<%BANNER%>

Quantum Dynamics of Finite Atomic and Molecular Systems Through Density Matrix Methods


PAGE 1

QUANTUM D YNAMICS OF FINITE A TOMIC AND MOLECULAR SYSTEMS THR OUGH DENSITY MA TRIX METHODS By BRIAN THORND YKE 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 2004

PAGE 2

Cop yrigh t 2004 b y Brian Thorndyk e

PAGE 3

T o m y father, Gerry Thorndyk e

PAGE 4

A CKNO WLEDGMENTS First of all, I w ould lik e to thank m y paren ts, Gerry and Carol Thorndyk e, whose un w a v ering supp ort and lo v e m y en tire life ha v e allo w ed me to pursue m y dreams. On the academic side, I w ould lik e to thank m y advisor, Dr. Da vid Mic ha, for his excellen t guidance and encouragemen t throughout m y do ctoral w ork. I w ould also lik e to thank the follo wing colleagues in Quan tum Theory Pro ject and the Ph ysics departmen t for their friendship and insigh ts during m y sta y in Gainesville: Jim Co oney Herb ert DaCosta, Alex P ac heco, Da v e Red, Andres Rey es, Akbar Salam, Alb erto San tana and Zhigang Yi. On a p ersonal lev el b ey ond the Ph ysics departmen t, I w ould b e remiss if I didn't express m y lo v e and gratitude to Natasha Lep or e. She has b een m y \soul sister" for o v er a decade, and I hop e our liv es will con tin ue to run with fascinating parallels and in tert wine for man y decades to come! I'd also lik e to recognize Alb ert V ernon who, since our early da ys in the Computer Science departmen t, has b een m y partner in our relen tless pursuit of aesthetically pleasing co de. His friendship has b oth con tributed to some of the b est of times and help ed me through some of the w orst o v er the last 8 y ears. Finally I'd lik e to express m y appreciation to Mik e Kuban and Rob Thorndyk e for b eing with me in spirit throughout m y Ph.D., and for alw a ys pro viding w onderful a v en ues of escap e and adv en ture y ear after y ear! iv

PAGE 5

T ABLE OF CONTENTS P age A CKNO WLEDGMENTS . . . . . . . . . . . . . . iv LIST OF T ABLES . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . x ABSTRA CT . . . . . . . . . . . . . . . . . . xiii CHAPTER1 INTR ODUCTION . . . . . . . . . . . . . . . 1 1.1 Ov erview of Classical and Quan tum Dynamics . . . . . 2 1.2 Appro ximations to Quan tum Dynamics . . . . . . . 5 1.2.1 W a v efunction-Based Approac hes . . . . . . . 5 1.2.2 Densit y Op erator Approac hes . . . . . . . . 7 1.3 Quan tum-Classical Liouville Equation . . . . . . . . 8 1.4 Our Approac h . . . . . . . . . . . . . . 10 1.5 Simple One-Dimensional Tw o-State Mo dels . . . . . . 11 1.6 Lithium-Helium Clusters . . . . . . . . . . . 11 1.7 Outline of the Dissertation . . . . . . . . . . . 12 2 QUANTUM-CLASSICAL LIOUVILLE EQUA TION: F ORMULA TION 14 2.1 In tro duction . . . . . . . . . . . . . . . 14 2.2 Quan tum Liouville Equation . . . . . . . . . . 14 2.3 Wigner Represen tation . . . . . . . . . . . . 15 2.4 Quan tum-Classical Liouville Equation . . . . . . . . 18 2.5 Eectiv e P oten tial . . . . . . . . . . . . . 20 3 QUANTUM CLASSICAL LIOUVILLE EQUA TION: COMPUT A TIONAL ASPECTS . . . . . . . . . . . . . . . . 22 3.1 In tro duction . . . . . . . . . . . . . . . 22 3.2 T ra jectory Solution . . . . . . . . . . . . . 22 3.3 Electronic Basis Set . . . . . . . . . . . . . 23 3.4 Nuclear Phase Space Grid . . . . . . . . . . . 24 3.5 Relax-and-Driv e Algorithm . . . . . . . . . . . 25 3.5.1 W Indep enden t of Time . . . . . . . . . 26 3.5.2 W Dep enden t on Time . . . . . . . . . . 26 3.5.3 V elo cit y V erlet for the Classical Ev olution . . . . . 31 v

PAGE 6

3.5.4 Algorithm Details . . . . . . . . . . . 32 3.6 Computing Observ ables . . . . . . . . . . . . 33 3.6.1 Op erators in an Orthonormal Basis . . . . . . 33 3.6.2 P opulation Analysis . . . . . . . . . . . 34 3.6.3 Exp ectation V alues . . . . . . . . . . . 35 3.6.4 Hamiltonian Eigenstates and Eigen v alues . . . . . 35 3.7 Programming Details . . . . . . . . . . . . 36 3.7.1 Orthogonalit y of Co de Dev elopmen t . . . . . . 37 3.7.2 Extensibilit y . . . . . . . . . . . . . 37 4 ONE-DIMENSIONAL TW O-ST A TE MODELS . . . . . . . 39 4.1 In tro duction . . . . . . . . . . . . . . . 39 4.2 Eectiv e-P oten tial QCLE in the Diabatic Represen tation . . 39 4.3 Near-Resonan t Electron T ransfer Bet w een an Alk ali A tom and Metal Surface . . . . . . . . . . . . . . 41 4.3.1 Mo del Details . . . . . . . . . . . . 41 4.3.2 Prop erties of In terest . . . . . . . . . . 42 4.3.3 Results . . . . . . . . . . . . . . 47 4.4 Binary Collision In v olving Tw o Av oided Crossings . . . . 59 4.4.1 Mo del Details . . . . . . . . . . . . 59 4.4.2 Prop erties of In terest . . . . . . . . . . 60 4.4.3 Results . . . . . . . . . . . . . . 62 4.5 Photoinduced Disso ciation of a Diatomic System . . . . . 63 4.5.1 Mo del Details . . . . . . . . . . . . 63 4.5.2 Prop erties of In terest . . . . . . . . . . 68 4.5.3 Results . . . . . . . . . . . . . . 70 4.6 Comparison Using V ariable and Constan t Timesteps . . . . 72 4.7 Conclusion . . . . . . . . . . . . . . . 72 5 ALKALI A TOM-RARE GAS CLUSTERS: GENERAL F ORMULA TION 80 5.1 In tro duction . . . . . . . . . . . . . . . 80 5.2 Ph ysical System . . . . . . . . . . . . . . 80 5.3 Prop erties of In terest . . . . . . . . . . . . 81 5.4 Hamiltonian for Alk ali-Rare Gas P airs . . . . . . . 81 5.5 Hamiltonian for the Alk ali-Rare Gas Cluster . . . . . . 84 5.6 Electronic Sp ectral Calculations . . . . . . . . . 86 5.7 Electronic Basis of Gaussian A tomic F unctions . . . . . 87 5.7.1 Equations of Motion . . . . . . . . . . . 87 5.7.2 Ov erlap Matrix Elemen ts . . . . . . . . . 89 5.7.3 Kinetic Energy Matrix Elemen ts . . . . . . . 90 5.7.4 Coulom b Matrix Elemen ts . . . . . . . . . 90 5.7.5 Momen tum Coupling Matrix Elemen ts . . . . . 91 5.7.6 Dip ole Matrix Elemen ts . . . . . . . . . . 91 5.7.7 Pseudop oten tial Matrix Elemen ts . . . . . . . 92 vi

PAGE 7

5.8 Computing the Quasiclassical T ra jectory . . . . . . . 93 5.9 Computational Details . . . . . . . . . . . . 93 5.10 Conclusion . . . . . . . . . . . . . . . 97 6 LITHIUM-HELIUM CLUSTERS . . . . . . . . . . . 99 6.1 In tro duction . . . . . . . . . . . . . . . 99 6.2 Description of the System . . . . . . . . . . . 99 6.3 Prop erties to b e In v estigated . . . . . . . . . . 101 6.4 Preparation of Lithium-Helium Clusters . . . . . . . 102 6.4.1 Bulk Helium . . . . . . . . . . . . . 102 6.4.2 Liquid Helium Droplets . . . . . . . . . . 109 6.4.3 Lithium-Helium In teractions . . . . . . . . 109 6.5 Results: Lithium Inside the Helium Cluster . . . . . . 119 6.6 Results: Lithium on the Helium Cluster Surface . . . . . 121 6.6.1 Dynamics of Li(2 p ) . . . . . . . . . . 124 6.6.2 Dynamics of Li(2 p ) . . . . . . . . . . 130 6.7 Conclusion . . . . . . . . . . . . . . . 137 7 CONCLUSION . . . . . . . . . . . . . . . . 140 7.1 Eectiv e P oten tial Quan tum-Classical Liouville Equation . . 140 7.2 One-Dimensional Tw o-State Mo dels . . . . . . . . 141 7.3 Alk ali-Rare Gas Clusters . . . . . . . . . . . 142 7.4 Soft w are Dev elopmen t . . . . . . . . . . . . 144 7.5 F uture W ork . . . . . . . . . . . . . . . 145 APPENDIXA THE CAULDRON PR OGRAM . . . . . . . . . . . . 146 A.1 Ov erview . . . . . . . . . . . . . . . . 146 A.2 Comp onen t Descriptions . . . . . . . . . . . 147 A.2.1 Read Input File . . . . . . . . . . . . 147 A.2.2 System: Get Dieren tial Equation Co ecien ts . . . 147 A.2.3 Propagation: Ev olv e Single Timestep . . . . . . 149 A.2.4 Prop erties: Output Prop erties . . . . . . . . 149 A.3 Subroutine Details . . . . . . . . . . . . . 150 B SPLIT OPERA TOR-F AST F OURIER TRANSF ORM METHOD . . 151 C THE QUALDRON PR OGRAM . . . . . . . . . . . . 154 C.1 Ov erview . . . . . . . . . . . . . . . . 154 C.2 Comp onen t Descriptions . . . . . . . . . . . 155 C.2.1 Read Input File . . . . . . . . . . . . 155 C.2.2 System: Get Hamiltonian Matrix Elemen ts . . . . 155 C.2.3 Propagation: Ev olv e Single Timestep . . . . . . 155 vii

PAGE 8

C.2.4 Prop erties: Output Prop erties . . . . . . . . 157 C.3 Subroutine Details . . . . . . . . . . . . . 157 REFERENCE LIST . . . . . . . . . . . . . . . . 158 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 168 viii

PAGE 9

LIST OF T ABLES T able page 4{1 P arameters used in the Na-surface and Li-surface mo dels. . . . 42 4{2 P arameters used in the dual a v oided crossing collision mo del. . . 60 4{3 P arameters used in the NaI complex mo del. . . . . . . . 68 5{1 Pseudop oten tial rotation for d -function mixing. . . . . . . 98 6{1 P arameters for the He-He in teraction from Aziz ( V Az iz ). . . . . 106 6{2 P arameters for the correction to the He-He in teraction ( V 0 ) . . . 106 6{3 P arameters for the e -Li in teraction. . . . . . . . . . 113 6{4 P arameters for the e -He in teraction. . . . . . . . . . 113 6{5 P arameters for the Li-He core in teraction. . . . . . . . . 114 ix

PAGE 10

LIST OF FIGURES Figure page 4{1 P oten tial curv es for Hamiltonian I: Na inciden t up on a metal surface. 43 4{2 P oten tial curv es for Hamiltonian I I: Li inciden t up on a metal surface. 44 4{3 ( R ) at t = 0 au, for the Na-surface mo del. This w a v efunction is ev olv ed through the SO-FFT algorithm. . . . . . . . 48 4{4 11 at t = 0 au, for the Na-surface mo del. This PWTDM is ev olv ed through the EP-QCLE metho d. . . . . . . . . . 49 4{5 ( R ) at t = 14000 au, for the Na-surface mo del. . . . . . 50 4{6 Phase space grid p oin ts at t = 14000 au, for the Na-surface mo del. 51 4{7 Na p opulations 1 and 2 vs. time. . . . . . . . . . 52 4{8 Li p opulations 1 and 2 vs. time. . . . . . . . . . 53 4{9 Coherence describ ed b y Re( 12 ) vs. time, for the Na-surface system. 54 4{10 Coherence describ ed b y Re( 12 ) vs. time, for the Li-surface system. 55 4{11 Exp ectation of p osition and disp ersion for the Na-surface system. . 56 4{12 Exp ectation of momen tum and disp ersion for the Na-surface system. 57 4{13 Densit y function ( R ) for the Na-surface system. . . . . . 58 4{14 P oten tial curv es for the dual a v oided crossing collision. . . . . 61 4{15 P opulations 1 and 2 vs. time for the dual crossing collision mo del. 64 4{16 Coherence describ ed b y R e ( 12 ) vs. time, for the dual crossing collision mo del. . . . . . . . . . . . . . . . . 65 4{17 Grid deformation at t = 1400 au, for the dual crossing collision mo del. 66 4{18 Probabilit y of transmission in the ground state, for the dual crossing collision mo del. . . . . . . . . . . . . . . 67 4{19 P oten tial curv es for the NaI complex. . . . . . . . . . 69 4{20 Ionic and neutral p opulations o v er time, for the NaI complex. . . 73 x

PAGE 11

4{21 Exp ectation of p osition and its deviance, for the NaI complex. . . 74 4{22 Coherence as a function of time, for the NaI complex. . . . . 75 4{23 Phase space grid at the end of the sim ulation, for the NaI complex. 76 4{24 Num b er of steps required b y the relax-and-driv e algorithm, compared to an estimated n um b er required for a xed timestep v ersion. . 77 6{1 Sc hematic of Li(2 p ) ab o v e a He surface. A) Li(2 p ). B) Li(2 p ). . 101 6{2 Radial distribution functions for bulk liquid helium. . . . . . 106 6{3 Comparison of the Aziz p oten tial with the eectiv e form. . . . 107 6{4 Eectiv e He-He p oten tial. . . . . . . . . . . . . 108 6{5 Constraining p oten tial used to k eep He atoms from ev ap orating. . 110 6{6 T emp erature ructations of the He droplet o v er time. . . . . . 111 6{7 Helium densit y prole from the cen ter-of-mass of the cluster. . . 112 6{8 Adiabatic energy for Li and He as a function of in tern uclear distance. 114 6{9 Adiabatic energies for Li and one or more He along the z-axis. . . 116 6{10 Adiabatic energy for Li and one or more He along the y-axis. . . 117 6{11 Adiabatic energy for Li and a surface of He atoms parallel to the x-y plane. . . . . . . . . . . . . . . . . . 118 6{12 Ev olution of ground state Li em b edded in the cen ter of a He cluster. A) Initial time t = 0 au. B) Final time t = 10,000 au. . . . 120 6{13 Comparison of Li and He motion within a He cluster. The time scale has b een reduced b y a factor of 100 for the He curv e. . . . 122 6{14 Electronic p opulation of Li as it emerges from the He cluster. . . 123 6{15 Ev olution of Li(2 p ) as it recedes from the He cluster surface. A) Initial time t = 0 au. B) Final time t = 33 ; 000 au. . . . . 125 6{16 Mixing of the Li(2 p ) and Li(2 p ) states at distances where Li(2 p ) is triply degenerate. . . . . . . . . . . . . . 126 6{17 Electronic p opulation of Li with and without a p erturbing electromagnetic eld, resonan t to the D line. . . . . . . . . 128 6{18 Dip ole emission sp ectra of Li(2 p ) as it recedes from the He cluster surface. . . . . . . . . . . . . . . . . 129 xi

PAGE 12

6{19 Snapshot of Li(2 p ) as it in teracts with the He cluster surface. A) Initial time t = 0 au. B) Final time t = 67 ; 000 au. . . . . 131 6{20 Electronic p opulation of Li(2 p ) as it in teracts with the He cluster surface. . . . . . . . . . . . . . . . . 132 6{21 Dip ole emission sp ectrum of Li(2 p ) during the rst 3000 au. . . 134 6{22 Dip ole emission sp ectrum of Li(2 p ) during the nal 3000 au. . . 135 6{23 Adiabatic curv es of Li surrounded b y a cubic lattice of He atoms. The parameter R refers to the half-length of the lattice edge. . . . 136 6{24 Deca y of Li(2 p ) surrounded b y surface He atoms, induced b y an EM eld with frequency resonan t to the Li(2 p 2 s ) transition. . 138 A{1 Flo w c hart describing the cauldron program. . . . . . . . 148 C{1 Flo w c hart describing the qualdron program. . . . . . . . 156 xii

PAGE 13

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 QUANTUM D YNAMICS OF FINITE A TOMIC AND MOLECULAR SYSTEMS THR OUGH DENSITY MA TRIX METHODS By Brian Thorndyk e Ma y 2004 Chair: Da vid A. Mic ha Ma jor Departmen t: Ph ysics W e dev elop a mixed quan tum-classical form ulation to describ e the dynamics of fewand man y-b o dy atomic systems b y applying a partial Wigner transform o v er the quan tum Liouville equation of motion. In this approac h, the densit y op erator b ecomes a function in quasiclassical phase space, while remaining an op erator o v er a subset of quan tal v ariables. By taking appropriate limits and in tro ducing an eectiv e p oten tial, w e deriv e equations of motion describing quasiclassical n uclear tra jectories coupled to quan tal electronic ev olution. W e also in tro duce a v ariable timestep pro cedure to accoun t for the disparit y b et w een slo w n uclear motion and fast electronic ructuations. Our mixed quan tum-classical metho d is applied to the study of three simple one-dimensional t w o-state mo dels. The rst mo del represen ts the photoinduced desorption of an alk ali atom from a metal surface, where near-resonan t electron transfer is imp ortan t. A second mo del explores a binary collision under conditions where t w o a v oided crossings are presen t. The third mo del follo ws the photoinduced disso ciation of the so dium io dide complex, whose long-range attractiv e surface results in xiii

PAGE 14

oscillations of in tern uclear distance. Quan tities suc h as state p opulations and quantum coherence are computed, and found to b e in excellen t agreemen t with precise quan tal results obtained through fast F ourier transform grid metho ds. Ha ving v alidated our approac h, w e turn to the study of alk ali atoms em b edded in rare gas clusters, treating the alk ali atom-rare gas in teractions with l -dep enden t semi-lo cal pseudop oten tials. Ligh t emission from the electronic motion of the alk ali atom is deriv ed in the semiclassical limit, and computational metho ds to render the sim ulation feasible for a man y-atom cluster are discussed. The formalism is applied to lithium atoms in helium clusters, where the cluster conguration and the electronic p opulation dynamics of the lithium atom are monitored o v er time. W e study b oth the ground and rst excited states of lithium, and in tro duce a resonan t electromagnetic eld to induce electronic transitions. Our results correlate w ell with other exp erimen tal and theoretical studies on dop ed helium droplets, and pro vide insigh t in to the dynamics of an excited lithium atom near a helium cluster surface. xiv

PAGE 15

CHAPTER 1 INTR ODUCTION This w ork is part of a broader eort to bring new insigh ts in to the time dep endence of fewand man y-b o dy molecular systems. In order to study these systems, w e are particularly in terested in mixed quan tum-classical metho ds. There are man y approac hes to com bining classical and quan tum mec hanics, 1 2 3 but the underlying theme is to use classical mec hanics where quan tal descriptions are not essen tial to the dynamics of the system. By doing so, w e can sa v e a tremendous amoun t of computational time with hop efully minimal exp ense in accuracy In our study w e fo cus on the densit y op erator treatmen t, whic h allo ws for a general in tro duction of semiclassical and classical limits for some degrees of freedom. 4 In the applications b eing considered, electronic v ariables are describ ed quan tally and the n uclear v ariables are propagated quasiclassically but our theoretical treatmen t of quan tum-classical coupling is more general. W e are able to v alidate our metho ds through extensiv e study of small test mo dels. These mo dels can b e ev olv ed through a fully quan tal propagation using fast F ourier grid metho ds, p ermitting a rigorous assessmen t of the accuracy of our approac h. As a realistic application, w e turn to the study of ground and excited alk ali atoms em b edded in clusters of rare gas atoms. 5 6 7 8 9 Rare gas clusters pro vide an in teresting bridge b et w een few-atom systems and bulk matter. A mixed quan tumclassical approac h allo ws us to follo w the n uclear motion and p opulation state dynamics as the alk ali atom in teracts with the cluster. By trac king the electronic and induced dip ole, w e are able to compute the electronic sp ectra of the alk ali atom; and b y explicitly in tro ducing an electromagnetic eld, w e are able to induce electronic transitions. 1

PAGE 16

2 While our formalism is applicable to arbitrary alk ali-rare gas com binations, w e ha v e concen trated on the dynamics of lithium em b edded in helium clusters, for whic h there has b een a surge of recen t exp erimen tal and theoretical activit y 10 11 12 13 9 14 Stable helium clusters dop ed with lithium atoms ha v e b een pro duced at ultralo w temp eratures, where the ground lithium atom has b een sho wn to preferen tially reside on the surface of the helium cluster. F urthermore, the b eha vior of the lithium atom subsequen t to electronic excitation dep ends hea vily on the orien tation of the excited state. Our mixed quan tum-classical approac h corrob orates these ndings, and leads to additional insigh t in to the dynamics of these in teractions. 1.1 Ov erview of Classical and Quan tum Dynamics The v ast ma jorit y of c hemical and biological pro cesses can b e describ ed, in principle, b y nonrelativistic quan tum mec hanics. Within this con text, the state of a system of n uclei and electrons is represen ted en tirely b y a w a v efunction or densit y op erator. 15 A Hamiltonian op erator describ es all in teractions b et w een the particles, and can b e extended to include en vironmen tal comp onen ts (for example, a b oundary or electromagnetic eld). When the Hamiltonian do es not dep end on time, its eigenstates are stationary (up to a phase), and represen t time-in v arian t congurations of the system. When the Hamiltonian con tains time dep endence, or the initial state is nonstationary the molecular system ev olv es through the action of the Hamiltonian. The solutions to the time indep enden t Sc hr odinger equation (TISE) are the eigenstates of the Hamiltonian at an y giv en time. The corresp onding eigen v alues are the energy lev els a v ailable to the system. While conceptually compact, the analytical solution of the TISE is not p ossible for more than t w o particles in the general case. An enormous b o dy of computational w ork in c hemical and molecular ph ysics is dev oted to the n umerical solution of the TISE, 16 17 18 and accurate ground state

PAGE 17

3 energies ha v e b een computed for molecules in v olving h undreds of atoms. 19 Kno wledge of the full sp ectrum of eigenstates w ould allo w one to follo w the dynamics of the system as w ell, but unfortunately it is v ery dicult to obtain accurate results for excited states, and in an y case, the n um b er of states required for accurate computations w ould b e prohibitiv ely exp ensiv e for most dynamics problems. An alternativ e is to follo w the dynamics directly Quan tum dynamics (QD) follo ws the ev olution of a system in real time. The w a v efunction ev olv es according to the time dep enden t Sc hr odinger equation (TDSE), 20 while the densit y op erator (DOp) ev olv es through the quan tum Liouville equation (QLE). 21 These formalisms are equiv alen t, although statistical ensem bles are more naturally describ ed b y the densit y op erator. Quan tum dynamical calculations are imp ortan t in understanding the path w a ys b et w een initial and nal states, for example in c hemical reactions or molecular collisions, and ha v e b ecome an imp ortan t complemen t to mo dern exp erimen ts that use picoor fem tosecond ligh t pulses to prob e ultrafast dynamics. 22 The computational complexit y of n umerical solutions to full QD, ho w ev er, sev erely limits the n um b er of degrees of freedom that can b e studied, and incorp oration of classical concepts is necessary for most realistic applications. F ull QD solutions can b e readily implemen ted b y discretizing the w a v efunction along a m ultidimensional grid. Since the Hamiltonian con tains nonlo cal op erators lik e kinetic energy metho ds suc h as nite dierence 23 24 25 26 or F ourier transform 27 28 are needed to ev aluate the action of the Hamiltonian on the w a v efunction. In the general case, these tec hniques require a dense grid to obtain accurate results. F urthermore, their nonlo cal nature can result in signican t shifts of the w a v efunction o v er time, so that ev en if the initial state is spatially compact, a large grid is required to accommo date translation. Sparse grid metho ds 29 can alleviate some of the computational burden of using large grids, and dynamically c hanging grids 30 can b etter

PAGE 18

4 follo w the distortion and translation of the w a v efunction o v er time. Alternativ ely discretization on a basis set instead of a grid can simplify deriv ativ e calculations. 31 Ultimately ho w ev er, the exp onen tial scaling of the n um b er of grid p oin ts or basis functions with the system size renders full QD solutions in tractable for more than a few degrees of freedom. Classical molecular dynamics (MD) can b e deriv ed from the QLE in the classical limit ( ~ 0). 32 In the con text of molecular sim ulations, the most basic MD treats n uclei as p oin t particles in phase space, and follo ws tra jectories according to the Hamilton equations of motion. 33 In tern uclear p oten tials are deriv ed from ab initio exp erimen tal and empirical results, and the forces on the n uclei are obtained b y summing o v er the partial deriv ativ es of the pair p oten tials. These classical force calculations are no w the b ottlenec k, and straigh tforw ard ev olution of N classical degrees of freedom has only O ( N 2 ) time complexit y 34 T ree metho ds based on o ctree spatial partitions 35 or m ultip ole p oten tial expansions 36 37 reduce this complexit y to O ( N log N ), with a prop ortionalit y constan t dep enden t on the desired accuracy of the sim ulation. Because of its in tuitiv e nature and computational eciency MD is routinely used to study molecular systems with 10 3 {10 6 atoms. 38 The problem with MD is its inabilit y to adequately describ e quan tum mec hanical phenomena, suc h as c harge transfer, electronic excitation, tunneling and zero-p oin t motion. These eects are ubiquitous in c hemical and biological pro cesses at thermal or lo w er energies, and cannot b e completely neglected in most cases. 3 An attractiv e alternativ e is to construct an appro ximate QD mo del that ideally com bines the accuracy of QD with the computational eciency and in tuitiv e simplicit y of MD. In general, this can b e done b y augmen ting MD metho ds to include quan tum features, or b y simplifying QD mo dels to incorp orate classical or quasiclassical tra jectories. 1 There are man y v ariations of this theme, and question remains as to whic h approac h is the most suitable under arbitrary conditions. In Section 1.2 w e

PAGE 19

5 surv ey some of the most common appro ximation sc hemes, dieren tiating b et w een metho ds based on the w a v efunction and those cen tered on the densit y op erator. 1.2 Appro ximations to Quan tum Dynamics 1.2.1 W a v efunction-Based Approac hes W a v efunction-based approac hes are most useful for the propagation of pure states in closed en vironmen ts, and ha v e b een studied theoretically and n umerically since the da wn of quan tum mec hanics. W e limit our surv ey to some principal metho ds that incorp orate some classical concepts to solv e the TSDE, including time dep enden t self-consisten t eld (TDSCF) calculations, 39 40 41 42 Gaussian w a v epac k et (GWP) propagation, 43 44 surface hopping metho ds, 45 electron n uclear dynamics (END), 46 47 and path in tegral metho ds. 48 49 The time dep enden t self-consisten t eld appro ximation b egins with the w a v efunction written as a pro duct of n uclear and electronic w a v efunctions, and uses time dep enden t v ariational principles to ev olv e eac h w a v efunction along a p oten tial a v eraged o v er all other w a v efunctions. When the n uclear v ariables are expressed in the classical limit, the TDSCF appro ximation reduces to a set of electronic w a v efunctions ev olving o v er n uclear tra jectories, the tra jectories propagating according to eectiv e forces from the quan tal system. Man y mixed quan tum-classical sc hemes retain this mean-eld ra v or, where classical tra jectories propagate along precomputed or sim ultaneously ev olving electronic states. On the other hand, quan tum dynamics has a v ery dieren t c haracter than classical propagation, and mixed quan tum-classical metho ds v ary tremendously in their treatmen t of quan tumclassical in teractions, initial conditions, and measuremen t of system observ ables. Gaussian w a v epac k et propagation, in its original form, tak es the system to b e a Gaussian w a v epac k et in n uclear space, that propagates along a single electronic surface. By lo cally expanding the p oten tial up to harmonic con tributions, equations of motion for the Gaussian w a v epac k et parameters are deriv ed, whic h lead to shifts

PAGE 20

6 and distortions of the Gaussian o v er time. The Gaussian cen ter follo ws classical equations of motion, while its distorting shap e results from quan tal corrections to the classical motion. The GWP metho d has b een extended to describ e nonadiabatic pro cesses with the m ultiple spa wning approac h, 50 where sev eral w a v epac k ets propagate on m ultiple electronic surfaces, and proliferate in to additional w a v epac k ets at crossing p oin ts. Surface hopping b egins with n uclear tra jectories ev olving on one or more electronic surfaces, and repro duces nonadiabatic ev en ts through statistically based jumps b et w een the surfaces. In its earliest v ersion, these transitions tak e place near a v oided crossings, but later generalizations allo w hops to o ccur at an y time during the simulation. 51 One dra wbac k to the surface hopping approac h is the need to rescale v elo cities after sto c hastic transitions, in order to conserv e energy Although accurate state transitions are often ac hiev ed, the dynamics are clearly not represen tativ e of the true system ev olution, except in a statistical sense. Electron n uclear dynamics is an en tirely dieren t approac h, that deriv es equations of motion b y minimizing the Maxw ell-Sc hr odinger Lagrangian densit y and treating the n uclear degrees of freedom as coheren t states 52 in the semiclassical limit. By expanding the electronic w a v efunction in a basis of Slater determinan ts, n uclear and electronic motion are com bined in a computationally accessible sc heme. P ath in tegral metho ds solv e the TDSE for the quan tum time ev olution op erator, exp ( i ^ H t= ~ ), in the co ordinate represen tation. 53 P ath in tegral equations are equiv alen t to full QD, but unfortunately they are computationally in tractable in the general case. Harmonic appro ximations to the in termolecular p oten tials substan tially reduce the complexit y of the calculations, making it p ossible to repro duce the dynamics of a quan tum system em b edded within an harmonic bath. 54 55 F or example, v ariations suc h as the initial v alue represen tation, 56 57 58 ha v e successfully describ ed the spin-b oson mo del of coupled electronic states in a condensed phase

PAGE 21

7 en vironmen t. While not applicable to arbitrary systems, these metho ds are quite useful when the system and its in teractions can b e cast in to the appropriate forms. 1.2.2 Densit y Op erator Approac hes There are a n um b er of adv an tages to using the densit y op erator. First, it pro vides a con v enien t represen tation of mixed rather than pure states. 59 Second, man y systems can b e naturally partitioned in to a primary subsystem of in terest and its surrounding bath. By taking the trace o v er the bath degrees of freedom, the QLE pro vides a reduced densit y description of the primary system in teracting with a bath. 21 59 Finally it is p ossible to include terms directly in the QLE that represen t energy dissipation in to the en vironmen t. 60 F or these reasons, the QLE is often used where initial conditions are sp ecied as statistical a v erages, or when departure from a full quan tum treatmen t is necessary due to the size of the system. Among mixed quan tum-classical solutions to the QLE, an early approac h splits the densit y op erator in to a pro duct of Gaussian w a v epac k ets, and deriv es equations of motions for the Gaussian parameters using self-consisten t eld appro ximations. 61 In the semiclassical limit, this metho d has similarities to generalized forms of GWP propagation, but b ecause it is based on the densit y matrix, it can naturally treat b oth op en and closed systems. Another metho d is similar in spirit to END, but uses densit y matrices rather than Slater determinan ts to represen t the electronic state. 62 The quan tum degrees of freedom are expanded in some basis set, while the classical degrees of freedom are written as coheren t states. The Lagrangian is minimized through the DiracF renk el v ariational principle, and one arriv es at equations of motion for a reduced densit y matrix, coupled to semiclassical equations of motion for the coheren t state parameters. The densit y matrix ev olution (DME) metho d 63 64 65 divides the system in to a quan tum and classical space. The quan tum subsystem is expanded o v er a nonlo cal,

PAGE 22

8 orthogonal basis set, while the classical co ordinates ev olv e along tra jectories according to the Hellmann-F eynman force. This metho d has w ork ed w ell for simple mo dels in v olving analytically accessible matrix elemen ts, but is not designed for arbitrary systems with nonlo cal and p ossibly nonorthogonal basis sets. Eik onal metho ds 66 67 68 exploit the limit of small n uclear w a v elengths to separate quan tal and classical motion. Through this formalism, the electronic densit y matrix propagates sim ultaneously with n uclear tra jectories, the latter guided b y eectiv e forces from the quan tum densit y Com bined with v ariable timestep metho ds and tra v elling atomic function basis sets, the eik onal approac h is particularly w ell suited for the study of binary collision problems. A recen t branc h of QLE metho ds uses the Wigner transform (WT) to cast the op erator-based QLE in to an equation of motion o v er phase space v ariables. The classical Liouville equation for the densit y function emerges in the classical limit, while judicious application of the WT to a subset of the system v ariables leads to w ell dened quan tum-classical sc hemes. In Section 1.3 w e outline some of the ma jor approac hes found in the literature. 1.3 Quan tum-Classical Liouville Equation It is fascinating that a phase space represen tation of quan tum mec hanics can b e dev elop ed, that is fully equiv alen t to the Hilb ert space represen tation. By applying the Wigner transform to b oth sides of the QLE, an equation of motion is deriv ed for a Wigner function, whic h is a function of classical p osition and momen tum v ariables. The Wigner function do es not ev olv e through simple classical equations of motion, ho w ev er, but in v olv es nonlo cal op erators in phase space. 4 One class of solutions exploits the similarit y of these equations of motion with h ydro dynamics found in classical ph ysics. So-called h ydro dynamic quan tum dynamics w as form ulated b y Bohm in 1952, and recen t dev elopmen ts in the eld ha v e revitalized in terest in phase space tra jectory metho ds for solving QD. 69 70 71 72 73 Unfortunately these metho ds,

PAGE 23

9 lik e those in v olving basis set or grid solutions to the TDSE or QLE, suer from exp onen tial gro wth with system size. A mixed quan tum-classical sc heme arises b y dividing the quan tum system in to quan tal and quasiclassical v ariables, the quasiclassical v ariables asso ciated with energies or masses m uc h greater than the quan tal v ariables. T aking the partial Wigner transform (PWT) of the QLE o v er only the quasiclassical v ariables pro duces an equation of motion for a partially transformed Wigner op erator (PTWDOp), whic h is a function of phase space in the quasiclassical v ariables, but remains a quan tal op erator in the space of the quan tal v ariables. After some appropriate appro ximations relating to the dieren t masses of the v ariables, w e arriv e at a dieren tial op erator equation referred to as the quan tum-classical Liouville equation (QCLE). 74 While the quan tal ev olution m ust still b e tac kled in Hilb ert space, the propagation of the quasiclassical phase space acquires a classical c haracter. If the ma jorit y of the system can b e describ ed with quasiclasical v ariables, the computation sa vings ma y b e considerable. One n umerical approac h represen ts the PWTDOp in phase space with a xed n um b er of delta functions. 75 76 77 Sp ecically the PWTDop is pro jected on a basis, resulting in a set of diagonal and o-diagonal functions in phase space. Eac h function is appro ximated b y a set of delta p eaks, that ev olv e along surfaces constructed from the densities and coherences. Nonadiabatic coupling is implemen ted through a mo died surface hopping pro cedure, where hops o ccur b et w een tra jectories at curv e crossings. An alternativ e solution uses the same delta p eak represen tation, but rather than use sto c hastic jumps b et w een tra jectories, new tra jectories are generated at curv e crossings. 78 A v ariation on this approac h, the m ultithread metho d, spa wns new tra jectory p oin ts at ev ery timestep, but com bines the newly generated p o ols

PAGE 24

10 of tra jectory p oin ts in to smaller n um b ers through energy and other conserv ation considerations. 79 80 81 Recen t eorts ha v e b een made to com bine Gaussian w a v epac k ets in phase space (GPPs) with the tra jectory solutions along diagonal and o-diagonal surfaces. 82 Rather than represen t quasiclassical functions through delta p eaks, the functions are expanded as a linear com bination of GPPs, thereb y more eectiv ely co v ering quasiclassical phase space. Sto c hastic jumps b et w een surfaces at crossing p oin ts repro duce nonadiabatic b eha vior, and a v oid the need to spa wn new GPPs. 1.4 Our Approac h In our approac h, w e in tro duce an eectiv e p oten tial to the QCLE, to pro vide a simple n umerical pro cedure for solving the equation. 83 Our eectiv e p oten tial quan tum-classical Liouville equation (EP-QCLE) p ermits solution in terms of full quan tal ev olution of the quan tal v ariables along quasiclassical tra jectories guided b y the quan tal state. W e in tro duce a quan tal basis and a quasiclassical grid to render the equations suitable for n umerical propagation, and implemen t a sc heme to ecien tly accoun t for the dieren t time scales of quasiclassical and quan tal motion. Our approac h shares sev eral p ositiv e features with other QCLE metho ds of solution. The QCLE rigorously com bines quan tum and classical motion, and pro vides systematic w a ys to incorp orate higher-order appro ximations and quan tum-classical coupling. The formalism is based on the densit y op erator, making it attractiv e for incorp orating thermo dynamical features or dissipativ e en vironmen ts. The QCLE also lends itself to tra jectory-based n umerical solutions that reduce to the classical Liouville equation in the classical limit, pro viding an in tuitiv e computational framew ork. Our EP-QCLE solution diers from other QCLE primarily in its generalit y and its scalabilit y The eectiv e p oten tial can tak e a v ariet y of forms, and while w e ha v e found the Hellmann-F eynman force to pro vide the b est results for our mo del

PAGE 25

11 systems, it is p ossible that other forms w ould pro vide ev en b etter results under dieren t circumstances. F urthermore, an electronic basis is only in tro duced once the EP-QCLE has b een dev elop ed in terms of partial Wigner op erators, and no assumptions are made on the form of the basis set. This rexibilit y has allo w ed a uniform treatmen t of problems ranging from one-dimensional mo dels expanded in a t w o-state diabatic basis, to fully three-dimensional atomic clusters in a basis of Gaussian atomic functions. Finally while the ma jorit y of QCLE solutions rely on in teractions b et w een the propagating tra jectories, our use of an eectiv e p oten tial results in completely indep enden t tra jectories whose only connection follo ws from initial conditions. Suc h indep enden t tra jectory ev olution maps optimally to certain parallel arc hitectures, and substan tially reduces the cost of propagation, that w ould otherwise b e required to compute tra jectory in teractions at eac h timestep. 1.5 Simple One-Dimensional Tw o-State Mo dels W e test our metho ds on a set of one-dimensional t w o-state mo dels, whic h are simple enough to b e ev aluated precisely through fast F ourier grid metho ds. Among these metho ds, w e study a mo del of an alk ali atom approac hing a metal surface, where near-resonan t electronic transfer is imp ortan t. 84 85 Secondly w e consider a system represen ting a molecular collision with t w o a v oided crossings, where imp ortan t in terference eects arise as one v aries the collision energy 79 Finally w e study a system represen ting the predisso ciation of the so dium io dide complex, where the long-range attraction of the excited state results in oscillatory n uclear motion. 86 1.6 Lithium-Helium Clusters Clusters are aggregates of atoms con taining b et w een three and a few thousand atoms. The smallest clusters, or micr o clusters ha v e b et w een 3 and 10 atoms, and are just small enough that molecular concepts still apply to some degree. The next range of clusters, or smal l clusters ha v e b et w een 10 and 100 atoms, a range where molecular concepts break do wn and the clusters form shells with nonempt y

PAGE 26

12 in teriors. L ar ge clusters b et w een 100 and 1,000 atoms, pro vide the nal transition from isolated molecules to the bulk condensed state. Rare gas clusters can b e prob ed b y doping them with a c hromophore and follo wing the c hromophore through v arious laser detection metho ds. 9 87 88 89 Our study used a lithium atom as the dopan t in a pure helium cluster. Semi-lo cal l -dep enden t pseudop oten tials are used to describ e the lithium-helium in teractions, b ecause they are found to accurately repro duce adiabatic energy curv es for the lithium-helium pair for s p and d symmetries. 90 91 92 93 94 By in tro ducing the pseudop oten tial formalism in to the EP-QCLE, w e are able to follo w the n uclear conguration o v er time. In addition, w e are able to prob e the ev olving electronic energy surface b y monitoring the electronic and induced dip ole, and computing the resulting dip ole emission sp ectrum. By applying a resonan t electromagnetic eld, w e are able to stim ulate the emission of ligh t b y excited lithium atoms near the cluster surface. Both the n uclear conguration of the cluster and the sp ectral prop erties of its dopan ts ha v e b een under in tense in v estigation o v er the recen t y ears, and our con tribution giv es insigh t in to the dynamics of these in teractions. 1.7 Outline of the Dissertation Chapter 2 presen ts the formalism w e ha v e used to explore the dynamics of mixed quan tum-classical systems. The ideas are presen ted for the general case of quan tal and quasiclassical v ariables, and it is sho wn ho w the EP-QCLE naturally emerges from the partial Wigner transform o v er the quasiclassical v ariables. Chapter 3 explores the computational asp ects of the EP-QCLE. In particular, the application of an electronic basis and a n uclear grid render the equations of motion suitable to n umerical solution. W e also sho w ho w observ able quan tities can b e calculated within this framew ork. Chapter 4 applies our formalism to three dieren t one-dimensional t w o-state systems, where the results can b e compared to exact quan tal sim ulations based on

PAGE 27

13 the Sc hr odinger equation and n umerically precise grid metho ds. Along the w a y v aluable insigh ts in to the accuracy and limitations of our metho ds are obtained. Chapter 5 presen ts the formalism for general alk ali-rare gas clusters. W e describ e the Hamiltonian for general alk ali-rare gas in teractions, and deriv e matrix elemen ts for a basis set of Gaussian atomic functions. W e further discuss the dip ole of the cluster and its matrix elemen ts in detail. Finally w e presen t computational metho ds used to render the n umerical propagation of the equations of motion feasible for small to large clusters. Chapter 6 presen ts sp ecic results for lithium-helium clusters. W e consider the thermo dynamic equilibration of the helium cluster in detail, follo w ed b y the ev olution of the lithium atom through the cluster and near its surface once the cluster has reac hed equilibrium. W e follo w the n uclear motion and discuss the in teraction of the excited lithium atom near surface helium atoms, in particular with regard to the n uclear conguration and the dip ole sp ectrum. Finally w e in tro duce a resonan t electromagnetic eld to stim ulate photon emission after the excited lithium atom em b eds itself within the surface. Chapter 7 summarizes the main conclusions obtained in this dissertation. App endix A discusses the program cauldron dev elop ed o v er the course of this dissertation, to sim ulate the EP-QCLE for the test mo del systems and the full lithium-helium cluster. App endix B deriv es the n umerical ev olution of the Sc hr odinger equation through the split op erator fast F ourier transform metho d. The solution obtained b y this approac h is exact to within n umerical precision, but is computationally exp ensiv e and th us prohibitiv e for more than a few degrees of freedom. App endix C presen ts the qualdron co de, dev elop ed alongside cauldron to compare the results of the test mo dels using mixed quan tum-classical dynamics with the full quan tal treatmen t.

