(f-)orbitals

for heavy atoms as well as an basis function with p-symmetry to H and He. This

level is denoted with two asterisks (such as 6:31G**)[49]. Again, this pattern can

be "'apl-v-d using larger split-valence bases, resulting in such combinations as

6:311G**, a basis that is commonly employed for correlated calculations [50].

The effect of including polarization functions has traditionally been observed

in structural properties, particularly in constrained systems where the electron

density is shifted away from the nuclear centers [34], and in systems subjected to

external electric fields [30]. In particularly, atoms which can be multiply bonded

will, in particular, require a greater degree of polarization. This is evident from the

valence-bond description of chemical bonding, in which the formation of "hybrid"

atomic orbitals (which are nothing more than polarized atomic orbitals) is the

underlying principle of chemical bond formation [51]. For this reason, it has long

been accepted that it is more important to include polarization basis sets on main









group elements rather than singly bonding species, such as Group 1 elements, as

indicated by the strong statement of Szabo and Ostlund [30] that it Ii.i' been

empirically determined that adding polarization functions to the heavy atoms is

more important than adding polarization functions to hydrogen." While this is

generally true for structure calculations, it cannot be accepted when dynamical

calculations are being performed. In particular, any atom that possesses non-zero

momentum will experience a polarization of its electronic orbitals due to the

motion of the atom. This effect will be present in all atoms, including H and He.

For this reason, it is of crucial importance to include polarizing p-functions on H

atoms for dynamical calculations.

Diffuse basis sets

The previously mentioned basis set structures do a good job of describing

various chemical processes, however, all of them locate the electronic density

relatively close to the nuclear centers. The split-valence structure allows for

increasing the size of orbitals, but the exponents are always close in magnitude

to the exponent in a comparable single-valence basis function. This limits the

ability of the orbital constructed from a split-valence basis to expand beyond

small fluctuations around the size of the orbital constructed from a single-valence

basis. Additionally, the polarization functions allow for shifting of the electron

density away from the nuclear center, say to a chemical bond. Again, this shift in

the density is not large. As a consequence, systems with large electron densities

that are located a significant distance from the nuclear center (such as anions

and systems involving Rydberg states) are not properly modeled using minimal,

split-valence, or polarization basis sets.

To properly describe such systems, diffuse basis sets must be employed [34, 44].

Diffuse basis sets are structured in a manner very similar to split-valence basis

sets. A minimal (or split-valence or polarization) basis set in constructed and









additional basis functions are included to provide for the diffuse atomic orbitals.

However, the exponents of these diffuse basis functions are much smaller than for

the valence basis functions, resulting in an electron density that is located much

further away from the nuclear center. In general, the diffuse functions are of the

same angular momentum as the valence basis functions. This means that a carbon

atom would incorporate one additional s- and one additional p-function. This

diffuse structure is denoted using a + symbol [34]. If a single diffuse s-function

is added to a hydrogen or helium atom, then this is denoted by two plus signs.

Thus, one can now begin to evpl']' a virtual alphabet soup of such combinations

as 3:21+G*, 6:31++G*, or 6:311+G**.

Even-tempered and universal even-tempered basis sets

Further advances in the building of basis sets were made when it was realized

that, as a basis set got larger, the orbital exponents within a given angular momen-

tum converged to a geometric sequence. This geometric sequence takes the form



(i = CoP, (3.33)

where Q( is the ith exponent in the sequence, (o is the largest exponent, and 3 is a

constant that is specific to the angular momentum. A basis set that is constructed

using this type of method is called an even-tempered basis set [39, 42]. The general

feature of an even-tempered basis set is that it limits the number of parameters

that must be optimized. Furthermore, even-tempering ensures that a GTO

expansion of an STO is well-spanned, with no regions in which the representation is

particularly poor. There is in general a small energy price that must be paid, but

this usually is on the order of several hundredths of a Hartree [42].

Furthermore, it has been postulated that, if enough even-tempered exponents

are include in a contraction, then this set of exponents would eventually become









identical over an entire row of the periodic table. This leads to a universal even-

tempered basis set [39, 42].

Other basis set structures

In this section, a brief discussion has been made of the general structure of

basis sets, with details given about those basis sets most commonly employed in

quantum chemical calculations. This is only a very small sampling of the basis

sets available for computation, however, most of these basis sets include the basic

principles listed above. It is beyond the scope of this work to provide an in depth

discussion of the different types of basis sets used in calculations. The comparisons

made in the forthcoming sections will, in general, be related to the types of basis

sets reviewed in this section.


3.3 Method for Constructing Basis Sets Consistent with Dynamical
Calculations

The structure of a basis set is heavily dependent upon the types of 1pr, -i-

cal properties that one desires to calculate. In some cases diffuse functions are

required, in others polarization functions are required. In most cases, some bal-

anced combination of all of the properties is needed. Because of this, many have

viewed basis set construction as an art (or black magic in some cases). Yet, no

matter what form the basis set takes, most have one trait in common: with very

few exceptions, basis sets must be optimized. Usually basis sets are optimized

with respect to the energy of the ground state of the system by means of the vari-

ational principle. However, the HF equations are non-linear, and therefore any

optimization process becomes computationally costly.

An additional feature that is common to most basis sets is that they are built

for use in stationary state calculations of the ground state of a given system. While

most of these basis sets do provide representations of unoccupied orbitals either due









to constructing a split-valence or diffuse basis set, these unoccupied orbitals bear

little or no resemblance to the virtual orbitals in the system.

