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## Material Information- Title:
- Electron-nuclear dynamics a theoretical treatment using coherent states and the time-dependent variational principle
- Creator:
- Diz, AgustÃn Carlos, 1961-
- Publication Date:
- 1992
- Language:
- English
- Physical Description:
- viii, 112 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Approximation ( jstor )
Atoms ( jstor ) Coordinate systems ( jstor ) Electronics ( jstor ) Electrons ( jstor ) Energy ( jstor ) Momentum ( jstor ) Orbitals ( jstor ) Protons ( jstor ) Trajectories ( jstor ) Dissertations, Academic -- Physics -- UF Electron-atom collisions ( lcsh ) Physics thesis Ph. D Quantum theory ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1992.
- Bibliography:
- Includes bibliographical references (leaves 104-110)
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by AgustÃn Carlos Diz.
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ELECTRON-NUCLEAR DYNAMICS: A THEORETICAL TREATMENT USING COHERENT STATES AND THE TIME-DEPENDENT VARIATIONAL PRINCIPLE By AGUSTfN CARLOS DIZ 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 1992 To Marcela, Agustin and Ximena ACKNOWLEDGMENTS I would like to thank Erik Deumens and Yngve Ohm, for their warmth as persons and for the excellent scientists they are. The combination of both of these qualities made working with them something very special. I cannot think of a better atmosphere in which to learn and develop new ideas. I thank my wife, Marcela, who gave up so much for me; my children, Agustfn and Ximena, for their love; my parents, for being there and knowing that I can count on them. Finally, I am grateful to Keith Runge for all those discussions we had. iii TABLE OF CONTENTS ACKNOWLEDGMENTS ................. LIST OF TABLES ........................ LIST OF FIGURES ...................... ABSTRACT ......................... CHAPTERS 1 2 INTRODUCTION ......................... AN OVERVIEW OF TIME-DEPENDENT METHODS Potential Energy Surfaces ............ Time-Dependent Methods on PES ........ Classical Trajectory Schemes ......... Semiclassical Schemes .............. Exact Quantum Mechanical Schemes . . . . Time-Dependent Methods Without PES . . . . Close-Coupling Methods . . . . . . . . . . . Methods with Nuclear Dynamics ....... Electron Translation Factors .......... 3 THE END FORMALISM ............ The Time-Dependent Variational Principle ... Coherent States .................... The Equations of Motion .............. Interpretive Tools ................. .. 1 ~4 ~8 ~8 ~9 ... .. .10 ... .. .14 ... .. .15 . . . . . . . . . 16 . . . . . . . . . 20 . . . . . . . . . 24 . . . . . . . . . 24 . . . . . . . . . 28 . . . . . . . . . 40 . . . . . . . . . 51 4 RESULTS ..... The p+H Collision Introduction . .. Results ...... 60 60 60 63 iv . vi . vii viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The a+He Collision ........................ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 CONCLUSIONS ............................ APPENDICES A DETAILS OF DERIVATION IN ORTHONORMAL BASIS B DETAILS OF DERIVATION IN ATOMIC BASIS ...... REFERENCES ................................ BIOGRAPHICAL SKETCH ............................ .. . . . .. 74 ....... 74 ....... 76 ....... 80 .. . . .. . 83 ....... 96 ...... 104 ...... 111 V LIST OF TABLES Table 4-1: Contraction coefficients c and exponents a for the basis used in this work. . 63 Table 4-2: Total transfer cross-sections for proton colliding with a hydrogen atom (x 10-16 cm 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Table 4-3: Total transfer cross-sections for collisions of a proton on hydrogen and deuterium atoms, as well as that of a D' ion on hydrogen, for energies below 1 eV, compared to results by Hunter and Kuriyan (H&K) (Cross-section units x10 -16 cm 2). . . . . . . . . . . . . . . . . . . . . . . 68 Table 4-4: Excitation and transfer cross-sections for 2s, 2px, 2pz states and total 2p cross-sections (x 10-16 cm2 ). . . . . . . . . . . . . . . . . . . . . . . . . . 69 Table 4-5: One-electron transfer cross sections using three different basis sets (see text for description of the basis sets) are compared with experiments by Afrosimov. (Cross-section units x 10-17 cm2.) ............... 77 Table 4-6: Two-electron transfer cross sections using three different basis sets (see text for description of the basis sets) are compared with experiments by Afrosimov. (Cross-section units x 10-17 cm2.) . . . . . . . . . . . . . . . 78 vi LIST OF FIGURES Figure 4-1: Weighted transition probabilities for total electron transfer at 2 keV as a function of impact parameter. All data in atomic units. Dots are the results by Grin et al. The full line are the results of calculations by Liidde and Dreizler. The dash dotted line are the results from DYNAMO. (See text for references). . . . . . . . . . . . . . . . . . . . . . 65 Figure 4-2: Total transfer cross sections from 0.02 eV to 4000 eV. Comparison of experiment to theory. DYNAMO computations are the larger open circles joined by solid line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Figure 4-3: n=2 excitation cross sections. Comparison of DYNAMO results with experiments. Total 2p excitation calculations are shown by open circles joined by a solid line. 2s excitation calculations are shown by solid circles joined by a dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . 70 Figure 4-4: n=2 transfer cross sections. The symbols and lines have the same meaning as in the previous figure. . . . . . . . . . . . . . . . . . . . . . . . 71 Figure 4-5: Computed integral alignment and experimental results, as a function of collision energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Figure 4-6: Expectation value of x component of electron momentum and nuclear momentum as a function of time. Middle curve is the electron expectation value, the upper curve is the momentum corresponding to the incoming proton and the lower curve is that of the proton originally at rest at the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 vii 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 ELECTRON-NUCLEAR DYNAMICS: A THEORETICAL TREATMENT USING COHERENT STATES AND THE TIME-DEPENDENT VARIATIONAL PRINCIPLE By AGUSTfN CARLOS DIZ August, 1992 Chairman: N. Yngve Ohm Major Department: Physics A new method for studying electron-nuclear dynamics in chemical processes is presented. The method is founded on the Time-Dependent Variational Principle and the description of the electronic wavefuntions via coherent states. The resulting equations have a symplectic structure resembling that of classical mechanics. A first model is developed with the nuclei treated in the limit of narrow Gaussian wavepackets and the electrons restricted to a single determinantal description. The equations of motion are analyzed to understand their physical significance. These equations are used to study the p+H and a+He collisions. The range of validity of the model is examined and future developments discussed. viii CHAPTER 1 INTRODUCTION Since the dawn of quantum mechanics, the greatest emphasis has been placed on the solution of the time-independent Schr6dinger equation. The time-dependent Schr6dinger equation, except in the simplest model systems, is far more difficult to solve, enhancing the bias toward the time-independent studies. Much interesting knowledge has been obtained from the work done in the time-independent picture, and more will come. Yet a time-dependent scheme gives important insight on the actual dynamics and specific mechanisms important for the evolution of a physical system. Recent advances in computational power have led to a growing interest in solving time-dependent problems. Among the phenomena which can be studied with a time-dependent method are molecular photodissociation, collision induced dissociation and gas phase scattering. Electron transfer can be analyzed in detail and reaction rates can be computed. Using spectral analysis, much information can be gleaned from a time-dependent wavefunction. At the same time, experiments are beginning to probe chemical reactions at the femtosecond time scale [1], which no time-independent method can hope to describe. Here I present a theoretical method to study the time evolution of a molecular system and the first results obtained from its use. The potentials considered here are only the Coulombic ones. This restriction can easily be lifted to include other couplings if necessary. Electrons are treated quantummechanically while nuclei can be treated in a similar fashion or semiclassically. It is also 1 2 possible to treat some of the nuclei classically and others quantum mechanically. This can be important for such processes as proton transfer through quantum tunneling for which it is not necessary to have a fully quantum description of all nuclei. The nonrelativistic molecular Hamiltonian is N M N M N N H~ + A1 L L i=1 A=1 i=1 A=1 =1 j>i (1.1) M M QQ A=1 B>ARAB where Hartree atomic units have been used; N is the number of electrons and M is the number of nuclei. The nuclei will be considered as point particles. The equation to be solved approximately is the time-dependent Schrudinger equation: dt i kb) =H|') . (1.2) In most cases, attention is focused on doing quantum, semiclassical or classical nuclear dynamics on electronic potential energy surfaces. This involves the work of up to three research groups working over a period of years. One group determines the potential energy surface at a number of discrete points. Another interpolates the results to get a full surface. Finally another group does the dynamics on the surface. However, it is possible to think in terms of a more general approach, to develop a method which in principle can be applied to any molecular system, regardless of the kind of properties that are to be analyzed, and in which the theme steps of the process just described are done in one step, with the additional bonus that the electron-nuclear dynamics is retained 3 fully. With such a method it is possible to study scattering problems, charge transfer, follow chemical reactions, predict and analyze vibrational spectra, Raman spectra, etc. in one unified manner. The work I present here is a new approach to such a general, electron-nuclear dynamics (END) method. The method has been developed over the past five years. It began as a simple model of an electron transfer process with classical nuclei moving on quartic surfaces, studied by Deumens, Ohm and Lathouwers [2]. Then I did some work with Deumens and Ohm to describe the nuclei quantum mechanically on such surfaces [3]. After this preliminary work, we generalized the method so that both electrons and nuclei could be treated dynamically. This led to a computer code called DYNAMO from which results of calculations on a water molecule and the collision process of a proton on a hydrogen atom have just been published [4]. The method is based on mathematical methods developed over the past 30 years in other branches of physics. Chapter 2 reviews prior time-dependent schemes. The theory of the method I use is discussed in detail in Chapter 3 along with the physical interpretation of the resulting equations. Chapter 4 discusses the results of this method when applied to two systems I have studied. The first is a proton colliding on a hydrogen atom, for collision energies varying over 6 orders of magnitude and including an analysis of the electron transfer and the state-to-state transfer and excitation probabilities. The second is the scattering of an alpha particle off a helium atom at low keV energies. CHAPTER 2 AN OVERVIEW OF TIME-DEPENDENT METHODS Potential Energy Surfaces A widely used procedure is the Born-Oppenheimer approximation (BOA). It is based on the fact that nuclei are heavier than electrons, and move more slowly than electrons under most circumstances. Thus, to a good approximation, one can think of the electrons as moving among fixed nuclei; the electrons have enough time to accommodate themselves to the changing nuclear degrees of freedom. With this concept, a particular expansion of the system wavefunction is used. The Hamiltonian is separated as M H=-V VA+Helec A=1 (2.1) N N M N N M M Hejec = -- IV?- _ E 2 +E E 1 + E E QAQB , 2s E riA L r L L RA i=1=1 A=1 r=1 j>s A=1 B>A AB A complete set of eigenstates of the electronic Hamiltonian is found and the system wavefunction is expanded in terms of these: A { } Rec( A) uc( A (2.2) He4ecq {ec({i}, {A}) = EfecQ{A)1{e(g}, {iRA}) No approximation is made up to this point. Only a particular expansion of the system wavefunction is chosen which I will refer to as a Born-Oppenheimer (BO) expansion of 4 5 states. Allowing the Hamiltonian to operate on this expansion of the system wavefunction, and projecting on a BO electronic state (geiec ~I I4ec)tkfluCI(AA) (2.3) - V + Iec 6+TI,+T; p2," where M T 4 V *,) - (--i)V A=1 (2.4) M = (geec S 2 VA A=I Eq. (2.3) defines the action of the Hamiltonian on the nuclear wavefunctions. The operators T1 and T2 are dynamic couplings which are the result of the nuclear momentum operator acting on the electronic BO states. These couplings can be written as M T = - (-i)VA A=I M(2.5) (IPA) . . = (Ol- .AIj and if the set of electronic states is complete, T = 2A (-iVA. (PA) -i. + (PA PA) . (2.6) The nuclear momentum operator is replaced by -iVA - -iVA + PA (2.7) 6 in Eq. (2.3) to yield (o,elec IHI (S a Tb~c(RA)) { M A2 (2.8) i A=I1 9 This Hamiltonian for the nuclear wavefunctions is no longer Hermitian, since (-iVi + PA) is not Hermitian. This derives from the expansion of the full wavefunction in terms of the electronic eigenstates [5]. Since the BO electronic states are instantaneously in equilibrium with the nuclear configuration, inertial effects between electrons and nuclei are described through PA. The PA couplings represent the change of the electronic basis functions with the nuclear configuration. They contain rotations, distortions, polarizations, change of character of the electronic basis functions as well as the change of the electronic basis functions due to simple displacement of the nuclei [6]. The BO approximation (BOA) neglects the PA matrices, effectively decoupling the equations for the nuclear wavefunctions. Each component of the nuclear wavefunction feels an effective potential given by Veffec (R}) E=-e (2.9) called Potential Energy Surface (PES). Another possibility is to keep the diagonal terms of the vector potential couplings in the definition of the PES. This is usually referred to as the adiabatic approximation. 