PAGE 28

CHAPTER 2 QUANTUM-CLASSICAL LIOUVILLE EQUA TION: F ORMULA TION 2.1 In tro duction Rather than fo cus on the w a v efunction of a molecular system, w e construct its densit y op erator and pro ceed from there. The DOp is more general than the w a v efunction, in that it naturally describ es a statistical ensem ble of quan tum states. This is particularly useful when the molecular system in teracts with its en vironmen t, and one do es not ha v e a complete kno wledge of that en vironmen t. The DOp also pro vides a con v enien t starting p oin t for deriving mixed quan tum-classical metho ds, b ecause the classical limit of the DOp is the classical densit y function. One particular route to a mixed quan tum-classical description of a molecular system is through the the Wigner transform. By applying a partial Wigner transform to the DOp and its equation of motion, one obtains a new represen tation that is a function in the phase space of the Wigner transformed v ariables, but remains an op erator in the remaining v ariables. If the PWT is judiciously applied to classical-lik e (or quasiclassic al ) v ariables, then appro ximations can b e made to the equations of motion that pro vide a classical ev olution of the quasiclassical v ariables coupled to a quan tal ev olution of the quan tal v ariables. In the case of molecular systems, the quasiclassical v ariables are t ypically the n uclear co ordinates, while the remaining v ariables describ e the electronic state. In this c hapter, the PWT is describ ed in detail, and its application to molecular systems is outlined. 2.2 Quan tum Liouville Equation A system of atoms or molecules is represen ted b y nonrelativistic quan tum mec hanics as a state v ector j i in Hilb ert space. This state v ector ev olv es in time 14

PAGE 29

15 according to Sc hr odinger's equation, i ~ @ j i @ t = ^ H j i ; (2.1) where ^ H is the Hamiltonian of the full system. If, ho w ev er, w e are studying an ensem ble of states fj i ig with statistical w eigh ts f w i g it is con v enien t to construct the densit y op erator, ^ X i w i j i ih i j : (2.2) T aking its time deriv ativ e and using Eq. 2.1 w e nd the densit y op erator to ev olv e in time according to the quan tum Liouville equation of motion, i ~ @ ^ @ t = [ ^ H ; ^ ] : (2.3) There are a n um b er of adv an tages to using the densit y op erator, but in our case the primary adv an tage is that it leads directly to the classical densit y function in the limit ~ 0. W e rst discuss the full Wigner transform and its application to the QLE, and then sho w ho w the partial Wigner transform can b e used to deriv e a mixed quan tum-classical represen tation in the appropriate limits. 2.3 Wigner Represen tation The Wigner transform pro vides a phase space represen tation of the DOp (the Wigner function ) and other quan tum mec hanical op erators. It is dened as the F ourier transform of an op erator pro jected on co ordinate space, 4 95 96 97 W ( r ; p ) = 1 (2 ~ ) n Z d n z exp ( ip z = ~ ) h r z = 2 j ^ j r + z = 2 i ; (2.4) A W ( r ; p ) = Z d n z exp ( ip z = ~ ) h r z = 2 j ^ A j r + z = 2 i ; (2.5) where ^ A is arbitrary The in tegration generates functions that are lo cal in b oth co ordinate and momen tum space, whic h is imp ortan t for the emergence of classical features in the dev elopmen t of our mixed quan tum-classical metho d. The prefactors

PAGE 30

16 are dened dieren tly for the densit y op erator than for other op erators, in order to pro vide con v enien t parallels b et w een the Wigner function and the classical Liouville densit y Classically the probabilit y distribution function is w ell dened. F or an N -b o dy system with co ordinates r ( r 1 ; r 2 ; : : : ; r 3 N ) and momen ta p ( p 1 ; p 2 ; : : : ; p 3 N ), the classical Liouville densit y = ( r ; p ) generates the exp ectation v alue of an y function A = A ( r ; p ), 32 h A i = Z dr dp ( r ; p ) A ( r ; p ) : (2.6) Quan tum mec hanically ev en the notion of phase space is problematic, as Heisenb erg's uncertain t y principle prohibits the sim ultan teous measuremen t of p osition and momen tum for a giv en degree of freedom. Ho w ev er, quasidistribution functions lik e the Wigner function pro vide an analogous form of the quan tum mec hanical exp ectation v alue. The exp ectation v alue of an arbitrary op erator is w ell kno wn, 59 h ^ A i = T r( ^ ^ A ) ; (2.7) whic h can readily b e seen b y expanding the trace o v er eigenstates of ^ A Ho w ev er, b y Wigner transforming b oth ^ and ^ A w e arriv e at a form for the exp ectation v alue similar to the classical case, h ^ A i = Z dr dp W ( r ; p ) A W ( r ; p ) : (2.8)

PAGE 31

17 This can b e seen b y expanding W and A W in Eq. 2.8 Z dr dp W ( r ; p ) A W ( r ; p ) = 1 (2 ~ ) 3 N Z dr dp Z dz exp ( ip z = ~ )( r + z = 2 ; r z = 2) Z dz 0 exp ( ip z 0 = ~ ) A ( r + z 0 = 2 ; r z 0 = 2) = 1 (2 ~ ) 3 N Z dr dpdz dz 0 exp [ ip ( z + z 0 ) = ~ ] ( r + z = 2 ; r z = 2) A ( r + z 0 = 2 ; r z 0 = 2) = Z dr dz ( r + z = 2 ; r z = 2) A ( r z = 2 ; r + z = 2) : (2.9) T ransforming v ariables q = r + z = 2, q 0 = r z = 2, w e retriev e the quan tum exp ectation v alue, Z dr dp W ( r ; p ) A W ( r ; p ) = Z dq dq 0 ( q ; q 0 ) A ( q 0 ; q ) = Z dq dq 0 h q j ^ j q 0 ih q 0 j ^ A j q i = Z dq h q j ^ ^ A j q i = T r( ^ ^ A ) : (2.10) Although W is not a probabilit y distribution, v arious prop erties justify its classication as a quasiprobabilit y function: j ( r ) j 2 = R dp W ( r ; p ). j ~ ( p ) j 2 = R dr W ( r ; p ). F or an y function f ( r ), h f ( r ) i = R dr dp W ( r ; p ) f ( r ) : F or an y function g ( p ), h g ( p ) i = R dr dp W ( r ; p ) g ( p ) : One can also deriv e the WT of pro ducts of op erators in terms of the WT of the op erators themselv es. 4 F or a general op erator ^ F = ^ A ^ B w e nd that F W = A W exp ( ~ = 2 i ) B W ; (2.11)

PAGE 32

18 where is the bidirectional op erator, @ @ p @ @ r @ @ r @ @ p : (2.12) Expanding the comm utator directly w e ha v e [ A; B ] W = ( AB ) W ( B A ) W = A W exp ( ~ = 2 i ) B W B W exp ( ~ = 2 i ) A W : (2.13) 2.4 Quan tum-Classical Liouville Equation W e ha v e seen that the densit y op erator ev olv es according to the QLE (Eq. 2.3 ). Rather than study the solution to the QLE, w e will examine the ev olution of the Wigner function. When the densit y op erator is transformed o v er all its v ariables, w e arriv e at equations of motion that, in the limit that ~ 0, reduce to the classical Liouville equation. Here w e are in terested in p erforming a partial Wigner transform of the densit y op erator o v er a subset of v ariables (those termed quasiclassic al ), while lea ving the remaining v ariables (those termed quantal ) in their original represen tation. By taking appropriate limits, w e deriv e equations of motion for the quasiclassical v ariables, coupled to quan tal equations of motion for the remainder. T o b e sp ecic, w e divide the degrees of freedom in to N quasiclassical v ariables Q ( Q 1 ; Q 2 ; : : : ; Q N ) and n quan tal electronic v ariables ^ q = ( ^ q 1 ; ^ q 2 ; : : : ; ^ q n ). The Wigner transform is p erformed o v er quasiclassical v ariables only ^ W ( R ; P ) = (2 ~ ) N Z d Z exp ( iP Z = ~ ) h R Z = 2 j ^ j R + Z = 2 i ; (2.14) ^ A W ( R ; P ) = Z d Z exp ( iP Z = ~ ) h R Z = 2 j ^ j R + Z = 2 i : (2.15) These are distinguished from fully Wigner transformed op erators b y virtue of remaining op erators in the quan tal space. By taking the PWT of b oth sides of the QLE, w e nd the equation of motion for the partially Wigner transformed densit y op erator

PAGE 33

19 ^ W ( R ; P ), 98 99 100 i ~ @ ^ W @ t = ^ H W exp ( ~ cl = 2 i ) ^ W ^ W exp ( ~ cl = 2 i ) ^ H W ; (2.16) where cl @ @ P @ @ R @ @ R @ @ P : (2.17) T o pro ceed further, w e appro ximate Eq. 2.16 to O ( ~ cl ), @ ^ W @ t = 1 i ~ [ ^ H W ; ^ W ] + 1 2 ( f ^ H W ; ^ W g f ^ W ; ^ H W g ) ; (2.18) where f A; B g is the P oisson brac k et, f A; B g A cl B : (2.19) This is an appro ximation within the quasiclassical space, and reduces quan tal motion of the quasiclassical degrees of freedom, but k eeps dep endence of the quasiclassical v ariables on the quan tal state through the in teraction p oten tial. It is an appropriate truncation when the quasiclassical v ariables are asso ciated with a m uc h greater mass than the quan tal v ariables. 75 101 80 79 F or a sp ecic example, consider a Hamiltonian for a molecular system comp osed of kinetic, p oten tial and in teraction terms, ^ H = ^ P 2 2 M + ^ V ( ^ Q ) + ^ p 2 2 m + ^ v ( ^ q ) + ^ V 0 ( ^ q ; ^ Q ) ; (2.20) where ^ P is the n uclear momen tum op erator, ^ V ( ^ Q ) the n uclear p oten tial, ^ p the electronic momen tum, ^ v ( ^ q ) the electronic p oten tial, and ^ V 0 ( ^ q ; R ) the electronic-n uclear coupling. By in terpreting the n uclear v ariables ^ Q as quasiclassical and taking the

PAGE 34

20 PWT o v er the n uclear v ariables, w e nd the partially Wigner transformed Hamiltonian, ^ H W = P 2 2 M + V ( R ) + ^ p 2 2 m + ^ v ( ^ q ) + ^ V 0 ( ^ q ; R ) : (2.21) Dening ^ H q ^ p 2 2 m + ^ v ( ^ q ) + ^ V 0 ( ^ q ; R ) ; (2.22) ^ V V ( R ) + ^ V 0 ( ^ q ; R ) ; (2.23) w e get the quan tum-classical Liouville equation of motion for ^ W ( R ; P ), @ ^ W @ t = 1 i ~ [ ^ H q ; ^ W ] P M @ ^ W @ R + 1 2 ( @ ^ V @ R @ ^ W @ P + @ ^ W @ P @ ^ V @ R ) : (2.24) 2.5 Eectiv e P oten tial The third term on the RHS of Eq. 2.24 presen ts a c hallenging obstacle to solving the equation n umerically One problem common among man y prop osed sc hemes is the requiremen t of computationally demanding algorithms to ev aluate the partial deriv ativ es of the PWTDOp. T o a v oid this problem, w e in tro duce an eectiv e p oten tial V ( R ; P ) in Eq. 2.24 @ ^ W @ t = 1 i ~ [ ^ H q ; ^ W ] P M @ ^ W @ R + @ V @ R @ ^ W @ P + 1 2 ( @ ^ V @ R @ V @ R @ ^ W @ P + @ ^ W @ P @ ^ V @ R @ V @ R !) : (2.25) The in tro duction of V b ecomes computationally useful insofar as w e can neglect the fourth term in Eq. 2.25 While the c hoice of V is clearly arbitrary w e ha v e found optimal results b y setting the exp ectation v alue o v er quan tal v ariables of ( @ R ^ V @ R V ) to zero, T r qu ^ W @ ^ V @ R @ V @ R !# = 0 : (2.26)

PAGE 35

21 In this w a y w e retriev e the Hellmann-F eynman force, @ V @ R = T r qu h ^ W @ ^ V =@ R i T r qu [ ^ W ] (2.27) = F H F ( R ; P ) : (2.28) The denominator in Eq. 2.27 is itself a function of quasiclassical phase space, whic h diers from eectiv e p oten tial approac hes in v olving a single normalized densit y op erator. Neglecting the fourth term in Eq. 2.25 w e nd an appro ximated but computationally adv an tageous eectiv e-p oten tial QCLE (EP-QCLE), @ ^ W @ t = 1 i ~ [ ^ H q ; ^ W ] P M @ ^ W @ R F H F @ ^ W @ P : (2.29) The appro ximations used to deriv e the EP-QCLE substan tially reduce the quan tal c haracter of the quasiclassical solution space. In con trast, the b est adiabatic metho ds, based on the Born-Opp enheimer separation of n uclear and electronic motion, retain full quan tum n uclear dynamics along adiabatic curv es. 102 A principal adv an tage to the EP-QCLE is its nonadiabatic c haracter, whic h is capable of describing nonadiabatic ev en ts when the Born-Opp enheimer limit no longer applies. A further b enet is its suitabilit y for solution using tra jectory metho ds, greatly increasing the size of problems amenable to n umerical analysis. The theoretical and computational asp ects of the tra jectory solution to the EP-QCLE are discussed in the next c hapter.

PAGE 36

CHAPTER 3 QUANTUM CLASSICAL LIOUVILLE EQUA TION: COMPUT A TIONAL ASPECTS 3.1 In tro duction The EP-QCLE is a partial dieren tial op erator equation in the quasiclassical v ariables and time. One w a y of solving this kind of problem is to represen t the op erators as matrices on a large grid, and ev olv e the matrices using nite dierence or sp ectral metho ds. The dra wbac k to this approac h is that a v ery dense grid is required for n umerical accuracy and a v ery large grid is necessary if the quasiclassical densit y shifts lo cation appreciably as it ev olv es in time. Since the grid dimension v aries directly with the classical degrees of freedom, a m ultiparticle system presen ts v ery serious n umerical diculties. Moreo v er, nite dierence and sp ectral grid solutions are inheren tly dicult to parallelize, as substan tial comm unication is required b et w een pro cessors regardless of the division of computational lab or. An alternativ e approac h, applicable to the class of partial dieren tial equations to whic h the EP-QCLE b elongs, is to follo w tra jectories in phase space as the system ev olv es. Only the imp ortan t tra jectory p oin ts are represen ted, so that the (mo ving) grid main tains a minimal size. In this c hapter, w e explore the tra jectory approac h and see ho w the EP-QCLE can b e solv ed in an ecien t and ev en parallel manner. 3.2 T ra jectory Solution W e can formally solv e Eq. 2.29 b y follo wing tra jectories in classical phase space, with R and P b ecoming functions of time, dR dt = P M ; (3.1) dP dt = F H F ( R ; P ) : (3.2) 22

PAGE 37

23 One could follo w any paths in phase space, but b y using those suggested in Eq. 3.2 w e are able to transform Eq. 2.29 in to an ordinary dieren tial equation in time. Inserting Eq. 3.2 in Eq. 2.29 and mo ving the partial deriv ativ es to the LHS, w e deriv e the c hange of the PWTDOp along the quasiclassical tra jectories, d ^ dt = 1 i ~ [ ^ H q ; ^ ] : (3.3) Note that w e ha v e omitted the subscript on the PWTDOp for notational con v enience, and from here on w ard will con tin ue to lab el all PWT op erators without this subscript. Eq. 3.3 remains a formal solution, and b efore solving it n umerically w e m ust discretize the equations in b oth quan tal and quasiclassical space. This is the sub ject of the next t w o sections. 3.3 Electronic Basis Set Let us in tro duce an arbitrary basis, fj i ig In man y c hemical ph ysics applications, a Gaussian basis is used, but for no w w e consider the basis to b e general, and not necessarily orthogonal or normalized. Con v erting to matrix notation, w e let j i b e the ro w matrix, j i ( j 1 i j 2 i j 3 i ) : (3.4) Then w e can expand our op erators, ^ = j i S 1 h j ^ j i S 1 | {z } h j ; (3.5) ^ A = j i S 1 h j ^ A j i | {z } A S 1 h j ; (3.6) where S is the o v erlap h j i Pro jecting Eq. 3.3 on this basis set, and setting ~ = 1, w e obtain d dt = ( i H q n y ) S 1 S 1 ( i H q + n ) ; (3.7)

PAGE 38

24 where w e ha v e used the notation, n h j @ =@ t j i + dR dt h j @ =@ R j i + dP dt h j @ =@ P j i : (3.8) 3.4 Nuclear Phase Space Grid Although w e'v e transformed the EP-QCLE in to a discrete represen tation in electronic space and are no w dealing with the partially Wigner transformed densit y matrix (PWTDM) instead of op erator, w e m ust still discretize the quasiclassical phase space. T o this end, w e c ho ose a set of initial grid p oin ts f ( R i ; P i ) g Their distribution should appro ximately co v er the domain of ( R ; P ), and should b e sufcien tly dense to w ell represen t the ev olution of the PWTDM. In practice, the grid should b e adjusted un til con v ergence is ac hiev ed. Once a grid is c hosen, the grid p oin ts follo w n uclear tra jectories f ( R j ( t ) ; P j ( t )) g according to Eqs. 3.1 and 3.2 : dR j dt = P j M ; (3.9) dP j dt = F H F ( R j ; P j ) : (3.10) A t the same time, Eq. 3.7 b ecomes a set of uncoupled equations, one for eac h tra jectory: d j dt = j ( i H qj n yj ) S 1 j S 1 j ( i H qj + n j ) j ; (3.11) where j ( R j ( t ) ; P j ( t )) ; (3.12) H j H ( R j ( t ) ; P j ( t )) ; (3.13) n j n ( R j ( t ) ; P j ( t )) : (3.14)

PAGE 39

25 Eac h tra jectory follo ws the ev olution of the PWTDM along that p ath indep enden t of the other tra jectories. While one cannot exp ect coherence b et w een the classical degrees of freedom to b e represen ted b y this approac h, there are some substan tial computational adv antages. In particular, the sc heme can b e optimally p orted to a parallel pro cessor, whereb y eac h pro cessor indep enden tly ev olv es a single tra jectory; comm unication b et w een the pro cessors is unnecessary 3.5 Relax-and-Driv e Algorithm Before the tra jectory solution can b e implemen ted, a detailed propagation sc heme needs to b e sp ecied. One w ould exp ect that with a sucien tly small timestep t all propagation metho ds w ould con v erge to the same results, pro vided roundo error w ere not signican t. Ho w ev er, the paths to con v ergence will certainly dier, in that metho ds with higher accuracy can use larger timesteps. The relaxand-driv e metho d, dev elop ed originally b y Mic ha and Runge, 66 67 68 incorp orates the rapid electronic oscillatory b eha vior with the relativ ely slo wly ev olving n uclear v ariables in an accurate, v ariable timestep sc heme. The relax-and-driv e pro cedure has b een sho wn to giv e excellen t results for a wide v ariet y of t w o-b o dy collision problems. F or completeness, w e review its details next. First of all, since all tra jectories are propagated analagously it suces to lo ok at the ev olution of a single tra jectory Accordingly let us rewrite the EP-QCLE for a single tra jectory i d ( t ) dt = W ( t ) ( t ) ( t ) W y ( t ) ; (3.15) where w e ha v e dened W S 1 ( H + i n ) : (3.16)

PAGE 40

26 W e wish to propagate from initial conditions W 0 W ( t 0 ), 0 ( t 0 ). If W is indep enden t of time, w e nd a n umerical solution that is exact up to mac hine precision. If W dep ends on time, but is slo wly v arying with resp ect to the timescale of w e can propagate to a high degree of accuracy b y linearizing Eq. 3.15 in time and incremen ting in small timesteps t = t 1 t 0 3.5.1 W Indep enden t of Time When W is indep enden t of time, W ( t ) = W 0 w e can formally solv e Eq. 3.15 ( t ) = U 0 ( t; t 0 ) 0 U y0 ( t; t 0 ) ; (3.17) where U ( t; t 0 ) exp [ i W 0 ( t t 0 )] : (3.18) If w e diagonalize W 0 W 0 = TT 1 ; (3.19) w e can rewrite Eq. 3.17 as, ( t ) = T [ T 1 U 0 ( t; t 0 ) T ] T 1 0 ( T y ) 1 [ T y U y0 ( t; t 0 )( T y ) 1 ] T y : (3.20) The exp onen tial matrices can b e formed analytically since T 1 U 0 ( t; t 0 ) T = exp [ i ( t t 0 )] ; (3.21) and ( t ) can b e computed in the time complexit y of the diagonalization of W 0 3.5.2 W Dep enden t on Time When W dep ends on time, w e separate in to a reference 0 and correction Q term, ( t ) = 0 ( t ) + Q ( t ) : (3.22)

PAGE 41

27 The reference densit y is propagated b y the time indep enden t W 0 of Section 3.5.1 i d 0 ( t ) dt = W 0 0 ( t ) 0 ( t ) W y 0 : (3.23) The ev olution of Q ( t ) is formed b y inserting Eq. 3.22 in to Eq. 3.15 i d dt ( 0 + Q ) = ( W 0 + W )( 0 + Q ) ( 0 + Q )( W 0 + W ) y ; (3.24) where W = W W 0 : (3.25) Using Eq. 3.23 w e obtain i d Q dt = W 0 + W 0 Q + W Q 0 W y QW y 0 Q W y : (3.26) T ransforming to the lo cal in teraction picture, where A U 0 A L U y0 ; (3.27) w e get i U 0 d Q L dt U y0 = U 0 W L U y0 U 0 0L U y0 U 0 0L U y0 U 0 W y L U y0 + U 0 W L U y0 U 0 Q L U y0 U 0 Q L U y0 U 0 W y L U 0 : (3.28) W e can simplify Eq. 3.28 b y m ultiplying on the left b y U 1 0 and on the righ t b y ( U y0 ) 1 to obtain i d Q L dt = D L + [ W L U y0 U 0 Q L Q L U y0 U 0 W y L ] ; (3.29) where D L ( t ) W L U y0 U 0 0L 0L U y0 U 0 W y L : (3.30)

PAGE 42

28 F ormally solving for Q L Q L ( t ) = Q DL ( t ) + 1 i t Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) Q L ( t 0 ) Q L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] ; (3.31) where Q DL ( t ) 1 i t Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) 0L ( t 0 ) 0L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] : (3.32) Solving b y iteration, Eq. 3.31 b ecomes Q L ( t ) = Q DL ( t ) + 1 i t Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) Q DL ( t 0 ) Q DL ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] + : (3.33) Neglecting the second term and higher for small timesteps t = t 1 t 0 Q L ( t 1 ) 1 i t 1 Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) 0L ( t 0 ) 0L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] : (3.34) Con v erting bac k to the original represen tation, A L = U 1 0 A ( U y0 ) 1 ; (3.35) w e ha v e U 1 0 ( t 1 ; t 0 ) Q ( t 1 ) U y0 ( t 1 ; t 0 ) 1 = 1 i t 1 Z t 0 dt 0 [ U 1 0 ( t 0 ; t 0 ) W ( t 0 ) 0 ( t 0 ) U y0 ( t 0 ; t 0 ) 1 U 1 0 ( t 0 ; t 0 ) 0 ( t 0 ) W y L ( t 0 ) U y0 ( t 0 ; t 0 ) 1 ] : (3.36)

PAGE 43

29 Multiplying Eq. 3.36 on the left b y U 0 ( t 1 ; t 0 ) and on the righ t b y U y0 ( t 1 ; t 0 ), and noting U 0 ( t 1 ; t 0 ) U 1 0 ( t 0 ; t 0 ) = exp [ i W 0 ( t 1 t 0 )] exp [ i W 0 ( t 0 t 0 )] = exp [ i W 0 ( t 1 t 0 )] = U 0 ( t 1 ; t 0 ) ; (3.37) U y0 ( t 0 ; t 0 ) 1 U y0 ( t 1 ; t 0 ) = exp [ i W y 0 ( t 0 t 0 )] exp [ i W y 0 ( t 1 t 0 )] = exp [ i W y 0 ( t 1 t 0 )] = U y0 ( t 1 ; t 0 ) ; (3.38) w e get an appro ximate correction term, Q ( t 1 ) = 1 i t 1 Z t 0 dt 0 U 0 ( t 1 ; t 0 ) D ( t 0 ) U y0 ( t 1 ; t 0 ) ; (3.39) where D ( t 0 ) = W ( t 0 ) 0 ( t 0 ) 0 ( t 0 ) W y ( t 0 ) : (3.40) W e can compute Q ( t 1 ) b y quadrature if w e assume D ( t ) = D ( t 1 = 2 ) ; t 0 t t 1 ; (3.41) where t 1 = 2 = t 0 + t= 2. W e then ha v e Q ( t 1 ) 1 i t 1 Z t 0 dt 0 U 0 ( t 1 ; t 0 ) D ( t 1 = 2 ) U y0 ( t 1 ; t 0 ) ; (3.42)

PAGE 44

30 whic h w e can rewrite as Q ( t 1 ) = 1 i t 1 Z t 0 dt 0 T [ T 1 U 0 ( t 1 ; t 0 ) T ] D T [ T y U y0 ( T y ) 1 ] T y = 1 i T 24 t 1 Z t 0 dt 0 exp [ i ( t 1 t 0 )] D T exp [ i y ( t 1 t 0 )] 35 T y ; (3.43) where D T T 1 D ( t 1 = 2 )( T y ) 1 : (3.44) Examining the elemen ts of Q Q j k ( t 1 ) = X l m 1 i T j l 24 t 1 Z t 0 dt 0 exp [ i l ( t 1 t 0 )] D T l m exp [ i ym ( t 1 t 0 )] 35 T y mk = X l m 1 i T j l 24 t 1 Z t 0 dt 0 exp [ i ( l ym )( t 1 t 0 )] D T l m 35 T y mk = X l m 1 i T j l 1 exp [ i ( l ym ) t ] i ( l ym ) D T l m T y mk ; (3.45) where w e ha v e used the notation i ii Rev erting bac k to matrix form, w e ha v e Q ( t 1 ) = TXT y ; (3.46) where X l m = exp [ i ( l ym ) t ] 1 l ym D T l m ; D T = T 1 D ( t 1 = 2 )( T y ) 1 : (3.47) The full densit y matrix at t 1 is then simply ( t 1 ) = 0 ( t 1 ) + Q ( t 1 ) : (3.48)

PAGE 45

31 3.5.3 V elo cit y V erlet for the Classical Ev olution In order to complete the relax-and-driv e algorithm, w e need to accoun t explicit y for the propagation of the classical v ariables. W e do this b y assuming that during eac h timestep, R and P are adv anced b y the reference densit y 0 and that the correction term con tributes negligibly to the classical propagation. The precise nature of the classical propagation is indep enden t of the relax-and-driv e algorithm, although to k eep accuracy to O ( t 2 ) it is necessary to in tegrate using an algorithm lik e v elo cit y V erlet or Runge-Kutta. W e c ho ose to use the v elo cit y V erlet metho d, whic h is accurate to O ( t 2 ) and is self-starting. W e pro ceed b y adv ancing the classical p ositions, R ( t + t ) = R ( t ) + P ( t ) M t + 1 2 M dP ( t ) dt t 2 : (3.49) The last term of Eq. 3.49 is the acceleration term, whic h comes from the eectiv e p oten tial. Ha ving adv anced the p ositions, w e calculate the acceleration at the new lo cation, and adv ance the momen ta, P ( t + t ) = P ( t ) + 1 2 M dP ( t ) dt + dP ( t + t ) dt t: (3.50) In the con text of the relax-and-driv e pro cedure, the classical co ordinates are adv anced in t w o steps of t= 2, in order to compute the correction term Q V ariable timestep. As the propagation pro ceeds, the initial timestep ma y no longer b e appropriate. This often o ccurs when the system en ters a region where the magnitude of p oten tial in teractions c hanges so that the reference densit y ructuates at a dieren t rate. An example is the collision of t w o atoms, where large timesteps can b e tak en at large distances, but small timesteps are required at close range. W e can monitor this ructuation b y observing the correction term Q As Q b ecomes to o small (large), w e need to increase (decrease) the timestep to main tain eciency

PAGE 46

32 (accuracy). T o this end, w e dene a correction measuremen t, max i;j Q ij 0ij 2 : (3.51) A t the end of eac h timestep, w e ev aluate If is less than some threshold, sa y l w e discard the step and use a new 3 If is greater than some threshold, sa y u w e discard the step and use a new = 2. By either m ultiplying b y 3 or dividing b y 2, w e a v oid an y oscillation (and th us innite lo op) b et w een a pair of timesteps.3.5.4 Algorithm Details The algorithm can b e divided in to an outer piece (sa y Main ), whic h calls an inner piece (sa y Propagate ). They are describ ed in p oin t form as follo ws: Main : T o propagate from t A to t B giv en t = t 0 and f l ; u g 1. t = min f t; t B t A g 2. Propagate with t 0 = t A 3. Compute = max i;j j Q ij = ij j 2 4. Is < l and t A + t < t B ? YES: Reset v ariables (matrices and functions return to their v alues at t 0 ), set t 3 t and return to step 1 5. Is > u ? YES: Reset v ariables to t 0 set t t= 2 and return to step 1 6. Is t A + t < t B ? YES: Set t A t A + t and return to step 1 Propagate : T o propagate from t 0 to t 1 = t 0 + t 1. Initialize v ariables: W 0 = W ( t 0 ), 0 = 0 ( t 0 ) = ( t 0 ), R ( t 0 ), P ( t 0 ), t 2. Diagonalize W 0 = T T 1 3. Adv ance classical v ariables b y half timestep, f R ( t 1 = 2 ) ; P ( t 1 = 2 ) g where t 1 = 2 = t + t= 2, using initial reference densit y 0 ( t 0 ) and W 0

PAGE 47

33 4. Compute W ( t 1 = 2 ) using R ( t 1 = 2 ) and P ( t 1 = 2 ). 5. Compute 0 ( t 1 = 2 ) = T exp[ i ( t= 2)] T 1 0 ( T y ) 1 exp [ i y ( t= 2)] T y 6. Compute Q ( t 1 ) b y assuming D ( t ) = D ( t 1 = 2 ), t 0 t t 1 (a) Calculate W ( t 1 = 2 ) = W ( t 1 = 2 ) W 0 (b) Calculate D ( t 1 = 2 ) = W ( t 1 = 2 ) 0 ( t 1 = 2 ) 0 ( t 1 = 2 ) W y ( t 1 = 2 ). (c) Compute D T = T 1 D ( t 1 = 2 )( T y ) 1 (d) Compute X where X l m = exp [ i ( l ym ) t ] 1 l ym D T l m : (e) Compute Q ( t 1 ) = TXT y 7. Adv ance classical v ariables to full timestep, f R ( t 1 ) ; P ( t 1 ) g using 0 ( t 1 = 2 ) and W ( t 1 = 2 ). 8. Compute W ( t 1 ) using f R ( t 1 ) ; P ( t 1 ) g 9. Compute 0 ( t 1 ) = T exp ( i t ) T 1 0 ( T y ) 1 exp ( i y t ) T y 10. Compute ( t 1 ) = 0 ( t 1 ) + Q ( t 1 ). 3.6 Computing Observ ables 3.6.1 Op erators in an Orthonormal Basis Up to this p oin t, w e ha v e considered the general basis j i without an y condition of orthogonalit y or normalit y W e can transform this basis to an orthonormal one, j 0 i through a L owdin transformation, j 0 i = j i S 1 = 2 : (3.52) Since quan tal traces are naturally form ulated in orthogonal bases, it is useful to express the relationship b et w een matrix represen tations of op erators in b oth bases.

PAGE 48

34 F or the densit y op erator, ^ = j 0 i 0 h 0 j = j i S 1 = 2 0 S 1 = 2 h j = j i h j : (3.53) Th us w e equate = S 1 = 2 0 S 1 = 2 : (3.54) Since the densit y matrix has b een dened dieren tly than the matrix represen tations for general op erators, w e also consider the represen tations for a general op erator ^ A ^ A = j 0 i A 0 h 0 j = j i S 1 = 2 A 0 S 1 = 2 h j = j i S 1 AS 1 h j : (3.55) Th us A = S 1 = 2 A 0 S 1 = 2 : (3.56) 3.6.2 P opulation Analysis The p opulation is naturally dened in an orthonormal basis, suc h that the p opulation of state i is the i th diagonal elemen t of the orthonormal represen tation of the densit y matrix, i = 0ii = [ S 1 = 2 S 1 = 2 ] ii : (3.57)

PAGE 49

35 In the case of the PWT represen tation w e m ust in tegrate o v er n uclear phase space as w ell, so that Eq. 3.57 b ecomes i = Z dR dP [ S 1 = 2 ( R ; P ) ( R ; P ) S 1 = 2 ( R ; P )] ii : (3.58) 3.6.3 Exp ectation V alues The exp ectation v alue of a general op erator ^ A W is found b y taking b oth the quan tal and classical trace of the pro duct of the op erator with the PWTDM, h ^ A W i = T r[ ^ A W ^ W ] = T r cl T r qu [ ^ A W ^ W ] = Z dR dP T r qu [ ^ A W ^ W ] : (3.59) The quan tal trace is naturally computed in the orthonormal basis, but as w e no w sho w, the nonorthonormal basis represen tation can also b e used: T r qu [ ^ A W ^ W ] = T r qu [ A 0W 0W ] = T r qu [ S 1 = 2 A W S 1 = 2 S 1 = 2 W S 1 = 2 ] = T r qu [ A W W ] : (3.60) 3.6.4 Hamiltonian Eigenstates and Eigen v alues Using the orthonormal represen tation of the Hamiltonian, H 0 = S 1 = 2 HS 1 = 2 ; (3.61) w e can compute its eigen v alues b y diagonalizing the matrix, L 1 H 0 L = H 0D ; (3.62)

PAGE 50

36 where H 0D is the diagonalized Hamiltonian and con tains the energy eigen v alues along its diagonal. Since the Hamiltonian is Hermitian, its eigen v alues are real, so that L y H 0 ( L 1 ) y = H 0D ; (3.63) and the diagonalizing matrix L is seen to b e unitary L y = L 1 : (3.64) The columns of a unitary matrix are orthogonal, and since the columns of L are the eigenstates of the Hamiltonian, w e see that the eigenstates pro duced b y diagonalizing the orthonormal represen tation of the Hamiltonian are also orthonormal. This is useful in the case of degenerate eigenstates, as they are automatically orthogonal and no additional pro cedures are needed to ensure orthogonalit y in the degenerate subspace. 3.7 Programming Details When designing a computational pac k age, it is desirable to ensure the co de remains orthogonal and extensible throughout the design and implemen tation. Mo dern programming languages use ob ject-orien ted concepts to ac hiev e these goals, 103 but unfortunately a great deal of computational w ork is built on older pro cedural languages and cannot b e readily incorp orated in to an ob ject-orien ted sc heme without substan tial eort. Indeed, as m uc h of this legacy co de has b een dev elop ed o v er man y y ears and has undergone extensiv e testing, it is en ticing to adhere to the older languages in whic h they w ere written and incorp orate them directly In the dev elopmen t of co de for this researc h, a compromise w as found b y using man y adv anced features found in F ortran 90, but main taining a co ding st yle whic h p ermit straigh tforw ard in tegration of legacy F ortran 77 co de. Details of the pac k age ( cauldron ) are found in App endix A and in the remainder of this c hapter w e briery o v erview the programming principles used in the dev elopmen t of the pac k age.

PAGE 51

37 3.7.1 Orthogonalit y of Co de Dev elopmen t By orthogonalit y of co de dev elopmen t, w e mean that dieren t asp ects of the co de can b e dev elop ed indep enden tly Th us one ma y decide to build a completely differen t algorithm than relax-and-driv e, for example, to propagate the mixed quan tumclassical system, but b e able to do so without c hanging asp ects of the co de whic h dene the system, compute its prop erties, read conguration les, generate output les, and so forth. This helps ensure that once a v ersion of the co de w orks w ell, c hanges to its comp onen ts will b e less lik ely to in tro duce errors. This asp ect of program dev elopmen t is crucial for soft w are designed in a team en vironmen t, but also v ery useful for the solitary designer when the problems and their solutions ma y rapidly c hange. Orthogonalit y has b een k ept within cauldron through judicious use of v ariable and subroutine naming con v en tions, and b y building a solid hierarc h y of directories and subroutines from the b eginning. 3.7.2 Extensibilit y In scien tic w ork, the systems studied and the solutions used are constan tly c hanging, as progress is made in understanding the solutions, and new problems arise. One w a y to help main tain rexibilit y whic h has b een used throughout cauldron is to ensure that systems are represen ted as generically and as dynamically as p ossible. Generic co de attempts to represen t the fundamen tal asp ects of all molecular systems, for example, b y the same set of v ariables and arra ys. When new comp onen ts (e.g., new kinds or n um b ers of n uclei) are added to the system, the same v ariables are used, and it is only the in terpretation of the results that diers from system to system. By dynamic represen tations, w e refer to the dening of v ariable size at run time rather than xing the size at compilation. The ma jor b enet in this comes from b eing able to implemen t mo dels of v arying sizes without recompiling the co de and creating a new executable for eac h system studied. Systems can then

PAGE 52

38 b e dened completely within input les, for example, preserving the p olished executable without mo dication. F ortran 90 encourages dynamic memory allo cation, whic h has b een used to great adv an tage in cauldron to pro vide v ery extensible co de.

PAGE 53

CHAPTER 4 ONE-DIMENSIONAL TW O-ST A TE MODELS 4.1 In tro duction While our ultimate goal is to study realistic three-dimensional mo dels of alk ali atoms em b edded in rare gas clusters, the complexit y of these systems places full quan tal solutions out of computational reac h. On the other hand, simple mo dels can sometimes capture elemen ts of larger and more realistic systems, and pro vide a rigorous basis for v alidating appro ximate n umerical metho ds. In this c hapter, w e study the dynamics of three simple mo dels in v olving t w o electronic states and one n uclear co ordinate. The rst represen ts photoinduced desorption of an alk ali atom from a metal surface, where near-resonan t electron transfer is imp ortan t. The second mo dels the collision b et w een t w o n uclei in a framew ork in v olving t w o a v oided crossings. The third mo dels the photoinduced disso ciation of the NaI complex, where oscillatory motion b et w een neutral and ionic states is observ ed. Because of the limited size of these mo dels, in eac h case w e are able to propagate a grid solution to the TDSE, and th us compare our EP-QCLE approac h to the dynamics of the full quan tal system. F or all three mo dels, w e will see that the mixed quan tum-classical metho ds pro vide qualitativ ely and often quan titativ ely similar results to the full quan tal ev olution. 4.2 Eectiv e-P oten tial QCLE in the Diabatic Represen tation In Chapter 3 w e deriv ed the EP-QCLE for an arbitrary basis. Here, w e consider the sp ecic case where the system is describ ed in an orthonormal diab atic basis. There are man y v arieties of diabatic bases, 104 105 but here w e refer to the strictly 39

PAGE 54

40 diab atic represen tation f di g where the momen tum coupling v anishes, 106 h d j @ =@ R j d i = 0 : (4.1) W e also assume that the basis do es not explicitly dep end on P or t so that n = 0. Since the basis is orthonormal, the o v erlap is unit y and Eq. 3.7 reduces to the simple form, d dt = i [ H qu ; ] : (4.2) In the diabatic represen tation, the eectiv e force is also simplied, since the op erators in the quan tal trace can b e replaced directly with their matrix represen tations, F H F = T r qu [ @ V =@ R ] T r qu [ ] : (4.3) F or our test mo dels, the partial deriv ativ es of the p oten tial can b e calculated analytically at eac h grid p oin t in phase space. Since the PWTDM is propagated along these grid p oin ts, the pro duct in the n umerator of Eq. 4.3 is computed through matrix m ultiplication, while the quan tal trace is calculated b y summing o v er the diagonal comp onen ts of this matrix pro duct. The quan tal trace in the denominator, on the other hand, is simply the sum o v er the diagonal comp onen ts of the PWTDM. Had w e used a dieren t basis, w e w ould not ha v e b een able to simplify the EP-QCLE in this w a y Ho w ev er, the real adv an tage to the diabatic represen tation is that for v ery small systems, it lends itself to a fully quan tal n umerical solution through the propagation of the TDSE on a grid. One sc heme, the split op erator fast F ourier transform (SO-FFT) metho d, splits the Hamiltonian in to its kinetic and p oten tial comp onen ts, and uses the fast F ourier transform to compute the ev olution due to the kinetic terms. 107 108 While this metho d is v ery accurate, it is also in tractable for systems with more than a few degrees of freedom. Ho w ev er, b ecause

PAGE 55

41 our mo dels are simple, the SO-FFT pro cedure pro vides an excellen t test of the accuracy of the EP-QCLE. A complete description of the SO-FFT is giv en in App endix B and the co de implemen ting the SO-FFT ( qualdron ) is describ ed in App endix C 4.3 Near-Resonan t Electron T ransfer Bet w een an Alk ali A tom and Metal Surface 4.3.1 Mo del Details In the rst of our test systems, w e consider a mo del describing the near-resonan t electron transfer b et w een an alk ali atom (Ak) and a metal surface (M) at thermal energies. The mo del consists of t w o diabatic surfaces corresp onding to a ground state of neutral comp onen ts Ak + M (state 1) and an excited state for ionic comp onen ts Ak + + M (state 2), whic h cross at short distance. The surfaces and in teraction term are giv en b y 85 84 H 11 ( R ) = U 0 f exp [ 2 ( R R 0 )] + 2 exp[ ( R R 0 )] g = 2 ; (4.4) H 22 ( R ) = U 0 f exp [ 2 ( R R 0 )] 2 exp[ ( R R 0 )] g + = 2 ; (4.5) H 12 ( R ) = exp[ 2 ( R R x ) 2 ] : (4.6) Here, R is the distance b et w een the metal surface and the n uclear cen ter of the Ak atom, and is the quasiclassical v ariable o v er whic h w e tak e the PWT. The ionic curv e, H 22 ( R ), is a Morse p oten tial with a binding energy U 0 The repulsiv e neutral curv e, H 11 ( R ), is oset relativ e to the ionic curv e to giv e an excitation p oten tial The strength of the coupling term, H 12 ( R ), is c haracterized b y and p eaks at the crossing R = R x b et w een H 11 and H 22 The initial state is formed b y appro ximating the ionic surface as a harmonic p oten tial around its minim um R = R 0 and nding the lo w est b ound vibrational state within that (harmonic) w ell, ( R ) = 1 2 1 = 4 exp ( R R 0 ) 2 2 2 exp [ iP 0 ( R R 0 )] : (4.7)

PAGE 56

42 T able 4{1. P arameters used in the Na-surface and Li-surface mo dels. Hamiltonian I Hamiltonian I I P arameter v alue (au) v alue (au) U 0 0.025 0. 184 0.4 0. 4 0.005 0. 147 R 0 5.0 5. 0 R x 12.5 9. 0 P 0 0.0 0. 0 0.233153 0. 1908559 0.15 0. 15 M 42,300 12,800 The PWTDOp is the PWT of Eq. 4.7 o v er R giving a Gaussian densit y in (P ,R), ^ ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # : (4.8) A t t = 0, the electronic state is promoted b y a sudden optical excitation to the repulsiv e neutral p oten tial, so that the PWTDM b ecomes 11 ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # ; (4.9) with 12 = 21 = 22 = 0. The sim ulation follo ws the sp on taneous deca y of this state. The parameters used in the calculation are sho wn in T able 4{1 where w e consider t w o mo del Hamiltonians: (I) Na-surface and (I I) Li-surface. The diabatic p oten tials for Hamiltonians I and I I are sho wn in Figures 4{1 and 4{2 4.3.2 Prop erties of In terest P opulations. W e can follo w the p opulations o v er time b y taking the full classical trace o v er either diagonal elemen t of the PWTDM, i = Z dR dP ii ( R ; P ) : (4.10)

PAGE 57

43 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0 5 10 15 20 25 30 Energy (au)R (au) H 11 H 22 H 12 Figure 4{1. P oten tial curv es for Hamiltonian I: Na inciden t up on a metal surface.