In this section a new method for the construction of basis sets will be intro-

duced. This method has a simple ].li,--i .,1 underlying justification. The method

does not require expensive and complex energy optimizations (and does not, in

fact, require any optimizations at all). Finally, this method allows for the construc-

tion of 1.1r, -i, .,lly meaningful virtual orbitals, a necessity for the computation of a

wide variety of dynamical properties.


3.3.1 Basis Set Properties for Dynamical Calculations

In stationary state calculations minimal basis sets are rarely, if ever, sufficient

for the description of the chemical species in question. As outlined in the previous

section, a variety of extra basis functions must be included to improve the descrip-

tion. In this case any set of functions added to the minimal basis set generally

demonstrates no 1ir, -i .,1 resemblance to atomic orbitals in the system. Rather,

they just act to provide a more complete spanning of the electronic Hilbert space,

increasing the flexibility of the basis set. These extra basis functions serve to lower

the ground state energy (due to an increase in the accuracy of the representation

of the occupied atomic orbitals and in some cases by partially accounting for the

correlation energy in the system) but they do little else.

For dynamical calculations, particularly charge transfer processes, the basis

set must be flexible enough to allow for correct description of electronic transi-

tions between atomic orbitals, either within a single atom or molecule or between

the collision pair. One specific aspect of this requirement is the fact that virtual

orbitals (atomic or molecular) must be available for occupation throughout the dy-

namical processes. To this point, little effort has been made by the computational

community to construct atomic basis sets that properly describe virtual atomic

orbitals, mainly due the fact that most basis sets are optimized with respect to the









total ground state energy of the system rather than optimized with respect to the

individual atomic orbital energies. For this reason, and with very few exceptions,

stock basis sets that are most commonly "mpl-v-id in computational chemistry are

inadequate for use in dynamical calculations.

Methods for improving stock basis sets traditionally follow along the lines of

increasing the size of the basis sets by the inclusion of more and more uncontracted

diffuse primitives. This method will, in the limit of infinite expansion sizes, lead

to correct virtual energy levels by virtue of the Hylleraas-Unsold separation the-

orem [52]. However, this brute force method is extremely inefficient for basis set

construction. As the number of basis function increases, so does computation time.

While this is extremely limiting in the area of structure theory, it is virtually im-

possible in dynamical methods such and END, where a large number of calculations

must be made per trajectory (with many trajectories required for a single collision

energy). For this reason, a new method must be devised for building basis sets that

include correct representations of virtual orbitals.

The construction of dynamically meaningful virtual orbitals is dependent

most strongly upon two properties of the atomic orbitals, the energetic of the

orbitals and the shapes of the orbitals. The energetic of the orbitals are the most

obvious concern. If the orbital energies are not correct, then the energy required

for electronic transitions within the basis set will not properly model the energy

differences in nature. Too small of an energy gap will result in increased transfer

probability while too large of a energy difference will have the opposite effect. In a

single determinantal treatment of the electrons, the orbital energies should mimic

the energetic of the system. Therefore, the correct energetic of the atomic orbitals

should be a measure of the ability to correctly simulate dynamical transitions.

The shape of the orbital wave function is also important for dynamical

calculations. The nodal structure of the wave function determines regions in which









electron density is zero or where it is non-zero. Again, a basis set must correctly

model the electron density in an atom.

More discussion about these properties will be made in the next sections.

However, at this point it will suffice to say that both of these properties can be

addressed quite effectively through the use of STO basis sets. Specifically, the

energetic of an orbital is largely dependent upon the structure of the tail of the

orbital wave function. As was mentioned in the previous section, STO bases

correctly describe the tail of hydrogenic orbitals and, likewise, reproduce the orbital

tails in many-electron atoms quite well. Additionally, while single STO functions

do not contain any nodal structure, linear combinations of STO functions can if

carefully built. For this reason, it seems most reasonable to construct basis sets

using STO functions, at least initially.


3.3.2 Physical Justification for the Basis Set Construction Method

As a part of the pli, -i. .,1 justification of the proposed method for basis set

construction, one must first return to the radial factor of a Slater-type orbital basis

function, given by the form

S(2)2n+rl1 1/2
RsO r- le-c r. (3.34)
(2n)! j

When one compares this form to the hydrogenic orbital functions, one finds that

the orbital exponent is related directly to the nuclear charge of the hydrogenic

atom in question, specifically
z-, (3.35)

where Z is the nuclear charge and n is the principal quantum number. In the

case of a hydrogenic system, only one electron is associated with the system and

therefore the electron will always feel the full nuclear charge (that is, there is no

nuclear shielding due to the presence of other electrons).