7 The definition of the PES and the use of the Born-Oppenheimer or adiabatic approximation is the starting point for most molecular dynamics today. Additional approximations are usually made, such as treating the nuclei purely as classical particles on these surfaces, or to treat the nuclei as quantum particles only insofar as to quantize the vibrational or rotational degrees of freedom of the nuclei. In the past decade, efforts have been made to describe the nuclei in terms of wavepackets moving along these PES. The breakdown of the BOA occurs when the assumption behind the approximation is no longer valid, i.e. when nuclear motion cannot be considered to be slower than electronic motion, or when PES become degenerate. This requires that dynamic couplings be retained to some degree. Processes in which this happens are avoided crossings of PES (if adiabatic potentials are used, or crossings for diabatic surfaces) where a change in the electronic character of the states may generate important dynamic couplings [7]. Another case is when nuclei are vibrating and rotating; as nuclei get closer, the moment of inertia decreases, increasing the angular velocity. Under these conditions the electrons may not have time to keep up with the nuclear motion, leading to predissociation. To generate a PES for dynamics is not simple. The work and time invested is very large. Sometimes it takes the work of three research groups working over a period of years to produce the electronic energies, the interpolated PES and do the dynamics. Even then, discrepancies between experiments and theory can become painfully obvious, as witnessed in the discussion in the journals over the existence of certain resonances in the H+H2 scattering problem [8, 9]. 8 Time-Dependent Methods on PES The methods used today to do dynamics on surfaces can be divided into classical trajectory, semiclassical and "exact" quantum mechanical. Couplings between surfaces and other non Born-Oppenheimer effects are sometimes treated as perturbations. Classical Trajectory Schemes The simplest method is the classical trajectory scheme [10-13] whereby nuclei are propagated as classical particles on a single PES. The greatest amount of work is finding the surface. The dynamics is simple to implement and the results are simple to interpret. Extensive sampling of phase space is required for a full final state resolution, i.e., to have reaction cross sections and rates. Classical trajectory methods are capable of producing good results. They are used to describe isolated encounters as well as processes in a condensed phase by using the generalized Langevin equation [14, 15]. The shortcomings of this approach are the neglect of quantum effects, such as zeropoint energy and tunneling, which may be important. The study of systems with a single potential energy surface of importance is also a severe restriction. However, the most severe limitation, which is shared by all methods that use PES, is the time it takes to obtain such a surface. The fitting procedure is also crucial, since errors in the fitting can lead to large quantitative errors in the dynamics. The limitations are more evident as the degrees of freedom of the system increase. The computational effort of finding points on 9 the surface and then fitting those points to some analytic function becomes unmanageable for any system having more than three or four atoms. Systems investigated with the classical trajectories include the H+H2 collision [16] and charge transfer in the H2+ + H2 collisions [17]. Semiclassical Schemes The semiclassical approaches try to correct for the deficiencies of the classical trajectories without going to the full-blown quantum mechanical approach. There are many different implementations, from time-independent, such as Miller's S-matrix approach [18, 19] in which the Feynman path integral is evaluated on the classical trajectory to find an approximation to the S-matrix, to Heller's Gaussian wavepacket path integral formulation of semiclassical dynamics [20-22]. In Heller's work, the wavepacket is assumed to remain Gaussian throughout the evolution, with the form exp[-a(x - q)2+ip(x - q)+c] and equations are derived for the parameters q, p, a, and c. The wavepackets are propagated on a potential energy surface. To realize the propagation, the potential is quadratically expanded about the instantaneous center of the wavepacket. The relationship between the S-matrix approach and the time-dependent wavepackets has been studied by Heller [23]. To correct for the divergence of semiclassical wavefunctions near caustics a generalization was developed [24, 25] which is similar to Klauder's global, uniform semiclassical approximation for wavefunctions [26] proposed earlier. In this work, the Gaussian wavepackets are generalized so that all the time-dependent parameters are complex and the manifold of states labelled by p and q is searched to find the most important states 10 which lead to the same final state. Other approaches along similar lines include the work of McDonald [27] and Littlejohn [28]. The eikonal approach to semiclassical time-dependent mechanics has been proposed by Micha [29] and compared to Heller's work [30, 31], yielding very similar results. Applications of these methods include work on the photodissociation of methyl iodide [30, 31]. A comparison between the semiclassical approximation and an "exact" approach indicates that the two methods agree closely for this system[31]. Another semiclassical computation uses Gaussian wavepackets in the interaction picture for the H+H2 collision [32]. It has been compared to the classical trajectory work [16]. The two methods give similar results, with some differences in the details. Exact Quantum Mechanical Schemes The so-called exact methods do use approximations. Their name derives from the fact that the errors can, in principle, be made as small as desired. There are two parts to consider: the representation of the wavefunction and the actual time propagation algorithm. The wavefunction representation determines how the operators, such as the Hamiltonian, are to be evaluated. Wavefunctions are represented in one of two ways, by expanding in a basis set or discretization on a grid of points. The discretization of space depends on the PES and the choice of the coordinates. All of these methods work only with internal coordinates of the molecular system, avoiding the calculation of global rotations or translations of the system. The effort put into finding good internal coordinates is nontrivial [6]. 11 Transformations to such coordinates can be quite a problem. It is much more convenient to do the calculations in Cartesian coordinates, including global translation and rotation. For this, however, it is necessary that the conservation laws related to these quantities remain valid within the approximations made. The Pseudo Spectral Fourier Approximation [33] uses the wavefunction represented on a grid in coordinate space and then uses the Discrete Fourier Transform to obtain the momentum space representation. Fast Fourier Transform codes allow for efficient switching from coordinate to momentum space. This permits fast evaluation of operators such as the potential and kinetic operators of the Hamiltonian. The most used propagation methods can be divided into four categories: Second-order differencing (SOD), the split-operator method (SO), the short-time iterative Lanczos (SIL) method and the Chebychev expansion (CE) method. For the SOD [34], the Schr6dinger equation is solved by approximating the time derivative by a second order difference. Short time steps are used, and the wavefunction at tN+1 is evaluated as (t+1) = (_) - 2iZAtHO(tn) . (2.10) This method is correct through second order in the time step. The other three methods use different approximations to the time evolution operator: U(e) = exp(-iHe) (2.11) 12 In the SO scheme [35] the exponential operator is decomposed symmetrically, to obtain an approximate evolution operator of the form U(E) ; exp(-i K)exp(-iEV)exp(-i K) (2.12) which is also correct through second order. The SIL propagation formula [36] is U(E) ; exp[-icA(HI (0))] (2.13) where U is a matrix operator in the Krylov subspace, which is generated by the Hamiltonian and the initial wavefunction, and A is the tridiagonal Lanczos matrix representing the Hamiltonian in the Krylov space. The Krylov subspace is generated through the action of the Hamiltonian operator on the initial wavefunction. Thus the N - 1 Krylov subspace is generated by the N vectors uj = Hi0(0) . (2.14) The Lanczos recurrence generates a set of orthogonal polynomials within the subspace which represent a finite polynomial approximation to the operator. Then the projected subspace representation of the Hamiltonian operator has a tridiagonal form. The operator is diagonalized and the propagation is performed with the diagonal eigenvalue matrix. The length of time dictates the size of the Krylov space needed for a predefined accuracy. Short time steps are used in order not to lose the advantage of the Lanczos reduction. The CE method [37] approximates the exact propagator by the expansion N U(t) = a,(t)T(-iHR) (2.15) n=1 13 where the Hamiltonian is renormalized so that its spectrum coincides with the domain of the Chebychev polynomials Tn. The normalization is carried out by dividing by the range of energies permitted in the evolution, AE = Emx - Emin, and shifting the energy level so that all the eigenvalues fall between -1 and 1. The coefficients an are proportional to Bessel functions an(t) = 2J, (tA) (2.16) and the Chebychev polynomials Tn are obtained from the recursion relation 00 = 0(0) 01 = -iHRO(O) (2.17) 0n+1 = -2iHRO, + -on-The CE method is a global propagator. It is such that the number of terms in the expansion does not decrease significantly for small time steps. Therefore, for efficiency, the time step should be large, sometimes to the point that a single time step completes the calculation. Restrictions of this method are that only time-independent Hamiltonians may be used. Differences with the Lanczos scheme include the fact that the coefficients are fixed for the CE while in Lanczos they depend on the initial state. Also, the SIL uses small time steps for efficiency compared to the long time steps of the CE method. Specific advantages of each of these methods are reviewed by Kosloff [38]. One of the limitations for all these methods is the number of atoms in a molecular system that can be studied due to computational complexities. At this point systems of up to 3 14 atoms are possible, with some work being started on systems of 4 atoms in which some internal degrees of freedom can be frozen. A few representative works with "exact" methods are reactions [39] and absorption spectra [40]. Because of the SOD's ease of implementation and satisfactory accuracy it has been used extensively for a variety of problems, including eigenspectra [41], nonadiabatically coupled systems [42], photodetachment spectra [431 and systems with timedependent Hamiltonians [44]. The CE method has been used on atom-diatom collisions [45], photodissociation [46] and computation of energy levels [47, 48]. To reduce the number of grid points that are required for the calculations, most work is done with absorbing potentials which are placed at the boundary of the coordinate space considered to be important for the dynamics of the reaction. Time-Dependent Methods Without PES Most chemistry does not happen on a single potential energy surface with only three or four atoms. Determining PES for higher numbers of atoms becomes a difficult, if not impossible task. These facts make the search for a method that does not use surfaces more urgent. The limitations of PES have generated interest in developing methods that avoid their use. One example in which methods using PES fail is the collision of a proton with a hydrogen atom at energies above 10 eV. In this case many PES would be required, since excitation to n=2 states becomes important and provides for interesting physical 15 phenomena. The collision has been studied on the lowest energy surfaces made up by the molecular lsu and 2pu states [49-51], but for energies below 10 eV. Close-Coupling Methods For collisions at high (above 1 keV) energies, the close-coupling methods developed by Fritsch, Lin, Kimura, Thorson and Lane and others [52-56] have been successful in describing collisions for systems with one electron, or in which all but one electron can be approximated by some potential. Work on systems which explicitly retain two electrons is new [57] and very time consuming. The close-coupling method projects the time-dependent Schr6dinger equation on a basis. The nuclei are not treated dynamically