PAGE 58

44 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0 5 10 15 20 Energy (au)R (au) H 11 H 22 H 12 Figure 4{2. P oten tial curv es for Hamiltonian I I: Li inciden t up on a metal surface.

PAGE 59

45 Since the system b egins in the repulsiv e (neutral) state, one exp ects a certain p ercen tage of the p opulation to fall to the attractiv e (ionic) state as the system passes through the region of nonnegligible p oten tial in teraction. Coherences. The coherence b et w een the neutral and ionic state is describ ed b y the real and imaginary comp onen ts of the o-diagonal terms of the PWTDM, for example ij = Z dR dP ij ( R ; P ) : (4.11) Coherence is a purely quan tum phenomenon, and one measure of the qualit y of a mixed quan tum-classical metho d is the degree to whic h it main tains coherence. P osition exp ectation v alues. One observ able w e can study is the exp ectation of p osition, h R i = T r [( R ; P ) R ] = Z dR dP T r qu [ ( R ; P )] R : (4.12) W e can also measure the disp ersion, R = [ h ( R h R i ) 2 i ] 1 = 2 = Z dR dP T r qu [ ( R ; P )]( R h R i ) 2 1 = 2 : (4.13) Momen tum exp ectation v alues. Similarly w e can compare the exp ectation and disp ersion of momen ta, h P i = T r[( R ; P ) P ] = Z dR dP T r qu [ ( R ; P )] P ; (4.14) P = [ h ( P h P i ) 2 i ] 1 = 2 = Z dR dP T r qu [ ( R ; P )]( P h P i ) 2 1 = 2 : (4.15)

PAGE 60

46 Probabilit y densit y W e can compute the probabilit y densit y ( R ) from the PWTDM b y taking the trace o v er quan tum v ariables and momen ta, ( R ) = 1 Z 1 dP T r qu [ ( R ; P )] : (4.16) In practice, since the grid in phase space quic kly deforms as the system ev olv es, w e m ust nd a w a y of appro ximating this in tegral. One pro cedure is to determine the supp ort of the PWTDM in quasiclassical phase space, [ R min ; R max ] [ P min ; P max ]. W e then divide this space in to N R N P equisized bins f b ij g suc h that bin b ij spans the rectangular region, [ R min + ( i 1) R max R min N R ; R min + i R max R min N R ] [ P min + ( j 1) P max P min N P ; P min + j P max P min N P ] : (4.17) W e then assign a v alue to eac h bin, ij whic h is the w eigh ted sum of all N ij tra jectory p oin ts whic h fall within that bin, ij = N ij P k =1 k ( R k ; P k ) N ij : (4.18) W e can determine the matrix probabilit y densit y from ij b y summing o v er all bins con taining a giv en p osition R ( R ) = X j ij ; 8 R 2 b ij : (4.19) Finally w e compute the probabilit y densit y b y taking the quan tal trace o v er the matrix probabilit y densit y ( R ) = T r qu [ ( R )] : (4.20)

PAGE 61

47 This probabilit y densit y can b e compared to the densit y function obtained from the SO-FFT sim ulation, ( R ) = j 1 ( R ) j 2 + j 2 ( R ) j 2 : (4.21) W a v efunction and PWTDM. Of course, the ev olution of the quan tum w a v efunction can b e con trasted directly with the ev olution of the PWTDM. Ho wev er, w e can also observ e the distortion in the phase space grid used b y the EP-QCLE metho d. While the grid is initially uniform, it c hanges shap e in an in teresting w a y b ecause of the action of the eectiv e p oten tial. 4.3.3 Results Figures 4{3 and 4{4 sho w the initial w a v efunction and its PWT, resp ectiv ely Note that the PWTDM formed from a Gaussian w a v epac k et is a Gaussian function itself, alb eit in t w o-dimensional phase space. Figure 4{5 sho ws the w a v efunction at t = 14000 au, ha ving b een propagated through the SO-FFT metho d. The PWTDM ev olv es in phase space through the EP-QCLE, and in Figure 4{6 w e sho w the grid in phase space at the nal time. While substan tially distorted from its initial uniform distribution, w e notice that the p oin ts are globally p ositioned along a straigh t line in phase space. This rerects the asymptotic state, where eac h p oin t is sub ject to a v anishing Hellmann-F eynman force, and th us propagates at constan t v elo cit y The observ ables are presen ted in Figures 4{7 to 4{13 There are a n um b er of in teresting things w e can glean from these plots. F rom Figures 4{7 and 4{8 w e see that as the atom mo v es a w a y from the metal surface, m uc h of its p opulation shifts from the neutral to ionic state. In the case of Na, appro ximately 2/3 of the neutral p opulation shifts, while for Li the transfer is total. This ma y rerect the stronger in teraction coupling in v olv ed with Li. Figures 4{9 and 4{10 describ e the coherence b et w een the states. F or the Na-surface system, the coherence remains large for long times, while for the Li-surface

PAGE 62

48 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 3 3.5 4 4.5 5 5.5 6 6.5 7 Re( Y1)R (au) Figure 4{3. ( R ) at t = 0 au, for the Na-surface mo del. This w a v efunction is ev olv ed through the SO-FFT algorithm.

PAGE 63

49 -15 -10 -5 0 5 10 15 P (au) 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 R (au) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 G 11 Figure 4{4. 11 at t = 0 au, for the Na-surface mo del. This PWTDM is ev olv ed through the EP-QCLE metho d.

PAGE 64

50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 20 25 30 35 40 Real( Y1)R (au) Figure 4{5. ( R ) at t = 14000 au, for the Na-surface mo del.

PAGE 65

51 22 24 26 28 30 32 34 36 60 65 70 75 80 85 90 95 100 R (au)P (au) Figure 4{6. Phase space grid p oin ts at t = 14000 au, for the Na-surface mo del.

PAGE 66

52 0 0.2 0.4 0.6 0.8 1 0 2000 4000 6000 8000 10000 12000 14000 Populations h1 and h2time (au) EP-QCLE: h 1 SO-FFT: h 1 EP-QCLE: h 2 SO-FFT: h 2 Figure 4{7. Na p opulations 1 and 2 vs. time.

PAGE 67

53 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 Populations h1 and h2time (au) EP-QCLE: h 1 SO-FFT: h 1 EP-QCLE: h 2 SO-FFT: h 2 Figure 4{8. Li p opulations 1 and 2 vs. time.

PAGE 68

54 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 2000 4000 6000 8000 10000 12000 14000 Coherence: Re( h12)time (au) EP-QCLE SO-FFT Figure 4{9. Coherence describ ed b y Re( 12 ) vs. time, for the Na-surface system.

PAGE 69

55 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 2000 Coherence: Re( h12)time (au) EP-QCLE SO-FFT Figure 4{10. Coherence describ ed b y Re( 12 ) vs. time, for the Li-surface system.

PAGE 70

56 0 5 10 15 20 25 30 35 0 2000 4000 6000 8000 10000 12000 14000 Position and dispersion (au)time (au) EP-QCLE: SO-FFT: EP-QCLE: s R SO-FFT: s R Figure 4{11. Exp ectation of p osition and disp ersion for the Na-surface system.

PAGE 71

57 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 14000 Momentum and dispersion (au)time (au) EP-QCLE:

SO-FFT:

EP-QCLE: s P SO-FFT: s P Figure 4{12. Exp ectation of momen tum and disp ersion for the Na-surface system.

PAGE 72

58 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 5 10 15 20 25 30 35 40 r (R)R (au) EP-QCLE SO-FFT Figure 4{13. Densit y function ( R ) for the Na-surface system.

PAGE 73

59 system, the coherence rapidly diminishes once the in teraction p oten tial is crossed. This can b e explained b y the size of the energy gap at asymptotic distances, as larger ionization energies lead to more rapidly v anishing coherences. Figure 4{11 sho ws the exp ectation of the p osition of Na steadily increases from its initial a v erage, while the disp ersion in p osition initially decreases and then increases again. On the other hand, Figure 4{12 sho ws a mark ed dierence in the b eha vior of the exp ectation of the momen tum, where it b egins at h P i = 0 au, rapidly increases and then b ecomes stationary around h P i = 78 au. W e also nd that the momen tum disp ersion rst increases, and then decreases. Finally Figure 4{13 presen ts the densit y function at the end of the sim ulation. Because of the distortion of the phase space grid, the densit y function obtained from the EP-QCLE had to b e calculated using bins and through appro ximations within these bins, and th us is sub ject to some noise. Nev ertheless, w e nd excellen t agreemen t b et w een the EP-QCLE and SO-FFT results. F or all observ ables, the EP-QCLE results are quan titativ ely similar to the SOFFT results to visual resolution. Ha ving studied b oth Na and Li approac hing a metal surface, w e see that the EP-QCLE can b e exp ected to yield v ery accurate results for these kinds of systems, ev en when coherence is main tained o v er long p erio ds. 4.4 Binary Collision In v olving Tw o Av oided Crossings 4.4.1 Mo del Details F or the second system, w e lo ok at a t w o-state collision mo del where the diabatic surfaces in tersect t wice. Because of the dual crossing and the coupling in this region, quan tum in terference and eects suc h as tunnelling pla y a substan tial role in the dynamics of the quasiclassical v ariable. As suc h, this mo del is quite demanding for mixed quan tum-classical metho ds, where one can exp ect deviations at lo w er energies.

PAGE 74

60 T able 4{2. P arameters used in the dual a v oided crossing collision mo del. P arameter V alue (au) R 0 -8 P 0 [10, 30] 2.5176 M 2000 The Hamiltonian elemen ts are 79 H 11 ( R ) = 0 ; (4.22) H 22 ( R ) = 0 : 1 exp ( 0 : 28 R 2 ) + 0 : 05 ; (4.23) H 12 ( R ) = 0 : 015 exp( 0 : 06 R 2 ) : (4.24) In the ab o v e, R is the n uclear-n uclear separation. The initial state is a ground state Gaussian w a v epac k et whic h b egins in the asymptotic region R 0 = 8 au. Its PWT o v er R giv es the Gaussian densit y 11 ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # ; (4.25) with 12 = 21 = 22 = 0. A t t = 0, the w a v epac k et propagates to w ard the region of coupling, where it is partially transmitted and partially rerected, no w with p opulations in b oth the ground and excited state. The parameters used in the calculation are sho wn in T able 4{2 The diabatic p oten tials used in the mo del are sho wn in Figure 4{14 4.4.2 Prop erties of In terest As in the alk ali atom-surface mo del, w e can follo w the p opulations o v er time. The system b egins in the ground state, and as it passes through the collision region, w e exp ect some of the p opulation to transfer to the excited state. The amoun t of transfer dep ends on the collision energy W e can also follo w the coherence once the

PAGE 75

61 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 -10 -5 0 5 10 Energy (au)R (au) H 11 H 22 H 12 Figure 4{14. P oten tial curv es for the dual a v oided crossing collision.

PAGE 76

62 collision has ended and the system reac hes asymptotic v alues. W e also consider the transmission of the ground state as the w a v epac k et passes through the in teraction. T ransmission. As the w a v epac k et passes through the collision, part of it con tin ues forw ard while the remainder is rerected bac k. W e can study this b y computing the probabilit y of transmission of the ground state, T 1 = 1 Z 0 dR dP 11 ( R ; P ) : (4.26) Although not included, w e could also compute the probabilit y of rerection of the ground state, as w ell as the asymptotic p opulations of the excited state. 4.4.3 Results In Figure 4{15 w e see the total p opulation transfer at energy P 0 = 30 au. In this energy region, the EP-QCLE matc hes the quan tum results to within a couple p ercen t. F or b oth the EP-QCLE and SO-FFT sim ulations, the rerection of the w a v efunction (or PWTDM) w as negligible, so that w e need only compare the transmission. This negligible rerection w as found throughout the energies studied. In Figure 4{16 w e see the coherence as a function of time. The v ariations are large initially and diminish asymptotically to regular oscillations. The EP-QCLE captures this b eha vior qualitativ ely and to within 5% quan titativ ely W e sho w the deformation of the phase space grid in Figure 4{17 W e see that it is primarily the inner part of the grid that undergo es deformation, while the enclosing p oin ts main tain a structured order. This is lik ely due to only a small range of v alues in phase space whic h are seriously aected b y the p oten tials. Outside the in teraction region, all Hellmann-F eynman forces are zero, so one w ould exp ect deformations only for p oin ts whose time in the in teraction region w as signican t. In Figure 4{18 w e displa y the transmission probabilit y for a wide range of energies. As exp ected, the EP-QCLE p erforms b etter for higher energies, and while the double-w ell is qualitativ ely repro duced, the EP-QCLE fails to giv e quan titativ ely

PAGE 77

63 accurate results for energies lo w er than (log E = -2). This is lik ely due to the quantum tunnelling eects describ ed earlier, whic h are not exp ected to b e repro duced w ell b y the EP-QCLE. W e also compare transmission probabilities obtained through surface hopping metho ds b y T ully and co w ork ers. 51 These surface hopping calculations sho w deviations from the quan tal results that are similar to the EP-QCLE probabilities at lo w er energies; at higher energies, the EP-QCLE is sligh tly sup erior to the surface hopping sc heme. 4.5 Photoinduced Disso ciation of a Diatomic System 4.5.1 Mo del Details F or the third test system, w e explore a mo del of the NaI complex. As in the previous mo dels, it in v olv es t w o diabatic surfaces and an in teraction around the a v oided crossing. A substan tially dieren t feature, ho w ev er, is a long-range Coulom bic attraction in an ionic state. As w e shall see, this attractiv e p oten tial results in the the complex oscillating b et w een a neutral and ionic state as the so dium and io dine separate and come bac k together, partially disso ciating at eac h crossing in to an asymptotically neutral state. This oscillatory motion is a go o d test for the EP-QCLE at long times in cases where asymptotic states are not reac hed quic kly The Hamiltonian elemen ts are 86 H 11 ( R ) = A 1 exp [ 1 ( R R 0 )] ; (4.27) H 22 ( R ) = [ A 2 + ( B 2 =R ) 8 ] exp ( R = ) 1 =R ( + + ) = 2 R 4 C 2 =R 6 2 + =R 7 + E 0 ; (4.28) H 12 ( R ) = A 12 exp [ 12 ( R R x ) 2 ] : (4.29) T o form the initial state, w e T a ylor expand (to rst order) the ionic p oten tial H 22 ab out its minim um R = R 0 and nd the lo w est energy state of this harmonic w ell. A t t = 0, the w a v epac k et undergo es a sudden optical promotion to the neutral curv e,

PAGE 78

64 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 1200 1400 Populations h1 and h2time (au) EP-QCLE: h 1 SO-FFT: h 1 EP-QCLE: h 2 SO-FFT: h 2 Figure 4{15. P opulations 1 and 2 vs. time for the dual crossing collision mo del.

PAGE 79

65 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0 200 400 600 800 1000 1200 1400 Coherence: Re( h12)time (au) EP-QCLE SO-FFT Figure 4{16. Coherence describ ed b y R e ( 12 ) vs. time, for the dual crossing collision mo del.

PAGE 80

66 0 5 10 15 20 25 30 35 40 45 20 22 24 26 28 30 32 34 36 R (au)P (au) Figure 4{17. Grid deformation at t = 1400 au, for the dual crossing collision mo del.

PAGE 81

67 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3.5 -3 -2.5 -2 -1.5 -1 h1 Tlog e (E) (au) EP-QCLE SO-FFT Tully et al. Figure 4{18. Probabilit y of transmission in the ground state, for the dual crossing collision mo del.

PAGE 82

68 T able 4{3. P arameters used in the NaI complex mo del. P arameter V alue (au) R 0 5.047 P 0 0.0 0.12462 A 1 0.02988 A 2 101. 43 A 12 0.00202 B 2 3.000 C 2 18. 950 + 2.756 12. 179 0.660 E 0 0.07626 1 2.158 12 0.194 R x 13. 24 M 35,482 so that the PWTDM b ecomes, 11 ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # ; (4.30) with 12 = 21 = 22 = 0. The sim ulation follo ws the resulting motion of this state. The mo del parameters are sho wn in T able 4{3 The diabatic p oten tials are displa y ed in Figure 4{19 4.5.2 Prop erties of In terest In addition to observing the coherence, exp ectation v alue of the p osition and its disp ersion, and the phase space grid, w e also consider b ound and free neutral and ionic p opulations as the NaI complex oscillates from its primarily ionic to primarily co v alen t state. Bound and free p opulations. It is in teresting to follo w the p opulations of the ionic and neutral states as the system ev olv es, giving insigh t in to the nature of the disso ciation in to an asymptotically free neutral system. F or this, w e dene three

PAGE 83

69 -0.1 -0.05 0 0.05 0 5 10 15 20 25 30 Energy (au)R (au) H 11 H 22 H 12 Figure 4{19. P oten tial curv es for the NaI complex.

PAGE 84

70 p opulations: the ionic, the b ound neutral and the free neutral. The ionic p opulation is the probabilit y of nding the system in the ionic state at an y in tern uclear separation, 2 = 1 Z 1 dR dP 22 ( R ; P ) : (4.31) W e dene the b ound neutral p opulation is the probabilit y of nding the NaI complex in the neutral state at n uclear separation up to the crossing p oin t R = R x b 1 = R x Z 0 dR dP 11 ( R ; P ) ; (4.32) while the free ionic p opulation is dened as the probabilit y of the neutral state from the crossing p oin t b ey ond, f 1 = 1 Z R x dR dP 11 ( R ; P ) : (4.33) This division b et w een b ound and free is motiv ated b y the observ ation that the ma jorit y of the transfer b et w een ionic and neutral p oten tials o ccurs at the a v oided crossing, and that an y part of the w a v epac k et whic h ends up in the neutral state but propagating to w ard innite separation has a negligible probabilit y of shifting to the ionic state m uc h b ey ond the a v oided crossing. 4.5.3 Results The ionic and co v alen t p opulations are displa y ed in Figure 4{20 W e see the oscillations in the p opulations b et w een ionic and co v alen t, rep eating appro ximately ev ery 40 000 au. This pattern can b e compared to the exp ectation v alues of p osition see in Figure 4{21 The p osition oscillates with the same frequency as the c hange in p opulation, sho wing that eac h time the w a v epac k et heads across the a v oided crossing from its b ound co v alen t state, it con v erts almost completely in to the ionic state, with a small amoun t escaping in to the free neutral state. Ov er time, the

PAGE 85

71 free neutral p opulation gradually increases at these crossings, and the NaI slo wly disso ciates. The EP-QCLE is quan titativ ely similar to the exact results for the rst half of the sim ulation, and main tains qualitativ e accuracy for the remainder. The disp ersion in Figure 4{21 sho ws an in teresting dierence b et w een the exact and the quan tum-classical algorithm. While the disp ersion con tin ues to rise in the SO-FFT sim ulation, it reac hes its rst p eak and then b egins to decline somewhat in the EP-QCLE sim ulation. This rerects the nature of the eectiv e p oten tial, where eac h p oin t is guided b y a com bination of excited and ground state forces. Because the ionic curv e do es not p ermit escap e, what w ould normally b e asymptotically free w a v epac k ets tend to b e pulled bac k to w ard the crossing b ecause of the attractiv e ionic p oten tial. Consequen tly the free neutral p opulation is alw a ys lo w er using the EP-QCLE equation than the SO-FFT, an observ ation supp orted b y Figure 4{20 Since the ma jorit y of the p opulation remains in the ionic or b ound neutral state, and these p opulations are w ell matc hed b et w een the exact and quan tum-classical sim ulations, it is not surprising that the exp ectation v alue of the p osition is quantitativ ely in the b eginning, and qualitativ ely for the remaining, similar for b oth metho ds. Ho w ev er, the disp ersion is m uc h more sensitiv e to the increased asymptotically free w a v epac k ets in the SO-FFT sim ulation, and for the reasons discussed, w e nd signican t div ergence b et w een the EP-QCLE and SO-FFT results. The coherence, sho wn in Figure 4{22 initially p eaks through the rst crossing, but through subsequen t crossings it is substan tially diminished. Ho w ev er, the EPQCLE sho ws quan titativ ely similar results to the SO-FFT calculations. The deformation of the phase space grid, plotted in Figure 4{23 has c haracteristics not seen in the other t w o mo dels. One line of p oin ts emits from the cen ter of the cluster, quic kly straigh tening and rerecting the negligible force on the p oin ts. These p oin ts corresp ond to the asymptotically free neutral comp onen ts of the PWTDM. The second group circles around, gaining v elo cit y and p osition, then

PAGE 86

72 turning. These ellipses are c haracteristic of the phase space of classical particles in a w ell, and indeed rerect the quasiclassical motion under the Hellmann-F eynman force of the PWTDM p oin ts as they follo w the ionic and b ound co v alen t curv es. 4.6 Comparison Using V ariable and Constan t Timesteps In this Section, w e ev aluate the usefulness of the v ariable timestep asp ect of the relax-and-driv e algorithm. T o do this, w e consider the Na-surface algorithm of Section 4.3 and sim ulate using v arying upp er and lo w er tolerances. The n um b er of steps tak en in eac h case is compared with the n um b er that w ould b e required of the same algorithm, but k eeping the timestep xed. The xed timestep w ould necessarily adv ance b y steps no greater than the smallest timestep used in the v ariable timestep algorithm, and it is based on this timestep that w e estimate the corresp onding steps required for the xed timestep approac h. The results are sho wn in Figure 4{24 W e see that as the tolerance decreases (and th us the accuracy increases), the fraction of steps sa v ed b y the in tro duction of the v ariable timestep increases sup erlinearly One concludes that while the v ariable timestep ma y not b e imp ortan t for lo w accuracy sim ulations, considerable computational sa vings can b e had for high accuracy propagation. 4.7 Conclusion By examining three simple t w o-state mo dels, w e w ere able to compare the EP-QCLE metho d with the exact SO-FFT quan tum mec hanical solution. F or all mo dels, w e found v ery go o d agreemen t b et w een the EP-QCLE and the SO-FFT results. The agreemen t w as at least qualitativ e, and in man y cases quan titativ e to visual precision. W e also sa w conditions under whic h the quan tum-classical mo del deviated from exact quan tal results. Finally w e compared a xed timestep v arian t of the relax-and-driv e algorithm with the v ariable timestep v ersion. F or the mo del of an alk ali atom approac hing a metal surface, w e examined probabilit y transitions, p osition exp ectation and its deviation, momen tum exp ectation

PAGE 87

73 0 0.5 1 1.5 2 0 20000 40000 60000 80000 100000 120000 Ionic and Neutral Populationstime (au) EP-QCLE: Ionic SO-FFT: Ionic EP-QCLE: Neutral Bound SO-FFT: Neutral Bound EP-QCLE: Neutral Free SO-FFT: Neutral Free Figure 4{20. Ionic and neutral p opulations o v er time, for the NaI complex.

PAGE 88

74 0 5 10 15 20 25 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 time (au) EP-QCLE: SO-FFT: EP-QCLE: s R SO-FFT: s R Figure 4{21. Exp ectation of p osition and its deviance, for the NaI complex.

PAGE 89

75 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 20000 40000 60000 80000 100000 120000 Coherence: Real( G12)time (au) EP-QCLE SO-FFT Figure 4{22. Coherence as a function of time, for the NaI complex.

PAGE 90

76 0 20 40 60 80 100 120 140 -80 -60 -40 -20 0 20 40 60 80 R (au)P (au) Figure 4{23. Phase space grid at the end of the sim ulation, for the NaI complex.

PAGE 91

77 0 500 1000 1500 2000 2500 3000 3500 10 -3 -10 -1 10 -5 -10 -3 10 -7 -10 -5 10 -9 -10 -7 number of timestepstolerance Variable Fixed Figure 4{24. Num b er of steps required b y the relax-and-driv e algorithm, compared to an estimated n um b er required for a xed timestep v ersion.

PAGE 92

78 and its deviation, densit y function and phase space grid ev olution. W e found that the EP-QCLE metho d repro duced quan titativ ely the exact v alues found through the quan tum mo del for all observ ables. W e also found the phase space grid to deform substan tially as the system ev olv ed, initially distorting but ev en tually lining up in a straigh t line as the eectiv e force v anished. F or the dual crossing diabatic surface collision, in addition to probabilit y transitions for a giv en energy w e calculated the transmission probabilit y of the ground state for a wide range of energies. This sho w ed the EP-QCLE to deviate at the lo w er energies, where n uclear in terference eects w ere imp ortan t. A t higher energies, the corresp ondence b et w een the mixed quan tum-classical results and the exact quan tum results w as go o d. W e also noted that the phase space grid deformed only for p oin ts whic h sp en t a signican t amoun t of time in the region of strong in teraction. This is reasonable, as the deformation comes from the eectiv e force, whic h is non-v arying (and in fact zero) outside the in teraction. The NaI mo del sho w ed in teresting bac k-and-forth transitions b et w een the ionic and co v alen t state, as the in tern uclear distance oscillated due to the long-range attractiv e ionic p oten tial. W e also sa w that eac h passing through the a v oided crossing led to a small amoun t of disso ciation in to the asymptotically free neutral state. Ionic, b ound neutral and free neutral p opulations w ere quan titativ ely similar for the EP-QCLE and SO-FFT algorithms for the b eginning of the sim ulations, and qualitativ ely similar for the remainder. Coherences w ere quan titativ ely similar, sho wing an initial p eak after the rst pass through the a v oided crossing, and m uc h smaller magnitudes thereafter. Finally the grid deformation sho w ed the grid to split in to t w o groups. The rst represen ted the asymptotically free neutral state, where the p oin ts form a straigh t line as the Hellmann-F eynman force v anishes and the particles propagate to innite distances. The second group circled in phase space, as the

PAGE 93

79 p oin ts oscillated bac k and forth in the p oten tial w ell formed b y the ionic attraction at large distances and the neutral repulsion at small distances. The EP-QCLE metho d w as sho wn to b e v ery robust under a wide v ariet y of conditions. Comparing its v ariable timestep v ersion (whic h w as used in all the sim ulations) to the xed timestep alternativ e, w e found that not only w as the EP-QCLE metho d accurate, but use of the relax-and-driv e propagation generated m uc h more ecien t sim ulations than could ha v e b een obtained without dynamically v arying the timestep.

PAGE 94

CHAPTER 5 ALKALI A TOM-RARE GAS CLUSTERS: GENERAL F ORMULA TION 5.1 In tro duction In this c hapter, w e describ e the in teractions of an alk ali atom (Ak) em b edded in a cluster of rare gas atoms (Rg N ). Cluster dynamics are in teresting insofar as they presen t a bridge b et w een isolated atoms and the (eectiv ely) innite bulk liquid, and can pro vide insigh t in to the dynamics of molecules on extended surfaces. The AkRg N clusters are among the simplest cluster systems to study b ecause of the presence of a single v alence electron in the alk ali atom, and the closed shell structure of the rare gas atoms. Pseudop oten tial descriptions of the in teractions b et w een the v alence electron ( e ), the Ak core and the Rg atoms can greatly simplify the description of the cluster with v ery mo dest p enalties in accuracy In the con text of mixed quan tumclassical mo dels, these clusters are w ell suited to sim ulation through the EP-QCLE b y treating the v alence electron quan tally and the n uclear cores quasiclassically F or the ma jorit y of the c hapter, w e consider the general case of an alk ali atom em b edded in a rare gas cluster, although our sim ulations in Chapter 6 fo cus on the sp ecic case of the lithium atom (Li) em b edded in a helium cluster (He N ). 5.2 Ph ysical System W e consider a cluster initially at thermal equilibrium, and in tro duce a ground or excited alk ali atom to the cen ter of the cluster. W e do not concern ourselv es with the means b y whic h the alk ali atom is excited or em b edded within the cluster, although t ypically the excitation is due to a laser pulse. F rom this initial setup, w e follo w the dynamics of the alk ali atom as it deexcites and mo v es within the cluster. W e further assume the cluster is isolated from an y en vironmen tal eects, so that sp ectral and conguration measuremen ts represen t 80

PAGE 95

81 those of an isolated Ak-Rg N cluster. As w e see later, w e in tro duce a con taining p oten tial to k eep the Rg atoms from disp ersing. This p oten tial is used strictly to main tain a cluster formation, and do es not represen t in teraction with an en vironmen t. 5.3 Prop erties of In terest First of all, w e are in terested in the structure of the Rg cluster in equilibrium, b efore the in tro duction of the Ak atom. W e need to repro duce kno wn densit y proles and pair-pair correlation functions to ensure the cluster is represen tativ e of a real ph ysical system. Secondly w e are in terested in follo wing the dynamics of the Ak atom as it mo v es within the cluster. In particular, w e are lo oking for migration from the cen ter of the cluster to its surface, as other exp erimen tal and theoretical studies indicate that the Ak atom tends to reside on the cluster surface. Finally w e wish to compute the time dep enden t emission sp ectra resulting from the deexcitation of the Ak atom, Ak ( n 0 l 0 ) + R g N Ak ( nl ) + R g N + ; (5.1) where is a photon with energy corresp onding to the electronic deca y 5.4 Hamiltonian for Alk ali-Rare Gas P airs F ollo wing the dynamics of an Ak-Rg N cluster through a full quan tal treatmen t w ould b e an extremely demanding problem computationally for more than a few Rg atoms. Instead, w e reduce the Ak-Rg in teraction to a three-b o dy problem b y using pseudop oten tial in teractions b et w een the n uclear cores and the electron. The pseudop oten tial treatmen t is explored extensiv ely for the single Ak-Rg pair in Rey es. 109 In this section, w e summarize this approac h. In order to describ e the three-b o dy in teraction, w e consider a xed n uclear conguration with R AB the p osition v ector from the alk ali core (A) to the rare gas atom (B), and r A ( r B ) the p osition v ector from the alk ali core (rare gas atom) to

PAGE 96

82 the electron e With this notation, w e can write the Hamiltonian in v e distinct comp onen ts, ^ H pair = 1 2 r 2r A + ^ V Ak A ( r A ) + ^ V Rg B ( r B ) + ^ V cr oss AB ( r A ; R AB ) + ^ V cor e AB ( R AB ) : (5.2) W e ha v e implicitly tak en the PWT o v er n uclear v ariables but not electronic v ariables. Because the p oten tials can all b e expressed as p olynomial functions of ^ Q the PWT amoun ts to replacing ^ Q b y R throughout the Hamiltonian. In what follo ws, w e will assume w e are w orking with partially Wigner transformed op erators, but drop the subscript 'W' for notational simplicit y Also note that the electronic v ariable remains quan tal, although w e will use the notation r rather than ^ q for consistency with common usage in the literature. The rst term on the RHS of Eq. 5.2 is, of course, the kinetic energy op erator of the v alence electron. The second and third terms are the p oten tials arising from the in teraction of the electron with the Ak core and the Rg atom, resp ectiv ely The ^ V cr oss AB term is a crossterm stemming from the p olarization of the Rg atom b y b oth the v alence electron and Ak core. Finally the last term ^ V cor e AB is the in teraction b et w een the Ak core and the Rg atom. W e examine eac h of these p oten tials in detail for the general Ak-Rg pair, and pro vide sp ecic parameters for the Li-He in teraction in Chapter 6. The e -Ak core p oten tial can b e divided in to three comp onen ts, ^ V Ak A ( r A ) = Z A r A + ^ V pol Ak A ( r A ) + ^ V sr A ( r A ) ; (5.3) where the rst term is the Coulom b in teraction b et w een the Ak core of c harge Z A and the electron. The second con tribution arises from the dip ole p olarization of the

PAGE 97

83 Ak core b y the v alence electron, ^ V pol Ak A ( r A ) = d A 2 r 4 A w ( r A ; A ) 2 ; (5.4) with d A the dip ole p olarizabilit y of the core and w a cuto function of the distance w ( r ; ) = 1 exp ( r 2 ) : (5.5) The nal con tribution to ^ V Ak A is an l -dep enden t short-range pseudop oten tial, ^ V sr X ( r X ) = X l ;i B l ;i exp ( l ;i r 2 X ) P l ;X ; (5.6) where B l ;i and l ;i are pseudop oten tial parameters adjusted to t exp erimen tal data, P l ;X is the pro jection op erator on angular symmetry l P l ;X = X m j Y l m ( ^ r X ) ih Y l m ( ^ r X ) j ; (5.7) and the states fj Y l m ( ^ r X ) ig are the spherical harmonic functions cen tered on the core X. This pseudop oten tial sim ulates repulsion due to the eects of the P auli exclusion principle when the v alence electron approac hes the core electrons, as w ell as the attraction due to the incomplete screening of the n uclear c harge. The third term in Eq. 5.2 is the e -Rg p oten tial, and can b e divided in to t w o terms, ^ V Rg B ( r B ) = ^ V pol Rg B ( r B ) + ^ V sr B ( r B ) : (5.8) The rst comp onen t stems from the p olarization of the Rg atom b y the electron, ^ V pol Rg B ( r B ) = d B 2 r 4 B w ( r B ; ) 4 q 0 B 2 r 6 B w ( r B ; ) 6 ; (5.9)

PAGE 98

84 where q 0 B = q B 6 1 q B b eing the quadrup ole p olarizabilit y of the Rg atom and 6 1 the dynamical correction to the static p olarizabilit y The short-range pseudop oten tial ^ V sr B is dened in Eq. 5.6 The fourth term in Eq. 5.2 is a crossterm arising from the p olarization of the Rg atom b y b oth the Ak core and the v alence electron, ^ V cr oss AB ( r A ; R AB ) = Z A d B P 1 ( cos ) R 2 AB r 2 B w ( r B ; ) 2 + Z A q B P 2 ( cos ) R 2 AB r 2 B w ( r B ; ) 3 : (5.10) Here, P 1 and P 2 are the Legendre p olynomials and is the angle b et w een the v ectors R AB and r B Finally the last term in Eq. 5.2 is the in teraction b et w een the alk ali core and the rare gas atom. It is assumed to in v olv e short-range, dip ole and quadrup ole con tributions, ^ V cor e AB ( R AB ) = ^ V sr AB ( R AB ) 1 2 d B ( R 2 AB + d 2B ) 2 1 2 q 00 B ( R 2 AB + d 2B ) 3 ; (5.11) where q 00 B = q B + 2 d B d 2B The short-range term ^ V sr AB has b een determined b y assuming it to ha v e the form, ^ V sr AB ( R AB ) = A exp ( bR AB ) ; (5.12) and adjusting the parameters A and b so that ^ V cor e AB ( R AB ) in Eq. 5.11 ts the most accurate curv es in the literature. This pro cedure has b een sho wn to accurately repro duce Ak-Rg energy curv es for a v ariet y of alk ali atom and rare gas com binations. 109 5.5 Hamiltonian for the Alk ali-Rare Gas Cluster The cluster is similar to the pair, in that eac h Ak-Rg pair in the cluster is treated as a three-b o dy problem in v olving alk ali core, v alence electron and rare gas atom, and describ ed using the same Hamiltonian elemen ts. In the case of the cluster, ho w ev er, w e also require that the Rg atoms main tain a cohesiv e structure

PAGE 99

85 rather than drift apart. T o ensure this, w e imp ose a constraining p oten tial on the system. Dening R AB i as the p osition v ector from the alk ali core to the Rg atom i and R B i the p osition v ector of the Rg atom from the origin, w e can write the cluster Hamiltonian as, ^ H cl uster = 1 2 r 2r A + ^ V Ak A ( r A ) + N X i =1 ^ V Rg B i ( r B ) + N X i =1 h ^ V cr oss AB i ( r A ; R AB i ) + ^ V cor e AB i ( R AB i ) + ^ V hol d i ( R B i ) i + N X j >i =1 ^ V cor e ( j R B i R B j j ) : (5.13) The nal term in Eq. 5.13 is the in teraction b et w een the Rg atoms for the quasiclassical motion. F ollo wing Aziz and co w ork ers, 110 w e tak e the Rg-Rg p oten tial to ha v e the form, for in tern uclear distance R ^ V cor e ( R ) = V ( x ) ; (5.14) where V ( x ) = A exp ( x + x 2 ) F ( x ) X j =0 2 C 2 j +6 x 2 j +6 ; (5.15) F ( x ) = 8><>: exp [ ( D =x 1) 2 ] ; x D 1 ; x > D : (5.16) This p oten tial has b een written in terms of the dimensionless distance x = R =R M The constraining p oten tial, ^ V hol d i is tak en to b e a sigmoidal function cen tered along the b oundary of a sphere of radius R hol d ^ V hol d i ( R B i ) = a 1 + exp [ b ( R B i R hol d )] : (5.17)

PAGE 100

86 This p oten tial sho ws asymptotes ^ V hol d i ( 1 ) = 0 and ^ V hol d i (+ 1 ) = a with midp oin t at the holding radius ^ V hol d i ( R hol d ) = a= 2. The steepness of the function is con trolled b y the parameter b and determines the strength and range of the holding force. It acts on the Rg atoms only k eeping them b ound roughly within a sphere of radius R hol d while p ermitting the alk ali atom free motion within the cluster. 5.6 Electronic Sp ectral Calculations When a molecular system undergo es electronic motion, the accelerating c harges emit electromagnetic radiation. A t distances large compared to the electronic motion, the rux of this radiation at p oin t r from the cen ter of the system is giv en b y the P o yn ting v ector, 111 S ( t ) = 1 0 ( E B ) = 0 16 2 c [ D ( t )] 2 sin 2 r 2 ^ r ; (5.18) where D ( t ) is the dip ole momen t of the system, 0 is the p ermittivit y constan t and c is the sp eed of ligh t. By in tegrating o v er all angles, w e obtain the p o w er emited b y the source, P ( t ) = Z n S ( t ) d a = j A ( t ) j 2 (5.19) where A ( t ) = r 1 6 0 c 3 D ( t ) : (5.20) F rom Eq. 5.20 w e can see that the computation of the dip ole momen t is essen tial to the sp ectral calculation. In a mixed quan tum-classical system, the dip ole momen t is obtained b y calculating the exp ectation v alue of the dip ole op erator ^ D D ( t ) = T r [ ^ ( t ) ^ D ] = T r[( t )] : (5.21)

PAGE 101

87 F or the Ak-Rg N cluster, the dip ole op erator can b e written with the origin at the Ak core, so that it b ecomes, ^ D = r + d A r A r 3A w ( r A ; A ) + d B N X i =1 r B i r 3B i w ( r B i ; B ) + Z A R AB R 3AB w ( R AB ; B ) # : (5.22) The rst term is the dip ole from the v alence electron. The second is the induced dip ole in the Ak core b y the v alence electron. The third and fourth terms sum o v er all Rg atoms, and represen t the induced dip ole in the Rg core b y the v alence electron and Ak core, resp ectiv ely In practice, the induced dip oles are m uc h smaller than the v alence electron dip ole. W e can calculate the emission sp ectrum b y taking the F ourier transform of Eq. 5.19 I ( ) = 1 p 2 Z P ( t ) exp( i! t ) dt = 1 p 2 Z j A ( t ) j 2 exp ( i! t ) dt: (5.23) If the emission sp ectrum c hanges o v er time, one can use a windo w ed F ourier transform cen tered ab out the time of in terest. In the case of Ak-Rg N clusters, this allo ws us to trace the ev olution of the emission sp ectrum as the alk ali atom mo v es from the cen ter of the cluster to its surface. 5.7 Electronic Basis of Gaussian A tomic F unctions 5.7.1 Equations of Motion W e recall the EP-QCLE in an arbitrary basis, Eq. 3.7 repro duced here for con v enience: d dt = ( i H q n y ) S 1 S 1 ( i H q + n ) ; (5.24)