This idea can be extended to the construction of a wave function for an orbital

in any chosen atom. It should be noted, however, that for a given electron in an

arbitrary orbital in a many-electron atom the nuclear charge felt by that electron

will not be the full nuclear charge. Rather, the full nuclear charge will be shielded

by electron density located between the orbital in question and the nucleus. This

gives rise to the concept of an effective nuclear charge, Zeff. The orbital exponent

now takes the form
S Zeff (3.36)

The effective nuclear charge will vary as a function of the principal quantum

number, in general an orbital with a smaller principal quantum number will have

a larger effective charge. Zener provided values for these effective charges based

on variational calculations [53]. Slater [54] determined an empirical method for

calculating the effective nuclear charge for any arbitrary orbital and presented the

equation

Zff = Z s, (3.37)

where Z in the full nuclear charge and s is a screening constant that is a function

of the orbital and the number of electrons. For s- and p-orbitals, the screening

constant was defined to be


s = 0.35N, + 0.85N,_1 + 1.OON,,2, (3.38)


where N, is the number of additional electrons in the same principal level, N,_1

is the number of electrons in the principal level immediately lower, and N,_2 is

the number of all remaining electrons in lower principal levels [54]. Having defined

the effective charge, Slater then -II.-.i -r construction of atomic orbitals as single

STO functions of the same form as Equation (3.31), with the exception that the

orbital exponent takes the form of Equation (3.36) and the principal quantum

number n is replaced by an effective quantum number, n*. The effective quantum









number deviates from the true quantum number only for n > 3 [54]. This, the

form of the wave function consists of a single STO for each set of n and 1 quantum

numbers, with the orbital exponent equal to the effective nuclear charge felt by the

corresponding atomic orbital divided by the effective principal quantum number

associated with the orbital.

Having discussed a specific method for the construction of basis sets for many-

electron atoms, it is now time to consider some 1ir, -i, .1l aspects of atomic orbitals

in a bit more detail. One property that a basis set should properly model is the

radial distribution of the electronic orbital, defined as [55]


D,(r) = r2 [Rn(r)2 (3.39)

In the above equation, r is the radial distance from the nucleus, n is the principal

quantum number, I is the azimuthal quantum number, and Rni is the radial

factor of the wave function for a given n and 1. The top panel of Figure 3.2

demonstrates the radial distribution functions for the first four s-orbitals in the

hydrogen atom. The humps in the radial density function for a given orbital

indicate regions in which the probability for the electron to exist is greatest. This

feature demonstrates that the electron density corresponding to an orbital takes

the form of concentric shells of electron density [29, 55]. From this it becomes clear

that an atomic orbital with principal quantum number n will posses n regions of

electron density, each becoming increasingly closer to the nucleus but with smaller

probability. This fact is mirrored in the nuclear screening expression advanced by

Slater and given in Equation (3.38). The shielding due to electrons in the next

lowest principal level do not completely shield the nucleus, rather they have an only

an e.i' effective shielding due to the penetration of the higher principal levels.

This ]1li,--i .l1 feature of atomic orbitals is not limited to s-orbitals, nor is

it limited to the description of the hydrogen atomic orbitals. In Figure 3.2, the












06


05


04


03


02


01
02




0 5 10 15 20 25 30
Radial Distance (a u)
02

0 18

0 16

014

0 12

01

0 08 -

006

004

002
0 -" - - .
0 5 10 15 20 25 30
Radial Distance (a u )

Figure 3.2: Top: Plot of the radial distribution function for the Is (), 2s (- -),

3s (. .), and 4s (- -) orbitals of the hydrogen atom. Bottom: Plot of the radial

distribution function for the 2p (-), 3p (-- -), and 4p (. ) orbitals of the hydro-

gen atom.



bottom panel demonstrates the radial density functions for the 2p-, 3p-, and


4p-orbitals of the hydrogen atom, where a similar shell structure is observed.


Furthermore, Figure 3.3 shows the radial distribution functions for the occupied


s- and p-orbitals in the Ar atom, in the top and bottom panel, respectively. In


both plots, the orbitals wave functions that are plotted are the double-zeta wave


functions of Clementi and Roetti [45]. Again, the shell structure is clearly evident.


At this point, the most significant pil,v-i, .1l attribute of these shells of electron


density becomes apparent. In the case of s- and p-orbitals in the atoms of the first












35

3

25




O 15




05


0 05 1 15 2 25 3
Radial Distance (a u )
25



2-



15 -






05 / ""--
05



0 05 1 15 2 25 3
Radial Distance (a u )

Figure 3.3: Top: Plot of the radial distribution function for the Is (), 2s (- -),
and 3s (. .) orbitals of the argon atom. The Is-orbital is scaled by a factor of
two-thirds. Bottom: Plot of the radial distribution function for the 2p (-) and
3p (- -) orbitals of the argon atom.



few rows of a periodic table, the radial location of the shells is largely independent

of the principal quantum number associated with an orbital. In other words, all

s-orbitals have a shell of electron density that has roughly the same radial location

as the shell of electron density due to the Is orbital. Likewise, all s-orbitals with

principle quantum number n have n-1 shells that have roughly the same radial

location as the n-1 lower energy s-orbitals. This is also true for p-orbitals. The

consequence is that any orbital of s symmetry possesses a partial character of all of

the lower energy s-orbitals. Further, each of these characteristic shells exists within









the same region of effective nuclear charge that is specific to that shell of electron

density.

Now, relating the regions of effective nuclear charge back to the idea of the

Slater orbital exponent, this now means that a given orbital can be constructed

as a linear combination of all of the previous orbital basis functions, each with a

specific orbital exponent that relates to the effective charge region experienced by

the corresponding shell of electron density. Specifically, the orbital wave function

can be written as

Sn, = cNN,,e- (3.40)
i=1
where N,,I is the appropriate normalization factor for the STO in question, the ci's

are the expansion coefficients, and the & is the orbital exponent (effective nuclear

charge) for the ith shell. On the surface, this is nothing new. Relating the orbital

exponent to an effective charge was proposed by Slater and the linear combination

is nothing more than a restatement of the superposition principle [1] through which

the HF method determines the HF eigenstates as a linear combination of the basis

vectors [29]. However, this plr, -i, .,l insight does serve as an important under-tone

for the basis set construction proposed in this work.