but trajectories are prescribed for them. These are usually either straight line or Coulomb trajectories. The nuclear motion with this approximation leads to a time-dependent potential for the electrons. It is only the expansion coefficients of the electronic states that are calculated for different times. Atomic or molecular orbitals with electron translation factors (see the end of the chapter for a description of these factors) are used as the basis. For more than one electron, the basis is made up of different determinants of atomic or molecular orbitals. The electronic states are described as a CI wavefunction with time-dependent CI coefficients. For this reason as the number of electrons grows, the method becomes more and more difficult to use. There are several implementations, such as Fritsch and Lin's AO+ [54, 55] or the AO-MO matching procedure of Kimura and Lin and Winter and Lane [58, 59]. The differences depend on the basis sets being used. In general the electronic wavefunction 16 is written as N T (t)) = ( )k(t)) (2.18) k=1 for a basis set of N functions. This basis set changes with time, since the basis functions follow the nuclei. Projecting the Schr6dinger equation on the truncated basis gives at (1kk(t)i-- HI'I) = 0 (2.19) where the prescribed nuclear trajectories make the electron-nuclear attraction and nuclearnuclear repulsion potentials time-dependent. The equations take the form [53] N da~) N 1 N,,k(t) d = i E Mjk(t)ak(t) k=1 k=1 (2.20) Njk(i) = (pbjh/'), Mik = ($ - H(tI}|kk) where the final expression for N and M will depend on the basis chosen and the prescribed trajectory. Since the close-coupling method does not account for nuclear dynamics it cannot be successfully used for collision energies below about 1 keV. The path followed by the nuclei is very important in determining transitions at lower energies[60]. For energies below 1 keV it is essential to include electron-nuclear dynamics. The close-coupling methods have never been used for systems in which more than two electrons are explicitly described. Methods with Nuclear Dynamics Several methods have been proposed in this field. One method, the Car-Parrinello approach [61], has gained much attention among theoretical chemists because of its 17 applicability to larger systems. The method uses an optimization technique to follow the ground state surface and uses the Hellman-Feynman theorem to compute the forces on the nuclei. It is an approximation to dynamics on a surface which requires calculating only the part of the PES needed. There is no firm theoretical work which shows fully why the method works, in particular why it works better than if an SCF calculation were to be performed at each position instead of using a molecular dynamics optimization procedure [62]. Hartke and Carter use this method on Na4 singlet and triplet states [63]. Field [64] has used the Time-Dependent Hartree-Fock method for a closed-shell restricted Hartree-Fock wavefunctions using the neglect of diatomic differential overlap (NDDO) approximation. In his approach, the TDHF equations drive the electronic degrees of freedom, subject to the constraint of normalized molecular orbitals: M M i I-Ecjic) = Fci-Z Fjic, j=1 =(2.21) Fji = cFci where the integration over spin has been performed, c are the expansion coefficients of the molecular orbitals in terms of an orthogonal basis and F is the Fock matrix in the basis. The nuclei are assumed to evolve as in classical dynamics, with the forces given by the gradient of the energy of the system dR1 _P1 dt M (2.22) dP1 OE dt 0R. 18 The energy E is expressed as 1 1:P E = 2 Z+V c Eij (2.23) Eeiec = 2jc* icviHp, + E c,icvicjcxj(2(palvA) - ({piolv})} pvi pu'aij Field has applied this method to the study of properties such as dipole moments and power spectra for lithium hydride, water and formaldehyde [64]. In Field's approach discussed up to this point, there is a curious omission. The electronic equations of motion are derived without considering the translation of the basis set with the nuclei. This neglects couplings in the equations, couplings which transform the electronic coefficients between basis sets at different times. Another approach has been developed by Runge, Micha and Feng[60]. Starting from the TDHF approximation, and making no further approximations such as the NDDO or restricted close-shell determinant of Field, they derive equations of motion for a basis which includes electron translation factors. In this case the transformation due to the translation of the basis is retained initially. However, by using electron translation factors it is eliminated because the coefficients are written in terms of a moving basis. They do neglect another coupling, namely the one between the nuclear accelerations and the electronic degrees of freedom, which derives from the electron translation factors. In their first implementation (Ref. 60) they also neglect the electron translation factors in the actual calculation of the Fock matrix. Thus, electron translation factors are used only to eliminate the basis transformation effects. Because of these approximations the equations 19 reduce to ones similar to those of Field but without the additional approximations of NDDO and restricted close-shell determinant. Runge et al solve an equation for the density matrix instead of the orbital coefficients. The equations to be solved are [60] iP- = S-1FP7-P7FS-1 PA,(t) = c -(t)ct )* (2.24) dR1 _P1 dt M, dP1 e9E where E is the total energy of the system. This method has been applied to systems with one electron, such as the p+H collision [60]. Work in progress by this group lifts the neglect of electron translation factors in the calculation of the Fock matrix. Another approach is taken by Meyer and Miller [65]. They too derive equations which couple the nuclear and electronic degrees of freedom. Their approach is formal and has not been used in an ab initio fashion on any real systems. They assume the existence of a complete diabatic basis for the electrons (i.e. one which does not depend on the nuclear positions). This has the effect of decoupling the electronic and nuclear degrees of freedom, except through the electron-nuclear attraction. The basis considered is not made up from an expansion of one-electron orbitals, but is a general N-electron basis. The electronic degrees of freedom are the expansion coefficients of the electronic state in the basis. The coefficients are written as a norm times a phase factor. The nuclei are approximated by point-like classical particles. Using the time-dependent variational 20 principle they derive the equations of motion. The equations for the electronic degrees of freedom look just like the action-angle equations of motion of classical mechanics. Thus the whole system of equations looks very "classical", although the electrons are treated quantum-mechanically. A canonical transformation can be performed to obtain the equations of motion for the electrons and nuclei using an adiabatic electronic basis. In this case the equations of motion for the electrons and nuclei are [65] OH ii = - = (pi + Pi)/Mi Opt . = H 9EK 0P (P + P) i= =- NKOx1 Ox, M, K A(2.25) OH OP (p+P) QK= =EK iONK M 'K aOH - - P (p+P) OQK OQK M with the wavefunction written as 1) = Zexp(-iQK)V1NKIK) (2.26) K where x, p are the classical nuclear position and momenta, P are the momentum couplings defined in the BO Hamiltonian (Eq. 2.8), 1K) are the adiabatic electronic states and the Hamiltonian has been taken to the classical limit for the nuclear degrees of freedom. Electron Translation Factors The Perturbed Stationary State method (PSS) [6, 66] is used for many atomic and molecular collisions calculations. Starting from the BO expansion of the Hamiltonian, Eq. (2.8), this method does not ignore the dynamic couplings like the BOA. Delos [6] 21 lists shortcomings of the PSS approach when the electronic as well as nuclear basis is truncated, i.e. when approximations are made to the BO expansion. He notes that the expression for T2 in terms of P in Eq. (2.6) is not valid for a truncated basis, though he states that it is usually sufficiently accurate within the approximation. More serious problems he notes are: 1. The individual terms in the BO expansion of the system state do not satisfy the scattering boundary conditions. 2. The P matrix elements contain couplings of infinite range. 3. P matrices contains fictitious "origin dependent " couplings. 4. Matrix elements in this formulation do not contain momentum transfer factors which are needed to describe Doppler shifts in the energy spectrum of moving systems. Delos suggests a solution via the so-called electron translation factors (ETF's), which eliminate many of these problems. The solution he proposes is essentially to include in each electronic basis function a complex phase factor which cancels some of the offensive terms listed above. This is effectively a correction to the BO expansion, accounting for the momentum of the electrons in the process, not through the dynamic couplings but through the basis. This correction is used in the close-coupling method for atomic collisions, and also in the approach by Runge, Micha and Feng to electron-nuclear dynamics. The following example should clarify the issue. If an H atom is moving with respect to an inertial frame (which I will call the lab frame), its ground state should satisfy the time-dependent Schr6dinger equation. It is a simple exercise to show that the orbital 22 exp(ime (v - r- v2t/2) - Eit)01.f- ) where v is the velocity of the moving atom, R its instantaneous position and 4ol the time-independent ground state solution with energy q solves the problem. To get a reasonable approximation of this orbital as an expansion in terms of a truncated set of the lab frame eigenstates of the H atom, then orbitals with n greater than 1 are necessary. The gmater the velocity, the greater the size of the basis needed. If the basis used for calculations is fixed to the lab (even if it "follows" the nuclear positions, i.e. using an instantaneous basis centered at the position of the nuclei, but which is not moving relative to the lab frame), a small basis will not appropriately describe the ground state of the moving H atom. Another way of viewing this is that the description is not Galilean invariant when using a truncated basis since calculations done with the basis fixed in a different frame gives different results. The solution to the problem is simple for atoms [6]. One just uses the phase factors as shown above on each basis function associated with an atom and Galilean invariance is recovered even for a truncated basis. If the calculations use molecular orbitals instead of atomic orbitals, the problem becomes a difficult one. In fact, no fully satisfactory solution has yet been found to this problem. The ETF's in this case have to be such that if the electron is close to a given nucleus it has the appropriate velocity for that nucleus, with a continuous transformation of the velocity between nuclear centers. Several forms have been proposed [56, 67], but each one has its drawbacks. Another problem with the MO-ETF's is that calculations depend on the origin of the electronic coordinates. After more then a decade of working on this problem, groups using close-coupling methods have settled on the AO-MO matching scheme [57]. In this scheme the orbitals used 23 outside a certain distance between nuclei are atomic and contain the electron translation factors. At the limiting distance these AO's are matched to molecular orbitals without electron translation factors to continue the calculations with the PSS. When the fragments separate, at the limiting distance the matching from the MO to the AO basis is again performed. In this way Galilean invariance is preserved outside the interaction region. An approach where the MO's are described in terms of an LCAO expansion with AO's having ETF's was attempted, but failed (A. Riera, personal communication). The failure stems from the fact that the Hamiltonian carries no information about the moving frame and so the MO's obtained undo the effects of the ETF's. When ETF's are used, there appears to be no work which has included the couplings that arise for accelerated nuclei. For close-coupling treatments with straight line trajectories and constant velocities these couplings are zero. CHAPTER 3 THE END FORMALISM The fully quantum, exact methods are setting benchmarks, but the use of the potential energy surfaces severely restricts them, a problem shared by all methods relying on PES. Even four atom systems are beyond reach, greatly limiting the chemical systems that can be studied. The move towards the elimination of potential energy surfaces is very recent. As already discussed, several proposals have been made. However, either the theory behind the methods is not well understood or the approximations made are many, and at times confusing. Here I present another method which does not rely on PES. The approximations made are few and simple to understand. The chapter is divided in four parts. The first is the description of the Time-Dependent Variational Principle. The second is the coherent state formalism. In the third part these tools are used to find the equations for electron-nuclear dynamics (the END equations). There is also a description of the approximations made in the first model developed. The fourth part deals with the tools developed to analyze the results of a calculation. The Time-Dependent Variational Principle Three main time-dependent variational principles are used today. They are the McLachlan variational principle [68], the Dirac-Frenkel variational principle [69, 70] and the Time-Dependent Variational Principle (TDVP) [71]. All of these have been shown to be equivalent [72] if the trial wavefunction is described with complex parameters and is 24 25 analytic in these parameters. Because the form of the wavefunction used here satisfies these two conditions, the three variational principles will give the same results. I will describe and use the TDVP. If we assume the existence of a set of electronic and nuclear parameters ( which completely define the quantum state of the system, I will label the state with these parameters: I(). Using Hartree atomic units so that h = 1, the following Lagrangian is proposed: L = (( - - HIC)((I) (3.1) =((IK () (C ()where the left-acting time derivative is noted as , and the operator K is defined. The action is then A = Ldt (3.2) = - ((( )C)and the TDVP requires that 6A = j [( - ((CI)IC) - ((IHI()]((I)dt = 0 . (3.3) The TDVP will result in the Schrtdinger equation if the trial wavefunction used has no restriction in the Hilbert space of the system. With a partial integration: 6A= t [(6(|i|() - (6(IHIC) at ((3.4) -((jK l()((j()-'(b(j() + c-c.] ((I()-ldt 26 Because the variation is arbitrary ( - H () = -)(CIKO (3.5) which is the Schr6dinger equation if the right hand side is zero. If the wavefunction is multiplied by global time-dependent phase factor exp(izy) with -y satisfying the equation d-' = ((IKK( (3.6) dt (WO( then it is simple to see that the variation of the action results in i-H) IT) = 0 (3.7) I) = exp (iy) 1) recovering the Schrtdinger equation. The global phase is important in time-correlation functions. In practical applications the trial wavefunction is restricted to a submanifold of the full Hilbert space and an approximation to the time-dependent equation is solved. Introducing the notations S(*, 0) (CIO (3.8) E(( = (IHI)/((() then z 9 9lnS((*, ) (3.9) = -Im ( 8aanS((*, )) E(C*, ). This differential equation can be integrated with a simple quadrature. 