PAGE 102

88 where w e ha v e used the notation, n h j @ =@ t j i + dR dt h j @ =@ R j i + dP dt h j @ =@ P j i : (5.25) F or the cluster, H q = H cl uster A con v enien t basis in v olv es Gaussian functions cen tered on the Ak core, and leads to matrix elemen ts whic h can b e ev aluated analytically T o this end, w e in tro duce primitiv e cartesian Gaussian states, dened b y an in teger triplet a = ( a x ; a y ; a z ), a Gaussian cen ter A = ( A x ; A y ; A z ) and an exp onen tial co ecien t a suc h that their pro jection on the electronic co ordinate r is h r j a ; A ; a i = ( r x A x ) a x ( r y A y ) a y ( r z A z ) a z exp [ a ( r A ) 2 ] : (5.26) F rom these primitiv es, w e can form Gaussian atomic functions (GAFs) whic h resemble the h ydrogenic orbitals. The lo w est three symmetries are j s ; R ; i j a = (0 ; 0 ; 0); R ; i = j 000; R ; i ; (5.27) j p ; R ; i 2 fj 100; R ; i ; j 010; R i ; j 001; R ; ig ; (5.28) j d ; R ; i 2 fj 110; R ; i ; j 011; R ; i ; j 101 ; R ; i ; j 200 ; R ; i j 020; R ; i ; 2 j 002; R ; i j 200; R ; i j 020; R ; ig : (5.29) Finally w e construct the basis elemen ts b y com bining linear com binations of these GAFs in segmen ted con tractions, j i = N X j c j j ; R ; j i ; (5.30) where 2 f s; p; d g f c j g are the con traction co ecien ts and N is a normalization factor. W e can write the full M -dimensional basis as a ro w v ector of these segmen ted

PAGE 103

89 con tractions, j i = ( j 1 i j 2 i : : : j M i ) ; (5.31) corresp onding to the notation used to deriv e Eq. 3.7 Since j i do es not dep end explicitly on time or n uclear momen ta, n reduces to a momen tum coupling term, n = dR dt h j @ @ R j i : (5.32) A non trivial matter is the analytical computation of these matrix elemen ts, atten tion to whic h w e turn presen tly 5.7.2 Ov erlap Matrix Elemen ts In computing matrix elemen ts, w e fo cus on the essen tial part of the computation, whic h is the matrix elemen t b et w een primitiv e cartesian Gaussians. F or the o v erlap S w e need to compute h a j b i whic h is p ossible using the recursion form ulae from Obara, 112 113 h a + 1 i j b i = ( P i A i ) h a j b i + 1 2 a i h a 1 i j b i + 1 2 b i h a j b 1 i i ; (5.33) h a j b + 1 i i = ( P i B i ) h a j b i + 1 2 h a j b 1 i i + 1 2 a i h a 1 i j b i ; (5.34) whhere 1 i is an in teger triplet ( ix ; iy ; iz ), = a + b and P = ( a A + b B ) = Recursion con tin ues to the base case, h s ; A ; a j s ; B ; b i = 2 exp [ ~ ( A B ) 2 ] ; (5.35) where ~ = a b = If computational eciency is a concern, some of the lo w er recursion relations can b e rolled out explicitly rather than computed recursiv ely

PAGE 104

90 5.7.3 Kinetic Energy Matrix Elemen ts F or the kinetic energy w e wish to compute h a j ^ K j b i = h a j 1 2 r 2r j b i W e can do this b y using the recursion relations, h a + 1 i j ^ K j b i = ( P i A i ) h a j ^ K j b i + 1 2 a i h a 1 i j ^ K j b i + 1 2 b i h a j ^ K j b 1 i i = 2 ~ h a + 1 i j b i 1 2 a a i h a 1 i j b i ; (5.36) h a j ^ K j b + 1 i i = ( P i B i ) h a j ^ K j b i + 1 2 b i h a j ^ K j b 1 i i + 1 2 a i h a 1 i j ^ K j b i = 2 ~ h a j b + 1 i i 1 2 b b i h a j b 1 i i : (5.37) The recursion con tin ues to the base case, h s ; A ; a j ^ K j s ; B ; b i = ~ (3 2 ~ ( A B ) 2 ) h s ; A ; a j s ; B ; b i = ~ (3 2 ~ ( A B ) 2 ) 2 exp [ ~ ( A B ) 2 ] : (5.38) 5.7.4 Coulom b Matrix Elemen ts The n uclear attraction in tegral is a three-cen ter in tegral, h a j ^ V C j b i = Z d r ( A ; a ) 1 j r C j ( B ; b ) : (5.39) F ollo wing Obara, 113 this can b e ev aluated using the recursion relations, h a + 1 i j ^ V C j b i ( m ) = ( P i A i ) h a j ^ V C j b i ( m ) ( P i C i ) h a j ^ V C j b i ( m +1) + 1 2 a i h h a 1 i j ^ V C j b i ( m ) h a 1 i j ^ V C j b i ( m +1) i + 1 2 b i h h a j ^ V C b 1 i i ( m ) hj ^ V C j b 1 i i ( m +1) i ; (5.40) and similarly for h a j ^ V C j b + 1 i i The recursion base is found to b e, h s ; A ; a j ^ V C j s ; B ; b i ( m ) = 2 ~ 1 = 2 h s ; A ; a j s ; B ; b i F m ( U ) ; (5.41)

PAGE 105

91 where U = ( P C ) 2 and F m is the sp ecial function, F m ( T ) = 1 Z 0 dt t 2 m exp ( T t 2 ) : (5.42) 5.7.5 Momen tum Coupling Matrix Elemen ts The momen tum coupling elemen ts h j @ =@ R j i are ev aluated in terms of the o v erlap elemen ts, h a j @ =@ B x j b i = b h a j b + 1 x i b 1 h a j b 1 x i ; (5.43) h a j @ =@ B y j b i = b h a j b + 1 y i b 2 h a j b 1 y i ; (5.44) h a j @ =@ B z j b i = b h a j b + 1 z i b 3 h a j b 1 z i : (5.45) Eac h elemen t on the RHS of Eqs. 5.43 to 5.45 is a straigh tforw ard o v erlap in tegral, whose ev aluation w e ha v e already deriv ed previously 5.7.6 Dip ole Matrix Elemen ts Recalling the dip ole for the Ak-Rg system, w e see that w e require the follo wing matrix elemen ts, h a j r A j b i ; (5.46) h a j r A r 3A (1 exp [ r A A ]) j b i ; (5.47) h a j r B r 3B (1 exp [ r B B ]) j b i ; (5.48) h a j b i : (5.49) The rst in tegral is a momen t in tegral, whic h can b e computed using the recursiv e pro cedure from Obara. 113 Consider the general momen t in tegral, h a jM ( ) j b i ; (5.50)

PAGE 106

92 where M ( ) = x x y y z z : (5.51) These in tegrals can b e found through the recursiv e form ula, h a + 1 i jM ( ) j b ) = ( P i A i ) h a jM ( ) j b i + 1 2 a i h a 1 i jM ( ) j b i + 1 2 b i h a jM ( ) j b 1 i i + 1 2 i h a jM ( 1 i ) j b i : (5.52) The recursion con tin ues to the base case, h s ; A ; a jM ( ~ 0 ) j s ; B ; b i = h s ; A ; a j s ; B ; b i = 2 exp [ ~ ( A B ) 2 ] : (5.53) The second and third in tegrals can b e calculated using a pro cedure presen ted b y McMurc hie and Da vidson, 114 and summarized in detail in Rey es. 109 The nal in tegral is a simple o v erlap in tegral, b ecause the op erator do es not dep end on electronic co ordinates. W e ha v e presen ted the calculation of the o v erlap in tegral previously 5.7.7 Pseudop oten tial Matrix Elemen ts In order to calculate the pseudop oten tial matrix, w e need the additional elemen ts, h a j 1 r 4 [1 exp ( c r 2 )] 2 j b i ; (5.54) h a j 1 r 4 [1 exp ( c r 2 )] 4 j b i ; (5.55) h a j 1 r 6 [1 exp ( c r 2 )] 6 j b i ; (5.56) h a j l X m = l j l m ih l m j j b i : (5.57)

PAGE 107

93 The rst three elemen ts are computed using a pro cedure from McMurc hie and Da vidson, 114 while the fourth elemen t is calculated based on metho ds in Sc h w erdtfeger and Silb erbac h. 115 These calculations are summarized in detail in Rey es. 109 5.8 Computing the Quasiclassical T ra jectory The c hange in p osition of core i (Ak or Rg) is simply calculated from the momen tum, d R i dt = P i M i : (5.58) Ho w ev er, the c hange in momen tum is substan tially more in v olv ed. Eac h quasiclassical v ariable k is sub jected to the Hellmann-F eynman force, F k = T r ^ @ ^ H @ R k # = T r[] ; (5.59) where w e ha v e used the abbreviated notation ^ H = ^ H cl uster Using our nonorthogonal basis j i this expression b ecomes, F k = T r h j @ ^ H @ R k j i # = T r[ S ] = 1 T r [ S ] T r[ @ H @ R k ] T r[ n yR k S 1 H ] T r [ HS 1 n R k ] : (5.60) The rst term on the RHS of Eq. 5.60 in v olv es the partial deriv ativ e of the cluster Hamiltonian matrix, and can b e ev aluated n umerically b y computing H at a p erturb ed co ordinate R k + @ H @ R k = H ( R k + ) H ( R k ) : (5.61) 5.9 Computational Details Examining the EP-QCLE, w e see that a n um b er of matrices can b e computed a single time at the b eginning of the sim ulation, suc h as the o v erlap, momen tum, electronic dip ole and kinetic energy Ho w ev er, the pseudop oten tial is dep enden t

PAGE 108

94 on the relativ e Ak-Rg congurations, and m ust b e computed at eac h timestep. In addition, the n umerical ev aluation of the Hellmann-F eynman force requires that the Hamiltonian for sev eral p erturb ed congurations b e ev aluated for eac h force calculation. Consequen tly it is imp ortan t that the pseudop oten tial calculations run quic kly This can b e done b y precomputing the pseudop oten tial matrix for sev eral in teratomic distances along the z-axis, extrap olating to the desired distance, and then rotating to the actual orien tation. In this section, w e demonstrate that the rotation is a simple matter of m ultiplication b y rotation matrices. Consider a single-cen ter basis j i and general op erator ^ O in the three-b o dy Ak-Rg system, where the Ak atom is cen tered at the origin, R is the p osition v ector from this core to the Rg atom, and r is the p osition v ector of the v alence electron to the Ak core. Supp ose that w e ha v e precomputed O = h j ^ O j i for sev eral v alues of R along the z-axis. Our task is to compute the new matrix O 0 for the general case where the Rg atom has b een rotated b y n = ( ; ), where and are the p olar and azim uthal angles, resp ectiv ely If the op erator is of the form, ^ O = ^ O ( j R j ; j r j ; j R r j ) ; (5.62) whic h is the case for the cluster Hamiltonian op erator, w e can compute O 0 through a rotation D (n), j 0 i = j i D (n) : (5.63) Algebraically w e can see this b y examining a single matrix elemen t with R along the z-axis, O = h j ^ O j i = Z d r ( r ) ^ O ( j R j ; j r j ; j R r j ) ( r ) : (5.64)

PAGE 109

95 When R is in the rotated p osition R 0 = D (n) R O 0 = Z dr ( r ) ^ O ( jD (n) R j ; j r j ; jD (n) R r j ) ( r ) : (5.65) Substituting r D (n) r = D r O 0 = Z dr ( D r ) ^ O ( jD R j ; jD r j ; jD R D r j ) ( D r ) : (5.66) Since a rotation of a v ector do es not c hange its magnitude, O 0 = Z dr ( D r ) ^ O ( j R j ; j r j ; j R r j ) ( D r ) (5.67) ) O 0 = Z dr ( D r ) ^ O ( j R j ; j r j ; j R r j ) ( D r ) = D y h j ^ O j i D : (5.68) T urning our atten tion to the construction of D w e use the fact that our basis in v olv es segmen ted con tractions of Gaussian atomic functions. In this case, basis elemen ts of same symmetry mix, while dieren t symmetries do not. W e will sho w the full deriv ation for s -t yp es and p -t yp es. Denoting the s -t yp e con traction, s ( r ) = X i exp ( i r 2 ) ; (5.69) w e w an t to calculate s ( D r ). Since D r implies the follo wing co ordinate transformations, x x cos cos + y sin sin z sin ; (5.70) y x sin + y cos ; (5.71) z x cos sin + y sin sin + z cos : (5.72)

PAGE 110

96 Then, s ( D r ) = X i c i exp ( i r 2 ) = s ( r ) : (5.73) Similarly consider the p -t yp e con tractions, p x ( r ) = X i c i x exp ( i r 2 ) ; (5.74) p y ( r ) = X i c i y exp ( i r 2 ) ; (5.75) p z ( r ) = X i c i z exp ( i r 2 ) : (5.76) Then p x ( D r ) = X i c i ( x cos cos + y sin cos z sin ) exp( i r 2 ) = p x ( r ) cos cos + p y ( r ) sin cos + p z ( r )( sin ) ; (5.77) p y ( D r ) = p x ( r )( sin ) + p y ( r ) cos ; (5.78) p z ( D r ) = p x ( r ) cos sin + p y ( r ) sin sin + p z ( r ) cos : (5.79) Supp ose our basis is ( r ) = [ s ( r ) p x ( r ) p y ( r ) p z ( r )] : (5.80) Then, ( D r ) = [ s ( D r ) p x ( D r ) p y ( D r ) p z ( D r )] = [ s ( r ) p x ( r ) p y ( r ) p z ( r )] D (n) = ( r ) D sp (n) ; (5.81)

PAGE 111

97 where D sp (n) = 0BBBBBBB@ 1 0 0 0 0 cos cos sin cos sin 0 sin cos cos sin sin 0 sin 0 cos 1CCCCCCCA : (5.82) W e can w ork out the mixing of d -functions similarly the results of whic h w e presen t in T able 5.9 5.10 Conclusion W e ha v e thoroughly describ ed the metho ds to ev olv e the Ak-Rg N cluster, starting with the Hamiltonian for the Ak-Rg pair and generalizing to a full cluster in three dimensions. W e ha v e treated the Ak v alence electron explicitly and ha v e computed the electron in teractions using semi-lo cal l -dep enden t pseudop oten tials. In the context of the EP-QCLE w e ha v e treated the electron as a quan tal v ariable, and the Ak and Rg cores as quasiclassical. By computing the dip ole of the molecular system, w e ha v e sho wn ho w the electronic emission sp ectra of an initially excited Ak atom can b e computed. In order to carry out the ev olution of the system n umerically w e ha v e in tro duced a basis of segmen ted con tractions of Gaussian atomic functions, and ha v e ev aluated all required matrix elemen ts explicitly The n umerical computation of the Hellmann-F eynman force guiding the quasiclassical motion of the n uclear cores has b een deriv ed, and the greatly accelerated computation of the pseudop oten tial matrices through table lo okup, n umerical in terp olation and rotation has b een discussed in detail.

PAGE 112

98 T able 5{1. Pseudop oten tial rotation for d -function mixing. D ij (n) elemen t v alue D xy ;xy (2 cos 2 1) cos D xy ;y z (2 cos 2 1) sin D xy ;xz 2 sin cos sin cos D xy ;x 2 y 2 2 sin cos cos 2 + sin cos sin 2 D xy ; 2 z 2 x 2 y 2 p 3 sin cos sin 2 D y z ;xy cos sin D y z ;y z cos cos D y z ;xz 2(cos 2 1) sin D y z ;x 2 y 2 sin sin cos D y z ; 2 z 2 x 2 y 2 p 3 sin sin cos D xz ;xy sin sin D xz ;y z sin cos D xz ;xz (2 cos 2 1) cos D xz ;x 2 y 2 cos sin cos D xz ; 2 z 2 x 2 y 2 p 3 cos sin cos D x 2 y 2 ;xy 2 sin cos cos D x 2 y 2 ;y z 2 sin cos sin D x 2 y 2 ;xz (2 cos 2 1) sin cos D x 2 y 2 ;x 2 y 2 (2 cos 2 1) cos 2 + 1 2 (2 cos 2 1) sin 2 D x 2 y 2 ; 2 z 2 x 2 y 2 q 3 4 (2 cos 2 1) sin 2 D 2 z 2 x 2 y 2 ;xy 0 D 2 z 2 x 2 y 2 ;y z 0 D 2 z 2 x 2 y 2 ;xz p 3 sin cos D 2 z 2 x 2 y 2 ;x 2 y 2 q 3 4 sin 2 D 2 z 2 x 2 y 2 ; 2 z 2 x 2 y 2 cos 2 1 2 sin 2

PAGE 113

CHAPTER 6 LITHIUM-HELIUM CLUSTERS 6.1 In tro duction In Chapter 5, w e explored all asp ects of the sim ulation of alk ali-rare gas clusters, in the con text of the EP-QCLE. In this c hapter, w e rene our description to the sp ecic case of a lithium atom em b edded in a cluster of helium atoms. The formalism is the same, in that the helium atoms and lithium core are treated quasiclassically while the lithium v alence electron is describ ed quan tally These clusters are only stable at v ery lo w temp eratures (a few degrees Kelvin), where core quan tal eects b ecome imp ortan t. Ho w ev er, b y appro ximating the helium-helium in teraction with an eectiv e p oten tial, w e are able to mo del bulk liquid helium classically suc h that its temp erature, densit y and radial distribution function matc h exp erimen tal and path in tegral results. F urthermore, b y extracting a sphere of helium atoms from this bulk, and imp osing an appropriate constraining p oten tial, w e are able to repro duce a liquid helium droplet whose densit y prole matc hes quan tum Mon te Carlo sim ulations. With an adequate classical description of the cluster, w e in tro duce a ground state lithium atom in to its cen ter and monitor the conguration and electronic p opulation dynamics o v er time. W e also study the ev olution of the excited lithium atom on a mo del of the cluster surface, in tro ducing electromagnetic elds to induce electronic transitions and ligh t emission. 6.2 Description of the System There are t w o stable isotop es of liquid helium, fermionic 3 He and b osonic 4 He. 116 W e are fo cused on the 4 He isotop e (whic h w e will designate as simply He), although in the classical appro ximation the only dierence resides in their mass. A t temp eratures b elo w 4.2 K, helium v ap or condenses in to the liquid state, 117 while b elo w 99

PAGE 114

100 2.17 K the liquid exhibits sup erruid prop erties. 118 117 Indeed, the thermal w a v elength at these energies exceeds the t ypical core-core separation, and quan tal eects suc h as zero-p oin t motion b ecome imp ortan t. 119 Ho w ev er, the excited electronic state of the He atom has an energy of 2 : 3 10 5 K ab o v e the ground state, making it v ery reasonable to assume the He atoms remain in their ground electronic conguration. 118 A t these temp eratures, the lo w kinetic energy p ermits the formation of stable clusters of helium con taining sev eral tens to sev eral h undreds of He atoms. These liquid droplets exhibit the bulk densit y and structure near their cen ter, with a decreasing densit y to w ard their surface. 120 When lithium atoms are in tro duced in to the cluster, they tend to reside on the surface, b ound b y a v ery shallo w w ell in the ground electronic state. 13 When the surface Li atom is excited, its subsequen t b eha vior dep ends hea vily on the excited state. The dierence in excited states is pictorally describ ed in Figure 6{1 Near a rat helium surface, the rst t w o excited states are degenerate, and rerect the p -orbital aligned parallel to the surface. This Li(2 p ) conguration minimizes the electronic o v erlap of the helium with the lithium, and results in an attraction to w ard the surface. The third excited state, 2 p has the p -orbital aligned orthogonally to the surface, where the electronic o v erlap induces repulsion b et w een the atoms. 121 While the Li(2 p ) tends to mo v e a w a y from the surface with minimal distortion of the He distribution, the Li(2 p ) mo v es to w ard the surface, where the He atoms resp ond b y clustering around the attractiv e Li atom. In exp erimen tal studies with excited Na, excimers of Na and He are found to desorb from the He surface within 70 to 700 ps, dep ending on the initial attractiv e state. 121 The desorb ed excimer is then found to emit to the red of the gas phase Na(3 p 3 s ) transition. Indeed, Li(2 p ) surrounded b y surface He atoms also emits to the red of the corresp onding

PAGE 115

101 A B Figure 6{1. Sc hematic of Li(2 p ) ab o v e a He surface. A) Li(2 p ). B) Li(2 p ). gas phase transition, but in terestingly no ligh t emission is observ ed when the Li(2 s ) is excited in the He bulk. 122 6.3 Prop erties to b e In v estigated First of all, w e are in terested in a classical description of the He atoms. While quan tal n uclear eects are clearly imp ortan t for ab initio descriptions of liquid helium, w e are fo cused on the in teraction of Li with the He, and seek only to reproduce the quan tum exp ectation of the He conguration. T o this end, it is imp ortan t to nd an eectiv e p oten tial for the He-He in teraction that generates an appro ximately correct He distribution through classical sim ulation, for a giv en He densit y and temp erature. Second, w e wish to mo del the liquid He droplet, where the He atoms hold together and exhibit a densit y distribution that approac hes the bulk helium densit y in the cen ter and tap ers o to w ard the edge. This can b e done b y using an appropriate constraining p oten tial whic h k eeps the atoms in a cluster. Ha ving formed a classical mo del of the He droplet, w e wish to examine the dynamics of a Li atom in tro duced in to the cluster. Sp ecically w e are in terested in

PAGE 116

102 follo wing the n uclear motion and electronic p opulations of ground state Li initially em b edded in the cen ter of the cluster, to b etter understand the mec hanism b y whic h Li atoms tend to reside on the surface of the He cluster. Once w e ha v e clearly established that the Li atom is not miscible with the He liquid, w e shift fo cus to the in teraction b et w een Li and the surface He atoms. In part, w e are in terested in the congurational ev olution of the He atoms and Li core once the Li atom is elev ated to an excited electronic state. Dep ending on whether the or state is p opulated, the dynamics are exp ected to b e quite dieren t, and these dierences are in v estigated. W e are also in terested in the c hange in electronic p opulations and energy lev els as the excited Li atom either approac hes or recedes from the He surface. By in troducing a classical electromagnetic eld resonan t to electronic transitions, w e are able to induce p opulation transitions in Li(2 p ), and stim ulate ligh t emission that can b e seen in the dip ole emission sp ectrum. In addition to pro viding sp ectral results for the Li-He N cluster, w e gain in teresting insigh ts in to the inclusion of an external electromagnetic eld within the mixed quan tum-classical con text. 6.4 Preparation of Lithium-Helium Clusters 6.4.1 Bulk Helium Our rst task is to generate a reasonable classical sim ulation of liquid He at ultralo w temp eratures. The four principal elemen ts in v olv ed in this kind of sim ulation are the system b oundaries, the propagation sc heme, the equilibration to attain the correct temp erature, and the in teratomic p oten tials. W e will discuss eac h of these in turn. In order to capture the eectiv ely innite spatial exten t of the liquid on the atomic scale, an attractiv e approac h is to use p erio dic b oundary conditions. 34 In this sc heme, a cen tral b o x is (imagined to b e) replicated at its sides and corners, so that there are 26 additional b o xes surrounding the cen tral v olume. Eac h of these

PAGE 117

103 b o xes con tains precisely the same conguration of He atoms as the cen tral b o x, and an y mo v emen t of an atom in the cen tral b o x (considered a r e al atom) is replicated b y the atoms in the surrounding b o xes (considered virtual atoms). If the in teratomic p oten tial is short range (as is the case for He atoms), then the minimum image con v en tion can b e used, whereb y a cuto distance of half the b o x length is used in the force calculations. This means that for a giv en He atom, only the neigh b oring atoms within a half b o x-length con tribute to the force on that He atom. In this w a y ev ery atom in the cen tral b o x con tributes to the force on ev ery other atom precisely once. If t w o atoms are within a half b o x-length of one another, then their in teraction is computed b et w een the corresp onding real atoms. If they are more than a half b o x-length from one another, then their in teraction o ccurs b et w een the real atom and the corresp onding virtual atom. In our sim ulations, w e b egan with a cen tral square b o x of length L = 40 : 46 au. This giv es a densit y = 0 : 00326 au 3 corresp onding to the kno wn bulk liquid He densit y 118 13 As for the initial state, w e w ould lik e to sim ulate the bulk He liquid in the canonical ensem ble, k eeping the n um b er of atoms, v olume and temp erature constan t. W e can pro vide a reasonable starting conguration b y randomly p ositioning the atoms within the cen tral b o x so that they are some minimal distance from one another. W e can also imp ose an initial temp erature b y assigning their momen ta acccording to the Maxw ellian distribution, p / exp p 2 2 mk B T ; (6.1) where k B is the Boltzmann constan t. W e p opulated our cen tral b o x using the minimal distance R min = 5 : 6 au for the conguration and temp erature T = 0 : 5 K for the momen tum distribution.

PAGE 118

104 There are man y metho ds of classical propagation, although w e found it useful to emplo y the v elo cit y V erlet algorithm, whic h is accurate to O ( t 2 ) and is selfstarting. In our case, it has the additional adv an tage of corresp onding directly to the propagation sc heme used for the quasiclassical v ariables in the relax-and-driv e pro cedure. While the c hoice of initial conditions ma y b e reasonable, it is necessary to equilibrate the system b y ev olving the atoms, and p erio dically rescaling the momen ta to obtain the desired temp erature, p p r T desir ed T actual : (6.2) There are other approac hes to temp erature calibration, but this rescaling sc heme a v oids c hanging the tra jectory directions so that the dynamics are b etter preserv ed. 34 Another imp ortan t issue in the equilibration is the timestep. T o k eep n umerical stabilit y w e imp osed a maxim um distance d max b y whic h an y atom could mo v e in a giv en timestep. This w as done b y nding the particle with maxim um sp eed v max and setting the (dynamically c hanging) timestep, t = d max v max : (6.3) Of course, this ignores the second order con tribution to the step in the v elo cit y V erlet algorithm, whic h dep ends on the particle accelerations. Ho w ev er, b y k eeping d max sucien tly small w e can neglect these higher order con tributions. W e found d max = 0 : 01 au to deliv er stable results without unnecessarily prolonging the sim ulation. Finally w e need to determine the eectiv e He-He in teratomic p oten tial to generate the correct particle distribution from classical sim ulations. A t higher temp eratures, the He-He p oten tial from Aziz and co w ork ers, 110 V Az iz ( R ), pro vides a

PAGE 119

105 v ery accurate description of the He-He in teraction. Its form w as describ ed in Chapter 5, and its parameters for helium are sho wn in T able 6{1 Ho w ev er, b ecause of the quan tal eects at lo w temp eratures, w e exp ect an eectiv e p oten tial V ef f ( R ) substan tially dieren t from V Az iz ( R ) to b e necessary for the classical sim ulation to describ e the bulk He distribution. This can b e seen in Figure 6{2 where w e plot the radial distribution function obtained using V Az iz in our classical sim ulation, and compare it to path in tegral results. 118 Not only is the true radial distribution substan tially rattened b y appro ximately an order of magnitude, but it is also shifted to the righ t b y roughly 1 au. W e found that b y adding a correction, V 0 ( R ), to the Aziz p oten tial, V ef f ( R ) = V Az iz ( R ) + V 0 ( R ) ; (6.4) w e w ere able to generate a similar radial distribution function to results from pathin tegral calculations. 118 The eectiv e p oten tial con v erges to the Aziz p oten tial at short distances, but has an attractiv e w ell whic h is shifted to the righ t and is signican tly more shallo w than V Az iz The correction term w as constructed b y taking the dierence b et w een a scaled, shifted Aziz p oten tial and the original Aziz p otential, and m ultiplying b y a sigmoidal function that is asymptotically zero at short distances, V 0 ( R ) = ( R )[ V Az iz ( R R M =R s ) V Az iz ( R )] ; (6.5) ( R ) = 1 1 + exp [ a ( R R )] : (6.6) The parameters for V 0 are sho wn in T able 6{2 The p oten tials are graphed in Figure 6{3 and the eectiv e p oten tial is sho wn with expanded axes in Figure 6{4 Note that while the eectiv e p oten tial is substan tially ratter than the original Aziz p oten tial, it nonetheless retains an attractiv e w ell. Sim ulations rev ealed that the radial distribution function is v ery sensitiv e to the heigh t and p osition of this w ell.

PAGE 120

106 T able 6{1. P arameters for the He-He in teraction from Aziz ( V Az iz ). R m A D C 6 C 8 C 10 5.6125 0.34648 1.922 10.735 -1.893 1.414 1.349 0.414 0.171 T able 6{2. P arameters for the correction to the He-He in teraction ( V 0 ) a R R S 4.0 4.0 0.003 6.62 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 g(R)R (au) V Aziz classical V Aziz path integral V eff classical 0 0.5 1 1.5 2 4 6 8 10 12 14 Figure 6{2. Radial distribution functions for bulk liquid helium.

PAGE 121

107 -0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 2 3 4 5 6 7 8 9 10 Energy (au)R (au) V Aziz V' V eff = V Aziz + V' Figure 6{3. Comparison of the Aziz p oten tial with the eectiv e form.

PAGE 122

108 -2e-07 -1.5e-07 -1e-07 -5e-08 0 5e-08 1e-07 1.5e-07 2e-07 4 5 6 7 8 9 10 Energy (au)R (au) V eff Figure 6{4. Eectiv e He-He p oten tial.

PAGE 123

109 6.4.2 Liquid Helium Droplets Ha ving formed an acceptable mo del of liquid He, w e are no w in a p osition to generate a mo del of a He droplet. W e b egin with a w ell-equilibrated p erio dic b o x of liquid He at = 0 : 00326 au 3 and T = 0 : 5 K, and extract a sphere of 100 He atoms. Although there are w eak v an der W aals forces holding these atoms together, one m ust remem b er that w e ha v e used an eectiv e p oten tial with a v ery shallo w w ell, so that ev en at these ultralo w temp eratures, the He atoms will ultimately escap e in to the gas phase. T o coun ter this, w e in tro duce the constraining p oten tial discussed in Chapter 5, whic h consists of a sigmoidal function near the edge of the cluster. The parameters of the sigmoidal function w ere adjusted un til w e obtained a stable cluster whose densit y distribution w as consisten t with path in tegral and diusion Mon te Carlo calculations. 118 13 F or the constraining function, V hol d ( R ) = a 1 + exp [ b ( R R hol d )] ; (6.7) w e w ere able to generate an acceptable pure He droplet using a = 5 10 4 b = 0 : 08 and R hol d = 80 : 0. In Figure 6{5 w e graph the constraining p oten tial. Figure 6{6 sho ws that the temp erature of the drop is prop erly calibrated to T = 0 : 5 K, and that ructuations are small. Finally Figure 6{7 displa ys the densit y prole from the cen ter-of-mass of the cluster, comparing our results with path in tegral and diusion Mon te Carlo calculations.6.4.3 Lithium-Helium In teractions The Li-He p oten tials are describ ed using the pseudop oten tial formalism discussed in the previous c hapter. The parameters are dened in T ables 6{3 to 6{5 F or the electronic basis, w e used the 5s5p4d basis set of Cartesian Gaussian atomic functions tak en from Czuc ha j and co w ork ers. 90 In the w ork b y Rey es, 109 adiabatic p oten tial curv es including the d symmetries w ere ev aluated as a function of basis

PAGE 124

110 0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 0 20 40 60 80 100 120 140 160 Energy (au)R (au) V hold Figure 6{5. Constraining p oten tial used to k eep He atoms from ev ap orating.

PAGE 125

111 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52 0 2e+06 4e+06 6e+06 8e+06 1e+07 Temperature (K)time (au) Figure 6{6. T emp erature ructations of the He droplet o v er time.

PAGE 126

112 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0 5 10 15 20 25 30 35 40 r (R)R (au) V eff classical V Aziz PIMC V Aziz DMC bulk 4 He Figure 6{7. Helium densit y prole from the cen ter-of-mass of the cluster.

PAGE 127

113 T able 6{3. P arameters for the e -Li in teraction. c c l B l l 0.1915 0.831 0 5.786 1.276 1 -1.065 1.607 T able 6{4. P arameters for the e -He in teraction. d B q B 1 l i B l l 1.3843 2.3265 0.706 0.79 0 1 0.83 1.30 0 2 2.27 0.50 0 2 2.27 0.50 1 1 -0.12 0.75 1 2 -1.87 1.00 size, and con v ergence w as found with the 5s5p4d basis. This basis pro vides an excellen t description of the lo w er adiabatic energies for the Li-He pair, as sho wn in Figure 6{8 F or this reason, w e emplo y ed this basis set for the remainder of our Li-He cluster sim ulations. F rom Figure 6{8 w e see that the 2 s ground state is essen tially repulsiv e. In fact, it con tains a v ery shallo w w ell near 11 au (not sho wn), whic h is resp onsible for the lo ose binding of Li atoms to the surface of He clusters. Ho w ev er, its depth is only 2 K, so that it will easily ev ap orate from the cluster. W e also see that the rst t w o excited states of lithium, 2 p and 2 p are substan tially dieren t in c haracter. In fact, the 2 p is a degenerate state of p -orbitals in the plane p erp endicular to the Li-He axis, while 2 p con tains the p orbital parallel to the Li-He axis. The electronic o v erlap in the 2 p results in the curv e sho wing repulsion at all distances, while the 2 p curv e has an attractiv e w ell with a minim um around 4 : 5 au. These dierences result in v ery dieren t Li-He dynamics for dieren t electronic states.

PAGE 128

114 T able 6{5. P arameters for the Li-He core in teraction. A b R D 27.26 2.29 1.93 -0.2 -0.15 -0.1 -0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s 3s s Figure 6{8. Adiabatic energy for Li and He as a function of in tern uclear distance.

PAGE 129

115 W e w ere also in terested in the eect of additional He atoms on the adiabatic energy giv en our in ten t to in tro duce the Li atom in to a cluster of He atoms. T o explore this eect, w e placed a single He atom at the origin, and then observ ed the c hange in adiabatic curv es as w e placed additional He atoms in the vicinit y of the origin. F or eac h conguration of helium atoms, w e mo v ed the lithium atom to w ard the origin along the z-axis, plotting the adiabatic energy as a function of the distance from the lithium atom to the origin. Initially w e added a single He atom at (0 ; 0 ; 6) au, and then another at (0 ; 0 ; 12) au. W e used a spacing of 6 au, since the radial distribution function p eak ed around this distance. The resulting adiabatic curv es are sho wn in Figure 6{9 where it is clear that the addition of the nal He atom aects the energy curv es negligibly In a second trial, w e b egan with a single He atom at the origin, and then added t w o along the y-axis: (0 ; 6 ; 0) au and (0 ; 6 ; 0) au. W e then added t w o more along the same line, (0 ; 12 ; 0) au and (0 ; 12 ; 0) au. The adiabatic curv es are plotted in Figure 6{10 where w e see the eects of additional He atoms are greater, but nonetheless the addition of the second set con tributes negligibly Finally w e lo ok ed at appro ximations to a cluster surface b y studying a 3 3 2 lattice of He atoms spanning [ 6 ; 6] [ 6 ; 6] [ 6 ; 0] au 3 with spacing of 6 au along eac h axis. The resulting adiabatic curv e w as compared to a larger 5 5 3 lattice of He atoms spanning [ 12 ; 12] [ 12 ; 12] [ 12 ; 0] au 3 again with 6 au spacing. The curv es are compared to the Li-He pair in Figure 6{11 The results rerect our exp ectations from the earlier curv es, in that there is v ery little dierence b et w een the smaller and larger surface in terms of adiabatic energies. These results giv e us condence that one can safely use a cuto in the order of 24 au (or ev en less) when computing the Li-He in teractions. This is esp ecially imp ortan t in larger cluster sim ulations, where the dominan t computational time is sp en t on matrix calculations.

PAGE 130

116 -0.2 -0.15 -0.1 -0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s One He Two He Three He -0.13176 -0.13168 -0.1316 4.2 4.3 4.4 4.5 4.6 4.7 4.8 2p p 2s s 2p p 2p s Figure 6{9. Adiabatic energies for Li and one or more He along the z-axis.

PAGE 131

117 -0.2 -0.15 -0.1 -0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s One He Three He Five He -0.1322 -0.13204 -0.13188 -0.13172 4.2 4.3 4.4 4.5 4.6 4.7 4.8 2p p 2s s 2p p 2p s Figure 6{10. Adiabatic energy for Li and one or more He along the y-axis.

PAGE 132

118 -0.2 -0.15 -0.1 -0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s One He 3x3x2 He 5x5x3 He -0.198115 -0.198095 -0.198075 9.5 10 10.5 11 11.5 12 12.5 13 2s s 2s s 2p p 2p s Figure 6{11. Adiabatic energy for Li and a surface of He atoms parallel to the x-y plane.

PAGE 133

119 6.5 Results: Lithium Inside the Helium Cluster Exp erimen tal and quan tum mec hanical structure calculations sho w that the Li atom in v ariably resides on the surface of the He cluster, rather than within its bulk. W e examined this phenomenon b y b eginning with a liquid helium droplet in thermal equilibrium at T = 0 : 5 K. Our droplet con tained 100 He atoms, and w as brough t to equilibrium through the v elo cit y rescaling describ ed earlier. A t this p oin t, the He atom closest to the cen ter-of-mass of the cluster w as replaced with a Li atom in the molecular ground state. This Li atom w as tak en to ha v e no momen tum initially so that all subsequen t dynamics w ere a result of in teractions with the surrounding He atoms. W e sho w the ev olution of the Li-He 99 cluster in Figure 6{12 where snapshots of the x-y plane are compared at initial and nal times. As exp ected, the Li atom mo v es to w ard the cluster surface. What is in teresting, ho w ev er, is the sp eed at whic h it is ejected from the cluster's in terior. The path is direct, and motion from the cen ter to the surface (corresp onding to the t w o snapshots) tak es only 10,000 au (241 fs). This is a far greater sp eed than w ould b e exp ected through thermal motion at 0 : 5 K, and indicates that not only is the Li atom ejected from the cluster, but is ejected violen tly The con v erse is seen in exp erimen ts where Li can only b e forced in to liquid He through laser ablation. This is explained b y the strong repulsion of the ground state, esp ecially at t ypical distances separating He atoms, and sho ws that while the ground Li atom migh t b e pic k ed up on the cluster surface, it is extremely unlik ely to p enetrate this surface in to the cluster in terior. T o v erify that the motion of the Li atom is drastically dieren t from the thermal motion of the He atoms, w e p erformed the same sim ulation, but without replacing the He atom b y the Li atom. W e then follo w the ev olution of the cen tral He atom as it slo wly w anders through the cluster, and compare the motion to the Li atom in Figure 6{13 There are t w o ma jor dierences in their motion, as seen from the

PAGE 134

120 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 Y (au)X (au) A Initial -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 Y (au)X (au) B Initial Final Figure 6{12. Ev olution of ground state Li em b edded in the cen ter of a He cluster. A) Initial time t = 0 au. B) Final time t = 10,000 au.

PAGE 135

121 gure. The rst is that the He atom mak es essen tially a random w alk through the cluster, and sta ys within a cen tral v olume of radius 11 au. The second is that the He atom mo v es m uc h more slo wly than the Li atom. In fact, on the time scale of the Li motion, the He atom app ears nearly stationary whic h is wh y w e ha v e had to scale the He time axis b y t w o orders of magnitude to sho w an y appreciable mo v emen t. In Figure 6{14 w e trac k the electronic p opulations of the Li as it emerges from the cluster. W e see that the p opulation remains almost exclusiv ely in the ground state, with a ground p opulation loss of less than 0 : 5% b y the time the Li atom reac hes the surface. This small loss is lik ely attributable to the higher collision energies a v ailable to the Li atom as it gains momen tum, so that close approac hes to He atoms resulting in p opulation transfer are p ossible. One should note that at times earlier than 2000 au, b efore the Li atom pic ks up m uc h momen tum, p opulation transfer is not visibly detectable on the scale of the graph. This is a recurren t theme that w e will see again as w e ev olv e excited Li atoms, in that the v ery lo w kinetic energies are insucien t to bring the Li and He atoms close enough where curv e crossing, and therefore state mixing, are appreciable. 6.6 Results: Lithium on the Helium Cluster Surface Ha ving established that the Li atom do es not rest within the cluster, w e wish to in v estigate its b eha vior on the cluster surface. Sp ecically w e are in terested in follo wing the dynamics once the Li has b een excited to either the 2 p or 2 p state. Ho w ev er, since our cluster has a rather diuse surface, w e instead mo del the surface as a square lattice of He atoms. Reviewing the adiabatic curv es computed earlier, w e nd a lattice of 5 5 3 He atoms to b e sucien tly large to mo del a droplet surface. Using a square lattice also p ermits us to b etter dene the inital electronic state of the Li atom (2 p or 2 p ), and ev aluate the resp onse of the He atoms to the presence and motion of the attac hed Li atom.

PAGE 136

122 5 10 15 20 25 30 0 2000 4000 6000 8000 10000 distance from c.o.m. (au)time (au) Li(2s s ) He Figure 6{13. Comparison of Li and He motion within a He cluster. The time scale has b een reduced b y a factor of 100 for the He curv e.

PAGE 137

123 0.98 1 1.02 1.04 1.06 1.08 1.1 0 2000 4000 6000 8000 10000 h (2s s )time (au) 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 1.0005 1.001 0 2000 4000 6000 8000 10000 Figure 6{14. Electronic p opulation of Li as it emerges from the He cluster.