3.3.3 Construction of the Basis Set

One begins construction of the basis set by defining a linear combination

of STO functions, each with orbital exponents derived from the effective charge

experienced by the orbital in question. The form of the effective charge may be

determined in any number of ways, the simplest of which is to employ Slater's

formulation for the screening constant [54]. However, Slater's screening constants

are empirically modeled and may not be as accurate as those determined by other

methods. Instead, one may consider the work of Clementi and Raimondi [56] as an

extension of the earlier work of Slater and Zener. Clementi and Raimondi made









a study of the elements through Kr, representing each atomic orbital as a single

STO function (that is a minimal basis set) with variable orbital exponents. The

exponents where then optimized (with respect to the ground state energy) using an

SCF procedure.

At this point, the proposed method has not deviated from older methods of

basis set construction. By eiply-,ving this method, one can construct a minimal

basis set for the ground state of the atom in question, however, no recourse is

available for construction of virtual orbitals. The optimization process could

be extended to determine the orbital coefficients for the virtual states, but this

would require calculations beyond the HF level to do so, as the transition energies

would be dependent upon electron correlation. The minimum level of theory that

could be eipl-1v- d would be configuration interaction. This would increase the

computational effort required to optimize the virtual orbital exponents.

To remedy this, a new approach to determining these virtual orbital exponents

is proposed in this work; a method that is extremely simplistic in its application,

yet has proven to be very powerful. The method begins through the investigation

of the behavior of Clementi's shielded orbital exponents as a function of the

atomic number. Figure 3.4 demonstrates the functional behavior for the Is orbital

exponents through Kr. As can be seen from the figure, the orbital exponents

exhibit a very linear dependence on the atomic number.

The top panel of Figure 3.5 shows the dependence of the 2s orbital exponents

on the atomic number. There are two data sets in this figure, the data of Clementi

and Raimondi [56], which are denoted using the plus symbols, and the data from

the present work, which are denoted using the open circles. Clementi's data still

demonstrates a linear dependence of the exponent on the atomic number, however,

it becomes clear at this point that more than one linear region is observed. As

an example, the orbital exponents for the elements from Li through Ne have a











4 0 ---- ---- ---- i ---- i ---- i ---- i ---- i ----




40


35-
35 + +



5+0
+ +
+









0 5 10 15 20 25 30 35 40
+

++
10 +

15







Atomic Number

Figure 3.4: Plot of the is orbital exponent for the atoms through Kr as a function
of atomic number. The data are from Clementi and Raimondi (+).



slightly different slope and intercept than for the remaining elements. Each block

of elements will have a slightly different slope. The data points current to this work

will be discussed at length later.

The data in the bottom panel of Figure 3.5 show the same relation for the 2p

orbital exponents. Likewise, the plots in Figure 3.6 demonstrate the dependence

for the 3s (top) and 3p (bottom) orbital exponents and those in Figure 3.7 present

the data for the 4s (top) and 4p (bottom) orbitals. The most striking feature

when comparing all of the previous plots is that, while the individual linear regions

become more distinct from one another as the principal and azimuthal quantum

numbers increase, the (local) linear dependence of the orbital exponent on the

atomic number is still quite strong. It is this feature that defines the proposed

method for virtual orbital construction.

Discussion must now be made with regards to the remaining sets of data

points, those denoted with the open circles. These data points are new to this work

and are derived from the data of Clementi and Raimondi. Specifically, these points

are orbital exponents corresponding to virtual orbitals for the atoms in question.
are orbital exponents corresponding to virtual orbitals for the atoms in question.














14


12


10

O+
8-
o +
IQ +

0








0 5 10 15 20 25 30 35 40
Atomic Number
18









14
12 + +


10
+














0 5 10 15 20 25 30 35 40
Atomic Number
16 + -
+
+























Figure 3.5: Plot of the 2s and 2p orbital exponents for the atoms through Kr as
a function of atomic number. Top: The 2s orbital exponents. Bottom: The 2p
+














orbitalexponents. he data are rom lementi and aimondi () and rom the











present work (o).



These virtual orbital exponents are determined by first considering the


hydrogen atom. As the hydrogen atom has only a single electron, there will never
be any nuclear shielding for that atom. This the electron will always experience

















the same effective nuclear charge (of unit magnitude) no matter in which orbital


the electron has probability for existing. This means that Zeff = 1 always, and


the orbital exponent for any orbital in the H atom is just equal to the reciprocal


of the principal quantum number. At this point, one makes reference to the linear
behavior of the occupied orbital exponents. The virtual orbital exponents are then
12 +





0 I


















7 -


65-















-4
+
+
++









0 5 10 15 20 25 30 35 40
7+







































Atomrc Number
6-
4 +

2 -
+








o o o + ao
0 5 10 15 20 25 30 35 40
Atomic Number

7




5 + +


4 -+
I+
S+
+
+

+












Figure 3.6: Plot of the 3s and 3p orbital exponents for the atoms through Kr as

a function of atomic number. Top: The 3s orbital exponents. Bottom: The 3p

orbital exponents. The data are from Clementi and Raimondi (+) and from the

present work (o).




determined by making a linear interpolation between the H atom virtual orbital


exponent and the exponent corresponding to the first available occupied orbital


in that symmetry. For example, in the case of the 3p orbitals, the interpolation


is made between the exponent corresponding to the H atom 3p orbital and the


exponent that corresponds to the 3p orbital of Al (atomic number = 13).