27 Because all time dependence of the state is through the parameters, Eq. (3.3) is equivalent to A=tJ [ InS - E dt S (a2lnS 2* -'nS ( 8C, 49246( -nE dE) ( (3.10) + i nS aE - bCl d The second step in Eq. (3.10) involves a partial integration of some terms with respect to t. Since all the b( and their complex conjugates are independent variations we obtain the dynamical equations OE iZCaiC, - , 9 (3.11) 8E with the elements of the metric matrix defined as 82lnS C.0 = . (3.12) The matrix C is clearly Hermitian. The phase factor does not influence the evolution of the other parameters, and can be computed afterwards. In practice, it is convenient to integrate (3.9) along with the equations (3.11) for the other parameters. 28 The equations (3.11) can be written in matrix form as . (8E(3.13) Furthermore, we can define a Poisson bracket (,) for two functions f and g depending on C and (* as {fg}=( df)(i .9 (3.14) With the symplectic structure defined by this Poisson bracket, the evolution equations (3.11) assume the standard form ={(, E}, (3.15) (* (*, E}, which shows that the energy E assumes the role of Hamiltonian or generator of time translations, while C and (* are conjugate variables, {(, C} = 0, {(*,(*} =0, (3.16) {C, (*} = -iC~1. The phase space is flat only when C is the unit matrix. Coherent States The parametrization of the wavefunction is of crucial importance. A poor parametrization will lead to a complicated phase space with a metric which may lead to integration problems and additionally make the equations difficult to understand. This problem was 29 first addressed by Thouless [73] for nuclear systems using the TDHF. The main problem he faced is that a single determinant is unchanged under a unitary transformation of its orbitals. This means that there are redundant parameters in the definition of a determinantal state. The parametrization he found turns out to be a special case of a more general scheme, referred to as coherent states. Schridinger first proposed the concept of coherent states in 1926 [74] in connection with the classical harmonic oscillator. Coherent states were later used by Glauber [75-77] and Sudarshan [78] in quantum optics in the early 60's. Since then there has been interest in them from both a mathematical [79] and physical [80] point of view. The representations of Lie groups can be studied with coherent states, and the range of physical systems treated with this technique goes from condensed matter and thermodynamics to elementary particle physics and path-integral developments, as well as to study the relationship between quantum and classical dynamics (for a compilation of these developments see Klauder and Skagerstam's book, Ref. 80). Coherent states are also useful in the study of unrestricted Hartme-Fock theory applied to molecules [81]. A coherent state satisfies two properties [80]: 1. A coherent state Iz) is a strongly continuous function of the label z. This means that lim I Iz) - z') |=0 (3.17) 2. There is a positive measure Sz on the space Q of labels z such that I = j Iz)(zIbz (3.18) 30 The first property indicates that there is a one-to-one mapping between coherent states and points on the label space Q. The second property shows that the coherent states are a complete set, since any state can be decomposed in terms of coherent states. Because of the continuous label, the set cannot be an orthogonal basis, but must be overcomplete. Although there are many forms of coherent states [80], I will restrict myself to ones related to compact Lie groups. Let U(g) be a continuous, irreducible unitary representation of a compact Lie group G and 10) be some normalized reference state, also called the extremal state. Then the states generated by the operation of the group on the reference, Ig) = U(g)I0) (3.19) satisfy the first property from (gig') = (OIU(g-Ig')10) and the continuity properties of Lie groups. To show that the second property is also satisfied, take P = jg)( gjdg (3.20) where dg is the group invariant measure. By virtue of this measure, it is clear that for any g', PU(g') = U(g')P. Schur's Lemma then requires that P be proportional to the identity. Renormalizing the invariant measure by a constant leads to P = I. The states Ig) may not all be distinct. To obtain a distinct states let h be an element of a subgroup H of G defined by the property that these group elements only change the reference by a phase factor, i.e. U(h)10) = exp [ic(h)]10) (3.21) 31 H is called the stability subgroup and the coset G/H provides a unique decomposition of any element g belonging to G [82]: g = ch (3.22) where c is a coset representative and h belongs to the stability subgroup. Then U(g)10) = U(c)U(h)0) = U(c)10) exp [ia(h)] (3.23) = c) exp [ia(h)] and Ic) is the coherent state, defined through the action of the unique coset representative C. For a single determinant, the stability subgroup is the subgroup made up of unitary transformations which only mix the occupied or unoccupied orbitals. These do not change the determinant. By using the coset parameter representation the possibility of parameters changing without changing the determinant is eliminated from the outset. I will return to the single determinant a little further on. First I will illustrate how to apply the above strategy to the simplest coherent state, the harmonic oscillator coherent state, starting from the Heisenberg-Weyl Lie algebra. These coherent states, also called canonical coherent states [80] can be used as a first approximation to treating nuclei quantum mechanically for molecular dynamics. 32 The Heisenberg-Weyl algebra is spanned by the operators I, a, at and n = ata with the commutation relations: [n, at]= at, [n, a] = -a, [n, I] = 0, [at, I] = [a,at] =I, [a,I]=0. The carrier space of irreducible representations for this He spanned by the number eigenstates of n: nlk) = kk), k = 0, 1, 2,. (at)k 1k) = (k! )1/210) (3.24) isenberg-Weyl group (H4) is (3.25) The state 10) will be used as the reference. It is the lowest weight state for the number operator n. The group elements h = exp [i(Sn + 4I)] of H4 only change the reference by a phase factor exp(io). These elements form the stability subgroup U(1) 0 U(1) of H4. A general element of H4 can be written as g = exp (aaf - a*a) exp [i(bn + 4)] (3.26) so that the coherent state is Ia) = exp (aaf - a*a) 0) = exp (-IaI2/2) exp (aat) exp (a*a)10) (3.27) = exp (aa ) 10) exp (-Ia12/2) 33 where the Baker-Cambell-Hausdorff theorem is used to get the second line and the property of the annihilation operator on the lowest weight state is used to get to the third line. The normalization exp (-IaI2/2) can be transferred to the metric defining in this way a coherent state that is not normalized, but which has the advantage of being analytic in a. The canonical, analytic coherent state is: 1a) = exp (aat) 10) . (3.28) In general I will work with analytic coherent states. This choice provides powerful mathematical properties, as can be noted from the fact that the three most common time-dependent variational principles are equivalent [72] with such trial wavefunctions. A general multiconfigurational wavefunction for the electrons has been formulated by Weiner, Deumens and Ohm [83] using the mathematics of vector coherent states [84, 85]. However, I will not describe that formulation here but restrict the details to the case of the single determinantal approximation for the electronic wavefunction, which is the model that has been developed into a working program. The coherent state description of the single determinant leads to the Thouless parametrization [73]. Assuming an orthonormal spin-orbital basis set made up of K vectors and N particles to be described by a single determinant, this determinant can be found as a unitary transformation acting on the orbital expansion coefficients of some 34 reference single determinant I|o). I will define this reference state as N I To) = j b Ivac) (3.29) 1=1 where bt (bi) are the fermion creation (annihilation) operators of the basis set used. The reference state is a lowest weight state for the irreducible representation [IN0(K] of the group U(K) with generators b bi because the weight operators btbi have eigenvalue 1 for i=1....N, and 0 for i=N+1,...K. The stability group of the reference state is U(N) 0 U(K N). Introducing a complex parametrization of the coset space U(K)/U(N) 0 U(K - N) an element g of the group U(k) is written as g = ch as previously discussed. Extending U(K) to GL(K, C) and using the Gauss factorization of c (see notational remarks below) c=(' 0 X2 0 1) (3.30) this leads to a parametrization of the coset space in terms of the complex parameters z, with x and y to be determined as functions of z from the condition that g is unitary. In the last expression and in what follows I use a notation intended to make the equations simpler to follow. In the atomic basis, there is a "hole" subspace generated by the first N orbitals which make up the reference single determinant and the "particle" space made up of the rest of the basis spin-orbitals. The field operators of the basis states which belong to the holes will be written with a bullet, i.e. bet, b*, and those associated to the particle states will have an open circle bn, b. In the same way, when referring to the molecular orbitals made up from the atomic orbitals, the occupied molecular orbitals have a bullet and the virtual orbitals have open circles. So bullets will always be used to describe hole states in a given basis and open circles will represent the 35 particle states. Matrices associated with the subspaces of holes or particles will also carry these symbols. Matrix blocks which carry a prime or a double prime indicate the upper or lower off-diagonal block respectively. Matrices associated with the complete one-particle space will not carry any symbols. So, for example, we may write the identity matrix as I = (0 I Let T be the unitary irreducible representation of U(K) in Fermi-Fock space. The coherent state is defined as IT.z) =T(c h)IWo) =T(c) T(h)j Iio) =T(c)I|o) =a T( P I*)po) (3.31) =a H bi~t + bt zji Ivac) i=1 j=N+l N K =a exp z ibjob *o) (i=1 j=N+1 The second step uses the fact that h is in the stability group of the reference state, and that the rightmost factor of c in (3.30) modifies only the virtual space and leaves the reference state unaffected. The middle factor of c only gives a constant a when acting on the reference state. The constant a can be determined from the normalization of the coherent state. Note that the representation is considered as acting on the orbital 36 coefficients, not on the basis functions. This is in accordance with the active point of view of coordinate transformations. I work with the unnormalized coherent state N K Iz) = exp zjib ObbiTO) i=1 j=N+1 N = j (c* ) Ivac) (3.32) i=1 N K =( bt + E bjozji|a) =JJ ~ 3vac) i=1 j=N+l This is the Thouless representation of a determinantal wave function with the elements of the (K - N) x N matrix z as time-dependent parameters. Thouless [73] worked in another direction to get to this result. Because his derivation clarifies the meaning of the z parameters I will include it here. If U is the matrix which transforms from the basis spin-orbitals to the occupied and virtual orbitals of the fermionic system, then (cot cot) ( b*f bo ) o ,, (3.33) 37 Following Thouless, N IT) =lcit vac) t=1 N N K = E (bjt U)+ I bUj'|vac) i=1 j=1 j=N+I N (N K N H 1 E byt + S E bt Uj'kUke-1J U0j* Ivac) (3.34) i=1 1=1 j=N+1 k=1 NK = i (it+ S S bjtUj'kU)1 |vac) i=1 j=N+1 k=1 N K N N =a (1 + E 5 bt U'kUI 1bi bot|vac), i=1 j=N+1 k=1 = where the invariance, up to a constant a, of a determinantal wave function under