PAGE 138

124 F or this lattice, w e nd a minim um energy along the ground state adiabatic curv e at R = 10 : 7 au, and th us place the Li atom initially at (0 ; 0 ; 10 : 7) au. Similar to the last section, w e b egin with the Li atom motionless, so that all n uclear motion comes from the Li-He in teractions. In the next t w o sections, w e follo w the dynamics of the Li atom once it has b een electronically excited. 6.6.1 Dynamics of Li( 2 p ) W e assume the Li atom has b een excited to 2 p and follo w its dynamics from this initial state. Snapshots of its ev olution are displa y ed in Figure 6{15 where w e see the motion of the He atoms and Li atom along the y-z plane. Since this is a repulsiv e state, the Li atom mo v es a w a y from the He surface. W e also see the He atoms rep el in w ard from the Li atom. The snapshots are tak en at the initial time t = 0 au, and at the nal time t = 33 ; 000 au. Similar desorption from the surface is seen in exp erimen ts where either Na or K is excited to the repulsiv e state, and then immediately lea v es the surface. 11 The excited state deca ys slo wly and in the time frame of our sim ulation the Li atom remains in the 2 p state throughout. Ho w ev er, as sho wn in Figure 6{16 w e notice that while the total 2 p p opulation remains unit y there is almost complete transfer b et w een the 2 p and 2 p p opulations at around t = 2000 au. This is explained b y the fact that the 2 p state b ecomes triply degenerate at large distances, so that the 2 p and 2 p p opulations mix completely Ho w ev er, their degeneracy means that v ery little ligh t emission can b e exp ected. On the other hand, w e can induce dip ole emission b y p erturbing the system with an electromagnetic (EM) eld, ^ H 0 = ^ H ^ D E 0 cos ( t + ) ; (6.8) where ^ D is the system dip ole. F or an isolated Li atom, w e nd an energy dierence E = 0 : 068 au b et w een the ground and rst excited state. An EM eld with

PAGE 139

125 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 25 30 Y (au)Z (au) A Li -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 25 30 Y (au)Z (au) B Li Figure 6{15. Ev olution of Li(2 p ) as it recedes from the He cluster surface. A) Initial time t = 0 au. B) Final time t = 33 ; 000 au.

PAGE 140

126 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 5000 10000 15000 20000 25000 30000 35000 htime (au) 2p p 2p s 2p p + 2p s Figure 6{16. Mixing of the Li(2 p ) and Li(2 p ) states at distances where Li(2 p ) is triply degenerate.

PAGE 141

127 frequency = 0 : 068 au 1 will stim ulate absorption and emission of ligh t b y the gas phase Li, as the Li atom exp eriences Li(2 p 2 s ) (called here the D ) transitions. As the EM frequency shifts from the electronic transition energy ligh t absorption and emission are still presen t, but with less magnitude. F or Li(2 p ) near the He surface, w e in tro duced an EM eld resonan t to the D line, with amplitude E 0 = 1 : 0 10 5 In addition, w e added the phase = 3 = 2 so that the electric eld rises from zero amplitude, rather than b eginning abruptly Finally w e c hose = (1 ; 1 ; 1) = p 3 so that there are comp onen ts along eac h axis, and the p erturbation has an eect regardless of the orien tation of the Li atom. In Figure 6{17 w e see the eect of in tro ducing the eld at t = 30 ; 000 au. By t = 32 ; 000 au, appro ximately half the total 2 p p opulation has deca y ed to the ground state. A t this p oin t, the 2 p p opulation b egins to rise again, corresp onding to ligh t absorption. This pattern of electronic deca y and reco v ery results from our semiclassical treatmen t of ligh t-matter in teractions. The eect of the EM eld is to induce the emission of dip ole radiation, whic h is measured from the second time deriv ativ e of the dip ole. Figure 6{18 sho ws the resulting dip ole sp ectrum for t w o time p erio ds, one when the p erturbing eld is applied during the initial 3000 au, and the second when the eld is applied during the nal 3000 au. W e also sho w the gas phase Li(2 p 2 s ) emission line, whic h w e compute from our asymptotic Li-He energies as 14903 cm 1 (and is extremely close to the kno wn v alue, 14904 cm 1 ). W e see that the Li(2 p ) emission p eak is initially blue shifted b y ab out 50 cm 1 ; at nal times, the p eak is substan tially broadened. This broadening can b e explained b y the motion of the He atoms, whic h ha v e acquired additional thermal energy from the repulsiv e in teraction with the Li atom.

PAGE 142

128 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 29000 30000 31000 32000 33000 h2ptime (au) without field with field Figure 6{17. Electronic p opulation of Li with and without a p erturbing electromagnetic eld, resonan t to the D line.

PAGE 143

129 13000 13500 14000 14500 15000 15500 16000 16500 17000 Intensity (arbitrary units)n (cm -1 ) initial time final time gas phase Li(2p <2s) Figure 6{18. Dip ole emission sp ectra of Li(2 p ) as it recedes from the He cluster surface.

PAGE 144

130 6.6.2 Dynamics of Li( 2 p ) The ev olution of the Li(2 p ) atom presen ts substan tially more in teresting features than the Li(2 p ) atom, as it is dra wn in to w ard the cluster surface. As suc h, it is not a matter of simply lea ving the cluster, but rather in v olv es non trivial in teractions with the He atoms. Snapshots are sho wn in Figure 6{19 again displa y ed as cross sections along the y-z plane. Because of the attractiv e nature of the 2 p surface, w e see the Li atom indeed mo v es in to w ard the cluster surface. The Li atom ev en tually em b eds itself in the surface, and b y t = 67 ; 000 au it is completely surrounded b y He atoms. The He densit y around the Li atom eviden tly increases, although the He-He repulsion coun ters this tendency once a certain He densit y is reac hed. Exp erimen tal results on excited Na atoms sho w the formation of Na-He N excimers, with N t ypically 3 or 4. According to the exp erimen tal results, once the excimers form, they desorb from the surface in 70 to 700 ps. 121 Our dynamics indicate that Li-He N excimers form on a m uc h shorter timescale (1.66 ps), alb eit w e are considering a dieren t alk ali atom. Ho w ev er, w e w ere not able to repro duce subsequen t desorption in longer sim ulation runs, partly b ecause of n umerical instabilities whic h ev en tually accum ulate. In Figure 6{20 w e sho w the p opulation of the 2 p state as the Li atom en ters the surface. P opulation transfer is exceptionally small, sho wing less than 0 : 01% o v er t = 67 ; 000 au, and once again reinforcing the conclusion that at these ultralo w temp eratures, collision-induced p opulation transfer is negligible. Ho w ev er, similar to the study of Li(2 p ), w e are able to prob e the electronic state b y in tro ducing an electromagnetic eld and observing the dip ole emission sp ectrum. In Figure 6{21 w e sho w the dip ole sp ectrum resulting from an EM eld resonan t to the D line, in tro duced during the rst 3000 au of the sim ulation. The p eak of this sp ectrum o ccurs at = 14895 cm 1 whic h is shifted to the red of the gas phase emission b y 8 cm 1 Laser-induced ruorescence exp erimen ts sho w that a

PAGE 145

131 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 25 30 Y (au)Z (au) A Li -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 25 30 Y (au)Z (au) B Li Figure 6{19. Snapshot of Li(2 p ) as it in teracts with the He cluster surface. A) Initial time t = 0 au. B) Final time t = 67 ; 000 au.

PAGE 146

132 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 0 10000 20000 30000 40000 50000 60000 70000 h2p ptime (au) Figure 6{20. Electronic p opulation of Li(2 p ) as it in teracts with the He cluster surface.

PAGE 147

133 Li(2 s ) attac hed to a He cluster surface emits at a frequency whic h is red shifted b y ab out 10 cm 1 13 agreeing with our results. A m uc h more substan tial shift, ho w ev er, is found when the same EM eld is in tro duced in the nal 3000 au of the sim ulation, while the Li atom is surrounded b y the He atoms. Sho wn in Figure 6{22 the dip ole sp ectrum is shifted to the red b y appro ximately 4000 cm 1 While this v ery large shift initially app ears susp ect, it is explained b y examining the adiabatic energy curv es for a Li atom surr ounde d b y He atoms, rather than approac hing a surface. In this calculation, w e ha v e k ept the Li atom at the origin, and surrounded it with a square lattice of 8 He atoms (one at eac h corner of the lattice). W e ha v e then computed the adiabatic energy as a function of the distance the He atoms are lo cated along eac h axis. The curv es for the ground and rst excited state are sho wn in Figure 6{23 W e nd that ev en substan tially b efore the minim um of the w ell is reac hed, the distance b et w een the t w o curv es corresonds to the energy p eak of the dip ole emission sp ectrum. It is also in teresting to note ho w close the t w o curv es come to one another at small distances. This ma y explain wh y radiativ e deca y is not observ ed for Li in bulk He, where the n uclear congurations are forcibly suc h that the Li atom is alw a ys surrounded b y He atoms. In this case, it ma y b e energetically p ossible for excited Li atoms to reac h the left side of the adiabatic curv e, and deca y through curv e mixing in to the ground state. If this transfer w ere to o ccur b efore the time scale of sp on taneous emission, there w ould b e v ery little ligh t emission, and certainly none in the frequency range of the D emission line. In order to v alidate these sp ectral and adiabatic curv e results for Li(2 p ), w e in tro duced an electromagnetic eld resonan t with the supp osed sp ectral p eak, = 10 ; 730 cm 1 W e k ept the amplitude and phase the same as b efore, and turned on the eld at t = 64 ; 000 au. The results are sho wn in Figure 6{24 One can see that the 2 p p opulation deca ys steadily to ab out 25%, while the 2 s p opulation rises

PAGE 148

134 13000 13500 14000 14500 15000 15500 16000 16500 17000 Intensity (arbitrary units)n (cm -1 ) initial time gas phase Li(2p <2s) Figure 6{21. Dip ole emission sp ectrum of Li(2 p ) during the rst 3000 au.

PAGE 149

135 7000 8000 9000 10000 11000 12000 13000 14000 Intensity (arbitrary units)n (cm -1 ) Figure 6{22. Dip ole emission sp ectrum of Li(2 p ) during the nal 3000 au.

PAGE 150

136 -0.2 -0.18 -0.16 -0.14 -0.12 -0.1 2 2.5 3 3.5 4 4.5 5 5.5 6 Energy (au)R (au) D E = 10730 cm -1 2s s 2p p Figure 6{23. Adiabatic curv es of Li surrounded b y a cubic lattice of He atoms. The parameter R refers to the half-length of the lattice edge.

PAGE 151

137 corresp ondingly The EM eld frequency is rerected in the small ondulations, but the net p opulation transfer o v er time can b e in terpreted as induced ligh t emission. This sho ws that the classical electromagnetic eld can indeed b e used to induce b oth absorption and emission of ligh t. 6.7 Conclusion W e found that w e could repro duce the distribution of bulk liquid He at ultralo w temp eratures through a classical sim ulation b y using an eectiv e p oten tial whic h w as signican tly scaled do wn as w ell as shifted from the p oten tial prop osed b y Aziz and co w ork ers. Sp ecically w e w ere able to qualitativ ely repro duce the radial distribution function in the bulk. F urthermore, b y extracting a sphere of He atoms from this bulk, and imp osing a sigmoidal con tainmen t p oten tial on the system, w e w ere able to pro duce a liquid helium droplet whose densit y prole matc hed path in tegral and diusion Mon te Carlo results. Ha ving sucien tly mo deled the He droplet, w e in tro duced a ground state Li atom in to its cen ter, and follo w ed its dynamics. W e found that it w as rapidly exp elled from the cluster, and con trasted this motion from the slo w random w alk tak en b y a t ypical He atom. W e also found that the p opulation of the Li atom remained predominan tly in the ground state. W e then examined the dynamics of an excited Li(2 p ) atom near a mo del He surface. W e found that it mo v ed a w a y from the surface, while the He atoms recoiled somewhat in to the cluster. As with the Li(2 s ), the p opulation remained primarily in the original 2 p state. In order to prob e the electronic structure, w e in tro duced an electromagnetic eld to the system, resonan t to the gas phase Li(2 p 2 s ) line. W e computed the resulting dip ole sp ectrum, and found it to b e initially shifted appro ximately 50 cm 1 to the blue of the D line. A t later times, the p eak broadened, o wing to the increased thermal motion of the He atoms.

PAGE 152

138 0 0.2 0.4 0.6 0.8 1 64000 64500 65000 65500 66000 66500 67000 htime (au) Li(2p p ) Li(2s s ) Figure 6{24. Deca y of Li(2 p ) surrounded b y surface He atoms, induced b y an EM eld with frequency resonan t to the Li(2 p 2 s ) transition.

PAGE 153

139 W e also studied the dynamics of the excited Li(2 p ) atom near the mo del surface. In con trast to the 2 p state, the Li(2 p ) atom mo v ed to w ard the surface, and ev en tually found itself surrounded b y a high densit y of He atoms. These excimers ha v e b een sho wn to form for Na atoms near He clusters, follo w ed b y desorption of the excimer from the cluster. W e w ere able to sho w the formation of the excimers, but could not repro duce the subsequen t desorption, p ossibly due to the v ery long times required for the desorption to tak e place. An external electromagnetic eld ga v e rise to a dip ole emission sp ectrum initially shifted to the blue of the gas phase transition for Li(2 p ), and shifted to the red for Li(2 p ). These shifts are in agreemen t with exp erimen tal and other computational results. When the Li w as surrounded b y surface He atoms, the dip ole sp ectrum w as found to b e shifted to the red b y ab out 4000 cm 1 This v ery large shift w as explained b y lo oking at the adiabatic curv es of Li surrounded b y a matrix of He atoms, whic h rev ealed that the ground and rst excited states came sucien tly close to one another to corresp ond to the dip ole sp ectrum p eak. The adiabatic curv es also rev ealed ho w excited Li atoms in bulk He migh t b e able to deca y to the ground state with minimal ligh t emission, giv en the pro ximit y of the curv es. Finally w e in tro duced an electromagnetic eld resonan t with the dip ole emission p eak for Li(2 p ) surrounded b y He atoms. This electromagnetic eld induced deca y in to the ground state, un til the 2 s p opulation reac hed appro ximately 75%. This v alidated the sp ectral results, and demonstrated that the classical electromagnetic eld can b e used in mixed quan tum-classical sim ulations to induce b oth electronic excitation and deexcitation.

PAGE 154

CHAPTER 7 CONCLUSION The purp ose of this study has b een to dev elop a computationally feasible time dep enden t formalism to describ e the dynamics of fewand man y-atom systems. W e ha v e dev elop ed a mixed quan tum-classical approac h whic h sho ws excellen t agreemen t with full quan tal calculations on small mo del systems, and is also capable of treating the ev olution of a realistic three-dimensional mo del of an alk ali atom em b edded within a cluster of rare gas atoms. The general pro cedure used for the tests mo dels in tro duces an eectiv e p oten tial to the mixed quan tum-classical formalism, and uses classical tra jectories coupled to quan tal ev olution to generate the dynamics. The eciency of the calculations w as impro v ed b y in tro ducing a propagation algorithm whic h tak es in to accoun t the dieren t time scales of the n uclear and electronic motion, and pro vides a means to ev olv e the system with v arying timesteps, maximizing eciency while ensuring a prescrib ed lev el of accuracy In order to describ e the alk ali-rare gas clusters, the general formalism w as extended to incorp orate pseudop oten tial in teractions b et w een the alk ali v alence electron and the n uclear cores, and n umerical in terp olation metho ds w ere used to render the calculation of the Hamiltonian matrix elemen ts computationally feasible. In this conclusion, w e review the principal results of eac h c hapter. 7.1 Eectiv e P oten tial Quan tum-Classical Liouville Equation Beginning with the quan tum Liouville equation of motion for the densit y op erator, w e divided the (general) system in to quan tal and quasiclassical degrees of freedom. By taking the partial Wigner transform o v er the quasiclassical degrees of freedom only and appro ximating for a large quasiclassical/quan tal mass ratio, 140

PAGE 155

141 w e deriv ed a mixed quan tum-classical equation of motion for the partially Wigner transformed densit y op erator. The transformed densit y op erator b ecomes a function of the quasiclassical phase space, but remains an op erator in quan tal space. By in tro ducing an eectiv e p oten tial to the mixed quan tum-classical equations of motion, w e generate a simplied v ersion, the eectiv e p oten tial quan tum-classical Liouville equation (EP-QCLE), whic h can b e solv ed b y follo wing tra jectories through the quasiclassical phase space. This sc heme is rendered n umerically accessible b y in tro ducing a general basis set for the quan tal space, and b y discretizing the quasiclassical space on to a grid. The grid p oin ts ev olv e through quasiclassical tra jectories whic h are coupled to the quan tal state through the eectiv e force. The quan tal state ev olv es sim ultaneously using the relax-and-driv e metho d, whic h divides the quan tal motion in to a relaxation term com bined with a driving con tribution from the ev olving quasiclassical v ariables. The relax-and-driv e sc heme w as mo died to pro vide a robust v ariable timestep whic h accoun ted for the t ypically fast quan tal motion compared to the slo w er quasiclassical tra jectory ev olution. In the EP-QCLE sc heme, measured quan tities w ere computed through an in tegration o v er quasiclassical phase space of the trace o v er quan tal v ariables. In this w a y exp ectation v alues of a general op erator, as w ell as p opulations of quan tal states, could b e computed analogously to the full quan tal calculations. 7.2 One-Dimensional Tw o-State Mo dels By sim ulationg three simple t w o-state mo dels, w e w ere able to compare the EP-QCLE metho d with the exact quan tum mec hanical solution, obtained through the split-op erator fast F ourier transform grid metho d. W e found the EP-QCLE to sho w excellen t agreemen t with the full quan tal results for man y observ ables, and at least qualitativ e agreemen t for all mo dels and energies. W e follo w ed a mo del of an alk ali atom aproac hing a metal surface, and ev aluated probabilit y transitions, p osition exp ectation and its deviation, momen tum

PAGE 156

142 exp ectation and its deviation, the densit y function and phase space grid ev olution. W e found the EP-QCLE metho d repro duced qualitativ ely the exact v alues obtained through the quan tal treatmen t for all observ ables, and sho w ed the phase space grid to deform due to the action of the eectiv e p oten tial. W e examined a collision mo del in v olving t w o a v oided crossings, and found the transmission probabilit y obtained b y the EP-QCLE to deviate from the quan tal results at lo w er energies. This problem w as observ ed in other theoretical mo dels whic h did not incorp orate n uclear coherence, and w as exp ected to arise b ecause of the imp ortance of n uclear coherence in the dual crossing mo del. Ho w ev er, w e found excellen t transmission probabilit y agreemen t at higher energies. Finally w e sim ulated a mo del of so dium in teracting with io dine, where the NaI complex oscillated b et w een a long range attractiv e ionic state and an asymptotically free neutral state. Disso ciation w as seen to o ccur at the diabatic crossing, as some of the ionic state w ould cross in to the neutral state and propagate to w ard free Na and I atoms. The oscillatory b eha vior w as repro duced qualitativ ely b y the EP-QCLE mo del, and coherence b et w een the states sho w ed agreemen t with the quan tal results for long sim ulation times. Ev aluation of the phase space grid rev ealed b oth the oscillatory motion of the b ound NaI complex, and the disso ciated neutral state of the free atoms. 7.3 Alk ali-Rare Gas Clusters W e ha v e applied the EP-QCLE formalism to the study of alk ali atoms em b edded within rare gas clusters. This has b een done b y treating the Ak-Rg in teraction as a three-b o dy problem comp osed of the alk ali core, its v alence electrons and the rare gas core. In the con text of the EP-QCLE, w e ha v e treated the electron as a quan tal v ariable and the atomic cores as quasiclassical. The electron-core in teractions ha v e b een describ ed using semi-lo cal l -dep enden t pseudop oten tials, while the Ak-Rg core in teractions ha v e b een describ ed with a tted parametric curv e. The

PAGE 157

143 Rg-Rg in teraction w as mo deled b y a parametric curv e, and the three-b o dy system w as generalized to the case of one Ak atom and sev eral Rg atoms. In order to propagate the system, w e in tro duced a basis of segmen ted contractions of Gaussian atomic functions cen tered on the alk ali core. W e ev aluated all required matrix elemen ts explicitly and describ ed in detail the n umerical ev aluation of the Hellmann-F eynman force guiding the atomic cores. In addition, w e ha v e implemen ted a sc heme in v olving lo ok-up tables, in terp olation and rotation to ecien tly compute the pseudop oten tial matrix elemen ts, a ma jor b ottlenec k in the n umerical propagation of the cluster. W e ha v e studied the dynamics and sp ectra of a lithium atom em b edded in a cluster of helium atoms at v ery lo w temp eratures ( 0 : 5 K). W e follo w ed the path of the ground state lithium atom, and found it to rapidly mo v e from the cen ter of the cluster to its exterior. This is in agreemen t with exp erimen tal results whic h indicate that alk ali atoms reside on the surface of rare gas clusters and not within their in terior. W e ha v e also examined the dynamics of excited lithium atoms near a mo del helium surface, and found that the b eha vior is strongly dep enden t on the symmetry of the excited state. In the case of Li(2 p ), the atom is immediately rep elled from the surface, while the He atoms recoil sligh tly On the other hand, Li(2 p ) is attracted to the surface, and ev en tually em b eds itself completely within the He atoms. F or b oth excited states, w e induce dip ole emission b y in tro ducing a p erturbing electromagnetic eld resonan t to the Li(2 p 2 s ) line. The computed dip ole sp ectra ha v e b een calculated for initial and nal times, where w e nd sp ectral shifts to generally matc h other theoretical and exp erimen tal results. In the case of Li(2 p ) surrounded b y surface He atoms, w e observ e a v ery high red shift in the sp ectrum, whic h can b e explained b y examining the pro ximit y of the ground and rst excited state adiabatic curv es for a lithium atom approac hed b y a surrounding lattice of

PAGE 158

144 He atoms. This curv e pro ximit y ma y also explain the absence of radiation detected from excited Li in bulk liquid He, with radiationless deca y b eing p ossible through state mixing at sucien tly lo w Li-He distances. Finally w e w ere able to induce deca y and reco v ery of Li(2 p ) b y applying an electromagnetic eld resonan t to the Li(2 p 2 s ) transition, conrming our sp ectral results as w ell as demonstrating the abilit y of a classical electromagnetic eld to induce these electronic state c hanges in the con text of a mixed quan tum-classical sim ulation. 7.4 Soft w are Dev elopmen t Ov er the course of this pro ject, w e ha v e dev elop ed a general program, cauldron from scratc h in F ortran 90, to sim ulate the dynamics of arbitrary systems whic h can b e cast in to the EP-QCLE formalism. Cauldron mak es no assumptions ab out and puts no limitations on the system size, and has b een dev elop ed in a mo dular fashion with the system, propagation and prop ert y calculations designed orthogonally Indeed, all three simple one-dimensional mo dels, as w ell as the full Li-He N cluster, w ere sim ulated with precisely the same relax-and-driv e propagation co de. This orthogonal approac h has enabled cauldron to b e extended as new questions arose and new systems w ere studied, without the need to mo dify already w orking co de. W e ha v e also dev elop ed a second program, qualdron analgous to cauldron but for sim ulating one-dimensional t w o-state systems using the split-op erator fast F ourier transform formalism. Qualdron has b een used primarily to generate precise results for comparison with the ( cauldron ) EP-QCLE results. Qualdron has also b een dev elop ed with orthogonal mo dules for the system, propagation and prop erties, but is limited to one-dimensional t w o-state systems describ ed in a diabatic basis. Ho w ev er, the extension to m ultistate systems w ould b e a trivial task, requiring the mo dication of a couple of subroutines in the propagation mo dule. Both cauldron and qualdron are large programs, in v olving o v er 35 ; 000 lines of co de and 200 subroutines. Both their use and in ternal functioning ha v e b een

PAGE 159

145 extensiv ely describ ed in t w o accompan ying man uals, where sp ecic proto cols for extending the co des to new problems ha v e b een suggested. 7.5 F uture W ork While w e ha v e fo cused on the application of the EP-QCLE to three small mo dels as w ell as the realistic three-dimensional Li-He N cluster, it w ould b e in teresting to ev aluate the p erformance and accuracy of the EP-QCLE for additional test mo dels where precise quan tal results can b e computed, in order to further assess the b enets and limitations of the metho d. It w ould also b e in teresting to examine the dynamics of Ak-Rg N clusters in v olving hea vier alk ali or rare gas atoms. Finally it w ould b e instructiv e to study the dynamics and sp ectra of collisions of Ak atoms with Rg atoms at higher temp eratures, where the Rg clusters v ap orize in to the gas phase. Ak-Rg collisions at thermal and h yp erthermal energies ha v e b een studied recen tly b y Rey es, and cauldron pro vides a means to study collisions at these energies where m ultiple rare gas atoms are in v olv ed.

PAGE 160

APPENDIX A THE CAULDRON PR OGRAM A.1 Ov erview Cauldron has b een written to n umerically solv e the EP-QCLE for an arbitrary n um b er of quan tal and classical degrees of freedom. F rom the programmer's p oin t of view, cauldron solv es the follo wing partial dieren tial matrix equation: @ @ t = a + b + ~ c @ @ ~ R + ~ d @ @ ~ P ; (A.1) where the matrices a b and ha v e dimensions of the electronic basis set, and the v ectors ~ c ~ d ~ R and ~ P ha v e dimensions rerecting the classical degrees of freedom. is a function of R P and t while the co ecien t v ectors and matrices can b e functions of these v ariables as w ell as itself (for example, when the Hellmann-F eynman eectiv e force is used). P osed this w a y Eq. A.1 can represen t man y systems and can b e solv ed using a v ariet y of propagation metho ds. T o this end, cauldron w as designed along three orthogonal branc hes, reminiscen t of classes used in ob ject-orien ted design. Ho wev er, while v ery mo dular, cauldron is not truly ob ject-orien ted in either design or implemen tation, and in the in terest of a v oiding confusion and misrepresen tation, ob ject-orien ted terminology will not b e used. The three orthogonal branc hes of cauldron are: 1. System: The system branc h pro vides the time dep enden t co ecien ts for Eq. A.1 completely dening the mo del under study All Hamiltonian terms (quan tal, classical and coupling), as w ell as the eectiv e p oten tial, are con tained within the co ecien ts and ev aluated en tirely within the system branc h. New mo dels 146

PAGE 161

147 are implemen ted b y dev eloping a system branc h sp ecic to the mo del, without reference to an y other branc h. 2. Propagation: The propagation branc h uses the co ecien ts computed b y the system branc h to ev olv e the system. This branc h is general, and a giv en metho d (e.g., relax-and-driv e) ma y b e applied to all systems without mo dication, regardless of system dimension. 3. Prop erties: A t eac h timestep, the prop erties branc h p erforms calculations on the system (e.g. exp ectation v alues of observ ables) and sends the results to appropriate output les for p ostpro cessing. The main cauldron executable calls the appropriate co de from eac h branc h, as sho wn in Figure A{1 Since these co de branc hes are orthogonal to one another, it is a simple matter to add a new system, propagation metho d or prop ert y calculation of arbitrary complexit y without aecting the rest of the co de. A.2 Comp onen t Descriptions A.2.1 Read Input File Eac h sim ulation is describ ed b y a single input le, cauldron.init comp osed of three namelists of data: one for the system, one for the propagation, and one for the prop ert y calculations. T o run a sim ulation, cauldron m ust b e executed in a directory con taining the desired cauldron.init le. A.2.2 System: Get Dieren tial Equation Co ecien ts A t this p oin t, the system branc h pro vides the dieren tial equation co ecien ts at a giv en time. Curren tly implemen ted systems are: Alk ali atom-metal surface. This is a one-dimensional t w o-state mo del of an alk ali atom approac hing a metal surface. Dual a v oided crossing. This system mo dels a collision b et w een t w o n uclei in v olving t w o electronic states and t w o a v oided crossings.

PAGE 162

148 Evolve single timestep Propagation: yes no end begin finished? Properties: Output properties Get diffeq coefficients System: read input file Figure A{1. Flo w c hart describing the cauldron program.

PAGE 163

149 NaI complex. Here w e mo del the NaI complex with the n uclear separation and t w o diabatic electronic surfaces. Bulk liquid helium. This is a three-dimensional mo del of an arbitrary n um b er of classical helium atoms in a p erio dically replicating cen tral v olume. Helium droplet. Here w e mo del a liquid droplet of classically in teracting helium atoms. Lithium-helium cluster. This system mo dels the in teraction of a lithium atom within a cluster of helium atoms in the framew ork of the EP-QCLE, treating the atomic cores quasiclassically and the lithium v alence electron quan tally Their in teraction is describ ed through l -dep enden t semi-lo cal pseudop oten tials and the eectiv e Hellmann-F eynman force. A.2.3 Propagation: Ev olv e Single Timestep The propagation branc h uses the the dieren tial equation co ecien ts to adv ance the system a single timestep. Curren tly implemen ted branc hes are: Relax-and-driv e. This is the mixed quan tum-classical relax-and-driv e algorithm, with the capabilit y to use either a xed or v ariable timestep. The propagation can also b e split b et w een the relaxation and the driving terms, omitting the driving term if desired. V elo cit y V erlet with p erio dic b oundaries. This is a classical v elo cit y V erlet propagation, but using p erio dic b oundary conditions so that particles lea ving the cen tral v olume reapp ear on the opp osite side. V elo cit y V erlet without p erio dic b oundaries. This is the classical v elo cit y V erlet algorithm with innite b oundaries, useful for propagating isolated clusters of helium. A.2.4 Prop erties: Output Prop erties The prop erties branc h outputs v arious quan tities of in terest to sp ecic and predened output les. Curren tly implemen ted prop erties are:

PAGE 164

150 Exp ectation v alue of p osition and momen tum. P osition and momen tum disp ersion. T race (b oth quan tum and classical) of the partially Wigner transformed densit y matrix. State probabilities. Nuclear conguration, for b oth original and smo othed surfaces. T emp erature, kinetic and p oten tial energy Nuclear densit y prole. P air distribution function. Dip ole exp ectation v alues. A.3 Subroutine Details Cauldron has b een written en tirely from scratc h in F ortran 90, and in v olv es nearly 10,000 lines of co de in o v er 100 les. It w ould b e a hop eless task to describ e the details in the space of an app endix. Ho w ev er, a detailed reference man ual 123 has b een written to accompan y cauldron whic h guides the programmer through all asp ects of b oth use and extension of the co de.

PAGE 165

APPENDIX B SPLIT OPERA TOR-F AST F OURIER TRANSF ORM METHOD Consider a quan tal system describ ed b y ( r ; R ; t ). The Born-Opp enheimer separation builds this w a v efunction in terms of a complete and orthonormal set f k ( r ; R ) g of electronic functions, ( r ; R ; t ) = X l l ( r ; R ) ( R ; t ) : (B.1) Beginning with the time dep enden t Sc hr odinger equation, i ~ @ @ t = ^ H ; (B.2) w e can insert a general Hamiltonian in v olving n uclear (N) and electronic (e) co ordinates, ^ H = ^ K N + ^ T e + ^ V N N + ^ V N e + ^ V ee | {z } ^ H e (B.3) so that i ~ @ @ t = [ ^ K N + ^ H e ] : (B.4) Switc hing to Dirac notation for con v enience, and setting ~ = 1, w e m ultiply on the left b y h k j h k j i @ @ t X l j l ij l i = h k j ^ K N + ^ H e ( j l ij l i ) (B.5) so that i @ @ t j k i = h k j ^ K N ( j l ij l i ) + h k j ^ H e j l i | {z } ^ V k l j l i : (B.6) 151

PAGE 166

152 Expanding the rst term on the RHS, h k j ^ K N ( j l ij l i ) = h k j 1 2 M r 2R ( j l ij l i ) = h k j 1 2 M r 2R j l ij l i + j l ir 2R j l i : (B.7) In the diabatic basis, the momen tum coupling h k jr R j l i disapp ears, so that i @ @ t k = 1 2 M r 2R k + ^ V k l l : (B.8) F or a t w o-state system, denoting the column v ector ( 1 ; 2 ), w e can write @ @ t = i 0B@ 1 2 M r 2R 0 0 1 2 M r 2R 1CA | {z } ^ K i 0B@ ^ V 11 ^ V 12 ^ V 21 ^ V 22 1CA | {z } ^ V : (B.9) F ormally w e can ev olv e a timestep t using the time-ordered op erator ^ T : ( t ) = ^ T exp 24 t Z 0 ( i ^ K ( t 0 ) i ^ V ( t 0 )) dt 0 35 (0) : (B.10) If the v ariation of the Hamiltonian is small on the timescale of t w e can write Eq. B.10 as ( t ) = exp h ( i ^ K (0) i ^ V (0)) t i (0) : (B.11) The split op erator metho d breaks up the exp onen tial, ( t ) = exp i t 2 ^ K (0) exp h i t ^ V (0) i exp i t 2 ^ K (0) (0) + O ( t 3 ) : (B.12)

PAGE 167

153 Since ^ K is diagonal but nonlo cal in the n uclear co ordinate space, w e can ev alute its exp onen tial b y computing the deriv ativ es in F ourier space, ~ ( k ) = r 1 2 1 Z 1 ( R ) e ik R dR ) @ 2 @ R 2 ~ ( k ) = k 2 ~ ( k ) On the other hand, ^ V is lo cal but nondiagonal. T o ev aluate its exp onen tial, w e need to diagonalize ^ V : ^ V = D 1 D ) exp [ i t ^ V ] = D exp[ i t ] D 1 = D 0B@ exp ( i t 1 ) 0 0 exp ( i t 2 ) 1CA D 1 : Th us, the split op erator fast F ourier transform metho d ev olv es the quan tal system a single timestep as follo ws: 1. Ev aluate exp [ i t= 2 ^ K ] ( t ) b y computing the second spatial deriv ativ e in F ourier space. 2. Apply exp [ i t ^ V ( t )] b y diagonalizing ^ V ( t ) and then con v erting the exp onential of the matrix in to a matrix of exp onen tials. 3. Apply exp [ i t= 2 ^ K ] once again using the fast F ourier transform. In practice, when sev eral timesteps are computed in a ro w, adjoining exp onen tials of ^ K (i.e., step 1/ follo wing step 3/) can b e com bined in to a single op eration with double the timestep.

PAGE 168

APPENDIX C THE QUALDRON PR OGRAM C.1 Ov erview Qualdron has b een written to n umerically solv e the time dep enden t Sc hr odinger equation for a t w o-state electronic system with a single classical degree of freedom. F rom the programmer's p oin t of view, qualdron solv es the follo wing partial dieren tial equation: i @ @ t = ( ^ K + ^ V ) ; (C.1) where is the column v ector of t w o electronic states, = ( 1 ; 2 ). The basis is assumed to b e diabatic, so that ^ K is diagonal and ^ V con tains the coupling b et w een states. While qualdron has b een written for the t w o-state case, expansion to an arbitrary n um b er of states w ould b e straigh tforw ard, and require the replacemen t of a single subroutine to rerect the m ultidimensional problem. P osed this w a y Eq. C.1 can represen t man y dieren t systems b y c hanging the v alue of the p oten tial coupling elemen ts ^ V in the Hamiltonian. T o this end, qualdron w as designed along three orthogonal branc hes, similar to the design of cauldron These branc hes are: 1. System: The system branc h pro vides the initial v alues and Hamiltonian for Eq. C.1 This section of co de denes the system, and can represen t an y one-dimensional t w o-state mo del that can b e describ ed b y a Hamiltonian in the diabatic represen tation. 2. Propagation: The propagation branc h uses the Hamiltonian computed b y the system branc h to ev olv e the system. The propagation co de is general, and 154

PAGE 169

155 a giv en metho d (e.g., split op erator) ma y b e applied to all dened systems without mo dication. 3. Prop erties: A t eac h timestep, the propagation co de calls the prop erties co de, whic h mak es system calculations and outputs appropriate les. The main qualdron executable calls the appropriate co de from eac h area to create the sim ulation, as sho wn in Figure C{1 Since these co de branc hes are for the most part orthogonal to one another, it is a simple matter to add a new system, a new propagation metho d, or new prop ert y calculations without aecting the rest of the co de. C.2 Comp onen t Descriptions C.2.1 Read Input File Eac h sim ulation is describ ed b y a single input le, qualdron.init comp osed of three namelists of data: one for the system, one for the propagator, and one for the prop erties. T o run a sim ulation, qualdron m ust b e executed in a directory con taining the desired qualdron.init le. C.2.2 System: Get Hamiltonian Matrix Elemen ts A system pro vides the Hamiltonian matrix elemen ts at a giv en time. Curren tly implemen ted systems are: Alk ali atom-metal surface. Dual a v oided crossing. NaI complex. C.2.3 Propagation: Ev olv e Single Timestep The propagation co de uses the Hamiltonian elemen ts to adv ance the system a single timestep. Curren tly the only implemen ted propagation is the split op erator fast F ourier transform metho d.

PAGE 170

156 Propagation: Evolve single trajectory Get Hamiltonian System: yes no end begin finished? Properties: Output properties read input file Figure C{1. Flo w c hart describing the qualdron program.

PAGE 171

157 C.2.4 Prop erties: Output Prop erties The prop erties co de outputs v arious quan tities of in terest to sp ecic and predened output les. Curren tly implemen ted prop erties are: Exp ectation v alue of p osition and momen tum. P osition and momen tum disp ersion. State probabilities. Nuclear congurations. C.3 Subroutine Details Qualdron has b een written en tirely from scratc h in F ortran 90, and in v olv es nearly 2,500 lines of co de in o v er 100 les. Since these details could not b e adequately describ ed b y an app endix, a detailed reference man ual 124 has b een written to accompan y qualdron whic h guides the programmer through all asp ects of b oth use and extension of the co de. In particular, instructions for c hanging qualdron from a t w o-state to a m ultistate system are presen ted in detail.

PAGE 172

REFERENCE LIST [1] Aatto Laaksonen and Y ao quan T u. Mole cular Dynamics: F r om Classic al to Quantum Metho ds v olume 7 of The or etic al and Computational Chemistry c hapter 1. Elsevier, Amsterdam, 1999. [2] J. Bric kmann and U. Sc hmitt. Mole cular Dynamics: F r om Classic al to Quantum Metho ds v olume 7 of The or etic al and Computational Chemistry c hapter 2. Elsevier, Amsterdam, 1999. [3] Nancy Makri. Time-dep enden t quan tum metho ds for large systems. A nnual R eview of Physic al Chemistry 50:167{191, 1999. [4] M. Hillery R. F. O'Connell, M. O. Scully and E. P Wigner. Distribution functions in ph ysics: F undamen tals. Physics R ep orts 106(3):121{167, 1984. [5] Chiac hin Tso o, Dario A. Estrin, and Sherwin J. Singer. Electronic energy shifts of a so dium atom in argon clusters b y sim ulated annealing. Journal of Chemic al Physics 93(10):7187{7200, 1990. [6] Kenneth Haug and Horia Metiu. The absorption sp ectrum of a p otassium atom in a Xe cluster. Journal of Chemic al Physics 95(8):5670{5680, 1991. [7] Kenneth Haug and Horia Metiu. Absorption sp ectrum calculations using mixed quan tum-Gaussian w a v e pac k et dynamics. Journal of Chemic al Physics 99(9):6253{6263, 1993. [8] Glenn Mart yna, Ching Cheng, and Mic hael L. Klein. Electronic states and dynamical b eha vior of LiXe n and CsXe n clusters. Journal of Chemic al Physics 95(2):1318{1336, 1991. [9] F. Stienk emeier, J. Higgins, C. Callegari, S. I. Kanorsky W. E. Ernst, and G. Scoles. Sp ectroscop y of alk ali atoms (Li, Na, K) attac hed to large helium clusters. Zeitschrift f ur Physik D 38:253{263, 1996. [10] Y ongkyung Kw on and K. Birgitta Whaley A tomic-scale quan tum solv ation structure in sup erruid helium-4 clusters. Physic al R eview L etters 83(20):4108{ 4111, 1999. [11] F rank Stienk emeier and Andrey F. Vileso v. Electronic sp ectroscop y in He droplets. Journal of Chemic al Physics 115(22):10119{10137, 2001. [12] R. N. Barnett and K. B. Whaley Molecules in helium clusters: SF 6 He N Journal of Chemic al Physics 99(12):9730{9744, 1993. 158

PAGE 173

159 [13] Akira Nak a y ama and Noic hi Y amashita. P ath in tegral Mon te Carlo study on the structure and absorption sp ecta of alk ali atoms (Li, Na, K) attac hed to sup erruid helium clusters. Journal of Chemic al Physics 114(2):780{790, 2001. [14] J. P eter T o ennies and Andrei F. Vileso v. Sp ectroscop y of atoms and molecules in liquid helium. A nnual R eview in Physic al Chemistry 49:1{41, 1998. [15] P W. A tkins and R. S. F riedman. Mole cular Quantum Me chanics Oxford Univ ersit y Press, Oxford, 3rd edition, 1997. [16] Ro dney J. Bartlett, editor. R e c ent A dvanc es in Couple d Cluster Metho ds v olume 3 of R e c ent A dvanc es in Computational Chemistry W orld Scien tic, Singap ore, 1997. [17] Mic hael Springb org. Metho ds of Ele ctr onic-Structur e Calculations: F r om Mole cules to Solids Wiley New Y ork, 2000. [18] A. Szab o and N. S. Ostlund. Mo dern Quantum Chemistry Do v er, Mineola, New Y ork, 1996. [19] Douglas L. Strout and Gusta v o E. Scuseria. A quan titativ e study of the scaling prop erties of the Hartree-Fo c k metho d. Journal of Chemic al Physics 102(21):8448{8452, 1995. [20] Da vid J. Griths. Intr o duction to Quantum Me chanics Pren tice Hall, Upp er Saddle Riv er, New Jersey 1994. [21] Karl Blum. Density Matrix The ory and Applic ations Plen um Press, New Y ork, 2nd edition, 1981. [22] John D. Simon, editor. Ultr afast Dynamics of Chemic al Systems v olume 7 of Understanding Chemic al R e activity Klu w er Academic Publishers, Dordrec h t, 1994. [23] R. Guan tes and S. C. F aran tos. High order nite dierence algorithms for solving the Sc hro dinger equation in molecular dynamics. Journal of Chemic al Physics 111(24):10827{10835, 1999. [24] D. A. Maziotti. Sp ectral dierence metho ds for solving dieren tial equations. Chemic al Physics L etters 299(5):473{480, 1999. [25] Mark Thac h uk and George C. Sc hatz. Time-dep enden t metho ds for calculating thermal rate co ecien ts using rux correlation functions. Journal of Chemic al Physics 97(10):7297{7313, 1992. [26] P atric k L. Nash and L. Y. Chen. Ecien t nite dierence solutions to the timedep enden t Sc hr odinger equation. Journal of Computational Physics 130:266{ 268, 1997.