This method for determining the exponents that correspond to virtual atomic


orbitals relies on well-documented trends exhibited by a parameter that is related


directly to a 1]li, -i. .1l property, namely the trend in the regions of effective charge














+


25
+


25
+




15



0 1 -
0
oo


0 5 10 15 20 25 30 35 40
Atomic Number
25


+

+
+
+
15 -++






0o











a function of atomic number. Top: The 4s orbital exponents. Bottom: The 4p

orbital exponents. The data are from Clementi and Raimondi (+) and from the

present work (o).



as demonstrated earlier in this section. However, it is important to note that


a number of severe assumptions have been made. Perhaps the two strongest


assumptions are that the virtual orbital exponents demonstrate the same linear


behavior as do the occupied orbital exponents and the assumption that there is not


a large change in the magnitude of the orbital exponent as one transitions from


the occupied orbital exponents to the unoccupied orbital exponents. While neither


of these assumptions can be tested without the construction of energy-optimized


virtual orbital wave functions, the severity of the assumptions is tolerated in lieu









of the ease of application. And, in spite of these assumptions, the calculations

using basis sets constructed from this starting point have yielded surprisingly

good results for both stationary state and dynamical calculations, as will be

demonstrated in the next section.

Once the interpolations have been accomplished, the next step is to construct

the atomic orbital wave functions. This is begun with the wave function for the Is

orbital, which is represented by a single Slater-type orbital,


is = Nls((ls)e-Clsr. (3.41)

The expansion coefficient in just the normalization coefficient for an STO basis

function with n = 1 and with orbital exponent (Ps. From this, the wave function

of the 2s orbital can then be constructed as a linear combination of the Is wave

function (providing for the cusp) and a single STO basis function with n = 2 that

is used to represent the tail portion of the orbital. The orbital wave function takes

the form

02s = Cls 1 1(C,)e-r c2 2N2~(,)re-2 (3.42)

Again, the terms N1, and N2s are the normalization coefficients for the specified

STOs. In Equation (3.42), two expansion coefficients must be determined. This

requires the simultaneous solution of a set of two equations. In this case the two

equations are the normalization condition for the 2s orbital ((2s i2s) = 1) and the

orthogonii.lil' of the Is orbital with the 2s orbital ((1is 2s) = 0).

It becomes clear at this point that the above process can be iterated over as

many s-orbitals as are required for an atomic basis set, be they occupied or virtual.

Specifically, the ns orbital wave function is defined as


'.. csNs((is)r-l e-('sr. (3.43)
i= 1









As before, the wave function has n undetermined coefficients and therefore requires

the solution of a set of n simultaneous equations. These equations take the form

the normalization of the ns wave function ({(Q, Qns,) = 1) and the orthogon. lil v of

the ns wave function with the other wave functions ((91.|. ..) = ( '.. '.) =.

({ ._-)s'..'. = 0). This forms a set of n simultaneous equations that can be used

to determine a set of expansion coefficients. The construction of the p-orbital wave

functions follows the same schema.

This proposed construction method is employed through a template designed

for any commercial computational package, such as Maple, into which the orbital

exponents are input. The program then calculates the expansion coefficients by

solving the orthonormality conditions for each set of orbitals. The power of this

method is that it does not require any expensive non-linear energy optimizations.

Furthermore, it allows for a general construction of any atomic orbital, either occu-

pied or virtual, provided that the orbital exponent is known or can be interpolated

using the above mentioned method.

Wave functions have been built for the atoms from He through Ne. Table 3.2

presents the STO exponents and coefficients for He, Li, and Be. These exponents

and coefficients were determined using the previously described method. Table 3.3

lists the wave function parameters for B, C, and N. Lastly, Table 3.4 contains the

exponents and coefficients for 0, F, and Ne.

As a final step, the STO wave functions must then be expanded in a basis

GTO functions to allow for computations to be performed. There are two main

methods by which this is accomplished. The first method is through use of a

linear least-squares fitting program which will determine the best set of GTO

exponents and expansion coefficients. The program that has been ei'pl-v-id for

some of the results reported in this work used a variation of the Amoeba program

from Numerical Recipes [37]. In this method the GTO orbital exponents were











Table 3.2: Slater Exponents and Coefficients for He, Li, and Be.