a linear transformation of its occupied spin orbitals is used. Then the coherent state is recovered as follows N K |z) =B + E bjozjibi |TO) =1 j=N+1 N K = H IH (1 + bjz1b) ITO) i=1 j=N+1 (3.35) N K exp(zjibb )|Wo) i=1 j=N+1 N K =exp ibjb ) (i=1 j=N+l 38 where k= U'U- = . In Eq. (3.35) we have used the nilpotency of operators botb. The determinantal wavefunction is expressed as det{X(xj)} where K Xi = Oi + 1 Ojzji, (1 < i < N). (3.36) j=N+l are nonorthogonal (but linearly independent) dynamical orbitals. The corresponding unoccupied dynamical orbitals may be chosen as N Xj = ?j - Z;i, (N +1 j K) (3.37) 1=1 and although mutually nonorthogonal they are orthogonal to the occupied space. The occupied dynamical orbitals can then be expressed as (* 0*) is (3.38) i.e. a partitioned row vector of basis spin orbitals times a partitioned rectangular matrix. Consider -Z I 0 z I 0 -z +z I* + zzt (3.39) (s + ztz 0 0 0 I + zztThis shows that I* -t) (3.40) defines dynamic occupied and virtual orbitals that span orthogonal spaces. It follows that the projectors on these spaces add up to the unit operator. I - PocC = Pjrit (3.41) 39 These relations are used many times when simplifying various expressions in the evolution equations. All of the above assumed a basis of orthonormal spin-orbitals. In practical applications it is preferable to work in the nonorthogonal atomic spin-orbital basis. In this case, since the anticommutation relations of the field operators is not the canonical one but includes the overlap matrix of the orbitals, it is not possible to write the coherent state as an exponential. However, it is still possible to express the molecular orbitals in the same way as in Eq. (3.36) with modified z's: K Xi = qi + E z (1 i < N) . (3.42) j=N+l The orbitals of the virtual space are not as simple to express as in the orthonormal basis. Assuming a form N Xj= j + 4Ov (N+1 < K) . (3.43) i=1 such that the (K-N)xN matrix v satisfies (P z)A( %) (V I z I0 vA + vA'z + At + Az AO + vA' + A'tvt + vAv*t ( (3.44) Ae + ztA't -*+ z + '+ 'A +0 40 i.e. the two subspaces are orthogonal, then vA* + vA'z = -A't - A0z (3.45) and hence V = - (A't + Aoz) (A. + A'z)~ (3.46) -t - (A0 + ZtAft) -1 (A/ + ZtAo). The Equations of Motion The preceding tools will now be applied to find specific equations. These equations are the ones that are written into the computer code DYNAMO. Here I will derive the general form of the equations and comment on their properties. The details can be found in the appendices. Three approximations are made: 1. In the TDVP the limit of narrow Gaussian wavepackets is taken. The nuclear parameters are then the position and momentum of the nuclei. 2. The electrons are described by means of a single determinant. 3. The spin-orbital electronic basis is truncated to a finite number of orbitals without electron translation factors. The truncated basis limitation is always imposed if an actual calculation is to be performed. The lack of ETF's is something which will be remedied in the near future to allow us to analyze collisions at high speeds. For lower speeds associated with most 41 chemical reactions, the velocities involved are sufficiently small and ETF's are not as critical. The single determinantal wavefunction can be improved upon by including more configurations. The theory to do this for the electronic states can be found in Ref. 83. For nuclei, there is the question of finding a good quantum representation for them. Presently work is being done in this direction. A first approach is to describe them via frozen Gaussians without taking the limit of narrow wavepackets. These are the canonical coherent states discussed in the previous section. However, in work done with Deumens and Ohrn [3], we have shown that a description in which the nuclear wavefunctions are not given the freedom to split, the dynamics becomes nearly identical to that done using classical nuclei. Currently, work is in progress to describe nuclei going beyond the Gaussian wavepacket approximation. Initially I will assume that the electronic basis is complete and is independent of the positions of the nuclei. Then I will show how to derive equations for two different electronic basis sets which are more adequate for calculations. The new equations can be obtained via a symplectic transformation. The basis set that does not depend on the nuclear positions will be referred to as a Non-Following Basis (NFB). Another basis of interest is one with orbitals that are instantaneously centered at the position of the nuclei, but which do not have associated with them the motion of the nuclei, i.e. they do not contain ETF's. This basis set will be referred to as the Static-Following Basis (SFB). A basis set centered at the position of the nuclei with ETF's will be called the Dynamic Following Basis (DFB). For practical calculations with a truncated basis it is desirable to use either the SFB or the DFB. 42 Otherwise, after some time, the nuclei may be far from the vicinity of the basis set location, which would be inadequate for a correct description of the system. There are two equivalent ways to derive the TDVP equations for narrow Gaussians. One is to find the equations of motion and then take the limit of zero width and the other is to take the limit before actually varying the action. I will do the latter. In this case the action becomes A= P,. N- N R, - P, +(|i-i -H) dt (3.47) where H is the molecular Hamiltonian, including the nuclear repulsion term, R1 and PA are the canonical positions and momenta of the nuclei and Iz) is the coherent state description of the electronic single determinant as defined in Eq. (3.32). Defining Zk = Rk +iPk (3.48) we can write Pkk - PkRk) = Zk - ZkZk (k - 2k Z'=Z) FZ1'* Z1 2 04 a~* I Iz;z = - Z IZZ)nS(Z'*, Z) 2k Zk -Z- *= (3.49) 43 The electronic overlap matrix is given by S (z'*, Z) =(Z' IZ) N N =(vacI c (z') I c (z)Ivac) 1=1 j=1 /K K = det ((0' + E z,kI + E 0zi))ij (3.50) k=N+1 I=N+1 K =det (bij + E Zkizk)ij k=N+1 = det (I* + z'iz) and if in the derivation of the TDVP equations the quantity S is replaced by S(('1*, o)SC, (Z'*, Z) (3.50) then Eq (3.11) can be used to get the equations of motion for the electronic and nuclear parameters: i5Cafip = ~ a 9z* k (3.51) aE - **= az i 6kZI* = O 44 where the unit matrix comes from S. In matrix form, these equations become iC 0 0 0\ (oE/az* 0 iI 0 0 Z _ 8E/Z* 0 0 -iC* 0 i* ~ E/z ' 0 0 0 -ii) * aE/aZ which, with the real form of the nuclear coordinates, can be expressed as iC 0 0 0 C OE/z* 0 -iC* 0 0 &E/z 0 0 0 -I E/t9R 0 0 I 0 ME/P (3.52) (3.53) A transformation that leaves a set of dynamical equations invariant, i.e. does not change the metric at all, is called a symplectic transformation and if the metric contains only ones and zeroes, the transformation is said to be canonical. We are interested in a generalized symplectic transformation that leaves the structure of the equations invariant, but changes the values of the matrix elements of the metric. The invariant is the Poisson bracket. First we consider the transformation from a NFB to a DFB. It has the general form = z(z,Z, Z*) (3.54) Z=Z(Z) or with the real form of the nuclear coordinates i = i(z, R, P) R= R(R) (3.55) P=P(P). With this notation I indicate that the i parameters are independent dynamical variables, i.e. they are independent of the "new" nuclear coordinates and momenta. In order to 45 transform the Poisson bracket, we need the matrix of partial derivatives, Jacobian, J g|l az * z*/ * 0 0 0 l/O z* al8z 0 az|ai 0 0 18|z (3.56) C =R az*IaR az/6R I 0 a/aR ' a|aP Oz*|OP Oz|aP 0 I O|8P such that c* 0 0 0 0 C 0 0 (3.57) r* r 1 0' p * P 0 I where c = az/ai, r = az/8R, and p = az/OP. Using the fact that the inverse of a Jacobian is the Jacobian of the inverse transformation on the Poisson bracket results in the transformation {f, g} = afM-1ag = {f,g}(3.58) = bf fk-g of the total phase space metric ic*CcT 0 0 icC*ct ir*CcT -irC*ct ip *Cci - ipC*ct = atf JIS~1 Jag A! = JMJt, with ic *CrT -ZcC*r ir*CrT - irC*rt I + ip*Cr-T _ ipC*rt ic*CPT -icC*pt -I + ir*Cp- irC* pt i*CPT- ipC*pt We then obtain for the dynamical equations in iC 0 iCR iCP 0 -iC* -iC-iC* iC -Rz CRR -I+ CRP iCP -iCT I -CRp Cpp the new coordinates the expression z aE/a* ) (3.60) R aE|aR' P ) OE|OP (3.59) 46 Here we define the matrices CXI = 2nS(zR', P, z, R, P) (3.61) ai*aXk IR'=R,P'=P and C = 21m2lnS(V*, R, P', , R, P) (3.62) aXp9Y R'=R,P'=P with X and Y standing for R or P. For the case of coherent states in terms of the atomic basis centered on the nuclei the calculation of the metric involves the overlap of two coherent states with different nuclear geometries. If, furthermore, electron translation factors are included in the orbitals, the overlap also depends on the velocity, and hence the momentum, of the nuclei. We find S(z'*, R', P', z, R, P) = det (A* + A'z + z'A" + ztAOz) (3.63) where the overlap matrix A(R', P', R, P) (3.64) depends on the nuclear positions and momenta. We have dropped the tilde on the z parameters, something we do from now on, since we will only deal with a following basis. Note that the overlap matrix of two different bases is not Hermitian, and that it becomes the unit matrix when the two nuclear configurations and momenta coincide. In the general derivation it is assumed that the basis set can depend on both the nuclear positions and momenta. This is the case for the DFB. If the dependence of the basis on the nuclear parameters is only through the nuclear positions, the equations become iC 0 iCR 0 zM/z* 0 -iC* -iCR 0 * (_ E/z 365) RiCL -iCR CRR -I R ~ yE/8) ( 0 0 I 0 ) P) E/OP 47 These are the equations that have been coded into DYNAMO, using the results found in Appendix B for a nonorthogonal basis The interpretation of these equations is very interesting. I will indicate in a general form the most salient features. The first point to make is that the equations are timereversible. This comes from the time-reversibility of the TDVP. Energy is conserved. This can be seen most easily using the Poisson brackets E = {E,E} = 0 (3.66) Another property that can be checked directly is conservation of total momentum of the system. The expectation value of the electronic momentum is fei = (zIPeilz)/(zlz) where the operator in brackets is the total electronic momentum operator. The time derivative of this quantity is ei- CAIR - CRPL) (3.67) k=1 since the time dependence only comes from the electronic and nuclear parameters. By inspecting Eq. (3.60), multiplying the third row of the metric and adding over all the nuclei gives -~ Nt. N(.68 E (-2ImCI z + CA"ARil + CRPPI Ef = E (3.68) k=1 k=1 k=1 I Because the basis set is centered at the position of the nuclei, the energy is invariant to a global translation of the nuclei and so the right hand side vanishes. Using the last two 48 expressions we get: PeI + PTot = 0 (3.69) which shows the conservation of total momentum. For a truncated NFB the preceding result is not valid since in that case it can be shown that Nt. Nat P,; = -E = i (z~eH|)(3.70) k=1 k=1 9R (zlz) but in a limited basis, Ehrenfest's theorem is not generally satisfied, which implies that (z[ei, H]Iz) Pe i ( Z IZ (3.71) PeI~Z (zlz) Thus, in a truncated non-following basis total momentum would not be conserved. This property is only satisfied in the basis sets which exhibit following. The next question I consider is the interpretation of the electronic equations of motion. The meaning is identical when the nonorthogonal atomic basis is used, but the more obscure mathematics coming from the nonorthogonality makes interpretation more difficult. Using the orthogonal basis and drawing from the results found in Appendix A to write down the equation of motion for the z's I+ z)- I*( + zz t 1 k=N+1,K (m(Ik Z+ (-Z *)+aE n=,Nt /kj 1(.7 (3.72) 49 where the vertical bar after the overlap matrix indicates that we are taking the derivative with respect to the unprimed coordinates in Eq. (3.64). This gives the following expression for the forces in matrix form li+i (-z I*)(RVq,A|+Pj1VpA (Io + I+t) "(- tzf) (3.73) -iz -i I *A zt)(RIV, + PJVPAt( t) = (I'+ztz I+ zz . which can be further reduced to izi (-I I*)(RVgAJ + Z~ = (-z I* )(h + Tr(V/b;a.b).) (* (3.74) =(-z I0 )F( ) The right hand side has a very simple interpretation: the Fock matrix acts on the occupied states and is then projected on the virtual space and only a nonzero projection on the virtual states gives rise to a change in the z coefficients. The Fock matrix in a basis which contains the ETF's of the form exp(ii- r) contains two extra terms in the h part of the Fock matrix derived from the operation of the electronic kinetic energy operator on the ETF's. One of them is associated with the kinetic energy v2/2, and the other exactly cancels the term coupling with the nuclear velocity in the metric. Thus, only 50 a term associated with the nuclear acceleration is left. If we assume, for a moment, that we are treating only a single atom and that the electron is initially in its ground state and that for some reason the nucleus is being accelerated, then the Fock matrix will not be generating any coupling, but the term with the nuclear acceleration generates a dipole coupling between the basis functions which couples to the acceleration of the nucleus. This generates dynamics in the electrons. What happens is that the electrons are being excited through the nuclear acceleration. These are the couplings I mention in the previous chapter which I have not seen implemented in the literature. If a basis without ETF's is used, then the Fock matrix reverts to the standard timeindependent form and the coupling to the nuclear velocity in the metric does not get