PAGE 174

160 [27] D. Koslo and R. Koslo. A Fourier metho d solution for the time dep enden t Sc hro dinger equation as a to ol in molecular dynamics. Journal of Computational Physics 52(1):35{53, 1983. [28] T. W. K orner. F ourier A nalysis Univ ersit y Press, Cam bridge, 1988. [29] C. Zenger. Par al lel A lgorithms for Partial Dier ential Equations v olume 31 of Notes on Numeric al Fluid Me chanics c hapter Sparse Grids. View eg, Braunsc h w eig, 1991. [30] Dmitrii V. Shalashilin and Mark S. Child. Time dep enden t quan tum propagation in phase space. Journal of Chemic al Physics 113(22):10028{10036, 2000. [31] Gert D. Billing. Time-dep enden t quan tum dynamics in a Gauss-Hermite basis. Journal of Chemic al Physics 110(12):5526{5537, 1999. [32] Herb ert Goldstein. Classic al Me chanics Addison-W esley Massac h usetts, 2nd edition, 1980. [33] Harv ey Gould and Jan T ob o c hnik. A n Intr o duction to Computer Simulation Metho ds Addison-W esley Reading, Massac h usetts, 2nd edition, 1996. [34] M. P Allen and D. J. Tildesley Computer Simulation of Liquids Oxford Univ ersit y Press, Oxford, 1987. [35] J. Barnes and P Hut. A hierarc hical O(N log N) force calculation algorithm. Natur e 324:446{449, 1986. [36] L. Greengard and V. Rokhlin. A fast algorithm for particle sim ulations. Journal of Computational Physics 73:325{348, 1987. [37] Leslie Greengard. F ast algorithms for classical ph ysics. Scienc e 265:909{914, 1994. [38] Celeste Sagui and Thomas A. Darden. Molecular dynamics sim ulations in biomolecules: Long-range electrostatic eects. A nnual R eview of Biophysics and Biomole cular Structur e 28:155{179, 1999. [39] Da vid A. Mic ha. Time-dep enden t man y-electron treatmen t of electronic energy and c harge transfer in atomic collisions. Journal of Physic al Chemistry A 103(38):7562{7574, 1999. [40] Da vid A. Mic ha. Time-ev olution of m ulticonguration densit y functions driv en b y n uclear motions. International Journal of Quantum Chemistry 60:109{118, 1996. [41] Da vid A. Mic ha. T emp oral rearrangemen t of electronic densities in slo w atomic collisions. International Journal of Quantum Chemistry 51:499{518, 1994.

PAGE 175

161 [42] Lic hang W ang and Anne B. McCo y Time-dep enden t studies of reaction dynamics: a test of mixed quan tum/classical time-dep enden t self-consisten t eld appro ximations. Physic al Chemistry and Chemic al Physics 1:1227{1235, 1999. [43] Eric J. Heller. Time-dep enden t approac h to semiclassical dynamics. Journal of Chemic al Physics 62(4):1544{1555, 1975. [44] Eric J. Heller. Time dep enden t v ariational approac h to semiclassical dynamics. Journal of Chemic al Physics 64(1):63{73, 1976. [45] John C. T ully T ra jectory surface hopping approac h to nonadiabatic molecular collisions: the reaction of H + with D 2 Journal of Chemic al Physics 55(2):562{ 572, 1971. [46] E. Deumens, A. Diz, H. T a ylor, and Y. Ohrn. Time-dep enden t dynamics of electrons and n uclei. Journal of Chemic al Physics 96(9):6820{6833, 1992. [47] E. Deumens, A. Diz, R. Longo, and Y. Ohrn. Time-dep enden t theoretical treatmen ts of the dynamics of electrons and n uclei in molecular systems. 66(3):917{983, 1994. [48] Nancy Makri. F eynman path in tegration in quan tum dynamics. Computer Physics Communic ations 63:389{414, 1991. [49] R. F eynman and A. R. Hibbs. Quantum Me chanics and Path Inte gr als McGra w-Hill, New Y ork, 1965. [50] M. Ben-Nun and T o dd J. Martinez. A m ultiple spa wning approac h to tunneling dynamics. Journal of Chemic al Physics 112(14):6113{6121, 2000. [51] John C. T ully Molecular dynamics with electronic transitions. Journal of Chemic al Physics 93(2):1061{1071, 1990. [52] John R. Klauder and Bo-Sture Sk agerstam. Coher ent States: Applic ations in Physics and Mathematic al Physics W orld Scien tic, Singap ore, 1985. [53] La wrence S. Sc h ulman. T e chniques and Applic ations of Path Inte gr ation Wiley New Y ork, 1981. [54] Nancy Makri. Numerical path in tegral tec hniques for long time dynamics of quan tum dissipativ e systems. Journal of Mathematic al Physics 36(5):2430{ 2457, 1995. [55] Nancy Makri and Dmitrii E. Mak aro v. T ensor propagator for iterativ e quantum time ev olution of reduced densit y matrices. I I. n umerical metho dology Journal of Chemic al Physics 102(11):4611{4618, 1995.

PAGE 176

162 [56] Xiong Sun, Haobin W ang, and William H. Miller. Semiclassical theory of electronically nonadiabatic dynamics: Results of a linearized appro ximation to the initial v alue represen tation. Journal of Chemic al Physics 109(17):7064{ 7074, 1998. [57] Xiong Sun and William H. Miller. Mixed semiclassical-classical approac hes to the dynamics of complex molecular systems. Journal of Chemic al Physics 106(3):916{927, 1997. [58] Xiong Sun and William H. Miller. Semiclassical initial v alue represen tation for electronically nonadiabatic molecular dynamics. Journal of Chemic al Physics 106(15):6346{6353, 1997. [59] U. F ano. Description of states in quan tum mec hanics b y densit y matrix and op erator tec hniques. R eviews of Mo dern Physics 29(1):74{93, 1957. [60] Bret Jac kson. Reduced densit y matrix description of gas-solid in teractions: Scattering, trapping and desorption. Journal of Chemic al Physics 108(3):1131{1139, 1998. [61] Jonathan Grad, Yi Jing Y an, Azizul Haque, and Shaul Muk amel. Reduced equations of motion for semiclassical dynamics in phase space. Journal of Chemic al Physics 86(6):3441{3454, 1987. [62] Y ang Zhao, Satoshi Y ok o jima, and GuanHua Chen. Reduced densit y matrix and com bined dynamics of electrons and n uclei. Journal of Chemic al Physics 113(10):4016{4027, 2000. [63] Herman J. C. Berendsen and Janez Ma vri. Quan tum sim ulation of reaction dynamics b y densit y matrix ev olution. Journal of Physic al Chemistry 97:13464{13468, 1993. [64] Herman J. C. Berendsen and Janez Marvi. Approac h to nonadiabatic transitions b y densit y matrix ev olution and molecular dynamics sim ulation. International Journal of Quantum Chemistry 57:975{983, 1996. [65] Marc F. Lensink, Janez Ma vri, and Herman J. C. Berendsen. Sim ultaneous in tegration of mixed quan tum-classical systems b y densit y matrix ev olution equations using in teraction represen tation and adaptiv e time step in tegrator. Journal of Computational Chemistry 17(11):1287{1295, 1996. [66] Da vid A. Mic ha and Keith Runge. Time-dep enden t man y-electron approac h to slo w ion-atom collisions: The coupling of electronic and n uclear motions. Physic al R eview A 50(1):322{336, 1994. [67] Keith Runge and Da vid A. Mic ha. Time-dep enden t approac h to slo w ion-atom collisions for systems with one activ e electron. Physic al R eview A 53(3):1388{ 1399, 1996.

PAGE 177

163 [68] Keith Runge and Da vid A. Mic ha. Time-dep enden t man y-electron approac h to slo w ion-atom collisions for systems with sev eral activ e electrons. Physic al R eview A 62(2):22703{1{22703{10, 2000. [69] Keith H. Hughes and Rob ert E. Wy att. T ra jectory approac h to dissipativ e quan tum phase dynamics: Application to barrier scattering. Journal of Chemic al Physics 120(9):4089{4097, 2004. [70] Irene Burghardt and G erard P arlan t. On the dynamics of coupled Bohmian and phase-space v ariables: A new h ybrid quan tum-classical approac h. Journal of Chemic al Physics 120(7):3055{3058, 2004. [71] I. Burghardt and L. S. Cederbaum. Hydro dynamic equations for mixed quan tum states. I. General form ulation. Journal of Chemic al Physics 115(22):10303{10311, 2001. [72] I. Burghardt and L. S. Cederbaum. Hydro dynamic equations for mixed quan tum states. I I. Coupled electronic states. Journal of Chemic al Physics 115(22):10312{10322, 2001. [73] Jerem y B. Maddo x and Eric R. Bittner. Quan tum relaxation dynamics using Bohmian tra jectories. Journal of Chemic al Physics 115(14):6309{6316, 2001. [74] Da vid A. Mic ha and Brian Thorndyk e. Dissipativ e dynamics in man y-atom systems: A densit y matrix treatmen t. International Journal of Quantum Chemistry 2002. T o b e published. [75] Arnoldo Donoso and Craig C. Martens. Sim ulation of coheren t nonadiabatic dynamics using classical tra jectories. Journal of Physic al Chemistry A 102:4291{4300, 1998. [76] Arnaldo Donoso and Craig C. Martens. Semiclassical m ultistate Liouville dynamics in the adiabatic represen tation. Journal of Chemic al Physics 112(9):3980{3989, 2000. [77] Arnaldo Donoso, Daniela Kohen, and Craig C. Martens. Sim ulation of nonadiabatic w a v e pac k et in terferometry using classical tra jectories. Journal of Chemic al Physics 112(17):7345{7354, 2000. [78] Stev e Nielsen, Ra ymond Kapral, and Gio v anni Ciccotti. Mixed quan tumclassical surface hopping dynamics. Journal of Chemic al Physics 112(15):6543{6553, 2000. [79] Ch un-Cheng W an and Jerem y Sc hoeld. Mixed quan tum-classical molecular dynamics: Asp ects of the m ultithreads algorithm. Journal of Chemic al Physics 113(17):7047{7054, 2000.

PAGE 178

164 [80] Ch un-Cheng W an and Jerem y Sc hoeld. Exact and asymptotic solutions of the mixed quan tum-classical Liouville equation. Journal of Chemic al Physics 112(10):4447{4459, 2000. [81] Ch un-Cheng W an and Jerem y Sc hoeld. Solutions of mixed quan tum-classical dynamics in m ultiple dimensions using classical tra jectories. Journal of Chemic al physics 116(2):494{506, 2002. [82] Illia Horenk o, Christian Salzmann, Burkhard Sc hmidt, and Christof Sc h utte. Quan tum-classical Liouville approac h to molecular dynamics: Surface hopping Gaussian phase-space pac k ets. Journal of Chemic al Physics 117(24):11075{ 11088, 2002. [83] D. A. Mic ha and B. Thorndyk e. Dissipativ e dynamics in man y-atom dynamics: A densit y matrix treatmen t. International Journal of Quantum Chemistry 90(2):759{771, 2001. [84] Jo el M. Cohen and Da vid A. Mic ha. Electronically diabatic atom-atom collisions: A self-consisten t eik onal appro ximation. Journal of Chemic al Physics 97(2):1038{1052, 1992. [85] Shin-Ic hi Sa w ada and Horia Metiu. A Gaussian w a v e pac k et metho d for studying time dep enden t quan tum mec hanics in a curv e crossing system: Lo w energy motion, tunneling, and thermal dissipation. Journal of Chemic al Physics 84(11):6293{6311, June 1986. [86] V olk er Engel and Horia Metiu. A quan tum mec hanical study of predisso ciation dynamics of NaI excited b y a fem tosecond laser pulse. Journal of Chemic al Physics 90(11):6116{6128, 1989. [87] J. P Visticot, P de Pujo, J. M. Mestdaugh, A. Lallemen t, J. Berlande, O. Sublemon tier, P Meynadier, and J. Cuv ellier. Exp erimen t v ersus molecular dynamics sim ulation: Sp ectroscop y of Ba-(Ar) n clusters. Journal of Chemic al Physics 100(1):158{164, 1994. [88] Thomas Sc hr oder, Reinhard Sc hink e, Suy an Liu, Zlatk o Bacic, and Jules W. Mosk o witz. Photo disso ciation of HF in Ar n HF ( n = 1 14 ; 54) v an der Waals clusters: Eects of the solv en t cluster size on the solute fragmen tation dynamics. Journal of Chemic al Physics 103(21):9228{9241, 1995. [89] M. A. Osb orne, M. A. Ga v eau, C. Gee, O. Sublemon tier, and J. M. Mestdagh. Dynamics of the deactiv ation and desorption of Ba atoms from Ar clusters. Journal of Chemic al Physics 106(4):1449{1462, 1997. [90] E. Czuc ha j, F. Reb en trost, H. Stoll, and H. Preuss. Pseudop oten tial calculations for the p oten tial energies of LiHe and BaHe. Chemic al Physics 196:37{46, 1995.

PAGE 179

165 [91] E. Czuc ha j, F. Reb en trost, H. Stoll, and H. Preuss. Semi-lo cal pseudop otential calculations for the adiabatic p oten tials of alk ali-neon systems. Chemic al Physics 136:79{94, 1989. [92] J. P ascale. Use of l -dep enden t pseudop oten tials in the study of alk ali-metal-He systems. the adiabatic molecular p oten tials. Physic al R eview A 28(2):632{643, 1983. [93] T. Grycuk, W. Behmen burg, and V. Staemmler. Quan tum calculation of the excitation sp ectra of Li He probing in teraction p oten tials and dip ole momen ts. Journal of Physics B 34, 2001. [94] W. Behmen burg, A. Mak onnen, A. Kaiser, F. Reb en trost, V. Staemmler, M. Jungen, G. P eac h, A. Devdariani, S. Tserk o vn yi, A. Zagrebin, and E. Cruc ha j. Optical transitions in excited alk ali + rare-gas collision molecules and related in teratomic p oten tials: Li + He. Journal of Physics B 29:3891{ 3910, 1996. [95] V. S. Filino v, Y u. V. Medv edev, and V. L. Kamskyi. Quan tum dynamics and Wigner represen tation of quan tum mec hanics. Mole cular Physics 85(4):711{ 726, 1995. [96] B. R. McQuarrie, T. A. Osb orn, and G. C. T abisz. Semiclassical Mo y al quantum mec hanics for atomic systems. Physic al R eview A 58(4):2944{2961, 1998. [97] E. Wigner. On the quan tum correction for thermo dynamic equilibrium. Physic al R eview 40, 1932. [98] K. Singer and W. Smith. Quan tum dynamics and the Wigner-Liouville equation. Chemic al Physics L etters 167(4):298{304, 1990. [99] Sarah John and E. A. Remler. Solution of the quan tum Liouville equation as a sto c hastic pro cess. A nnals of Physics 180:152{165, 1987. [100] Sarah John and John W. Wilson. Quan tum dynamics as a sto c hastic pro cess. Physic al R eview E 49(1):145{156, 1994. [101] Ra ymond Kapral and Gio v anni Ciccotti. Mixed quan tum-classical dynamics. Journal of Chemic al Physics 110(18):8919{8929, 1999. [102] J. M. Thijssen. Computational Physics Cam bridge Univ ersit y Press, Cambridge, United Kingdom, 1999. [103] P aul A. Fish wic k. Simulation Mo del Design and Exe cution Pren tice Hall, Upp er Saddle Riv er, New Jersey 1995. [104] Mic hael Baer and Rob ert Engelman. A study of the diabatic electronic represen tation within the Born-Opp enheimer appro ximation. Mole cular Physics 75(2):293{303, 1992.

PAGE 180

166 [105] T. P ac her, L. S. Cederbaum, and H. K opp el. Appro ximately diabatic states from blo c k diagonalization of the electronic Hamiltonian. Journal of Chemic al Physics 89(12):7367{7381, 1988. [106] A. Thiel and H. K opp el. Prop osal and n umerical test of a simple diabatization sc heme. Journal of Chemic al Physics 110(19):9371{9383, 1999. [107] M. D. F eit, J. A. Flec k, and A. Steiger. Solution of the Sc hr odinger equation b y a sp ectral metho d. Journal of Computational Physics 47:412{433, 1982. [108] C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. F eit, R. F riesner, A. Guldb erg, A. Hammeric h, G. Jolicard, W. Karrlein, H.-D. Mey er, N. Lipkin, O. Roncero, and R. Koslo. A comparison of dieren t propagation sc hemes for the time dep enden t Sc hr odinger equation. Journal of Computational Physics 94:59{80, 1991. [109] Andr es Rey es. Density Matrix The ory and Computational Asp e cts of A tomic Col lisions Including Spin-Orbit R e c oupling PhD thesis, Univ ersit y of Florida, Gainesville, FL, 2003. [110] Ronald A. Aziz, M. J. Slaman, A. Koide, A. R. Allnatt, and William J. Meath. Exc hange-Coulom b p oten tial energy curv es for He-He, and related ph ysical prop erties. Mole cular Physics 77(2):321{337, 1992. [111] Da vid J. Griths. Intr o duction to Ele ctr o dynamics Pren tice-Hall, Englew o o d Clis, New Jersey 2nd edition, 1989. [112] S. Obara and A. Saik a. Ecien t recursiv e computation of molecular in tegrals o v er Cartesian Gaussian functions. Journal of Chemic al Physics 84(7):3963{ 3974, 1986. [113] S. Obara and A. Saik a. General recurrence form ulas for molecular in tegrals o v er Cartesian Gaussian functions. Journal of Chemic al Physics 89(3):1540{ 1559, 1988. [114] L. E. McMurc hie and E. R. Da vidson. Calculation of in tegrals o v er ab initio pseudop oten tials. Journal of Computational Physics 44:289, 1981. [115] P Sc h w erdtfeger and H. Silb erbac h. Multicen ter in tegrals o v er long-range op erators using cartesian Gaussian functions. Physic al R eview A 37:2834, 1988. [116] J. A. North b y Exp erimen tal studies of helium droplets. Journal of Chemic al Physics 115(22):10065{10077, 2001. [117] Martin H. M user and Erik Luijten. On quan tum eects near the liquid-v ap or transition in helium. Journal of Chemic al Physics 116(4):1621{1628, 2002.

PAGE 181

167 [118] D. M. Cep erley P ath in tegrals in the theory of condensed helium. R eviews of Mo dern Physics 67(2):279{355, 1995. [119] K. Birgitta Whaley Structure and dynamics of quan tum clusters. International R eviews in Physic al Chemistry 13(1):41{84, 1994. [120] R. N. Barnett and K. B. Whaley V ariational and diusion Mon te Carlo tec hniques for quan tum clusters. Physic al R eview A 47(5):4082{4098, 1993. [121] F. Stienk emeier, F. Meier, A. H agele, H. O. Lutz, E. Sc hreib er, C. P Sc h ultz, and I. V. Hertel. Coherence and relaxation in p otassium-dop ed helium droplets studied b y fem tosecond pump-prob e sp ectroscop y Physic al R eview L etters 83(12):2320{2323. [122] James Reho, Carlo Callegari, John Higgins, W olfgang E. Ernst, Kevin K. Lehmann, and Giancin to Scoles. Spin-orbit eects in the formation of the na-he excimer on the surface of He clusters. F ar aday Discussions (108):161{ 174, 1997. [123] B. Thorndyk e. Cauldr on r efer enc e manual (unpublishe d) 2004. [124] B. Thorndyk e. Qualdr on r efer enc e manual (unpublishe d) 2004.

PAGE 182

BIOGRAPHICAL SKETCH Brian Thorndyk e w as b orn on Octob er 17 th 1969, in Calgary Alb erta, Canada. As he grew up, his mother, Carol Thorndyk e, w as an elemen tary sc ho ol teac her while his father, Gerry Thorndyk e, taugh t high-sc ho ol biology and general science. Their p ositiv e academic inruence w as notable, as Brian ended up graduating high sc ho ol with an adv anced In ternational Baccalaureate diploma, and indeed w as oered a full y ear's credit at the Univ ersit y of T oron to in the Departmen t of Ph ysics. Rather than mo v e to T oron to, Brian c hose to remain an additional y ear in Calgary immerse himself in v arious F renc h programs at the Univ ersit y of Calgary and then en ter in to a bac helor's program in ph ysics at an all-F renc h univ ersit y in Mon treal. He remained in Mon treal for b oth his bac helor's and master's degrees. After his time in Mon treal, he sp en t a y ear at the Univ ersit y of British Colum bia, w orking on pro jects in the Computer Science and Electrical Engineering Departmen ts. He w ould ha v e remained in computer science in V ancouv er had he not b een en ticed to mo v e to Florida, and complete a master's degree in computer science with Dr. P aul Fish wic k. Tw o y ears later, ho w ev er, Brian decided he had to return to his rst lo v e, ph ysics, and transferred to the Ph ysics Departmen t in the Quan tum Theory Pro ject with Dr. Da vid Mic ha. Tw o y ears b efore graduation, Brian's father w as diagnosed with terminal cancer, and Brian to ok a y ear of semi-lea v e to sp end time with his paren ts and supp ort them emotionally When his father passed a w a y in Jan uary of 2003, Brian returned to nish his Ph.D., and has since accepted a p ostdo ctorate p osition in the Departmen t of Radiation Oncology Division of Radiation Ph ysics at Stanford Univ ersit y 168


Permanent Link: http://ufdc.ufl.edu/UFE0004287/00001

Material Information

Title: Quantum Dynamics of Finite Atomic and Molecular Systems Through Density Matrix Methods
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0004287:00001

Permanent Link: http://ufdc.ufl.edu/UFE0004287/00001

Material Information

Title: Quantum Dynamics of Finite Atomic and Molecular Systems Through Density Matrix Methods
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0004287:00001


This item has the following downloads:


Full Text











QUANTUM DYNAMICS OF FINITE ATOMIC AND MOLECULAR SYSTEMS
THROUGH DENSITY MATRIX METHODS
















By

BRIAN THORNDYKE


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


2004

































Copyright 2004

by

Brian Thorndyke

















To my father,

Gerry Thorndyke















ACKNOWLEDGMENTS

First of all, I would like to thank my parents, Gerry and Carol Thorndyke,

whose unwavering support and love my entire life have allowed me to pursue my

dreams.

On the academic side, I would like to thank my advisor, Dr. David Micha, for his

excellent guidance and encouragement throughout my doctoral work. I would also

like to thank the following colleagues in Quantum Theory Project and the Physics

department for their friendship and insights during my stay in Gainesville: Jim

Cooney, Herbert DaCosta, Alex Pacheco, Dave Red, Andres Reyes, Akbar Salam,

Alberto Santana and Zhigang Yi.

On a personal level beyond the Physics department, I would be remiss if I

didn't express my love and gratitude to Natasha Lepor6. She has been my -..11i

sister" for over a decade, and I hope our lives will continue to run with fascinating

parallels and intertwine for many decades to come! I'd also like to recognize Albert

Vernon who, since our early days in the Computer Science department, has been

my partner in our relentless pursuit of aesthetically pleasing code. His friendship

has both contributed to some of the best of times and helped me through some of

the worst over the last 8 years. Finally, I'd like to express my appreciation to Mike

Kuban and Rob Thorndyke for being with me in spirit throughout my Ph.D., and

for always providing wonderful avenues of escape and adventure year after year!















TABLE OF CONTENTS
Page

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

LIST OF TABLES ........... ...... ........ ...... ix

LIST OF FIGURES ................................ x

ABSTRACT ................... ............. xiii

CHAPTER

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

1.1 Overview of Classical and Quantum Dynamics ........... 2
1.2 Approximations to Quantum Dynamics ...... ........ 5
1.2.1 Wavefunction-Based Approaches ....... ..... ... 5
1.2.2 Density Operator Approaches ...... .......... 7
1.3 Quantum-Classical Liouville Equation ... . 8
1.4 Our Approach ....... ......... ... ...... 10
1.5 Simple One-Dimensional Two-State Models . ... 11
1.6 Lithium-Helium Clusters ................ .... .. 11
1.7 Outline of the Dissertation ................. .. 12

2 QUANTUM-CLASSICAL LIOUVILLE EQUATION: FORMULATION 14

2.1 Introduction .................. ........... .. 14
2.2 Quantum Liouville Equation ............ .. .. 14
2.3 W igner Representation .................. ... 15
2.4 Quantum-Classical Liouville Equation . 18
2.5 Effective Potential .................. ........ .. 20

3 QUANTUM CLASSICAL LIOUVILLE EQUATION: COMPUTATIONAL
ASPECTS ................ .. ........ ..... 22

3.1 Introduction .................. ........... .. 22
3.2 Trajectory Solution .................. .. .... .. .. 22
3.3 Electronic Basis Set .................. ..... 23
3.4 Nuclear Phase Space Grid ................... 24
3.5 Relax-and-Drive Algorithm ................. .. 25
3.5.1 W Independent of Time .... .. 26
3.5.2 W Dependent on Time ....... . ... 26
3.5.3 Velocity Verlet for the Classical Evolution ... 31











3.5.4 Algorithm Details .. ...................
3.6 Computing Observables .. ....................
3.6.1 Operators in an Orthonormal Basis .. ..........
3.6.2 Population Analysis .. ...................
3.6.3 Expectation Values .. ..................
3.6.4 Hamiltonian Eigenstates and Eigenvalues .. ........
3.7 Programming Details .. .....................
3.7.1 Orthogonality of Code Development .. ...........
3.7.2 Extensibility . . . .

4 ONE-DIMENSIONAL TWO-STATE MODELS .. ...........

4.1 Introduction . . . .
4.2 Effective-Potential QCLE in the Diabatic Representation .....


4.3 Near-Resonant Electron Transfer Between an Alkali
Metal Surface ..... ..............
4.3.1 M odel Details ... .. .. .. .. ..
4.3.2 Properties of Interest .........
4.3.3 Results. . .
4.4 Binary Collision Involving Two Avoided Crossings .
4.4.1 M odel Details .............
4.4.2 Properties of Interest .........
4.4.3 Results... . .
4.5 Photoinduced Dissociation of a Diatomic System .
4.5.1 M odel Details ... .. .. .. .. ..
4.5.2 Properties of Interest .........
4.5.3 Results...... . .
4.6 Comparison Using Variable and Constant Timesteps .
4.7 C conclusion . . .


Atom and
. 41
. 41
. 42
. 47
. 59
. 59
. 60
. 62
. 63
. 63
. 68
. 70
. 72
. 72


5 ALKALI ATOM-RARE GAS CLUSTERS: GENERAL FORMULATION 80


Introduction . . .
Physical System .. ... .. .. .. .. ..
Properties of Interest ...........
Hamiltonian for Alkali-Rare Gas Pairs ...
Hamiltonian for the Alkali-Rare Gas Cluster .
Electronic Spectral Calculations ......
Electronic Basis of Gaussian Atomic Functions .
5.7.1 Equations of Motion .........
5.7.2 Overlap Matrix Elements ......
5.7.3 Kinetic Energy Matrix Elements ..
5.7.4 Coulomb Matrix Elements ......
5.7.5 Momentum Coupling Matrix Elements .
5.7.6 Dipole Matrix Elements .......
5.7.7 Pseudopotential Matrix Elements ..


. 80
. 80
. 81
. 81
. 84
. 86
. 87
. 87
. 89
. 90
. 90
. 91
. 91
. 92










5.8 Computing the Quasiclassical Trajectory .
5.9 Computational Details ...........
5.10 C conclusion . . .

6 LITHIUM-HELIUM CLUSTERS .........

6.1 Introduction . . .
6.2 Description of the System ..........
6.3 Properties to be Investigated ........
6.4 Preparation of Lithium-Helium Clusters ..
6.4.1 Bulk Helium ... .. .. .. .. ..
6.4.2 Liquid Helium Droplets .......
6.4.3 Lithium-Helium Interactions ....
6.5 Results: Lithium Inside the Helium Cluster .
6.6 Results: Lithium on the Helium Cluster Surface .
6.6.1 Dynamics of Li(2pa) ..
6.6.2 Dynamics of Li(2p7) ..
6.7 C conclusion . . .


. 93
. 93
. 97

. 99

. 99
. 99
. 101
. 102
. 102
. 109
. 109
. 119
. 121
. 124
. 130
. 137


7 CONCLUSION .. .............


Effective Potential Quantum-Classical Liouville Equation 140
One-Dimensional Two-State Models ... . 141
Alkali-Rare Gas Clusters .... . 142
Software Development .................. ...... 144
Future Work ....... ...................... 145


APPENDIX


A THE CAULDRON PROGRAM .................. ..... 146

A.1 Overview ........................ ....... 146
A.2 Component Descriptions ................ .... 147
A.2.1 Read Input File ... . ..... 147
A.2.2 System: Get Differential Equation Coefficients 147
A.2.3 Propagation: Evolve Single Timestep . ... 149
A.2.4 Properties: Output Properties . 149
A.3 Subroutine Details ................... 150


B SPLIT OPERATOR-FAST FOURIER TRANSFORM METHOD .

C THE QUALDRON PROGRAM. . .....

C 1 O verview . . .. ..
C.2 Component Descriptions . .....
C.2.1 Read Input File . .
C.2.2 System: Get Hamiltonian Matrix Elements .
C.2.3 Propagation: Evolve Single Timestep . .


. 151

. 154

. 154
. 155
. 155
. 155
. 155









C.2.4 Properties: Output Properties ... . 157
C.3 Subroutine Details .................. ..... 157

REFERENCE LIST .................. ............. 158

BIOGRAPHICAL SKETCH .................. ......... 168

















































viii















LIST OF TABLES
Table page

4-1 Parameters used in the N..-- n f. .:e and Li-surface models. ...... ..42

4-2 Parameters used in the dual avoided crossing collision model. 60

4-3 Parameters used in the Nal complex model. ............ ..68

5-1 Pseudopotential rotation for d-function mixing. ..... 98

6-1 Parameters for the He-He interaction from Aziz (VA). ... 106

6-2 Parameters for the correction to the He-He interaction (V) 106

6-3 Parameters for the e--Li interaction. ................. ..113

6-4 Parameters for the e--He interaction. ................ ..113

6-5 Parameters for the Li-He core interaction. .. . ..... 114















LIST OF FIGURES
Figure page

4-1 Potential curves for Hamiltonian I: Na incident upon a metal surface. 43

4-2 Potential curves for Hamiltonian II: Li incident upon a metal surface. 44

4-3 T(R) at t = 0 au, for the N..--ii if.,.e model. This wavefunction is
evolved through the SO-FFT algorithm. ............. ..48

4-4 F11 at t = 0 au, for the N..--II. f.i.:e model. This PWTDM is evolved
through the EP-QCLE method. ................. 49

4-5 T(R) at t = 14000 au, for the N..--11. f.,.!e model. . 50

4-6 Phase space grid points at t = 14000 au, for the N..--iii f.,.e model. 51

4-7 Na populations qi and 7q2 vs. time. ................. 52

4-8 Li populations mq and 7q2 vs. time. ................. 53

4-9 Coherence described by Re(iq12) vs. time, for the ..- iii f.i:e system. 54

4-10 Coherence described by Re(M12) vs. time, for the Li-surface system. 55

4-11 Expectation of position and dispersion for the N..-- ,i f.,.:e system. .56

4-12 Expectation of momentum and dispersion for the N..--i if.i.:e system. 57

4-13 Density function p(R) for the N..-- ii f.,:e system. . ... 58

4-14 Potential curves for the dual avoided crossing collision. ... 61

4-15 Populations piq and 7q2 vs. time for the dual crossing collision model. 64

4-16 Coherence described by Re(r12) vs. time, for the dual crossing collision
m odel. ........... ...... ........ ..... 65

4-17 Grid deformation at t = 1400 au, for the dual crossing collision model. 66

4-18 Probability of transmission in the ground state, for the dual crossing
collision model. .................. ..... 67

4-19 Potential curves for the Nal complex. ................. 69

4-20 Ionic and neutral populations over time, for the Nal complex. 73









4-21 Expectation of position and its deviance, for the Nal complex. 74

4-22 Coherence as a function of time, for the Nal complex. ... 75

4-23 Phase space grid at the end of the simulation, for the Nal complex. 76

4-24 Number of steps required by the relax-and-drive algorithm, compared
to an estimated number required for a fixed timestep version. 77

6-1 Schematic of Li(2p) above a He surface. A) Li(2pw). B) Li(2p(r). 101

6-2 Radial distribution functions for bulk liquid helium .... 106

6-3 Comparison of the Aziz potential with the effective form. ...... .107

6-4 Effective He-He potential. ........ . ..... 108

6-5 Constraining potential used to keep He atoms from evaporating. 110

6-6 Temperature fluctuations of the He droplet over time .... 111

6-7 Helium density profile from the center-of-mass of the cluster. 112

6-8 Adiabatic energy for Li and He as a function of internuclear distance. 114

6-9 Adiabatic energies for Li and one or more He along the z-axis. 116

6-10 Adiabatic energy for Li and one or more He along the y-axis. 117

6-11 Adiabatic energy for Li and a surface of He atoms parallel to the x-y
plane. . . .. .. .......118

6-12 Evolution of ground state Li embedded in the center of a He cluster.
A) Initial time t = 0 au. B) Final time t = 10,000 au. ...... .120

6-13 Comparison of Li and He motion within a He cluster. The time scale
has been reduced by a factor of 100 for the He curve. ..... ..122

6-14 Electronic population of Li as it emerges from the He cluster. 123

6-15 Evolution of Li(2pa) as it recedes from the He cluster surface. A)
Initial time t = 0 au. B) Final time t = 33, 000 au. ... 125

6-16 Mixing of the Li(2pa) and Li(2pr) states at distances where Li(2p) is
triply degenerate. ................ ........ 126

6-17 Electronic population of Li with and without a perturbing electro-
magnetic field, resonant to the D line. . 128

6-18 Dipole emission spectra of Li(2pa) as it recedes from the He cluster
surface. .... .. .. .... ..... ...... 129









6-19 Snapshot of Li(2pr) as it interacts with the He cluster surface. A)
Initial time t = 0 au. B) Final time t = 67, 000 au. ... 131

6-20 Electronic population of Li(2p7) as it interacts with the He cluster
surface. . .. .. .. .. 132

6-21 Dipole emission spectrum of Li(2pr) during the first 3000 au. .... 134

6-22 Dipole emission spectrum of Li(2pr) during the final 3000 au. .... 135

6-23 Adiabatic curves of Li surrounded by a cubic lattice of He atoms. The
parameter R refers to the half-length of the lattice edge. 136

6-24 Decay of Li(2pr) surrounded by surface He atoms, induced by an EM
field with frequency resonant to the Li(2pr <-- 2sra) transition. .138

A-1 Flowchart describing the cauldron program. . 148

C-1 Flowchart describing the qualdron program. . 156















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

QUANTUM DYNAMICS OF FINITE ATOMIC AND MOLECULAR SYSTEMS
THROUGH DENSITY MATRIX METHODS

By

Brian Thorndyke

M.I.- 2004

Chair: David A. Micha
M., i r Department: Physics

We develop a mixed quantum-classical formulation to describe the dynamics of

few- and hm..iii -body atomic systems by applying a partial Wigner transform over

the quantum Liouville equation of motion. In this approach, the density operator

becomes a function in quasiclassical phase space, while remaining an operator over

a subset of quantal variables. By taking appropriate limits and introducing an

effective potential, we derive equations of motion describing quasiclassical nuclear

trajectories coupled to quantal electronic evolution. We also introduce a variable

timestep procedure to account for the disparity between slow nuclear motion and

fast electronic fluctuations.

Our mixed quantum-classical method is applied to the study of three simple

one-dimensional two-state models. The first model represents the photoinduced des-

orption of an alkali atom from a metal surface, where near-resonant electron transfer

is important. A second model explores a binary collision under conditions where two

avoided crossings are present. The third model follows the photoinduced dissocia-

tion of the sodium iodide complex, whose long-range attractive surface results in









oscillations of internuclear distance. Quantities such as state populations and quan-

tum coherence are computed, and found to be in excellent agreement with precise

quantal results obtained through fast Fourier transform grid methods.

Having validated our approach, we turn to the study of alkali atoms embedded

in rare gas clusters, treating the alkali atom-rare gas interactions with i-dependent

semi-local pseudopotentials. Light emission from the electronic motion of the alkali

atom is derived in the semiclassical limit, and computational methods to render

the simulation feasible for a many-atom cluster are discussed. The formalism is

applied to lithium atoms in helium clusters, where the cluster configuration and the

electronic population dynamics of the lithium atom are monitored over time. We

study both the ground and first excited states of lithium, and introduce a resonant

electromagnetic field to induce electronic transitions. Our results correlate well with

other experimental and theoretical studies on doped helium droplets, and provide

insight into the dynamics of an excited lithium atom near a helium cluster surface.















CHAPTER 1
INTRODUCTION

This work is part of a broader effort to bring new insights into the time depen-

dence of few- and many-body molecular systems. In order to study these systems,

we are particularly interested in mixed quantum-classical methods. There are many

approaches to combining classical and quantum mechanics,1'2'3 but the underlying

theme is to use classical mechanics where quantal descriptions are not essential to

the dynamics of the system. By doing so, we can save a tremendous amount of

computational time with hopefully minimal expense in accuracy.

In our study, we focus on the density operator treatment, which allows for a

general introduction of semiclassical and classical limits for some degrees of freedom.4

In the applications being considered, electronic variables are described quantally and

the nuclear variables are propagated quasiclassically, but our theoretical treatment

of quantum-classical coupling is more general. We are able to validate our methods

through extensive study of small test models. These models can be evolved through

a fully quantal propagation using fast Fourier grid methods, permitting a rigorous

assessment of the accuracy of our approach.

As a realistic application, we turn to the study of ground and excited alkali

atoms embedded in clusters of rare gas atoms.5'6'7'8'9 Rare gas clusters provide an

interesting bridge between few-atom systems and bulk matter. A mixed quantum-

classical approach allows us to follow the nuclear motion and population state

dynamics as the alkali atom interacts with the cluster. By tracking the electronic and

induced dipole, we are able to compute the electronic spectra of the alkali atom; and

by explicitly introducing an electromagnetic field, we are able to induce electronic

transitions.









While our formalism is applicable to arbitrary alkali-rare gas combinations, we

have concentrated on the dynamics of lithium embedded in helium clusters, for which

there has been a surge of recent experimental and theoretical activity. 10,11,12,13,9,14

Stable helium clusters doped with lithium atoms have been produced at ultralow

temperatures, where the ground lithium atom has been shown to preferentially reside

on the surface of the helium cluster. Furthermore, the behavior of the lithium atom

subsequent to electronic excitation depends heavily on the orientation of the excited

state. Our mixed quantum-classical approach corroborates these findings, and leads

to additional insight into the dynamics of these interactions.

1.1 Overview of Classical and Quantum Dynamics

The vast majority of chemical and biological processes can be described, in

principle, by nonrelativistic quantum mechanics. Within this context, the state

of a system of nuclei and electrons is represented entirely by a wavefunction or

density operator.15 A Hamiltonian operator describes all interactions between the

particles, and can be extended to include environmental components (for example,

a boundary or electromagnetic field). When the Hamiltonian does not depend on

time, its eigenstates are stationary (up to a phase), and represent time-invariant

configurations of the system. When the Hamiltonian contains time dependence, or

the initial state is nonstationary, the molecular system evolves through the action

of the Hamiltonian.

The solutions to the time independent Schr6dinger equation (TISE) are the

eigenstates of the Hamiltonian at any given time. The corresponding eigenvalues

are the energy levels available to the system. While conceptually compact, the ana-

lytical solution of the TISE is not possible for more than two particles in the general

case. An enormous body of computational work in chemical and molecular ]1li,-i, ,

is devoted to the numerical solution of the TISE,16,17,18 and accurate ground state









energies have been computed for molecules involving hundreds of atoms.19 Knowl-

edge of the full spectrum of eigenstates would allow one to follow the dynamics of

the system as well, but unfortunately it is very difficult to obtain accurate results for

excited states, and in any case, the number of states required for accurate computa-

tions would be prohibitively expensive for most dynamics problems. An alternative

is to follow the dynamics directly.

Quantum dynamics (QD) follows the evolution of a system in real time. The

wavefunction evolves according to the time dependent Schridinger equation (TDSE),20

while the density operator (DOp) evolves through the quantum Liouville equation

(QLE).21 These formalisms are equivalent, although statistical ensembles are more

naturally described by the density operator. Quantum dynamical calculations are

important in understanding the pathways between initial and final states, for exam-

ple in chemical reactions or molecular collisions, and have become an important

complement to modern experiments that use pico- or femtosecond light pulses to

probe ultrafast dynamics.22 The computational complexity of numerical solutions

to full QD, however, severely limits the number of degrees of freedom that can

be studied, and incorporation of classical concepts is necessary for most realistic

applications.

Full QD solutions can be readily implemented by discretizing the wavefunc-

tion along a multidimensional grid. Since the Hamiltonian contains nonlocal opera-

tors like kinetic energy, methods such as finite dill. n -1,25,26 or Fourier trans-

form27,28 are needed to evaluate the action of the Hamiltonian on the wavefunction.

In the general case, these techniques require a dense grid to obtain accurate results.

Furthermore, their nonlocal nature can result in significant shifts of the wavefunction

over time, so that even if the initial state is spatially compact, a large grid is required

to accommodate translation. Sparse grid methods29 can alleviate some of the com-

putational burden of using large grids, and dynamically changing grids30 can better









follow the distortion and translation of the wavefunction over time. Alternatively,

discretization on a basis set instead of a grid can simplify derivative calculations.31

Ultimately, however, the exponential scaling of the number of grid points or basis

functions with the system size renders full QD solutions intractable for more than

a few degrees of freedom.