Atom: Helium
Configuration: He : 1s22s02p03s03po4so4po


s-Functions


p-Functions


Exponents Exponents
is 2s 3s 4s 2p 3p 4p
1.6875 0.5698 0.3836 0.2847 0.6777 0.4185 0.2935
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.29929 -1.04383 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.22486 -0.97897 1.37212 0.00000 3p 0.75715 -1.25430 0.00000
4s 0.17763 -0.8-.''.O 1.84082 -1.72886 4p 0.58192 -1.40551 1.52534


Atom: Lithium
Configuration: Li : 1s22s12p03so3po4so4p0

s-Functions p-Functions
Exponents Exponents
Is 2s 3s 4s 2p 3p 4p
2.6906 0.6396 0.4338 0.3193 0.S..4 0.5036 0.3370
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.16487 -1.01350 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.13471 -0.90398 1.36057 0.00000 3p 0.69166 -1.21589 0.00000
4s 0.04890 -0.38438 1.11657 -1.56430 4p 0.48801 -1.20388 1.42094


Atom: Beryllium
Configuration: Be : 1s22s22p03so3po4so4p0

s-Functions p-Functions
Exponents Exponents
is 2s 3s 4s 2p 3p 4p
3.6848 0.9560 0.4841 0.3540 1.0330 0.5888 0.3805
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.18228 -1.01832 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.09188 -0.54041 1.13214 0.00000 3p 0.65076 -1.19310 0.00000
4s 0.03755 -0.23713 0.87282 -1.46603 4p 0.43111 -1.08063 1.35846











Table 3.3: Slater Exponents and Coefficients for B, C, and N.

Atom: Boron
Configuration: B : 1s22s22p13so3po4so4p


s-Functions


p-Functions


Exponents Exponents
is 2s 3s 4s 2p 3p 4p
4.6795 1.2881 0.5343 0.3886 1.2107 0.6740 0.4241
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.21294 -1.02242 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.07096 -0.37100 1.06382 0.00000 3p 0.62253 -1.17794 0.00000
4s 0.03208 -0.17493 0.7--.1 -1.42357 4p 0.39304 -0.99801 1.31754


Atom: Carbon
Configuration: C : 1s22s22p23so3po4so4p

s-Functions p-Functions
Exponents Exponents
Is 2s 3s 4s 2p 3p 4p
5.6727 1.6083 0.5846 0.4233 1.5679 0.7591 0.4676
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.22393 -1.02477 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.05798 -0.28600 1.i 's ,, 0.00000 3p 0.47',, -1.10861 0.00000
4s 0.02788 -0.14169 0.75459 -1.40332 4p 0.29066 -O.,.',) 1.27113


Atom: Nitrogen
Configuration: N : 1s22s22p33so3po4so4p

s-Functions p-Functions
Exponents Exponents
is 2s 3s 4s 2p 3p 4p
6.6651 1.9237 0.6348 0.4580 1.9170 0.8443 0.5111
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.23080 -1.02629 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.04925 -0.23459 1.02580 0.00000 3p 0.39933 -1.07678 0.00000
4s O.-'2.1 -0.14023 0.90396 -1.52899 4p 0.23578 -0.79242 1.24296











Table 3.4: Slater Exponents and Coefficients for 0, F, and Ne.


Atom:
Configuration:


Oxygen
0 : s22s22p43s03p04s04po


s-Functions


p-Functions


Exponents Exponents
is 2s 3s 4s 2p 3p 4p
7.6579 2.2458 0.6851 0.4926 2.2266 0.9294 0.5546
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.23710 -1.02772 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.04298 -0.19877 1.01854 0.00000 3p 0.35976 -1.06274 0.00000
4s 0.01173 -0.05606 0.46221 -1.27638 4p 0.20750 -0.75025 1.22490


Atom: Fluorine
Configuration: F : 1s22s22p53s03p04s4p0

s-Functions p-Functions
Exponents Exponents
Is 2s 3s 4s 2p 3p 4p
8.6501 2.5638 0.7353 0.5273 2.5500 1.0146 0.5981
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.24136 -1.02871 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.03835 -0.17388 1.01419 0.00000 3p 0.32735 -1.05222 0.00000
4s 0.02018 -0.09298 0.71660 -1.37637 4p 0.18516 -0.71747 1.21094


Atom: Neon
Configuration: Ne : ls22s22p63s03po4s04p0

s-Functions p-Functions
Exponents Exponents
is 2s 3s 4s 2p 3p 4p
9.6421 2.8792 0.77 0.5619 2.8792 1.0997 0.6416
Coefficients Coefficients
is 1.00000 0.00000 0.00000 0.00000
2s 0.24439 -1.02943 0.00000 0.00000 2p 1.00000 0.00000 0.00000
3s 0.03483 -0.15570 1.01138 0.00000 3p 0.30169 -1.04452 0.00000
4s 0.01858 -0.08429 0.71010 -1.37115 4p 0.16788 -0.69204 1.20000









constrained to be even-tempered. Depending on the use of the basis set, any

number of GTO functions (up to ten) were used for each STO fit.

While the first fitting method employs a fairly simple linear optimization to

determine exponents and coefficients for GTO expansions of STO orbitals, the

second method does not require any such optimizations. This method uses the

results of Stewart [41]. Stewart performed least squares calculations to determine

the best set of parameters that allows for the construction of an expansion of up to

six GTO functions for a specified single STO function. As previously mentioned,

this method does not require tedious optimizations, as they have already been

performed. This allows for a simple template to be constructed in a numerical

spreadsheet program. The orbital exponents are input into this spreadsheet

template, along with the required parameters from Stewart's paper, and the GTO

orbital exponents and expansion coefficients are the resulting output.

While the method for basis set construction outlined in this section is very

simplistic in its application, the ]li',-i .,l underpinning of the construction is quite

strong and maintained throughout the method. Furthermore, this formalism is easy

to inrpl-v, with no costly or time-consuming optimizations required if engineered

correctly. In the following section, results that have been obtained with basis sets

constructed using this method are presented for comparison with several common

stock basis sets that were built using energy optimization methods.