cancelled. Analyzing this term in a complete basis and assuming that the nucleus moves with a constant velocity shows that this term in the metric is associated with the translation operator. After a time bt the nucleus has moved bR = R - 61. The electronic states are to be described in terms of a basis centered at the new position. The old basis functions must be expressed in terms of the new ones, so the transformation matrix: (O(r - R - 6R) ( - F?) = (#1exp (br. V i) (3.75) should be evaluated to apply on the expansion coefficients. For a small translation the electron translation operator is exp (IbtVr) a 1 + RbtVr (3.76) and its matrix elements in a basis which is centered on the nucleus (so that derivatives 51 with respect to the electron and nuclear coordinates can be interchanged) gives 6,j + izRtVAAI (3.77) Because we use an unnormalized state, the changes in the z's are computed as projected from the occupied to the unoccupied states through the translation operator. Dividing by bt leads to a time rate of change of the z's which is precisely the second term found in Eq 3.74. So the action of this coupling is to take care of the fact that the electron coefficients are being continually defined with respect to a basis which has been translated. This coupling, which is sometimes called "unphysical" [6], is just ensuring the correct description of the electronic state in a translating basis. Interpretive Tools Tools must be developed to understand the results of a time-dependent calculation. I have developed two such tools which will be described here. The first one is used to analyze state-to-state transfer probabilities for the p+H collision. The results found in the static following basis must be projected on eigenstates of the moving hydrogen atom. The eigenstates, which were described in Chapter 2 when discussing ETF's, are 4IM3- A) = exp (i' i(62/2 + Enlm)t)qnim B) (3.78) where D indicates the dynamic nature of the orbital. The transformation from the static 52 to the dynamic basis is determined from: R@) =: 1|k )ak k = | If)(4Pk)ak L.k (3.79) L I )/Tj~'ak L~k = I )ak l.k It is easily seen that the part in the dynamic phase which is multiplied by the time t is cancelled in the projection E IOD)(ODI since it is independent of the electronic coordinates. What is required, therefore, is the transformation matrix defined by (exp(i6- rnimfqn'zimn) = (#nim exp(-iid F)jqn/pm,) (3.80) which when applied to the expansion coefficients in the SFB gives the dynamic expansion coefficients. These will be called the boosted coefficients. To compute such a transformation is not trivial in a hydrogenic or STO basis. However, if the basis set is written in terms of Gaussians, the computation can be done analytically. DYNAMO relies on the integral package ABACUS written by Helgaker, Jensen and Jorgensen [86] using Gaussian basis functions. The basis functions used in the calculations of the p+H collision involve the hydrogen is, 2s and 2p functions written as contractions of Gaussians. The s- and p-type normalized Gaussian functions have the form #8(F a) = (2a/7r)311 exp (-ar2) (3.81) a) = xi(128a'/7r') / exp (-ar2) 53 Then, the fundamental matrix elements from which to construct the boost transformation matrix are (4(a)Iexp(-ig - F)|4()) = (2( )1/ 3/) a +/ C (-4(3)) (o, (a) I exp -) -zVX 2(ap) 1/2 3/2 V/-exp - 2 + #3 a + (-4(a +/3) (0, (a)|Iexp(-i*6- rj)I Op.(0)) (2(0)1/2 3/2 v2 I exp 4(a + #) (op. (a) Iexp(-X-rI o. ,(fl)) ) 5/2 2 2 a + /2(a+) exp 4(a+ ) (Op.(a)j exp(-iv-. )jjy(#)) -- V'Vy ~ 5/2 ep V 2(a+/) + # \4(a+)) (3.82) and from the fixed linear combinations of Gaussians describing the atomic basis it is a simple matter to evaluate the projection from the SFB to the boosted basis. These results are implemented in a program called BOOST. It uses the output from DYNAMO, performs the transformations and computes state-to-state probabilities. 54 For a many-electron molecular process in which the final product is made up of several distinct fragments we want to analyze the output of a DYNAMO run and find the probability of having a given number of electrons in each fragment. In general we will also want state-to-state probabilities, even for the case of a single molecule, not many fragments. For a many-electron system these states are timeindependent ones which approximate the eigenstates of the fragments or molecule. I have developed the following method to obtain such information from the single determinantal output from DYNAMO. As discussed in the previous section, our starting point is a single determinant made up of molecular spin-orbitals (MSO's) which are a linear combination of nonorthogonal atomic spin-orbitals (ASO's). This determinant is some final or even intermediate result of a DYNAMO run. I will call the MO's that are obtained in this way dynamic MSO's. The determinant is T' = 11....ON I K (3.83) i= Xi+ E Xj Zi j=N+l We can interpret the final result in terms of time-independent multiconfigurational states (I will use MC to denote multiconfigurational, not Monte Carlo) which are an approximation to the exact eigenstates of the system. If there are several fragments then we will want to interpret the result in terms of MC states corresponding to each moiety. The simplest example is that of a collision process: a fragment hits another, a reaction occurs and then fragments separate. Another might be two systems coming 55 in close contact, some reaction occurring and then separating. In all these cases there are two aspects: 1. There are different possible reaction channels, i.e. a different number of electrons associated with each fragment. 2. We want to understand the chemistry in terms of "local" states, where by "local" I mean molecular states written only in terms of the ASO's of the fragment of interest. The general procedure is to do a MC calculation for each of the fragments and for all the channels of interest within a fragment. This will become clearer in a moment. For a given fragment and channel (where by channel I mean a given number of electrons in the fragment) a standard SCF is performed to determine a set of MSO's which are used to perform a MC calculation to a given level of interest. These I will call reference MSO's for the fragment and channel. In principle, it is not necessary to use a HF state as the reference state. A more general reference state can be used if needed. One possibility is to use natural orbitals. What is needed is the transformation between the reference MSO's and the nonorthogonal basis functions. Calling this transformation U, we have Oj = xiUij (3.84) Xi =4 (SUt ( where 4 are the reference MSO's, and the relation USUt = I (3.85) is used, with S the overlap matrix of the ASO's. 56 Next we need to expand the state we have in terms of the fragments. To make this clearer assume there are only two fragments. It is straightforward to generalize to more fragments. The fragments contain a well defined subset of basis ASO's. The ASO's in each fragment ideally have no overlap with those of the other fragment. If there is some overlap I assume it is small and can be ignored for the calculation. Each dynamic MSO is a sum of the form =ZXCjs+ZXL i = + VB (3.86) jrA IcB The single determinant is then written, using the fact that a determinant is multilinear in its columns or rows, as (we also normalize the total wavefunction to 1 here) A B A B.. A B I/V -k-T (3.87) where each brace contains (N) single determinants corresponding to all the possible ways of getting (N - M) distinct states of fragment A and M distinct states of fragment B, (M = 0, 1...N). Another way of understanding these states is that they are all the possible determinants with (N - M) electrons in fragment A and M electrons in fragment B. In general, most of these will be zero, or very unimportant compared to others. For example, if a fragment with L electrons comes in contact with another of M electrons, there might be one electron transferred from A to B, so that the relevant 57 terms of the expansion are only those with either L electrons in A or L - 1 electrons in A. Otherwise, there will be 2N determinants in the preceding expansion, which might be necessary in some cases (such as in the a + He collision in which one determinant represents 2-electron transfer, another represents 2-electron excitation and the other two determinants represent only 1-electron transfer). The scheme just described sets the stage for the MC calculations. According to the channels of interest in the system, a MC calculation is performed to find the ground and excited states for the fragments. Perhaps at most a CI Singles and Doubles (CISD) will be possible due to computational limitations in the size of the system, and even then it might be truncated to only a certain number of excited states. This choice will depend on the system being studied. The next step is to expand the determinants of Eq. (3.87) in terms of the MSO transformation U for each fragment and appropriate channel. Redefining U as UC, where C refers to the fragment and M to the number of electrons in the fragment, and defining the reference MSO's of fragment C with M-electrons as OCM;i, the dynamic MSO's are written as
V1 M /MJC0Mj i |

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PAGE 1 ELECTRON-NUCLEAR DYNAMICS: A THEORETICAL TREATMENT USING COHERENT STATES AND THE TIME-DEPENDENT VARIATIONAL PRINCIPLE By AGUSTIN CARLOS DIZ 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 1992 PAGE 2 To Marcela, Agustfn and Ximena PAGE 3 ACKNOWLEDGMENTS I would like to thank Erik Deumens and Yngve Ohrn, for their warmth as persons and for the excellent scientists they are. The combination of both of these qualities made working with them something very special. I cannot think of a better atmosphere in which to learn and develop new ideas. I thank my wife, Marcela, who gave up so much for me; my children, Agustfn and Ximena, for their love; my parents, for being there and knowing that I can count on them. Finally, I am grateful to Keith Runge for all those discussions we had. iii PAGE 4 TABLE OF CONTENTS ACKNOWLEDGMENTS iii LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT viii CHAPTERS 1 INTRODUCTION 1 2 AN OVERVIEW OF TIME-DEPENDENT METHODS 4 Potential Energy Surfaces 4 Time-Dependent Methods on PES 8 Classical Trajectory Schemes 8 Semiclassical Schemes 9 Exact Quantum Mechanical Schemes 10 Time-Dependent Methods Without PES 14 Close-Coupling Methods 15 Methods with Nuclear Dynamics 16 Electron Translation Factors 20 3 THE END FORMALISM 24 The Time-Dependent Variational Principle 24 Coherent States 28 The Equations of Motion 40 Interpretive Tools 51 4 RESULTS 60 The p+H Collision 60 Introduction 60 Results 63 iv PAGE 5 The a+He Collision 74 Introduction 74 Results 76 5 CONCLUSIONS 80 APPENDICES A DETAILS OF DERIVATION IN ORTHONORMAL BASIS 83 B DETAILS OF DERIVATION IN ATOMIC BASIS 96 REFERENCES 104 BIOGRAPHICAL SKETCH Ill V PAGE 6 LIST OF TABLES Table 4-1: Contraction coefficients c and exponents a for the basis used in this work. . 63 Table 4-2: Total transfer cross-sections for proton colliding with a hydrogen atom (xlO" 16 cm 2 ) 66 Table 4-3: Total transfer cross-sections for collisions of a proton on hydrogen and deuterium atoms, as well as that of a D + ion on hydrogen, for energies below 1 eV, compared to results by Hunter and Kuriyan (H&K) (Cross-section units xlO " 16 cm 2 ) 68 Table 4-4: Excitation and transfer cross-sections for 2s, 2px, 2pz states and total 2p cross-sections (x 10" 16 cm 2 ) 69 Table 4-5: One-electron transfer cross sections using three different basis sets (see text for description of the basis sets) are compared with experiments by Afrosimov. (Crosssection units x 10" 17 cm 2 .) 77 Table 4-6: Two-electron transfer cross sections using three different basis sets (see text for description of the basis sets) are compared with experiments by Afrosimov. (Crosssection units x 10" 17 cm 2 .) 