Classical molecular dynamics (\!1)) can be derived from the QLE in the classical

limit (h -- 0).32 In the context of molecular simulations, the most basic MD treats

nuclei as point particles in phase space, and follows trajectories according to the

Hamilton equations of motion.33 Internuclear potentials are derived from ab initio,

experimental and empirical results, and the forces on the nuclei are obtained by

summing over the partial derivatives of the pair potentials. These classical force

calculations are now the bottleneck, and straightforward evolution of N classical

degrees of freedom has only O(N2) time complexity.34 Tree methods based on octree

spatial partitions35 or multiple potential .::p.in-i. .I- '. 37 reduce this complexity to

O(N log N), with a proportionality constant dependent on the desired accuracy of

the simulation. Because of its intuitive nature and computational efficiency, MD is

routinely used to study molecular systems with 103 106 atoms.38 The problem with

MD is its inability to adequately describe quantum mechanical phenomena, such as

charge transfer, electronic excitation, tunneling and zero-point motion. These effects

are ubiquitous in chemical and biological processes at thermal or lower energies, and

cannot be completely neglected in most cases.3

An attractive alternative is to construct an approximate QD model that ideally

combines the accuracy of QD with the computational efficiency and intuitive sim-

plicity of MD. In general, this can be done by augmenting MD methods to include

quantum features, or by simplifying QD models to incorporate classical or quasiclas-

sical trajectories.1 There are many variations of this theme, and question remains as

to which approach is the most suitable under arbitrary conditions. In Section 1.2, we









survey some of the most common approximation schemes, differentiating between

methods based on the wavefunction and those centered on the density operator.

1.2 Approximations to Quantum Dynamics

1.2.1 Wavefunction-Based Approaches

Wavefunction-based approaches are most useful for the propagation of pure

states in closed environments, and have been studied theoretically and numeri-

cally since the dawn of quantum mechanics. We limit our survey to some prin-

cipal methods that incorporate some classical concepts to solve the TSDE, includ-

ing time dependent self-consistent field (TDSCF) calculations,39,40,41'42 Gaussian

wavepacket (GWP) propagation,43'44 surface hopping methods,45 electron nuclear

dynamics (END),46'47 and path integral methods.48'49

The time dependent self-consistent field approximation begins with the wave-

function written as a product of nuclear and electronic wavefunctions, and uses

time dependent variational principles to evolve each wavefunction along a potential

averaged over all other wavefunctions. When the nuclear variables are expressed in

the classical limit, the TDSCF approximation reduces to a set of electronic wave-

functions evolving over nuclear trajectories, the trajectories propagating accord-

ing to effective forces from the quantal system. Many mixed quantum-classical

schemes retain this mean-field flavor, where classical trajectories propagate along

precomputed or simultaneously evolving electronic states. On the other hand,

quantum dynamics has a very different character than classical propagation, and

mixed quantum-classical methods vary tremendously in their treatment of quantum-

classical interactions, initial conditions, and measurement of system observables.

Gaussian wavepacket propagation, in its original form, takes the system to be

a Gaussian wavepacket in nuclear space, that propagates along a single electronic

surface. By locally expanding the potential up to harmonic contributions, equations

of motion for the Gaussian wavepacket parameters are derived, which lead to shifts









and distortions of the Gaussian over time. The Gaussian center follows classical

equations of motion, while its distorting shape results from quantal corrections to

the classical motion. The GWP method has been extended to describe nonadiabatic

processes with the multiple spawning approach,50 where several wavepackets prop-

agate on multiple electronic surfaces, and proliferate into additional wavepackets at

crossing points.

Surface hopping begins with nuclear trajectories evolving on one or more elec-

tronic surfaces, and reproduces nonadiabatic events through statistically based jumps

between the surfaces. In its earliest version, these transitions take place near avoided

crossings, but later generalizations allow hops to occur at any time during the sim-

ulation.51 One drawback to the surface hopping approach is the need to rescale

velocities after stochastic transitions, in order to conserve energy. Although accu-

rate state transitions are often achieved, the dynamics are clearly not representative

of the true system evolution, except in a statistical sense.

Electron nuclear dynamics is an entirely different approach, that derives equa-

tions of motion by minimizing the Maxwell-Schrodinger Lagrangian density and

treating the nuclear degrees of freedom as coherent states52 in the semiclassical

limit. By expanding the electronic wavefunction in a basis of Slater determinants,

nuclear and electronic motion are combined in a computationally accessible scheme.

Path integral methods solve the TDSE for the quantum time evolution opera-

tor, exp(-iHt/h), in the coordinate representation.53 Path integral equations are

equivalent to full QD, but unfortunately they are computationally intractable in

the general case. Harmonic approximations to the intermolecular potentials sub-

stantially reduce the complexity of the calculations, making it possible to reproduce

the dynamics of a quantum system embedded within an harmonic bath.54'55 For

example, variations such as the initial value representation,56'57'58 have successfully

described the spin-boson model of coupled electronic states in a condensed phase









environment. While not applicable to arbitrary systems, these methods are quite

useful when the system and its interactions can be cast into the appropriate forms.

1.2.2 Density Operator Approaches

There are a number of advantages to using the density operator. First, it

provides a convenient representation of mixed rather than pure states.59 Second,

many systems can be naturally partitioned into a primary -i, -1i--i IiI of interest and

its surrounding bath. By taking the trace over the bath degrees of freedom, the

QLE provides a reduced density description of the primary system interacting with

a bath.21'59 Finally, it is possible to include terms directly in the QLE that represent

energy dissipation into the environment.60 For these reasons, the QLE is often used

where initial conditions are specified as statistical averages, or when departure from

a full quantum treatment is necessary due to the size of the system.

Among mixed quantum-classical solutions to the QLE, an early approach splits

the density operator into a product of Gaussian wavepackets, and derives equations

of motions for the Gaussian parameters using self-consistent field approximations.61

In the semiclassical limit, this method has similarities to generalized forms of GWP

propagation, but because it is based on the density matrix, it can naturally treat

both open and closed systems.

Another method is similar in spirit to END, but uses density matrices rather

than Slater determinants to represent the electronic state.62 The quantum degrees

of freedom are expanded in some basis set, while the classical degrees of freedom

are written as coherent states. The Lagrangian is minimized through the Dirac-

Frenkel variational principle, and one arrives at equations of motion for a reduced

density matrix, coupled to semiclassical equations of motion for the coherent state

parameters.

The density matrix evolution (DME) method63,64'65 divides the system into a

quantum and classical space. The quantum -1i-,I -1. iIl is expanded over a nonlocal,









orthogonal basis set, while the classical coordinates evolve along trajectories accord-

ing to the Hellmann-Feynman force. This method has worked well for simple models

involving analytically accessible matrix elements, but is not designed for arbitrary

-I-' II- with nonlocal and possibly nonorthogonal basis sets.

Eikonal methods66'67'68 exploit the limit of small nuclear wavelengths to sepa-

rate quantal and classical motion. Through this formalism, the electronic density

matrix propagates simultaneously with nuclear trajectories, the latter guided by

effective forces from the quantum density. Combined with variable timestep meth-

ods and travelling atomic function basis sets, the eikonal approach is particularly

well suited for the study of binary collision problems.

A recent branch of QLE methods uses the Wigner transform (WT) to cast the

operator-based QLE into an equation of motion over phase space variables. The

classical Liouville equation for the density function emerges in the classical limit,

while judicious application of the WT to a subset of the system variables leads to

well defined quantum-classical schemes. In Section 1.3, we outline some of the major

approaches found in the literature.

1.3 Quantum-Classical Liouville Equation

It is fascinating that a phase space representation of quantum mechanics can be

developed, that is fully equivalent to the Hilbert space representation. By applying

the Wigner transform to both sides of the QLE, an equation of motion is derived for

a Wigner function, which is a function of classical position and momentum variables.

The Wigner function does not evolve through simple classical equations of motion,

however, but involves nonlocal operators in phase space.4 One class of solutions

exploits the similarity of these equations of motion with hydrodynamics found in

classical li-, -i. So-called hydrodynamic quantum dynamics was formulated by

Bohm in 1952, and recent developments in the field have revitalized interest in phase

space trajectory methods for solving QD.69'70'71'72'73 Unfortunately, these methods,









like those involving basis set or grid solutions to the TDSE or QLE, suffer from

exponential growth with system size.

A mixed quantum-classical scheme arises by dividing the quantum system into

quantal and quasiclassical variables, the quasiclassical variables associated with ener-

gies or masses much greater than the quantal variables. Taking the partial Wigner

transform (PWT) of the QLE over only the quasiclassical variables produces an

equation of motion for a partially transformed Wigner operator (PTWDOp), which

is a function of phase space in the quasiclassical variables, but remains a quantal

operator in the space of the quantal variables. After some appropriate approxi-

mations relating to the different masses of the variables, we arrive at a differential

operator equation referred to as the quantum-classical Liouville equation (QCLE).74

While the quantal evolution must still be tackled in Hilbert space, the propagation

of the quasiclassical phase space acquires a classical character. If the majority of

the system can be described with quasiclasical variables, the computation savings

may be considerable.

One numerical approach represents the PWTDOp in phase space with a fixed

number of delta functions.75,76'77 Specifically, the PWTDop is projected on a basis,

resulting in a set of diagonal and off-diagonal functions in phase space. Each function

is approximated by a set of delta peaks, that evolve along surfaces constructed

from the densities and coherences. Nonadiabatic coupling is implemented through a

modified surface hopping procedure, where hops occur between trajectories at curve

crossings.

An alternative solution uses the same delta peak representation, but rather

than use stochastic jumps between trajectories, new trajectories are generated at

curve crossings.78 A variation on this approach, the multithread method, spawns

new trajectory points at every timestep, but combines the newly generated pools









of trajectory points into smaller numbers through energy and other conservation

considerations.79,80,81

Recent efforts have been made to combine Gaussian wavepackets in phase space

(GPPs) with the trajectory solutions along diagonal and off-diagonal surfaces.82

Rather than represent quasiclassical functions through delta peaks, the functions

are expanded as a linear combination of GPPs, thereby more effectively covering

quasiclassical phase space. Stochastic jumps between surfaces at crossing points

reproduce nonadiabatic behavior, and avoid the need to spawn new GPPs.

1.4 Our Approach

In our approach, we introduce an effective potential to the QCLE, to provide

a simple numerical procedure for solving the equation.83 Our effective potential

quantum-classical Liouville equation (EP-QCLE) permits solution in terms of full

quantal evolution of the quantal variables along quasiclassical trajectories guided

by the quantal state. We introduce a quantal basis and a quasiclassical grid to

render the equations suitable for numerical propagation, and implement a scheme to

efficiently account for the different time scales of quasiclassical and quantal motion.

Our approach shares several positive features with other QCLE methods of solu-

tion. The QCLE rigorously combines quantum and classical motion, and provides

-1,-1.I'., i.il ways to incorporate higher-order approximations and quantum-classical

coupling. The formalism is based on the density operator, making it attractive for

incorporating thermodynamical features or dissipative environments. The QCLE

also lends itself to trajectory-based numerical solutions that reduce to the classi-

cal Liouville equation in the classical limit, providing an intuitive computational

framework.

Our EP-QCLE solution differs from other QCLE primarily in its generality and

its scalability. The effective potential can take a variety of forms, and while we

have found the Hellmann-Feynman force to provide the best results for our model









-'-1i -I.-. it is possible that other forms would provide even better results under

different circumstances. Furthermore, an electronic basis is only introduced once

the EP-QCLE has been developed in terms of partial Wigner operators, and no

assumptions are made on the form of the basis set. This flexibility has allowed a

uniform treatment of problems ranging from one-dimensional models expanded in

a two-state diabetic basis, to fully three-dimensional atomic clusters in a basis of

Gaussian atomic functions. Finally, while the majority of QCLE solutions rely on

interactions between the propagating trajectories, our use of an effective potential

results in completely independent trajectories whose only connection follows from

initial conditions. Such independent trajectory evolution maps optimally to certain

parallel architectures, and substantially reduces the cost of propagation, that would

otherwise be required to compute trajectory interactions at each timestep.

1.5 Simple One-Dimensional Two-State Models

We test our methods on a set of one-dimensional two-state models, which are

simple enough to be evaluated precisely through fast Fourier grid methods. Among

these methods, we study a model of an alkali atom approaching a metal surface,

where near-resonant electronic transfer is important.84s85 Secondly, we consider a

system representing a molecular collision with two avoided crossings, where impor-

tant interference effects arise as one varies the collision energy.79 Finally, we study

a system representing the predissociation of the sodium iodide complex, where the

long-range attraction of the excited state results in oscillatory nuclear motion.86

1.6 Lithium-Helium Clusters

Clusters are .-: --regates of atoms containing between three and a few thousand

atoms. The smallest clusters, or microclusters, have between 3 and 10 atoms, and

are just small enough that molecular concepts still apply to some degree. The

next range of clusters, or small clusters, have between 10 and 100 atoms, a range

where molecular concepts break down and the clusters form shells with nonempty









interiors. Large clusters, between 100 and 1,000 atoms, provide the final transition

from isolated molecules to the bulk condensed state.

Rare gas clusters can be probed by doping them with a chromophore and follow-

ing the chromophore through various laser detection methods.9'87'88'89 Our study

used a lithium atom as the dopant in a pure helium cluster. Semi-local i-dependent

pseudopotentials are used to describe the lithium-helium interactions, because they

are found to accurately reproduce adiabatic energy curves for the lithium-helium

pair for s, p and d symmetries.90'91'92'93'94 By introducing the pseudopotential for-

malism into the EP-QCLE, we are able to follow the nuclear configuration over time.

In addition, we are able to probe the evolving electronic energy surface by monitor-

ing the electronic and induced dipole, and computing the resulting dipole emission

spectrum. By applying a resonant electromagnetic field, we are able to stimulate

the emission of light by excited lithium atoms near the cluster surface. Both the

nuclear configuration of the cluster and the spectral properties of its dopants have

been under intense investigation over the recent years, and our contribution gives

insight into the dynamics of these interactions.

1.7 Outline of the Dissertation

Chapter 2 presents the formalism we have used to explore the dynamics of

mixed quantum-classical systems. The ideas are presented for the general case of

quantal and quasiclassical variables, and it is shown how the EP-QCLE naturally

emerges from the partial Wigner transform over the quasiclassical variables.

Chapter 3 explores the computational aspects of the EP-QCLE. In particular,

the application of an electronic basis and a nuclear grid render the equations of

motion suitable to numerical solution. We also show how observable quantities can

be calculated within this framework.

Chapter 4 applies our formalism to three different one-dimensional two-state

-,-1 in,-. where the results can be compared to exact quantal simulations based on









the Schridinger equation and numerically precise grid methods. Along the way,

valuable insights into the accuracy and limitations of our methods are obtained.

Chapter 5 presents the formalism for general alkali-rare gas clusters. We describe

the Hamiltonian for general alkali-rare gas interactions, and derive matrix elements

for a basis set of Gaussian atomic functions. We further discuss the dipole of the

cluster and its matrix elements in detail. Finally, we present computational meth-

ods used to render the numerical propagation of the equations of motion feasible for

small to large clusters.

Chapter 6 presents specific results for lithium-helium clusters. We consider

the thermodynamic equilibration of the helium cluster in detail, followed by the

evolution of the lithium atom through the cluster and near its surface once the cluster

has reached equilibrium. We follow the nuclear motion and discuss the interaction

of the excited lithium atom near surface helium atoms, in particular with regard to

the nuclear configuration and the dipole spectrum. Finally, we introduce a resonant

electromagnetic field to stimulate photon emission after the excited lithium atom

embeds itself within the surface.

Chapter 7 summarizes the main conclusions obtained in this dissertation.

Appendix A discusses the program cauldron developed over the course of this

dissertation, to simulate the EP-QCLE for the test model systems and the full

lithium-helium cluster.

Appendix B derives the numerical evolution of the Schridinger equation through

the split operator fast Fourier transform method. The solution obtained by this

approach is exact to within numerical precision, but is computationally expensive

and thus prohibitive for more than a few degrees of freedom.

Appendix C presents the qualdron code, developed alongside cauldron to com-

pare the results of the test models using mixed quantum-classical dynamics with the

full quantal treatment.















CHAPTER 2
QUANTUM-CLASSICAL LIOUVILLE EQUATION: FORMULATION

2.1 Introduction

Rather than focus on the wavefunction of a molecular system, we construct its

density operator and proceed from there. The DOp is more general than the wave-

function, in that it naturally describes a statistical ensemble of quantum states.

This is particularly useful when the molecular system interacts with its environ-

ment, and one does not have a complete knowledge of that environment. The DOp

also provides a convenient starting point for deriving mixed quantum-classical meth-

ods, because the classical limit of the DOp is the classical density function. One

particular route to a mixed quantum-classical description of a molecular system is

through the the Wigner transform. By applying a partial Wigner transform to the

DOp and its equation of motion, one obtains a new representation that is a function

in the phase space of the Wigner transformed variables, but remains an operator in

the remaining variables. If the PWT is judiciously applied to classical-like (or qua-

siclassical) variables, then approximations can be made to the equations of motion

that provide a classical evolution of the quasiclassical variables coupled to a quantal

evolution of the quantal variables. In the case of molecular systems, the quasiclas-

sical variables are typically the nuclear coordinates, while the remaining variables

describe the electronic state. In this chapter, the PWT is described in detail, and

its application to molecular systems is outlined.

2.2 Quantum Liouville Equation

A system of atoms or molecules is represented by nonrelativistic quantum

mechanics as a state vector IT) in Hilbert space. This state vector evolves in time









according to Schridinger's equation,

ih Ot = HI), (2.1)

where H is the Hamiltonian of the full system. If, however, we are studying an

ensemble of states {l|i)} with statistical weights {', }, it is convenient to construct

the density operator,

= IQ|') |. (2.2)

Taking its time derivative and using Eq. 2.1, we find the density operator to evolve

in time according to the quantum Liouville equation of motion,

ih = [H, F]. (2.3)

There are a number of advantages to using the density operator, but in our case

the primary advantage is that it leads directly to the classical density function in

the limit h -- 0. We first discuss the full Wigner transform and its application to

the QLE, and then show how the partial Wigner transform can be used to derive a

mixed quantum-classical representation in the appropriate limits.

2.3 Wigner Representation

The Wigner transform provides a phase space representation of the DOp (the

Wigner function) and other quantum mechanical operators. It is defined as the

Fourier transform of an operator projected on coordinate space,4'95'96'97

Fw(r, p) (2Ih) dzexp(ip z/)(r z/2|r + z/2), (2.4)

Aw(r,p) = d'zexp(ip z/h)(r- /2AIr + /2), (2.5)

where A is arbitrary. The integration generates functions that are local in both

coordinate and momentum space, which is important for the emergence of classical

features in the development of our mixed quantum-classical method. The prefactors









are defined differently for the density operator than for other operators, in order to

provide convenient parallels between the Wigner function and the classical Liouville

density.

Classically, the probability distribution function is well defined. For an N-body

-v-1, in with coordinates r = (ri, r,..., r3N) and moment p = (pl, I_,... ,P3N), the

classical Liouville density p = p(r, p) generates the expectation value of any function

A A(r, p),32

(A) drdpp(r,p)A(r,p). (2.6)

Quantum mechanically, even the notion of phase space is problematic, as Heisen-

berg's uncert.,iiii' principle prohibits the simultaneous measurement of position

and momentum for a given degree of freedom. However, quasidistribution functions

like the Wigner function provide an analogous form of the quantum mechanical

expectation value. The expectation value of an arbitrary operator is well known,59


(A) Tr(FA), (2.7)

which can readily be seen by expanding the trace over eigenstates of A. However,

by Wigner transforming both P and A, we arrive at a form for the expectation value

similar to the classical case,

(A) drdpfw(r,p)Aw(r,p). (2.8)








This can be seen by expanding Fw and Aw in Eq. 2.8,


I drdpFw(r, p)Aw(r, p)


S(2h) Ndrd I dzexp(-ip- z/h)(r +z/2,r- /2)
x f dz' exp(-ip z'/h)A(r + z'/2, r z'/2)


(2h)3N / drdpdzdz' exp[-ip (z +z')/h]
xF(r + z/2, r z/2)A(r + z'/2, r z'/2)
/ drdzF(r + z/2, r z/2)A(r z/2, r + z/2).


(2.9)


Transforming variables q
value,


I drdpFw(


r+z/2, q' = r-z/2, we retrieve the quantum expectation



r, p)Aw(r,p) = dqdq'F(q, q')A(q', q)

= dqdq'(q|Pfq')(q'A q'q)


f dq(qfPA q)
Tr(FA).


(2.10)


Although Fw is not a probability distribution, various properties justify its
classification as a quasiprobability function:
I|(r)|2 dpFw(r,p).
|(p)2 'drFw(r,p).
For any function f(r), (f(r)) drdpFw(r,p)f(r).
For any function g(p), (g(p)) = fdrdpw(r, p)g(p).
One can also derive the WT of products of operators in terms of the WT of the
operators themselves.4 For a general operator P = AB, we find that


Aw exp(hA /2i)Bw,


(2.11)









where A is the bidirectional operator,

A -- (2.12)
Op Or Or Op

Expanding the commutator directly, we have


[A, B]w = (AB)w (BA)w

S Awexp(h A/2i)Bw Bw exp( A/2i)Aw. (2.13)


2.4 Quantum-Classical Liouville Equation

We have seen that the density operator evolves according to the QLE (Eq. 2.3).

Rather than study the solution to the QLE, we will examine the evolution of the

Wigner function. When the density operator is transformed over all its variables, we

arrive at equations of motion that, in the limit that h 0, reduce to the classical

Liouville equation. Here we are interested in performing a partial Wigner trans-

form of the density operator over a subset of variables (those termed quasiclassical),

while leaving the remaining variables (those termed quantal) in their original rep-

resentation. By taking appropriate limits, we derive equations of motion for the

quasiclassical variables, coupled to quantal equations of motion for the remainder.

To be specific, we divide the degrees of freedom into N quasiclassical variables

Q (Qi,Q2, .* QN) and n quantal electronic variables q = (q, q2,..., ). The
Wigner transform is performed over quasiclassical variables only,


w(R,P) (27h)-N / dZexp(iP Z/h)(R- Z/21IR+ Z/2), (2.14)

Aw(R, P) dZexp(iP Z/h)(R- Z/2PR+ Z/2). (2.15)

These are distinguished from fully Wigner transformed operators by virtue of remain-

ing operators in the quantal space. By taking the PWT of both sides of the QLE, we

find the equation of motion for the partially Wigner transformed density operator









Sw(R, P),98,99, 100

ih Hw exp(hAf/2i)w Fw exp(hA f/2i)Hw, (2.16)


where


Ac (2.17)
OP OR OR OP

To proceed further, we approximate Eq. 2.16 to O(h A cl),

1Fw I I
S- [I^, F] + -({Hw, Fw} {Fw, Hw}), (2.18)
Ot ih 2

where {A, B} is the Poisson bracket,


{A, B} -A A ciB. (2.19)

This is an approximation within the quasiclassical space, and reduces quantal motion

of the quasiclassical degrees of freedom, but keeps dependence of the quasiclassical

variables on the quantal state through the interaction potential. It is an appropriate

truncation when the quasiclassical variables are associated with a much greater mass

than the quantal variables.75'101'80'79

For a specific example, consider a Hamiltonian for a molecular system composed

of kinetic, potential and interaction terms,

p2 2
H +V(Q)+ ()+ '(4, + Q), (2.20)
2M 2m

where P is the nuclear momentum operator, V(Q) the nuclear potential, p the elec-

tronic momentum, v(q) the electronic potential, and V'(q, R) the electronic-nuclear

coupling. By interpreting the nuclear variables Q as quasiclassical and taking the









PWT over the nuclear variables, we find the partially Wigner transformed Hamil-
tonian,


2 + V() + v() +V'(,).


(2.21)


Defining


fH +p + ) + V'(4, R),
2m
V V(R) + V'(,R),


(2.22)

(2.23)


we get the quantum-classical Liouville equation of motion for Fw(R, P),


OFw
Ot


[I q p Fw] I+ + (2.24)
ih M 2 OOR OP aP aR


2.5 Effective Potential

The third term on the RHS of Eq. 2.24 presents a challenging obstacle to solving

the equation numerically. One problem common among many proposed schemes is

the requirement of computationally demanding algorithms to evaluate the partial

derivatives of the PWTDOp. To avoid this problem, we introduce an effective

potential V(R, P) in Eq. 2.24,

9Pw 1 P avP w ap 1V OE w
[H i, Fw ] +
Ot ih M OR OR OP
{ ( R aR a _R aR (2.25)
2 OR OR OP OP iR ORf v

The introduction of V becomes computationally useful insofar as we can neglect

the fourth term in Eq. 2.25. While the choice of V is clearly arbitrary, we have found

optimal results by setting the expectation value over quantal variables of (aRVa-Rv)

to zero,


Trqu 1W ( RV a VI
OR OR


(2.26)









In this way, we retrieve the Hellmann-Feynman force,

9 Trq, q['wwv/9Rj
0- Trq u[P (2.27)
OR Trq. [fW]
-FHF(R,P). (2.28)

The denominator in Eq. 2.27 is itself a function of quasiclassical phase space, which

differs from effective potential approaches involving a single normalized density oper-

ator. Neglecting the fourth term in Eq. 2.25, we find an approximated but compu-

tationally advantageous effective-potential QCLE (EP-QCLE),

0f1w 1 P 80w 0 aw
Ot i[H F M OR F P

The approximations used to derive the EP-QCLE substantially reduce the

quantal character of the quasiclassical solution space. In contrast, the best adia-

batic methods, based on the Born-Oppenheimer separation of nuclear and electronic

motion, retain full quantum nuclear dynamics along adiabatic curves.102 A princi-

pal advantage to the EP-QCLE is its nonadiabatic character, which is capable of

describing nonadiabatic events when the Born-Oppenheimer limit no longer applies.

A further benefit is its suitability for solution using trajectory methods, greatly

increasing the size of problems amenable to numerical analysis. The theoretical and

computational aspects of the trajectory solution to the EP-QCLE are discussed in

the next chapter.















CHAPTER 3
QUANTUM CLASSICAL LIOUVILLE EQUATION: COMPUTATIONAL
ASPECTS

3.1 Introduction

The EP-QCLE is a partial differential operator equation in the quasiclassical

variables and time. One way of solving this kind of problem is to represent the

operators as matrices on a large grid, and evolve the matrices using finite difference

or spectral methods. The drawback to this approach is that a very dense grid is

required for numerical accuracy, and a very large grid is necessary if the quasiclassi-

cal density shifts location appreciably as it evolves in time. Since the grid dimension

varies directly with the classical degrees of freedom, a multiparticle system presents

very serious numerical difficulties. Moreover, finite difference and spectral grid solu-

tions are inherently difficult to parallelize, as substantial communication is required

between processors regardless of the division of computational labor. An alterna-

tive approach, applicable to the class of partial differential equations to which the

EP-QCLE belongs, is to follow trajectories in phase space as the system evolves.

Only the important trajectory points are represented, so that the (moving) grid

maintains a minimal size. In this chapter, we explore the trajectory approach and

see how the EP-QCLE can be solved in an efficient and even parallel manner.

3.2 Trajectory Solution

We can formally solve Eq. 2.29 by following trajectories in classical phase space,

with R and P becoming functions of time,

dR P
(3.1)
dt M'
dP
d FHF(R,P). (3.2)
dt









One could follow any paths in phase space, but by using those -II'.-.. -. .1 in Eq. 3.2

we are able to transform Eq. 2.29 into an ordinary differential equation in time.

Inserting Eq. 3.2 in Eq. 2.29 and moving the partial derivatives to the LHS, we

derive the change of the PWTDOp along the quasiclassical trajectories,

dF 1
Sfi[HF] (3.3)
dtih

Note that we have omitted the subscript on the PWTDOp for notational conve-

nience, and from here onward will continue to label all PWT operators without this

subscript.

Eq. 3.3 remains a formal solution, and before solving it numerically we must

discretize the equations in both quantal and quasiclassical space. This is the subject

of the next two sections.

3.3 Electronic Basis Set

Let us introduce an arbitrary basis, {|
tions, a Gaussian basis is used, but for now we consider the basis to be general, and

not necessarily orthogonal or normalized. Converting to matrix notation, we let 1I4)

be the row matrix,


14) (1 I) 1 ). (3.4)

Then we can expand our operators,


PF I4) S-l (FIIl4)S-1 (4,, (3.5)
F
A 4) S-1 (4 AI4) S- I(4 (3.6)
A

where S is the overlap (4|4). Projecting Eq. 3.3 on this basis set, and setting

h 1, we obtain

dT
r(iH q- _t)S-1 S-l(iH' + )F (3.7)
dt









where we have used the notation,

dR dP
S (fl (| //&R|4) + (4d|/&P|4). (3.8)
dt dt

3.4 Nuclear Phase Space Grid

Although we've transformed the EP-QCLE into a discrete representation in

electronic space and are now dealing with the partially Wigner transformed density

matrix (PWTDM) instead of operator, we must still discretize the quasiclassical

phase space. To this end, we choose a set of initial grid points {(Ri, Pi)}. Their

distribution should approximately cover the domain of F(R, P), and should be suf-

ficiently dense to well represent the evolution of the PWTDM. In practice, the grid

should be adjusted until convergence is achieved.

Once a grid is chosen, the grid points follow nuclear trajectories {(Rj(t), Pj(t))}

according to Eqs. 3.1 and 3.2:

dRj P1
S- (3.9)
dt M'
dP
FHF(RJ, P). (3.10)
dt

At the same time, Eq. 3.7 becomes a set of uncoupled equations, one for each

trajectory:


dr F(iHq- Q)S S 1 (iH + (3.11)
dt r^H s H (3.11)

where

r r((Rj(t),Pj(t)), (3.12)

Hj H(Rj(t),Pj(t)), (3.13)

Qfj F(R,(t),Pj(t)). (3.14)









Each trajectory follows the evolution of the PWTDM along that path, independent

of the other trajectories.

While one cannot expect coherence between the classical degrees of freedom to

be represented by this approach, there are some substantial computational advan-

tages. In particular, the scheme can be optimally ported to a parallel processor,

whereby each processor independently evolves a single trajectory; communication

between the processors is unnecessary.

3.5 Relax-and-Drive Algorithm

Before the trajectory solution can be implemented, a detailed propagation

scheme needs to be specified. One would expect that with a sufficiently small

timestep At, all propagation methods would converge to the same results, provided

roundoff error were not significant. However, the paths to convergence will certainly

differ, in that methods with higher accuracy can use larger timesteps. The relax-

and-drive method, developed originally by Micha and Runge,66'67'68 incorporates

the rapid electronic oscillatory behavior with the relatively slowly evolving nuclear

variables in an accurate, variable timestep scheme. The relax-and-drive procedure

has been shown to give excellent results for a wide variety of two-body collision

problems. For completeness, we review its details next.

First of all, since all trajectories are propagated analagously, it suffices to look

at the evolution of a single trajectory. Accordingly, let us rewrite the EP-QCLE for

a single trajectory,


S =. W(t)F(t)- F(t)Wt(t), (3.15)
dt

where we have defined


W S-1(-H + i+).


(3.16)









We wish to propagate from initial conditions Wo = W(to), Fo = r(to). If W

is independent of time, we find a numerical solution that is exact up to machine

precision. If W depends on time, but is slowly varying with respect to the timescale

of F, we can propagate to a high degree of accuracy by linearizing Eq. 3.15 in time

and incrementing in small timesteps At = tl to.

3.5.1 W Independent of Time

When W is independent of time, W(t) = Wo, we can formally solve Eq. 3.15,


F(t) Uo(t,to)roUt(t, to), (3.17)

where


U(t, to) = exp[-iWo(t-to)]. (3.18)

If we diagonalize Wo,


Wo = TAT-1, (3.19)


we can rewrite Eq. 3.17 as,


F(t) T[T-Uo(t, to)T]T- o(Tt)-1[TtUt(t, to)(Tt)-1]Tt. (3.20)

The exponential matrices can be formed analytically, since


T-1Uot, to)T exp[-iA(t-to)], (3.21)


and F(t) can be computed in the time complexity of the diagonalization of Wo.

3.5.2 W Dependent on Time

When W depends on time, we separate F into a reference F and correction Q

term,


r(t) r(t) + Q(t).


(3.22)









The reference density is propagated by the time independent Wo of Section 3.5.1,


dro(t)
dt


The evolution of Q(t) is formed by inserting Eq. 3.22 into Eq. 3.15,


(3.23)


.d
i-(ro + Q)
dt


(Wo + AW)(r Q)


(r + Q)(Wo + AW)t,


(3.24)


where


AW


W Wo.


(3.25)


Using Eq. 3.23 we obtain


AWro + WoQ + AWQ


FOAWt QWt QAWt. (3.26)


Transforming to the local interaction picture, where

A UoALU ,


UoAWLUoUoFrLU


+UoAWLUoUoQLUt


UoFiUoUoAW Uo


UoQLUoUoAWLUo. (3.28)


We can simplify Eq. 3.28 by multiplying on the left by U 1 and on the right by

(U )-1 to obtain


DL + [AWLU UoQL


QLUoUoAWj],


(3.29)


where


AWLUoUoFL FLUoUoAWL.


dQ
dt


we get


iU dQL
iUo-- Uo
dt 0


(3.27)


dQL
dt


WoF(t) ro(t)Wt0


DL(t)


(3.30)









Formally solving for QL,


Q(t) (t) + dt'[AWL(t')U(t', to)Uo(t', to)QL(t')
to
-QL(t')Uo(t', to)Uo(t', to)AWt(t', to)], (3.31)

where
t
Qf (t) f dt'[AWL(t')U(t', to)Uo to)F (t')
to
-ro t')U (t', to)Uo(t', to)AW (t', to)]. (3.32)


Solving by iteration, Eq. 3.31 becomes
t
Q t) Qf() f + dt'[AWL(t')Utt',to)Uo(t', to) Qf(t)
to
Qf (t')U (t', to)Uo(t', to)AW (t', to)] (3.33)


Neglecting the second term and higher for small timesteps At = ti to,
tl
QL(tf) dt'[AWL(t')Ut(t', to)Uo(t1', 10) (tl')
to
-ro (t')Uto(t', to)Uo(t', to) AW (t', to)]. (3.34)


Converting back to the original representation,


AL = UoA(U)-1, (3.35)


we have
t1
Uo-(tl, to)Q(t1)Uo(t1, to)-1 dt'[Uo l(t, to)AW(t')ro(t')Ut(t', to)1
to
-Uo1 (t', to)Fo(t')AW (t')U (t', to)-1].

(3.36)









Multiplying Eq. 3.36 on the left by Uo(ti, to) and on the right by Ut(ti, to), and

noting


Uo(t, to)Uo (t', to)


exp[-iWo(ti

exp[-iWo(ti

Uo(tl, t'),

exp[-iWV(t'

exp[iW(ti -


to)] exp[iWo(t'


(3.37)


U (ti, t'),


(3.38)


we get an approximate correction term,


Q(ti)


to
dt'Uog(t t')D(t')U (tl, t'),
to


where


D(t') AW(t')F(t') F(t')AWt(t').


We can compute Q(tl) by quadrature if we assume


D(t) = D(t1/2), to t < ti


where t1/2


to + At/2. We then have


Q(ti)


tl
S dt'Uo(t, t')D(tl/2)U (t t'),
to


(3.39)


(3.40)


(3.41)


(3.42)


Ut (t, to)-I u (ti, to)


to)] exp [^iWo (ti









which we can rewrite as


Q(ti)


-T Idt'exp[-iA(ti
-to


t')]DT exp[iAt(ti


T- D(tl/2)(Tt)-1


Examining the elements of Q,


t')] T k


Sdt'exp[-iAo(tl
oi


f dt' exp[
to


A )(ti t')]DL Ttk


1 exp[-i(Ai A )At] T
(A m A )k


where we have used the notation Ai Aii. Reverting back to matrix form, we have


Q(ti)


TXTt,


exp[-i(A A )At]- 1 T
A, (TAm
T- D(tl/2) )-1.


The full density matrix at tl is then simply,


(3.48)


tl
dtT[T- Uo(t, t')T]DT[TU(Tt)-1]T
to


where


t')] Tt,


(3.43)


Qjk(tl)


(3.44)


Til
Tm
Im



i
-Ti1
l ZTn


-i(Al


(3.45)


where


(3.46)


XDm

DT


(3.47)


t')]DI exp[iA (t,


F(ti) r(ti) + Q(ti).









3.5.3 Velocity Verlet for the Classical Evolution

In order to complete the relax-and-drive algorithm, we need to account explicit

for the propagation of the classical variables. We do this by assuming that during

each timestep, R and P are advanced by the reference density F, and that the

correction term contributes negligibly to the classical propagation. The precise

nature of the classical propagation is independent of the relax-and-drive algorithm,

although to keep accuracy to O(At2) it is necessary to integrate using an algorithm

like velocity Verlet or Runge-Kutta. We choose to use the velocity Verlet method,

which is accurate to O(At2) and is self-starting. We proceed by advancing the

classical positions,

P (t) 1 dP(t) 2
R(t + At) = R(t) + At + t2. (3.49)
M 2M dt

The last term of Eq. 3.49 is the acceleration term, which comes from the effective

potential. Having advanced the positions, we calculate the acceleration at the new

location, and advance the moment,


P(t + At) P(t) + At. (3.50)

In the context of the relax-and-drive procedure, the classical coordinates are advanced

in two steps of At/2, in order to compute the correction term Q.

Variable timestep. As the propagation proceeds, the initial timestep may no

longer be appropriate. This often occurs when the system enters a region where the

magnitude of potential interactions changes so that the reference density fluctuates

at a different rate. An example is the collision of two atoms, where large timesteps

can be taken at large distances, but small timesteps are required at close range. We

can monitor this fluctuation by observing the correction term Q. As Q becomes

too small (large), we need to increase (decrease) the timestep to maintain efficiency









(accuracy). To this end, we define a correction measurement,


if 2
e i max (3.51)


At the end of each timestep, we evaluate e. If c is less than some threshold, say cl,

we discard the step and use a new c 3 x c. If c is greater than some threshold,

-.1,v c, we discard the step and use a new -- e/2. By either multiplying by 3 or

dividing by 2, we avoid any oscillation (and thus infinite loop) between a pair of

timesteps.

3.5.4 Algorithm Details

The algorithm can be divided into an outer piece (say, Main), which calls an

inner piece (say, Propagate). They are described in point form as follows:

Main: To propagate F from tA to tB, given At = Ato and {fl, ej},

1. At min{At, tB tA}.

2. Propagate, with to = tA.

3. Compute = -n.i:: ; |Qj/ | l2.

4. Is e < I and tA At < tB?

YES: Reset variables (matrices and functions return to their values at

to), set At -- 3 x At and return to step 1.

5. Is e > c,?

YES: Reset variables to to, set At -+ At/2 and return to step 1.

6. IstA At
YES: Set tA -` tA + At, and return to step 1.

Propagate: To propagate F from to to tl = to + At,

1. Initialize variables: Wo = W(to), Fo = ro(to) = r(to), R(to), P(to), At.

2. Diagonalize Wo = TAT-.

3. Advance classical variables by half timestep, {R(tl/2), P(tl/2)}, where t1/2

t + At/2, using initial reference density Fo(to) and Wo.









Compute W(tl/2) using R(ti/2) and P(tl/2).
Compute or(tl/2) Texp[-iA(At/2)]T-lro(Tt)-1 exp[iAt(At/2)]Tt.

Compute Q(ti) by assuming D(t) = D(tl/2), to < t < ti.

(a) Calculate AW(tl/2) = W(t/2) Wo.

(b) Calculate D(tl/2) AW(tl/2)Fr(tl/2)- F(tl/2)AWt(tl/2).

(c) Compute DT = T-1D(tl/2)(Tt)-1

(d) Compute X, where


XIm.


exp[-i(A A )At] -
tA DlT-
A, Amtn


(e) Compute Q(ti) = TXTt.

7. Advance classical variables to full timestep, {R(ti), P(ti)}, using Fo(tl/2) and

W(ti/2).

8. Compute W(ti) using {R(ti), P(ti)}.

9. Compute Fr(ti) Texp(-iAAt)T-lro(Tt)-1 exp(iAtAt)Tt.

10. Compute F(ti) F (ti) + Q(ti).

3.6 Computing Observables

3.6.1 Operators in an Orthonormal Basis

Up to this point, we have considered the general basis |4) without any condition

of orthogonality or normality. We can transform this basis to an orthonormal one,

V'), through a L6wdin transformation,


I4)S-1/2


(3.52)


Since quantal traces are naturally formulated in orthogonal bases, it is useful to

express the relationship between matrix representations of operators in both bases.









For the density operator,


r I\'}r'{^'

I=)S-1/2r'S-1/2(4

) | F)r 1|. (3.53)

Thus we equate

S S-1/2F'S-1/2. (3.54)

Since the density matrix has been defined differently than the matrix representations

for general operators, we also consider the representations for a general operator A,

A ')A'({'|

= 1)S-1/2A'S-1/2(

I4)S-'AS- (). (3.55)

Thus

A S1/2A'S1/2. (3.56)

3.6.2 Population Analysis

The population is naturally defined in an orthonormal basis, such that the

population of state i is the ith diagonal element of the orthonormal representation

of the density matrix,


ri [Sii

[S1/2rsl/2]ii. (3.57)









In the case of the PWT representation we must integrate over nuclear phase space
as well, so that Eq. 3.57 becomes

S= dRdP[S1/2(R, P)F(R, P)S1/2 R, P) (3.58)


3.6.3 Expectation Values

The expectation value of a general operator Aw is found by taking both the

quantal and classical trace of the product of the operator with the PWTDM,

(Aw) = Tr[AwFw]

STrciTrqu[AwFw]

dRdPTrqu[Aww]. (3.59)

The quantal trace is naturally computed in the orthonormal basis, but as we
now show, the nonorthonormal basis representation can also be used:

Trqu[AwFw] = Trqu[A' F']

STrqu[S-1/2AwS-1/2S1/2FrS1/2]

STrqu[AwFw]. (3.60)

3.6.4 Hamiltonian Eigenstates and Eigenvalues

Using the orthonormal representation of the Hamiltonian,

H' = S-1/2HS-1/2, (3.61)

we can compute its eigenvalues by diagonalizing the matrix,


L-1H'L = H',


(3.62)









where H' is the diagonalized Hamiltonian and contains the energy eigenvalues along

its diagonal. Since the Hamiltonian is Hermitian, its eigenvalues are real, so that


LtH'(L-')t = H, (3.63)


and the diagonalizing matrix L is seen to be unitary,


Lt = L-. (3.64)


The columns of a unitary matrix are orthogonal, and since the columns of L

are the eigenstates of the Hamiltonian, we see that the eigenstates produced by

diagonalizing the orthonormal representation of the Hamiltonian are also orthonor-

mal. This is useful in the case of degenerate eigenstates, as they are automatically

orthogonal and no additional procedures are needed to ensure orthogonality in the

degenerate subspace.