3.4 Comparative Results

This section will provide comparisons between basis sets constructed using

the method proposed in the previous section and with some of the more commonly

used stock basis sets, such as the 3-21G, the 6-31G, and the 6-31G* basis sets. The

comparisons will be made such that a computation performed with, for example, a

6-31G basis set will be compared with a newly constructed basis set with the same

size parameters (same number of expansions per orbital). Comparisons will be









made for several types of ]li, -i, .,1 properties, specifically for the excitation energies

within the specific atom, for charge transfer probabilities and cross sections, and

for the vibrational and electronic properties of several diatomic and triatomic

molecules. These first two comparisons are for properties that are important to

dynamical processes. The third comparison relates properties that are traditionally

calculated based on energy (such as diagonalization of the Hessian to obtain

vibrational frequencies). This will offer comparison between energy optimized

basis sets with the basis sets constructed from the current method (with no energy

optimizations).


3.4.1 Atomic Energetics

In this section, the relative accuracy of electronic excitations within atoms

will be compared. Six types of basis sets will be used in this comparison, including

three stock basis sets (the 3-21G, 6-31G, and 6-31G** basis sets) and three

comparable basis sets built using the previously proposed method (here called the

3-21B, 6-31B, and 6-31B** basis sets). The new basis sets have been constructed

such that the same number of primitive Gaussians are used for each orbital

contraction. This will help to ensure that meaningful comparisons can be made

between the energy optimized stock basis sets and the newly constructed basis sets.

The structure of the new, dynamically consistent basis sets can be found in the

basis set library located in the Appendix.

The data in this section are comprised of excitation energies as calculated

using the Gaussian 98 computational suite [57]. The excitation energies are

calculated as the absolute energy difference between the two states in question.

The absolute energies are calculated using the multi-configurational capabilities

of Gaussian 98, as found within the CASSCF routine [44]. The complete active

space is defined to be the set of valence shell electrons and the valence shell and all













Table 3.5: Atomic energies and electronic excitations in Helium

Stock Basis Sets

Atomic HF Energy
(Experimental: -2.9031840 a.u.)
3-21G 6-31G 6-31G**
(a.u.) (a.u.) (a.u.)
-2.8505767 -2.8701621 -2.8873650
Excitation Energies
Expt. 3-21G % 6-31G % 6-31G** %
Transition Excitation Excitation Error Excitation Error Excitation Error
(cm-1) (cm-1) (cm-1) (cm-1)
Is2(1S)-+ s2s(3S) 159843.3 442932.5 177 322814.3 102 326589.9 104
1s2(1S)- ls2s(1S) 166264.7 557807.8 235 421708.2 154 425452.4 156
1s2(1S)- ls2p(3P) 169074.1 N/A N/A 483411.3 183
12(1S)- ls2p(1P) 171122.2 N/A N/A 558949.8 227

Dynamically Consistent Basis Sets

Atomic HF Energy
(Experimental: -2.9031840 a.u.)
3-21B 6-31B 6-31B**
(a.u.) (a.u.) (a.u.)
-2.7123567 -2.8118598 -2.8119307
Excitation Energies
Expt. 3-21B % 6-31B % 6-31B** %
Transition Excitation Excitation Error Excitation Error Excitation Error
(cm-1) (cm-1) (cm-1) (cm-1)
Is2(1S)- ls2s(3S) 159843.3 146507.5 -8.34 155196.7 -2.91 155212.3 -2.90
1s2(1S)- ls2s(1S) 166264.7 155066.0 -6.74 162821.9 2.07 162778.0 -2.10
1s2(1S)- ls2p(3P) 169074.1 N/A N/A 169004.0 -0.04
1s2(1S)- ls2p(1P) 171122.2 N/A N/A 171592.9 0.28












Table 3.6: Atomic energies and electronic excitations in Lithium

Stock Basis Sets

Atomic HF Energy
(Experimental: -7.8848995 a.u.)
3-21G 6-31G 6-31G**
(a.u.) (a.u.) (a.u.)
-7.3815132 -7.4312358 -7.4313723
Excitation Energies
Expt. 3-21G % 6-31G % 6-31G** %
Transition Excitation Excitation Error Excitation Error Excitation Error
(cm-1) (cm-1) (cm-1) (cm-1)
2s'(2S)+ 2pl(2P) 14903.8 14620.3 -1.90 15626.9 4.85 15647.1 4.99
2s'(2S)+ 3sl (2S) 27205.8 43415.8 59.6 46379.9 70.5 48098.6 76.8
2s1(2S)- 3p (2P) 30925.9 43662.6 41.2 49549.6 60.2 46408.9 50.1
2s1(2S) 3d1(2D) 31283.2 N/A N/A 112671.0 260

Dynamically Consistent Basis Sets

Atomic HF Energy
(Experimental: -7.8848995 a.u.)
3-21B 6-31B 6-31B**
(a.u.) (a.u.) (a.u.)
-7.3252176 -7.4161167 -7.4161167
Excitation Energies
Expt. 3-21B % 6-31B % 6-31B** %
Transition Excitation Excitation Error Excitation Error Excitation Error
(cm-1) (cm-1) (cm-1) (cm-1)
2s' (2S) 2pl(2P) 14903.8 14810.7 -0.62 15678.5 5.20 15678.5 5.20
2s (2S)+ 3 s (2S) 27205.8 26096.7 -4.08 26795.2 -1.51 26795.2 -1.51
2s1(2S)- 3p (2P) 30925.9 43823.9 41.7 44431.2 43.7 44431.2 43.7
2s1(2S)+ 3d (2D) 31283.2 N/A N/A 35503.5 13.5









virtual orbitals in the given basis set. By using the CASSCF methodology, essential

correlation can be accounted for in the basis set comparisons.