78 vi PAGE 7 LIST OF FIGURES Figure 4-1: Weighted transition probabilities for total electron transfer at 2 keV as a function of impact parameter. All data in atomic units. Dots are the results by Grim et al. The full line are the results of calculations by Ludtie and Dreizler. The dash dotted line are the results from DYNAMO. (See text for references) 65 Figure 4-2: Total transfer cross sections from 0.02 eV to 4000 eV. Comparison of experiment to theory. DYNAMO computations are the larger open circles joined by solid line 67 Figure 4-3: n=2 excitation cross sections. Comparison of DYNAMO results with experiments. Total 2p excitation calculations are shown by open circles joined by a solid line. 2s excitation calculations are shown by solid circles joined by a dotted line 70 Figure 4-4: n=2 transfer cross sections. The symbols and lines have the same meaning as in the previous figure 71 Figure 4-5: Computed integral alignment and experimental results, as a function of collision energy 72 Figure 4-6: Expectation value of x component of electron momentum and nuclear momentum as a function of time. Middle curve is the electron expectation value, the upper curve is the momentum corresponding to the incoming proton and the lower curve is that of the proton originally at rest at the origin 73 vii PAGE 8 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 ELECTRON-NUCLEAR DYNAMICS: A THEORETICAL TREATMENT USING COHERENT STATES AND THE TIME-DEPENDENT VARIATIONAL PRINCIPLE By AGUSTfN CARLOS DIZ August, 1992 Chairman: N. Yngve Ohm Major Department: Physics A new method for studying electron-nuclear dynamics in chemical processes is presented. The method is founded on the Time-Dependent Variational Principle and the description of the electronic wavefuntions via coherent states. The resulting equations have a symplectic structure resembling that of classical mechanics. A first model is developed with the nuclei treated in the limit of narrow Gaussian wavepackets and the electrons restricted to a single determinantal description. The equations of motion are analyzed to understand their physical significance. These equations are used to study the p+H and a+He collisions. The range of validity of the model is examined and future developments discussed. viii PAGE 9 CHAPTER 1 INTRODUCTION Since the dawn of quantum mechanics, the greatest emphasis has been placed on the solution of the time-independent Schrodinger equation. The time-dependent Schrddinger equation, except in the simplest model systems, is far more difficult to solve, enhancing the bias toward the time-independent studies. Much interesting knowledge has been obtained from the work done in the time-independent picture, and more will come. Yet a time-dependent scheme gives important insight on the actual dynamics and specific mechanisms important for the evolution of a physical system. Recent advances in computational power have led to a growing interest in solving time-dependent problems. Among the phenomena which can be studied with a time-dependent method are molecular photodissociation, collision induced dissociation and gas phase scattering. Electron transfer can be analyzed in detail and reaction rates can be computed. Using spectral analysis, much information can be gleaned from a time-dependent wavefunction. At the same time, experiments are beginning to probe chemical reactions at the femtosecond time scale [1], which no time-independent method can hope to describe. Here I present a theoretical method to study the time evolution of a molecular system and the first results obtained from its use. The potentials considered here are only the Coulombic ones. This restriction can easily be lifted to include other couplings if necessary. Electrons are treated quantummechanically while nuclei can be treated in a similar fashion or semiclassically. It is also 1 PAGE 10 2 possible to treat some of the nuclei classically and others quantum mechanically. This can be important for such processes as proton transfer through quantum tunneling for which it is not necessary to have a fully quantum description of all nuclei. The nonrelativistic molecular Hamiltonian is where Hartree atomic units have been used; N is the number of electrons and M is the number of nuclei. The nuclei will be considered as point particles. The equation to be solved approximately is the time-dependent SchrOdinger equation: In most cases, attention is focused on doing quantum, semiclassical or classical nuclear dynamics on electronic potential energy surfaces. This involves the work of up to three research groups working over a period of years. One group determines the potential energy surface at a number of discrete points. Another interpolates the results to get a full surface. Finally another group does the dynamics on the surface. However, it is possible to think in terms of a more general approach, to develop a method which in principle can be applied to any molecular system, regardless of the kind of properties that are to be analyzed, and in which the three steps of the process just described are done in one step, with the additional bonus that the electron-nuclear dynamics is retained N t M , N M ~ N N . ,=1 1 A =l LMA i=l A=l r ' A .=1 Â», r " (l.D 4w = w> (1.2) PAGE 11 3 fully. With such a method it is possible to study scattering problems, charge transfer, follow chemical reactions, predict and analyze vibrational spectra, Raman spectra, etc. in one unified manner. The work I present here is a new approach to such a general, electron-nuclear dynamics (END) method. The method has been developed over the past five years. It began as a simple model of an electron transfer process with classical nuclei moving on quartic surfaces, studied by Deumens, Ohrn and Lathouwers [2]. Then I did some work with Deumens and Ohrn to describe the nuclei quantum mechanically on such surfaces [3]. After this preliminary work, we generalized the method so that both electrons and nuclei could be treated dynamically. This led to a computer code called DYNAMO from which results of calculations on a water molecule and the collision process of a proton on a hydrogen atom have just been published [4]. The method is based on mathematical methods developed over the past 30 years in other branches of physics. Chapter 2 reviews prior time-dependent schemes. The theory of the method I use is discussed in detail in Chapter 3 along with the physical interpretation of the resulting equations. Chapter 4 discusses the results of this method when applied to two systems I have studied. The first is a proton colliding on a hydrogen atom, for collision energies varying over 6 orders of magnitude and including an analysis of the electron transfer and the state-to-state transfer and excitation probabilities. The second is the scattering of an alpha particle off a helium atom at low keV energies. PAGE 12 CHAPTER 2 AN OVERVIEW OF TIME-DEPENDENT METHODS Potential Energy Surfaces A widely used procedure is the Born-Oppenheimer approximation (BOA). It is based on the fact that nuclei are heavier than electrons, and move more slowly than electrons under most circumstances. Thus, to a good approximation, one can think of the electrons as moving among fixed nuclei; the electrons have enough time to accommodate themselves to the changing nuclear degrees of freedom. With this concept, a particular expansion of the system wavefunction is used. The Hamiltonian is separated as M 2M A (2.1) iÂ»l 1 ,=i A=l TlA i=l j>x 13 A=l B>A UAB A complete set of eigenstates of the electronic Hamiltonian is found and the system wavefunction is expanded in terms of these: *({*}, {Ra}) = Erf*^}! {^})^ nucl ({^}) (2.2) ^^({r,},^}) = Â£, elec ({^})^ lec ({r;},{^}) No approximation is made up to this point Only a particular expansion of the system wavefunction is chosen which I will refer to as a Born-Oppenheimer (BO) expansion of 4 PAGE 13 5 states. Allowing the Hamiltonian to operate on this expansion of the system wavefunction, and projecting on a BO electronic state where wf'i^D^wr 1 ({Â£Â»})) = E { (-E ^ + * + T h + m i (2.3) ^Â• = -(^ lec iE^ v ii^ lec ) Â• (2.4) Eq. (2.3) defines the action of the Hamiltonian on the nuclear wavefunctions. The operators T 1 and T 2 are dynamic couplings which are the result of the nuclear momentum operator acting on the electronic BO states. These couplings can be written as M (2.5) and if the set of electronic states is complete, j\ Â— 1 The nuclear momentum operator is replaced by -iV A -Â» -Â«V A + (2.7) PAGE 14 in Eq. (2.3) to yield (*fWÂ£i*?~>vr*({Â£4 (2.8) This Hamiltonian for the nuclear wavefunctions is no longer Hermitian, since ^-i'V/ + P,^ is not Hermitian. This derives from the expansion of the full wavefunction in terms of the electronic eigenstates [5]. Since the BO electronic states are instantaneously in equilibrium with the nuclear Â— * configuration, inertial effects between electrons and nuclei are described through P^. The P,4 couplings represent the change of the electronic basis functions with the nuclear configuration. They contain rotations, distortions, polarizations, change of character of the electronic basis functions as well as the change of the electronic basis functions due to simple displacement of the nuclei [6]. The BO approximation (BOA) neglects the P^ matrices, effectively decoupling the equations for the nuclear wavefunctions. Each component of the nuclear wavefunction feels an effective potential given by called Potential Energy Surface (PES). Another possibility is to keep the diagonal terms of the vector potential couplings in the definition of the PES. This is usually referred to as the adiabatic approximation. (2.9) PAGE 15 The definition of the PES and the use of the Born-Oppenheimer or adiabatic approximation is the starting point for most molecular dynamics today. Additional approximations are usually made, such as treating the nuclei purely as classical particles on these surfaces, or to treat the nuclei as quantum particles only insofar as to quantize the vibrational or rotational degrees of freedom of the nuclei. In the past decade, efforts have been made to describe the nuclei in terms of wavepackets moving along these PES. The breakdown of the BOA occurs when the assumption behind the approximation is no longer valid, i.e. when nuclear motion cannot be considered to be slower than electronic motion, or when PES become degenerate. This requires that dynamic couplings be retained to some degree. Processes in which this happens are avoided crossings of PES (if adiabatic potentials are used, or crossings for diabatic surfaces) where a change in the electronic character of the states may generate important dynamic couplings [7]. Another case is when nuclei are vibrating and rotating; as nuclei get closer, the moment of inertia decreases, increasing the angular velocity. Under these conditions the electrons may not have time to keep up with the nuclear motion, leading to predissociation. To generate a PES for dynamics is not simple. The work and time invested is very large. Sometimes it takes the work of three research groups working over a period of years to produce the electronic energies, the interpolated PES and do the dynamics. Even then, discrepancies between experiments and theory can become painfully obvious, as witnessed in the discussion in the journals over the existence of certain resonances in the H+H2 scattering problem [8, 9]. PAGE 16 8 Time-Dependent Methods on PES The methods used today to do dynamics on surfaces can be divided into classical trajectory, semiclassical and "exact" quantum mechanical. Couplings between surfaces and other non Born-Oppenheimer effects are sometimes treated as perturbations. Classical Trajectory Schemes The simplest method is the classical trajectory scheme [10-13] whereby nuclei are propagated as classical particles on a single PES. The greatest amount of work is finding the surface. The dynamics is simple to implement and the results are simple to interpret. Extensive sampling of phase space is required for a full final state resolution, i.e., to have reaction cross sections and rates. Classical trajectory methods are capable of producing good results. They are used to describe isolated encounters as well as processes in a condensed phase by using the generalized Langevin equation [14, 15]. The shortcomings of this approach are the neglect of quantum effects, such as zeropoint energy and tunneling, which may be important. The study of systems with a single potential energy surface of importance is also a severe restriction. However, the most severe limitation, which is shared by all methods that use PES, is the time it takes to obtain such a surface. The fitting procedure is also crucial, since errors in the fitting can lead to large quantitative errors in the dynamics. The limitations are more evident as the degrees of freedom of the system increase. The computational effort of finding points on PAGE 17 9 the surface and then fitting those points to some analytic function becomes unmanageable for any system having more than three or four atoms. Systems investigated with the classical trajectories include the H+H2 collision [16] and charge transfer in the H2 + + H2 collisions [17]. Semiclassical Schemes The semiclassical approaches try to correct for the deficiencies of the classical trajectories without going to the full-blown quantum mechanical approach. There are many different implementations, from time-independent, such as Miller's 5-matrix approach [18, 19] in which the Feynman path integral is evaluated on the classical trajectory to find an approximation to the 5-matrix, to Heller's Gaussian wavepacket path integral formulation of semiclassical dynamics [20-22]. In Heller's work, the wavepacket is assumed to remain Gaussian throughout the evolution, with the form exp[Â— a(x Â— q) 2 + ip(x Â— q)+c] and equations are derived for the parameters q,p,a, and c. The wavepackets are propagated on a potential energy surface. To realize the propagation, the potential is quadratically expanded about the instantaneous center of the wavepacket. The relationship between the 5-matrix approach and the time-dependent wavepackets has been studied by Heller [23]. To correct for the divergence of semiclassical wavefunctions near caustics a generalization was developed [24, 25] which is similar to Klauder's global, uniform semiclassical approximation for wavefunctions [26] proposed earlier. In this work, the Gaussian wavepackets are generalized so that all the time-dependent parameters are complex and the manifold of states labelled by p and q is searched to find the most important states PAGE 18 10 which lead to the same final state. Other approaches along similar lines include the work of McDonald [27] and Littlejohn [28]. The eikonal approach to semiclassical time-dependent mechanics has been proposed by Micha [29] and compared to Heller's work [30, 31], yielding very similar results. Applications of these methods include work on the photodissociation of methyl iodide [30, 31]. A comparison between the semiclassical approximation and an "exact" approach indicates that the two methods agree closely for this system[31]. Another semiclassical computation uses Gaussian wavepackets in the interaction picture for the H+H 2 collision [32]. It has been compared to the classical trajectory work [16]. The two methods give similar