3.7 Programming Details

When designing a computational package, it is desirable to ensure the code

remains orthogonal and extensible throughout the design and implementation. Mod-

ern programming languages use object-oriented concepts to achieve these goals,103

but unfortunately a great deal of computational work is built on older procedural

languages and cannot be readily incorporated into an object-oriented scheme with-

out substantial effort. Indeed, as much of this legacy code has been developed over

many years and has undergone extensive t. -l iii. it is enticing to adhere to the older

languages in which they were written and incorporate them directly. In the devel-

opment of code for this research, a compromise was found by using many advanced

features found in Fortran 90, but maintaining a coding style which permit straight-

forward integration of legacy Fortran 77 code. Details of the package (cauldron)

are found in Appendix A, and in the remainder of this chapter we briefly overview

the programming principles used in the development of the package.









3.7.1 Orthogonality of Code Development

By orthogonality of code development, we mean that different aspects of the

code can be developed independently. Thus one may decide to build a completely dif-

ferent algorithm than relax-and-drive, for example, to propagate the mixed quantum-

classical system, but be able to do so without changing aspects of the code which

define the system, compute its properties, read configuration files, generate output

files, and so forth. This helps ensure that once a version of the code works well,

changes to its components will be less likely to introduce errors. This aspect of

program development is crucial for software designed in a team environment, but

also very useful for the solitary designer when the problems and their solutions may

rapidly change. Orthogonality has been kept within cauldron through judicious use

of variable and subroutine naming conventions, and by building a solid hierarchy of

directories and subroutines from the beginning.

3.7.2 Extensibility

In scientific work, the systems studied and the solutions used are constantly

(1I.ii,-il,-. as progress is made in understanding the solutions, and new problems

arise. One way to help maintain flexibility, which has been used throughout cauldron,

is to ensure that systems are represented as generically and as dynamically as pos-

sible. Generic code attempts to represent the fundamental aspects of all molecular

- -1. ii, -. for example, by the same set of variables and .iili.-,. When new com-

ponents (e.g., new kinds or numbers of nuclei) are added to the system, the same

variables are used, and it is only the interpretation of the results that differs from

,-i-1-. I, to system. By dynamic representations, we refer to the defining of variable

size at runtime rather than fixing the size at compilation. The major benefit in this

comes from being able to implement models of varying sizes without recompiling

the code and creating a new executable for each system studied. Systems can then







38

be defined completely within input files, for example, preserving the polished exe-

cutable without modification. Fortran 90 encourages dynamic memory allocation,

which has been used to great advantage in cauldron to provide very extensible code.















CHAPTER 4
ONE-DIMENSIONAL TWO-STATE MODELS

4.1 Introduction

While our ultimate goal is to study realistic three-dimensional models of alkali

atoms embedded in rare gas clusters, the complexity of these systems places full

quantal solutions out of computational reach. On the other hand, simple models

can sometimes capture elements of larger and more realistic systems, and provide

a rigorous basis for validating approximate numerical methods. In this chapter,

we study the dynamics of three simple models involving two electronic states and

one nuclear coordinate. The first represents photoinduced desorption of an alkali

atom from a metal surface, where near-resonant electron transfer is important. The

second models the collision between two nuclei in a framework involving two avoided

crossings. The third models the photoinduced dissociation of the Nal complex,

where oscillatory motion between neutral and ionic states is observed. Because of

the limited size of these models, in each case we are able to propagate a grid solution

to the TDSE, and thus compare our EP-QCLE approach to the dynamics of the full

quantal system. For all three models, we will see that the mixed quantum-classical

methods provide qualitatively, and often quantitatively similar results to the full

quantal evolution.

4.2 Effective-Potential QCLE in the Diabatic Representation

In Chapter 3, we derived the EP-QCLE for an arbitrary basis. Here, we consider

the specific case where the system is described in an orthonormal diabetic basis.

There are many varieties of diabetic bases,104'105 but here we refer to the strictly









diabetic representation {()d} where the momentum coupling vanishes,106


({Qd =/Ol d) = 0. (4.1)

We also assume that the basis does not explicitly depend on P or t, so that Q = 0.

Since the basis is orthonormal, the overlap is unity, and Eq. 3.7 reduces to the simple

form,

dF
S- -[Hqu, ]. (4.2)

In the diabetic representation, the effective force is also simplified, since the opera-

tors in the quantal trace can be replaced directly with their matrix representations,

Trqu [FOV/OR]
FHF (4.3)
Trqu [r]

For our test models, the partial derivatives of the potential can be calculated ana-

lytically at each grid point in phase space. Since the PWTDM is propagated along

these grid points, the product in the numerator of Eq. 4.3 is computed through

matrix multiplication, while the quantal trace is calculated by summing over the

diagonal components of this matrix product. The quantal trace in the denominator,

on the other hand, is simply the sum over the diagonal components of the PWTDM.

Had we used a different basis, we would not have been able to simplify the

EP-QCLE in this way. However, the real advantage to the diabetic representation

is that for very small systems, it lends itself to a fully quantal numerical solution

through the propagation of the TDSE on a grid. One scheme, the split operator

fast Fourier transform (SO-FFT) method, splits the Hamiltonian into its kinetic

and potential components, and uses the fast Fourier transform to compute the evo-

lution due to the kinetic terms.107'108 While this method is very accurate, it is also

intractable for systems with more than a few degrees of freedom. However, because









our models are simple, the SO-FFT procedure provides an excellent test of the accu-

racy of the EP-QCLE. A complete description of the SO-FFT is given in Appendix

B, and the code implementing the SO-FFT (qualdron) is described in Appendix C.

4.3 Near-Resonant Electron Transfer Between an Alkali Atom and
Metal Surface

4.3.1 Model Details

In the first of our test systems, we consider a model describing the near-resonant

electron transfer between an alkali atom (Ak) and a metal surface (I\) at thermal

energies. The model consists of two diabetic surfaces corresponding to a ground state

of neutral components Ak + M (state 1) and an excited state for ionic components

Ak+ + M- (state 2), which cross at short distance. The surfaces and interaction

term are given by,85,84


H11(R) = Uo{exp[-2a(R Ro)]+ 2exp[-a(R Ro)]} /2, (4.4)

H2(R) Uo{exp[-2a(R -Ro)]- 2exp[-o(R -Ro)]} + /2, (4.5)

H12(R) = cexp[-a2(R- 1) 2]. (4.6)

Here, R is the distance between the metal surface and the nuclear center of the Ak

atom, and is the quasiclassical variable over which we take the PWT. The ionic

curve, H22(R), is a Morse potential with a binding energy Uo. The repulsive neutral

curve, H11(R), is offset relative to the ionic curve to give an excitation potential Ac.

The strength of the coupling term, H12(R), is characterized by 3 and peaks at the

crossing R = R, between H11 and H22.

The initial state is formed by approximating the ionic surface as a harmonic

potential around its minimum R = Ro, and finding the lowest bound vibrational

state within that (harmonic) well,


(R) = ( ) exp R2 exp[iPo(R Ro)]. (4.7)
7ff2 $U










Table 4-1. Parameters used in the N..--U1 f...:e and Li-surface models.

Hamiltonian I Hamiltonian II
Parameter value (au) value (au)

Uo 0.025 0.184
a 0.4 0.4
e 0.005 0.147
Ro 5.0 5.0
R, 12.5 9.0
Po 0.0 0.0
a 0.233153 0.1i
f 0.15 0.15
M 42,300 12,800

The PWTDOp is the PWT of Eq. 4.7 over R, giving a Gaussian density in (P,R),


F(P,R) I Ro-R 2( _Po)2 (4.8)


At t = 0, the electronic state is promoted by a sudden optical excitation to the

repulsive neutral potential, so that the PWTDM becomes


l(P, R) = 1 (R Ro 2 U(P PO P (4.9)

with F12 F21 = 22 0. The simulation follows the spontaneous decay of this

state.

The parameters used in the calculation are shown in Table 4-1, where we con-

sider two model Hamiltonians: (I) N..--II[ f.,:e and (II) Li-surface. The diabetic

potentials for Hamiltonians I and II are shown in Figures 4-1 and 4-2.

4.3.2 Properties of Interest

Populations. We can follow the populations over time by taking the full

classical trace over either diagonal element of the PWTDM,


r] J = fdRdPri(R, P). (4.10)
(7








43















0.02

H22
22 ro -------
0.015 \H12

0.01

0.005
s -7 --- --- -- -- -- -- -- -- -- -- --
H0 ------- ---- ------------------- "- < ^--- ----------------------------------------------------------
CT 0

C -0.005
LU
-0.01 -

-0.015

-0.02 -

-0.025
0 5 10 15 20 25 30
R (au)


Figure 4-1. Potential curves for Hamiltonian I: Na incident upon a metal
surface.








44
















0.15
H11
\H22
H12
0.1



0.05

-5 'i ---------- "----
) 0



-0.05



-0.1



-0.15
0 5 10 15 20
R (au)



Figure 4-2. Potential curves for Hamiltonian II: Li incident upon a metal
surface.








Since the system begins in the repulsive (neutral) state, one expects a certain per-
centage of the population to fall to the attractive (ionic) state as the system passes
through the region of nonnegligible potential interaction.
Coherences. The coherence between the neutral and ionic state is described
by the real and imaginary components of the off-diagonal terms of the PWTDM,
for example

i= j dRdPiFj(R, P). (4.11)

Coherence is a purely quantum phenomenon, and one measure of the 11,.,lil of a
mixed quantum-classical method is the degree to which it maintains coherence.
Position expectation values. One observable we can study is the expecta-
tion of position,

(R) =Tr[F(R,P)R]
I dRdPTrqu[F(R, P)]R. (4.12)

We can also measure the dispersion,

aR [((R (R))2)]1/2
[f dRdPTrqu[F(R, P)](R- (R))2 (4.13)

Momentum expectation values. Similarly, we can compare the expecta-
tion and dispersion of moment,

(P) = Tr[F(R, P)P]
f dRdPTrqu[F(R, P)]P, (4.14)
p = [((p (P))2)]1/2
/ dRdPTrqu[F(R, P)](P- (P))2 (4.15)
/ 1/









Probability density. We can compute the probability density p(R) from the

PWTDM by taking the trace over quantum variables and moment,
00
p(R) = dPTrq[F(R, P)]. (4.16)
-00

In practice, since the grid in phase space quickly deforms as the system evolves, we

must find a way of approximating this integral. One procedure is to determine the

support of the PWTDM in quasiclassical phase space, [Rmin, Rmx] x [Pmin, PFia].

We then divide this space into NR x Np equisized bins {bi}, such that bin bij spans

the rectangular region,

Rn (i 1 Rmax Rmin Rmax Rmin
[Rmin + (i 1) Rmin + i ]
NR NR

x [PTi + (j ) Pmin + J ]. (4.17)
Np Np

We then assign a value to each bin, pij, which is the weighted sum of all Nij trajec-

tory points which fall within that bin,

Nij
r F(Rk, Pk)
P 1 (4.18)

We can determine the matrix probability density from pai by summing over all bins

containing a given position R,


p(R) = VpR, VR b. (4.19)


Finally, we compute the probability density by taking the quantal trace over the

matrix probability density,


p(R) = Trqu[p(R)].


(4.20)









This probability density can be compared to the density function obtained from the

SO-FFT simulation,


p(R) = I(R) 2 + 2( R)2. (4.21)

Wavefunction and PWTDM. Of course, the evolution of the quantum

wavefunction can be contrasted directly with the evolution of the PWTDM. How-

ever, we can also observe the distortion in the phase space grid used by the EP-QCLE

method. While the grid is initially uniform, it changes shape in an interesting way

because of the action of the effective potential.

4.3.3 Results

Figures 4-3 and 4-4 show the initial wavefunction and its PWT, respectively.

Note that the PWTDM formed from a Gaussian wavepacket is a Gaussian function

itself, albeit in two-dimensional phase space. Figure 4-5 shows the wavefunction at

t = 14000 au, having been propagated through the SO-FFT method. The PWTDM

evolves in phase space through the EP-QCLE, and in Figure 4-6 we show the grid in

phase space at the final time. While substantially distorted from its initial uniform

distribution, we notice that the points are globally positioned along a straight line

in phase space. This reflects the .i-i. ,,,.1, -. i' state, where each point is subject to a

vanishing Hellmann-Feynman force, and thus propagates at constant velocity.

The observables are presented in Figures 4-7 to 4-13. There are a number of

interesting things we can glean from these plots. From Figures 4-7 and 4-8 we see

that as the atom moves away from the metal surface, much of its population shifts

from the neutral to ionic state. In the case of Na, approximately 2/3 of the neutral

population shifts, while for Li the transfer is total. This may reflect the stronger

interaction coupling involved with Li.

Figures 4-9 and 4-10 describe the coherence between the states. For the

N.- -ii f.i.e system, the coherence remains large for long times, while for the Li-surface











































3 3.5 4 4.5 5 5.5 6 6.5 7
R (au)


Figure 4-3. T(R) at t = 0 au, for the N..--i i f.-.e model. This wavefunction
is evolved through the SO-FFT algorithm.





























0.35
0.3
0.25
0.2
0.15
0.1
0.05
0


Figure 4-4. F1n at t = 0 au, for the N..--II. f.i.e model. This PWTDM is
evolved through the EP-QCLE method.





























I I I


0.4


0.3


0.2


0.1


0


-0.1


-0.2


-0.3


-0.4
20


Figure 4-5. T/(R) at t = 14000 au, for the N..--11. f.,.e model.


25 30 35
R (au)













































60 65 70 75 80
P (au)


85 90 95


Figure 4-6.


Phase space grid points at t
model.


14000 au, for the N..--II. f.T.e


100













































0 2000 4000


6000


8000 10000 12000 14000


time (au)


Figure 4-7. Na populations p]q and rq2 vs. time.























1 I ii SO-FFT: rii

/ EP-QCLE: 2
SO-FFT: q2
0.8


"0
S0.6



.5 0.4
o


0.2




2000
n-m1 .500
0 1000
0 time (au)




Figure 4-8. Li popul:lltin r and 'l2 vs. time.
























0.4
EP-QCLE
SO-FFT
0.3 -


0.2


0.1


0


-0.1


-0.2


-0.3


-0.4
0 2000 4000 6000 8000 10000 12000 14000
time (au)



Figure 4 9. Coherence described by Re(r/12) vs. time, for the N..-- u1 f.,.:e sys-
tem.

























0.45 u-I- I

0.4

0.35

0 0.3
S 0.25

S 0.2

o 0.15

0.1

0.05

0

-0.05
0 500 1000 1500 2000
time (au)


Figure 4-10. Coherence described by Re(lq12) vs. time, for the Li-surface sys-
tem.













































0 2000


4000


6000 8000 10000 12000 14000


time (au)


Figure 4-11. Expectation of position and dispersion for the N..--i f.,.:e sys-
tem.























EP-QCLE:


SO-FFT:


EP-QCLE: Gp
SO-FFT: op


0 2000 4000


6000


8000 10000 12000 14000


time (au)


Figure 4-12. Expectation of momentum and dispersion for the N..--i fi .e
system.







58















0.16
EP-QCLE
SO-FFT
0.14


0.12


0.1


- 0.08


0.06


0.04 -


0.02 -


0
0 5 10 15 20 25 30 35 40
R (au)


Figure 4-13. Density function p(R) for the N..--U1 f[i.:e system.









-i,-I-1 iii. the coherence rapidly diminishes once the interaction potential is crossed.

This can be explained by the size of the energy gap at ..-1i.. .1l i' distances, as

larger ionization energies lead to more rapidly vanishing coherences.

Figure 4-11 shows the expectation of the position of Na steadily increases

from its initial average, while the dispersion in position initially decreases and then

increases again. On the other hand, Figure 4-12 shows a marked difference in the

behavior of the expectation of the momentum, where it begins at (P) = 0 au, rapidly

increases and then becomes stationary around (P) = 78 au. We also find that the

momentum dispersion first increases, and then decreases.

Finally, Figure 4-13 presents the density function at the end of the simulation.

Because of the distortion of the phase space grid, the density function obtained

from the EP-QCLE had to be calculated using bins and through approximations

within these bins, and thus is subject to some noise. Nevertheless, we find excellent

agreement between the EP-QCLE and SO-FFT results.

For all observables, the EP-QCLE results are quantitatively similar to the SO-

FFT results to visual resolution. Having studied both Na and Li approaching a

metal surface, we see that the EP-QCLE can be expected to yield very accurate

results for these kinds of systems, even when coherence is maintained over long

periods.

4.4 Binary Collision Involving Two Avoided Crossings

4.4.1 Model Details

For the second system, we look at a two-state collision model where the diabetic

surfaces intersect twice. Because of the dual crossing and the coupling in this region,

quantum interference and effects such as tunnelling play a substantial role in the

dynamics of the quasiclassical variable. As such, this model is quite demanding

for mixed quantum-classical methods, where one can expect deviations at lower

energies.










Table 4-2. Parameters used in the dual avoided crossing collision model.

Parameter Value (au)

Ro -8
Po [10, 30]
a 2.5176
M 2000

The Hamiltonian elements are79


H1 (R) = 0, (4.22)

H22(R) = -0.1exp(-0.28R2) + 0.05, (4.23)

H12(R) = 0.015 exp(-0.06R2). (4.24)

In the above, R is the nuclear-nuclear separation. The initial state is a ground state

Gaussian wavepacket which begins in the .i-.,mptotic region R = 8 au. Its PWT

over R gives the Gaussian density,


ll(P, R) = 1 [ R) 2p 2 (4.25)

with F12 F21 F22 = 0. At t 0, the wavepacket propagates toward the

region of coupling, where it is partially transmitted and partially reflected, now

with populations in both the ground and excited state. The parameters used in the

calculation are shown in Table 4-2. The diabetic potentials used in the model are

shown in Figure 4-14.

4.4.2 Properties of Interest

As in the alkali atom-surface model, we can follow the populations over time.

The system begins in the ground state, and as it passes through the collision region,

we expect some of the population to transfer to the excited state. The amount of

transfer depends on the collision energy. We can also follow the coherence once the
























H -
H22
H12







-











-10 -5 0 5 10
R (au)


Figure 4-14. Potential curves for the dual avoided crossing collision.


0.08


0.06


0.04


0.02


0


-0.02


-0.04


-0.06









collision has ended and the system reaches ..i mptotic values. We also consider the

transmission of the ground state as the wavepacket passes through the interaction.

Transmission. As the wavepacket passes through the collision, part of it

continues forward while the remainder is reflected back. We can study this by

computing the probability of transmission of the ground state,



0
^ = dRdPTn{RP). (4.26)


Although not included, we could also compute the probability of reflection of the

ground state, as well as the .1- i.l.1 .tic populations of the excited state.

4.4.3 Results

In Figure 4-15 we see the total population transfer at energy Po = 30 au. In

this energy region, the EP-QCLE matches the quantum results to within a cou-

ple percent. For both the EP-QCLE and SO-FFT simulations, the reflection of

the wavefunction (or PWTDM) was negligible, so that we need only compare the

transmission. This negligible reflection was found throughout the energies studied.

In Figure 4-16 we see the coherence as a function of time. The variations are

large initially and diminish .i ,'iiil' .tically to regular oscillations. The EP-QCLE

captures this behavior qualitatively, and to within 5% quantitatively.

We show the deformation of the phase space grid in Figure 4-17. We see

that it is primarily the inner part of the grid that undergoes deformation, while

the enclosing points maintain a structured order. This is likely due to only a small

range of values in phase space which are seriously affected by the potentials. Outside

the interaction region, all Hellmann-Feynman forces are zero, so one would expect

deformations only for points whose time in the interaction region was significant.

In Figure 4-18 we display the transmission probability for a wide range of ener-

gies. As expected, the EP-QCLE performs better for higher energies, and while the

double-well is qualitatively reproduced, the EP-QCLE fails to give quantitatively









accurate results for energies lower than (log E = -2). This is likely due to the quan-

tum tunnelling effects described earlier, which are not expected to be reproduced

well by the EP-QCLE. We also compare transmission probabilities obtained through

surface hopping methods by Tully and coworkers.51 These surface hopping calcu-

lations show deviations from the quantal results that are similar to the EP-QCLE

probabilities at lower energies; at higher energies, the EP-QCLE is slightly superior

to the surface hopping scheme.

4.5 Photoinduced Dissociation of a Diatomic System

4.5.1 Model Details

For the third test system, we explore a model of the Nal complex. As in

the previous models, it involves two diabetic surfaces and an interaction around

the avoided crossing. A substantially different feature, however, is a long-range

Coulombic attraction in an ionic state. As we shall see, this attractive potential

results in the the complex oscillating between a neutral and ionic state as the sodium

and iodine separate and come back together, partially dissociating at each crossing

into an .i-, ,iii'l.itically neutral state. This oscillatory motion is a good test for the

EP-QCLE at long times in cases where ..i- mptotic states are not reached quickly.

The Hamiltonian elements .i "''


H11(R) = Aiexp[-/3(R- Ro)], (4.27)

H22R) [A2 (B2/R)] exp(-R/p) /R- (+ -)/2R4

-C2/R6 2+-/R7 + AEo, (4.28)

H12(R) A12 exp[- 12(R R)2]. (4.29)

To form the initial state, we Taylor expand (to first order) the ionic potential H22

about its minimum R = Ro, and find the lowest energy state of this harmonic well.

At t = 0, the wavepacket undergoes a sudden optical promotion to the neutral curve,






















EP-QCLE: rl
SO-FFT: l
EP-QCLE: 12
SO-FFT: 12


0 200 400 600 800 1000 1200 1400
time (au)


Figure 4-15. Populations qrl and rl2 vs. time for the dual crossing collision
model.























0.1



0.05



0


-0.05



-0.1



-0.15



-0.2
0 200 400


Figure 4-16.


EP-QCLE
SO-FFT


600 800 1000 1200 1400
time (au)


Coherence described by Re(riq2) vs. time, for the dual crossing
collision model.











































ue 4-1


(rid ,, rMu/ 3) 34


S400 a, for le dualoss

0coli,..i,







67














S" / \ EP-QCLE
S------- SO-FFT -
S0.9- Tully et al. -


0.8


0.7
I- \\
F-

0.6 -


0.5


0.4


0.3
-4 -3.5 -3 -2.5 -2 -1.5 -1
loge(E) (au)


Figure 4-18. Probability of transmission in the ground state, for the dual
crossing collision model.










Table 4-3. Parameters used in the Nal complex model.

Parameter Value (au)

Ro 5.047
Po 0.0
a 0.12462
A1 0 i111 i ',
A2 101.43
A12 0.00202
B2 3.000
C2 18.950
A+ 2.756
A- 12.179
p 0.660
AEo 0.07626
/1 2.158
/12 0.194
R, 13.24
M 35,482

so that the PWTDM becomes,


rl(P, R) 1 R RO 2 2(p o)2 (4.30)

with F12 F21 F22 = 0. The simulation follows the resulting motion of this

state. The model parameters are shown in Table 4-3. The diabetic potentials are

displayed in Figure 4-19.

4.5.2 Properties of Interest

In addition to observing the coherence, expectation value of the position and its

dispersion, and the phase space grid, we also consider bound and free neutral and

ionic populations as the Nal complex oscillates from its primarily ionic to primarily

covalent state.

Bound and free populations. It is interesting to follow the populations of

the ionic and neutral states as the system evolves, giving insight into the nature of

the dissociation into an ..I-.i.l.1 i ..lly free neutral system. For this, we define three













































5 10 15 20 25
R (au)


Figure 4-19. Potential curves for the Nal complex.


0.05




0


-0.05










populations: the ionic, the bound neutral and the free neutral. The ionic popula-

tion is the probability of finding the system in the ionic state at any internuclear

separation,
00
/2 = dRdP22(R, P). (4.31)
-00

We define the bound neutral population is the probability of finding the Nal complex

in the neutral state at nuclear separation up to the crossing point R = R,,

R
f = fdRdPI (R, P), (4.32)
0

while the free ionic population is defined as the probability of the neutral state from

the crossing point beyond,


J{ = dRdPFi(R, P). (4.33)
R

This division between bound and free is motivated by the observation that the

majority of the transfer between ionic and neutral potentials occurs at the avoided

crossing, and that any part of the wavepacket which ends up in the neutral state

but propagating toward infinite separation has a negligible probability of shifting to

the ionic state much beyond the avoided crossing.

4.5.3 Results

The ionic and covalent populations are displayed in Figure 4-20. We see the

oscillations in the populations between ionic and covalent, repeating approximately

every 40 000 au. This pattern can be compared to the expectation values of position

see in Figure 4-21. The position oscillates with the same frequency as the change

in population, showing that each time the wavepacket heads across the avoided

crossing from its bound covalent state, it converts almost completely into the ionic

state, with a small amount escaping into the free neutral state. Over time, the









free neutral population gradually increases at these crossings, and the Nal slowly

dissociates. The EP-QCLE is quantitatively similar to the exact results for the first

half of the simulation, and maintains qualitative accuracy for the remainder.

The dispersion in Figure 4-21 shows an interesting difference between the exact

and the quantum-classical algorithm. While the dispersion continues to rise in the

SO-FFT simulation, it reaches its first peak and then begins to decline somewhat in

the EP-QCLE simulation. This reflects the nature of the effective potential, where

each point is guided by a combination of excited and ground state forces. Because

the ionic curve does not permit escape, what would normally be ..i- .mptotically free

wavepackets tend to be pulled back toward the crossing because of the attractive

ionic potential. Consequently, the free neutral population is always lower using the

EP-QCLE equation than the SO-FFT, an observation supported by Figure 4-20.

Since the majority of the population remains in the ionic or bound neutral state,

and these populations are well matched between the exact and quantum-classical

simulations, it is not surprising that the expectation value of the position is quan-

titatively in the beginning, and qualitatively for the i. is,.iinii.-. similar for both

methods. However, the dispersion is much more sensitive to the increased ..I-i.mp-

totically free wavepackets in the SO-FFT simulation, and for the reasons discussed,

we find significant divergence between the EP-QCLE and SO-FFT results.

The coherence, shown in Figure 4-22 initially peaks through the first crossing,

but through subsequent crossings it is substantially diminished. However, the EP-

QCLE shows quantitatively similar results to the SO-FFT calculations.

The deformation of the phase space grid, plotted in Figure 4-23, has char-

acteristics not seen in the other two models. One line of points emits from the

center of the cluster, quickly straightening and reflecting the negligible force on the

points. These points correspond to the .,i- ,,iiilitically free neutral components of

the PWTDM. The second group circles around, gaining velocity and position, then









turning. These ellipses are characteristic of the phase space of classical particles in

a well, and indeed reflect the quasiclassical motion under the Hellmann-Feynman

force of the PWTDM points as they follow the ionic and bound covalent curves.

4.6 Comparison Using Variable and Constant Timesteps

In this Section, we evaluate the usefulness of the variable timestep aspect of

the relax-and-drive algorithm. To do this, we consider the N..--II f.,:e algorithm of

Section 4.3, and simulate using varying upper and lower tolerances. The number

of steps taken in each case is compared with the number that would be required of

the same algorithm, but keeping the timestep fixed. The fixed timestep would nec-

essarily advance by steps no greater than the smallest timestep used in the variable

timestep algorithm, and it is based on this timestep that we estimate the corre-

sponding steps required for the fixed timestep approach.

The results are shown in Figure 4-24. We see that as the tolerance decreases

(and thus the accuracy increases), the fraction of steps saved by the introduction of

the variable timestep increases superlinearly. One concludes that while the variable

timestep may not be important for low accuracy simulations, considerable compu-

tational savings can be had for high accuracy propagation.

4.7 Conclusion

By examining three simple two-state models, we were able to compare the

EP-QCLE method with the exact SO-FFT quantum mechanical solution. For all

models, we found very good agreement between the EP-QCLE and the SO-FFT

results. The agreement was at least qualitative, and in many cases quantitative to

visual precision. We also saw conditions under which the quantum-classical model

deviated from exact quantal results. Finally, we compared a fixed timestep variant

of the relax-and-drive algorithm with the variable timestep version.

For the model of an alkali atom approaching a metal surface, we examined prob-

ability transitions, position expectation and its deviation, momentum expectation








73

















2
EP-QCLE: Ionic
SO-FFT: Ionic
EP-QCLE: Neutral Bound
SO-FFT: Neutral Bound
EP-QCLE: Neutral Free
C 1.5 SO-FFT: Neutral Free
0

0




0
0z
.--" ------


"oz- -- -- -- --


o 0.5 .

-- --- --- -



0 T-- --- T- ---------- I-- I'^ 'T '''
0 20000 40000 60000 80000 100000 120000
time (au)



Figure 4-20. Ionic and neutral populations over time, for the Nal complex.







74














25
EP-QCLE:
SO-FFT: -:i-
EP-QCLE: GR
SO-EFT: o -:
20



15
A
v

10 / -.---"



5



0 -I I I I I I I I
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
time (au)


Figure 4-21. Expectation of position and its deviance, for the Nal complex.

























0.35
EP-QCLE
SO-FFT
0.3


0.25
(1)


0 0.2
rr

o
| 0.15

0
t-

0.1


0.05 -


0
0 20000 40000 60000 80000 100000 120000
time (au)


Figure 4-22. Coherence as a function of time, for the Nal complex.






















140


120


100


80

rr
60


40


20 ..


0
-80 -60 -40 -20 0 20 40 60 80
P (au)


Figure 4-23. Phase space grid at the end of the simulation, for the Nal
complex.























3500
Variable
Fixed --x
3000


2500
0-
0)
a 2000
E
0
5 1500 -
E


1000


500 -



103-10-1 10-5-10-3 10 -5 10-9- 10-7
tolerance



Figure 4-24. Number of steps required by the relax-and-drive algorithm,
compared to an estimated number required for a fixed timestep
version.









and its deviation, density function and phase space grid evolution. We found that

the EP-QCLE method reproduced quantitatively the exact values found through the

quantum model for all observables. We also found the phase space grid to deform

substantially as the system evolved, initially distorting but eventually lining up in

a straight line as the effective force vanished.

For the dual crossing diabetic surface collision, in addition to probability tran-

sitions for a given energy, we calculated the transmission probability of the ground

state for a wide range of energies. This showed the EP-QCLE to deviate at the lower

energies, where nuclear interference effects were important. At higher energies, the

correspondence between the mixed quantum-classical results and the exact quantum

results was good. We also noted that the phase space grid deformed only for points

which spent a significant amount of time in the region of strong interaction. This is

reasonable, as the deformation comes from the effective force, which is non-varying

(and in fact zero) outside the interaction.

The Nal model showed interesting back-and-forth transitions between the ionic

and covalent state, as the internuclear distance oscillated due to the long-range

attractive ionic potential. We also saw that each passing through the avoided cross-

ing led to a small amount of dissociation into the ..i-1.mptotically free neutral state.

Ionic, bound neutral and free neutral populations were quantitatively similar for the

EP-QCLE and SO-FFT algorithms for the beginning of the simulations, and quali-

tatively similar for the remainder. Coherences were quantitatively similar, showing

an initial peak after the first pass through the avoided crossing, and much smaller

magnitudes thereafter. Finally, the grid deformation showed the grid to split into

two groups. The first represented the .i -1'mptotically free neutral state, where the

points form a straight line as the Hellmann-Feynman force vanishes and the parti-

cles propagate to infinite distances. The second group circled in phase space, as the







79

points oscillated back and forth in the potential well formed by the ionic attraction

at large distances and the neutral repulsion at small distances.

The EP-QCLE method was shown to be very robust under a wide variety of

conditions. Comparing its variable timestep version (which was used in all the simu-

lations) to the fixed timestep alternative, we found that not only was the EP-QCLE

method accurate, but use of the relax-and-drive propagation generated much more

efficient simulations than could have been obtained without dynamically varying the

timestep.















CHAPTER 5
ALKALI ATOM-RARE GAS CLUSTERS: GENERAL FORMULATION

5.1 Introduction

In this chapter, we describe the interactions of an alkali atom (Ak) embedded in

a cluster of rare gas atoms (RgN). Cluster dynamics are interesting insofar as they

present a bridge between isolated atoms and the (effectively) infinite bulk liquid, and

can provide insight into the dynamics of molecules on extended surfaces. The Ak-

RgN clusters are among the simplest cluster systems to study because of the presence

of a single valence electron in the alkali atom, and the closed shell structure of the

rare gas atoms. Pseudopotential descriptions of the interactions between the valence

electron (e-), the Ak core and the Rg atoms can greatly simplify the description of

the cluster with very modest penalties in accuracy. In the context of mixed quantum-

classical models, these clusters are well suited to simulation through the EP-QCLE

by treating the valence electron quantally and the nuclear cores quasiclassically. For

the majority of the chapter, we consider the general case of an alkali atom embedded

in a rare gas cluster, although our simulations in Chapter 6 focus on the specific

case of the lithium atom (Li) embedded in a helium cluster (HeN).

5.2 Physical System

We consider a cluster initially at thermal equilibrium, and introduce a ground

or excited alkali atom to the center of the cluster. We do not concern ourselves

with the means by which the alkali atom is excited or embedded within the cluster,

although typically the excitation is due to a laser pulse.

From this initial setup, we follow the dynamics of the alkali atom as it deexcites

and moves within the cluster. We further assume the cluster is isolated from any

environmental effects, so that spectral and configuration measurements represent









those of an isolated Ak-RgN cluster. As we see later, we introduce a containing

potential to keep the Rg atoms from dispersing. This potential is used strictly to

maintain a cluster formation, and does not represent interaction with an environ-

ment.

5.3 Properties of Interest

First of all, we are interested in the structure of the Rg cluster in equilibrium,

before the introduction of the Ak atom. We need to reproduce known density profiles

and pair-pair correlation functions to ensure the cluster is representative of a real

ll,-i, .1i system. Secondly, we are interested in following the dynamics of the Ak

atom as it moves within the cluster. In particular, we are looking for migration from

the center of the cluster to its surface, as other experimental and theoretical studies

indicate that the Ak atom tends to reside on the cluster surface. Finally, we wish

to compute the time dependent emission spectra resulting from the deexcitation of

the Ak atom,


Ak(n'l') +RgN Ak(nl) + RgN + (5.1)


where 0 is a photon with energy corresponding to the electronic decay.

5.4 Hamiltonian for Alkali-Rare Gas Pairs

Following the dynamics of an Ak-RgN cluster through a full quantal treatment

would be an extremely demanding problem computationally for more than a few Rg

atoms. Instead, we reduce the Ak-Rg interaction to a three-body problem by using

pseudopotential interactions between the nuclear cores and the electron. The pseu-

dopotential treatment is explored extensively for the single Ak-Rg pair in Reyes.109

In this section, we summarize this approach.

In order to describe the three-body interaction, we consider a fixed nuclear

configuration with RAB the position vector from the alkali core (A) to the rare gas

atom (B), and rA (rB) the position vector from the alkali core (rare gas atom) to









the electron e-. With this notation, we can write the Hamiltonian in five distinct

components,


pair 2 r Ak(r)
f- = -VA A A)+VBgrB)+VA cro A, AB)

+V1ore(RAB). (5.2)


We have implicitly taken the PWT over nuclear variables but not electronic variables.

Because the potentials can all be expressed as polynomial functions of Q, the PWT

amounts to replacing Q by R throughout the Hamiltonian. In what follows, we

will assume we are working with partially Wigner transformed operators, but drop

the subscript 'W' for notational simplicity. Also note that the electronic variable

remains quantal, although we will use the notation r rather than q for consistency

with common usage in the literature.

The first term on the RHS of Eq. 5.2 is, of course, the kinetic energy operator

of the valence electron. The second and third terms are the potentials arising from

the interaction of the electron with the Ak core and the Rg atom, respectively. The

VA'O term is a crossterm stemming from the polarization of the Rg atom by both

the valence electron and Ak core. Finally, the last term Vo" is the interaction

between the Ak core and the Rg atom. We examine each of these potentials in

detail for the general Ak-Rg pair, and provide specific parameters for the Li-He

interaction in Chapter 6.

The e--Ak core potential can be divided into three components,
V k (AZA ^tAk
k(rAk ) -Z VA (-A) V+T(rA), (5.3)
TA

where the first term is the Coulomb interaction between the Ak core of charge ZA

and the electron. The second contribution arises from the dipole polarization of the









Ak core by the valence electron,
d
VT^'j -A A) A- w(A, 6A)2, (5.4)
2rA

with a$ the dipole polarizability of the core and w a cutoff function of the distance

6,

w(r, 6) = exp(-6r2). (5.5)

The final contribution to VAk is an i-dependent short-range pseudopotential,


r (rx) B1, exp(- / )P1x, (5.6)
l,i

where B,,i and /3,i are pseudopotential parameters adjusted to fit experimental data,

Pix is the projection operator on angular symmetry 1,

Pix = Y Yim(rx))(Yim(x)I, (5.7)

and the states { Yim(x))} are the spherical harmonic functions centered on the core

X. This pseudopotential simulates repulsion due to the effects of the Pauli exclusion

principle when the valence electron approaches the core electrons, as well as the

attraction due to the incomplete screening of the nuclear charge.

The third term in Eq. 5.2 is the e--Rg potential, and can be divided into two

terms,
V""(rB) pol-(r) + *(rB). (5.8)
v g^rB VToTM (5.8)

The first component stems from the polarization of the Rg atom by the electron,
d q4
tR(r) (rB, 6)4 (r, 6) (5.9)
VB ()r









where aC a/ 6/1, a' being the quadrupole polarizability of the Rg atom

and -6/1 the dynamical correction to the static polarizability. The short-range

pseudopotential VJ' is defined in Eq. 5.6.

The fourth term in Eq. 5.2 is a crossterm arising from the polarization of the

Rg atom by both the Ak core and the valence electron,

Ir Pi (cos) q P2 PCos0)
VAo(rA, RAB) ZA 2 2 WB, 6)2 ZAa 2 2 -B 6)3. (5.10)
IAB B 1)ABrB

Here, P1 and P2 are the Legendre polynomials and 0 is the angle between the vectors

RAB and rB.

Finally, the last term in Eq. 5.2 is the interaction between the alkali core and

the rare gas atom. It is assumed to involve short-range, dipole and quadrupole

contributions,

I1 1 '1
VA re(RAB) r V(RAB) 2 B 2 2 BB 3 (5.11)

where ac' = a + 2a' d2. The short-range term VjI' has been determined by

assuming it to have the form,


VI (RAB) = Aexp(-bRAB), (5.12)


and adjusting the parameters A and b so that VAI'e(RAB) in Eq. 5.11 fits the most

accurate curves in the literature. This procedure has been shown to accurately

reproduce Ak-Rg energy curves for a variety of alkali atom and rare gas combina-

tions.109

5.5 Hamiltonian for the Alkali-Rare Gas Cluster

The cluster is similar to the pair, in that each Ak-Rg pair in the cluster is

treated as a three-body problem involving alkali core, valence electron and rare

gas atom, and described using the same Hamiltonian elements. In the case of the

cluster, however, we also require that the Rg atoms maintain a cohesive structure










rather than drift apart. To ensure this, we impose a constraining potential on the

;- 1 I III

Defining RABi as the position vector from the alkali core to the Rg atom i, and

RB, the position vector of the Rg atom from the origin, we can write the cluster

Hamiltonian as,


icluster 1 2 Ak NR B
i=1
2 A (rA) + vY, rB)
N

i=1
N
+ VCore(R RB, ). (5.13)
j>i=1

The final term in Eq. 5.13 is the interaction between the Rg atoms for the

quasiclassical motion. Following Aziz and coworkers,110 we take the Rg-Rg potential

to have the form, for internuclear distance R,


Veore(R) = V*(x), (5.14)


where


V*(x) A* exp(-a*x + *2) F(x) 2 2 (5.15)
2 -;+ '6
j=0


F (x) = exp[-(D/x 1)2], x < D (5.16)
F(x) (5.16)
1, x >D

This potential has been written in terms of the dimensionless distance x = R/RM.

The constraining potential, Vhold is taken to be a sigmoidal function centered

along the boundary of a sphere of radius Rhold,


hold (RB ) a ex(5.17)
1+ exp [-b(RB, Rhold)]'









This potential shows .'.-mptotes Vhold(-oo) = 0 and Vzold(+oo) = a, with midpoint

at the holding radius Vhod(Rhold) = a/2. The steepness of the function is controlled

by the parameter b, and determines the strength and range of the holding force. It

acts on the Rg atoms only, keeping them bound roughly within a sphere of radius

Rhold, while permitting the alkali atom free motion within the cluster.
5.6 Electronic Spectral Calculations

When a molecular system undergoes electronic motion, the accelerating charges

emit electromagnetic radiation. At distances large compared to the electronic motion,

the flux of this radiation at point r from the center of the system is given by the

Poynting vector,111

1
S(t) -(Ex B)
Pto
to 2sin2 2
I-i[t)2 r (5.18)

where D(t) is the dipole moment of the system, [to is the permittivity constant and

c is the speed of light. By integrating over all angles, we obtain the power emited

by the source,

P(t) S(t) da

I |A (t)|2 (5.19)

where

A(t) (t). (5.20)

From Eq. 5.20, we can see that the computation of the dipole moment is essential

to the spectral calculation. In a mixed quantum-classical system, the dipole moment

is obtained by calculating the expectation value of the dipole operator D,


D(t) Tr[F(t)D]/Tr[P(t)].


(5.21)