The data in Table 3.5 demonstrates the excitations for the helium atom. The

top part of the table relates the experimental ground state atomic energy and the

excitation energies using the stock basis sets. The bottom half of the table shows

the same values calculated in the newly constructed and dynamically consistent

basis sets. It becomes clear immediately upon inspection of the data that the stock

basis sets provide a much more accurate ground state atomic energy. However,

this is to be expected, as the stock basis sets were constructed in such a way as

to minimize the Hartree-Fock atomic energy and the dynamically consistent basis

sets were not. It should be noted that, even in the worst case, the dynamically

consistent basis sets are no more than 6.57 percent in error with the experimental

energy, taken from Moore [58].

The most striking feature of the table is the excitation energy comparisons.

The experimental excitation energies are taken from Bacher and Goudsmit [59].

The stock basis sets do not represent the orbital-to-orbital excitation energies

well. One's attention can first be directed to the percent errors of the calculated

excitation energies (using stock basis sets) with respect to the experimental values.

The smallest error is about 100 percent. Even more important is the fact that all of

the excitations correspond to virtual orbitals that are bound states in the He atom.

The excitation energies are all greater in magnitude than the ionization energy of

the atom (about 197000 cm-1) [59]. This fact is denoted by the red coloration of

the excitation energy values.

By contrast, the dynamically consistent basis sets constructed in this work

show a remarkable improvement in the excitation energetic. The percent errors

are reduced from a minimum of 102 percent in the stock basis sets to a maximum









of 8.34 percent in the newly constructed basis sets. Moreover, all of the newly

constructed orbitals represent bound states in the atom.

Table 3.6 relates the same data, though for the lithium atom. The experimen-

tal atomic energy value is calculated in this case by considering the correlation

energy obtained by Eggarter and Eggarter using second-order perturbation meth-

ods [60], and the excitation energies are from Bacher and Goudsmit [59]. The

atomic energies demonstrate the same pattern as the helium atom: the stock basis

sets offer good representations of the atomic energy, while the newly constructed

basis sets are not as good (though now the largest percent error is only 7.1 per-

cent). The excitation energy data offers a considerably different comparison than

in the case of helium. Particularly one finds that in this case the stock basis sets

do allow for a good representation of the 2p virtual orbital excitation which are, in

fact, better than the orbitals arising from the new basis sets. However, the 3s, 3p,

and 3d orbitals are all unbound. The dynamically consistent basis sets all provide

good 2p, 3s, and 3d virtual orbitals, yet perform very poorly in the description of

the 3p orbitals. This can be attributed to either of the principal assumptions made

in the interpolation method (c.f. Section 3.3.3).


3.4.2 Charge Transfer Results

From the preceding data, it becomes clear that a basis set can be constructed

that has a relatively small size and 1.li-,-i .,lly meaningful excitations into virtual

orbitals. While this is a ].li,--i .,1 characteristic that is very important in many

dynamical properties, one must specifically investigate the behavior of the basis

sets when calculating such dynamical properties. In this section, charge transfer

probabilities are calculated using the 6-31G stock basis set and several basis sets

constructed using the newly proposed methodology. The contraction schemes for

each of the basis sets are given in the Appendix.









Three charge transfer systems will be investigated. The first is the near-

resonant charge transfer in the H+/Li collision system. In this process, the n = 2

orbitals are in near-resonance with the 2p orbitals in Li (with orbital energies of

-0.125 a.u. and -0.130 a.u., respectively) [61]. Likewise, the n = 3 orbitals of H

are in near-resonance with the 3p orbitals in Li (with orbital energies of -0.055

a.u. and 0.057 a.u., respectively) [61]. These near-resonances should result in a

few regions of large probability for transfer, as the electron can be excited into

one of the virtual orbitals in Li and then transfer over to the H atom. The second

-v-1 inII is the resonant charge transfer in the Li+/Li collision system. In this case,

the orbitals in both collision species have the same energies, promoting strong

resonances in the charge transfer process. Finally, the resonant transfer between

He+ and He is investigated.

The minimal END formalism has been applied successfully to investigations

of resonant charge transfer processes, particularly the collision of H+/H [62]. The

same methods are employed in this investigation. Specifically, the probability for

electron transfer is calculated to be the difference in Mulliken population between

the incident projectile and the fastest particle after collision, the differential cross

section is calculated using the distinguishable or identical particle scattering

amplitudes (as the case warrants), the scattering amplitude is calculated from the

small angle Schiff Approximation, and the total cross section for resonant transfer

is calculated using the semi-classical formula


oRT = bP(b)db, (3.44)


where b is the impact parameter and P(b) is the transfer probability. All of the

calculations in this section were performed using ENDyne, version 5 [63].

The first comparison to be made is the transfer probability. Figure 3.8

demonstrates the near-resonant charge transfer probability in the H+/Li collision