results, with some differences in the details. Exact Quantum Mechanical Schemes The so-called exact methods do use approximations. Their name derives from the fact that the errors can, in principle, be made as small as desired. There are two parts to consider: the representation of the wavefunction and the actual time propagation algorithm. The wavefunction representation determines how the operators, such as the Hamiltonian, are to be evaluated. Wavefunctions are represented in one of two ways, by expanding in a basis set or discretization on a grid of points. The discretization of space depends on the PES and the choice of the coordinates. All of these methods work only with internal coordinates of the molecular system, avoiding the calculation of global rotations or translations of the system. The effort put into finding good internal coordinates is nontrivial [6]. PAGE 19 11 Transformations to such coordinates can be quite a problem. It is much more convenient to do the calculations in Cartesian coordinates, including global translation and rotation. For this, however, it is necessary that the conservation laws related to these quantities remain valid within the approximations made. The Pseudo Spectral Fourier Approximation [33] uses the wavefunction represented on a grid in coordinate space and then uses the Discrete Fourier Transform to obtain the momentum space representation. Fast Fourier Transform codes allow for efficient switching from coordinate to momentum space. This permits fast evaluation of operators such as the potential and kinetic operators of the Hamiltonian. The most used propagation methods can be divided into four categories: Second-order differencing (SOD), the split-operator method (SO), the short-time iterative Lanczos (SIL) method and the Chebychev expansion (CE) method. For the SOD [34], the Schrodinger equation is solved by approximating the time derivative by a second order difference. Short time steps are used, and the wavefunction at *n+i is evaluated as 0(t n+1 ) = V(<Â»-i)-2tA PAGE 20 12 In the SO scheme [35] the exponential operator is decomposed symmetrically, to obtain an approximate evolution operator of the form U(e) w exp(-i|A')exp(-ieV)exp(-t'|/i:) (2.12) which is also correct through second order. The SIL propagation formula [36] is U(e) Â«exp[-ieA(#,V>(0))] (2-13) where U is a matrix operator in the Krylov subspace, which is generated by the Hamiltonian and the initial wavefunction, and A is the tridiagonal Lanczos matrix representing the Hamiltonian in the Krylov space. The Krylov subspace is generated through the action of the Hamiltonian operator on the initial wavefunction. Thus the N I Krylov subspace is generated by the N vectors Uj = H j xl>(0) . (2.14) The Lanczos recurrence generates a set of orthogonal polynomials within the subspace which represent a finite polynomial approximation to the operator. Then the projected subspace representation of the Hamiltonian operator has a tridiagonal form. The operator is diagonalized and the propagation is performed with the diagonal eigenvalue matrix. The length of time dictates the size of the Krylov space needed for a predefined accuracy. Short time steps are used in order not to lose the advantage of the Lanczos reduction. The CE method [37] approximates the exact propagator by the expansion N U(t) = ^^(t)T n (-iH R ) (2.15) n=l PAGE 21 13 where the Hamiltonian is renormalized so that its spectrum coincides with the domain of the Chebychev polynomials TÂ„. The normalization is carried out by dividing by the range of energies permitted in the evolution, AE = E max Â— Â£min, and shifting the energy level so that all the eigenvalues fall between -1 and 1. The coefficients aÂ„ are proportional to Bessel functions a n (t) = 2J n (^lp) (2.16) and the Chebychev polynomials T n are obtained from the recursion relation to = V>(0) fa = -iH R il>(0) (2.17) PAGE 22 14 atoms are possible, with some work being started on systems of 4 atoms in which some internal degrees of freedom can be frozen. A few representative works with "exact" methods are reactions [39] and absorption spectra [40]. Because of the SOD's ease of implementation and satisfactory accuracy it has been used extensively for a variety of problems, including eigenspectra [41], nonadiabatically coupled systems [42], photodetachment spectra [43] and systems with timedependent Hamiltonians [44]. The CE method has been used on atom -diatom collisions [45], photodissociation [46] and computation of energy levels [47, 48]. To reduce the number of grid points that are required for the calculations, most work is done with absorbing potentials which are placed at the boundary of the coordinate space considered to be important for the dynamics of the reaction. Time-Dependent Methods Without PES Most chemistry does not happen on a single potential energy surface with only three or four atoms. Determining PES for higher numbers of atoms becomes a difficult, if not impossible task. These facts make the search for a method that does not use surfaces more urgent. The limitations of PES have generated interest in developing methods that avoid their use. One example in which methods using PES fail is the collision of a proton with a hydrogen atom at energies above 10 eV. In this case many PES would be required, since excitation to n=2 states becomes important and provides for interesting physical PAGE 23 15 phenomena. The collision has been studied on the lowest energy surfaces made up by the molecular ls PAGE 24 16 is written as N !*(<)> = E fl *(')W)) (2.18) for a basis set of N functions. This basis set changes with time, since the basis functions follow the nuclei. Projecting the Schrodinger equation on the truncated basis gives where the prescribed nuclear trajectories make the electron-nuclear attraction and nuclearnuclear repulsion potentials time-dependent. The equations take the form [53] where the final expression for N and M will depend on the basis chosen and the prescribed trajectory. Since the close-coupling method does not account for nuclear dynamics it cannot be successfully used for collision energies below about 1 keV. The path followed by the nuclei is very important in determining transitions at lower energies[60]. For energies below 1 keV it is essential to include electron-nuclear dynamics. The close-coupling methods have never been used for systems in which more than two electrons are explicitly described. Methods with Nuclear Dynamics Several methods have been proposed in this field. One method, the Car-Parrinello approach [61], has gained much attention among theoretical chemists because of its (v*(oi4 #i*> = Â° (2.19) (2.20) Njk{i) = tohfo), M jk = WjlijH(t)\^ k ) PAGE 25 17 applicability to larger systems. The method uses an optimization technique to follow the ground state surface and uses the Hellman-Feynman theorem to compute the forces on the nuclei. It is an approximation to dynamics on a surface which requires calculating only the part of the PES needed. There is no firm theoretical work which shows fully why the method works, in particular why it works better than if an SCF calculation were to be performed at each position instead of using a molecular dynamics optimization procedure [62]. Hartke and Carter use this method on Na4 singlet and triplet states [63]. Field [64] has used the Time-Dependent Hartree-Fock method for a closed-shell restricted Hartree-Fock wavefunctions using the neglect of diatomic differential overlap (NDDO) approximation. In his approach, the TDHF equations drive the electronic degrees of freedom, subject to the constraint of normalized molecular orbitals: where the integration over spin has been performed, c are the expansion coefficients of the molecular orbitals in terms of an orthogonal basis and F is the Fock matrix in the basis. The nuclei are assumed to evolve as in classical dynamics, with the forces given by the gradient of the energy of the system (2.21) Fji = c]Fc, dRj_ dt Â— * Mi (2.22) dPj dt dE d&i PAGE 26 18 The energy E is expressed as (2.23) #ekc = 22^cJ i PAGE 27 19 reduce to ones similar to those of Field but without the additional approximations of NDDO and restricted closeshell determinant. Runge et al solve an equation for the density matrix instead of the orbital coefficients. The equations to be solved are [60] iP^ = S1 F 7 P 7 -P 7 F 7 S 1 i (2-24) dRj_ Pj_ dt ~ Mi dP] dE dt ~ &R 7 where E is the total energy of the system. This method has been applied to systems with one electron, such as the p+H collision [60]. Work in progress by this group lifts the neglect of electron translation factors in the calculation of the Fock matrix. Another approach is taken by Meyer and Miller [65]. They too derive equations which couple the nuclear and electronic degrees of freedom. Their approach is formal and has not been used in an ab initio fashion on any real systems. They assume the existence of a complete diabatic basis for the electrons (i.e. one which does not depend on the nuclear positions). This has the effect of decoupling the electronic and nuclear degrees of freedom, except through the electron-nuclear attraction. The basis considered is not made up from an expansion of one-electron orbitals, but is a general TV-electron basis. The electronic degrees of freedom are the expansion coefficients of the electronic state in the basis. The coefficients are written as a norm times a phase factor. The nuclei are approximated by point-like classical particles. Using the time-dependent variational PAGE 28 20 principle they derive the equations of motion. The equations for the electronic degrees of freedom look just like the action-angle equations of motion of classical mechanics. Thus the whole system of equations looks very "classical", although the electrons are treated quantum-mechanically. A canonical transformation can be performed to obtain the equations of motion for the electrons and nuclei using an adiabatic electronic basis. In this case the equations of motion for the electrons and nuclei are [65] dpi _V/v (p + p) Pl ~ dxi~ V dx > dx > M Â« (2.25) Â• dH dP (p + P) Qk = -KTT = E K + N K = dN K dN K Mi dH dP (p + P) dQ K dQ K M, with the wavefunction written as |0) = X^xpHQ*)^!*) (2-26) K where x, p are the classical nuclear position and momenta, P are the momentum couplings defined in the BO Hamiltonian (Eq. 2.8), |A') are the adiabatic electronic states and the Hamiltonian has been taken to the classical limit for the nuclear degrees of freedom. Electron Translation Factors The Perturbed Stationary State method (PSS) [6, 66] is used for many atomic and molecular collisions calculations. Starting from the BO expansion of the Hamiltonian, Eq. (2.8), this method does not ignore the dynamic couplings like the BOA. Delos [6] PAGE 29 21 lists shortcomings of the PSS approach when the electronic as well as nuclear basis is truncated, i.e. when approximations are made to the BO expansion. He notes that the expression for T 2 in terms of P in Eq. (2.6) is not valid for a truncated basis, though he states that it is usually sufficiently accurate within the approximation. More serious problems he notes are: 1. The individual terms in the BO expansion of the system state do not satisfy the scattering boundary conditions. 2. The P matrix elements contain couplings of infinite range. 3. P matrices contains fictitious "origin dependent " couplings. 4. Matrix elements in this formulation do not contain momentum transfer factors which are needed to describe Doppler shifts in the energy spectrum of moving systems. Delos suggests a solution via the so-called electron translation factors (ETF's), which eliminate many of these problems. The solution he proposes is essentially to include in each electronic basis function a complex phase factor which cancels some of the offensive terms listed above. This is effectively a correction to the BO expansion, accounting for the momentum of the electrons in the process, not through the dynamic couplings but through the basis. This correction is used in the close-coupling method for atomic collisions, and also in the approach by Runge, Micha and Feng to electron-nuclear dynamics. The following example should clarify the issue. If an H atom is moving with respect to an inertial frame (which I will call the lab frame), its ground state should satisfy the time-dependent SchrOdinger equation. It is a simple exercise to show that the orbital PAGE 30 22 exp(im e (u Â• fv 2 t/2) eit) PAGE 31 23 outside a certain distance between nuclei are atomic and contain the electron translation factors. At the limiting distance these AO's are matched to molecular orbitals without electron translation factors to continue the calculations with the PS S. When the fragments separate, at the limiting distance the matching from the MO to the AO basis is again performed. In this way Galilean invariance is preserved outside the interaction region. An approach where the MO's are described in terms of an LCAO expansion with AO's having ETF's was attempted, but failed (A. Riera, personal communication). The failure stems from the fact that the Hamiltonian carries no information about the moving frame and so the MO's obtained undo the effects of the ETF's. When ETF's are used, there appears to be no work which has included the couplings that arise for accelerated nuclei. For close-coupling treatments with straight line trajectories and constant velocities these couplings are zero. PAGE 32 CHAPTER 3 THE END FORMALISM The fully quantum, exact methods are setting benchmarks, but the use of the potential energy surfaces severely restricts them, a problem shared by all methods relying on PES. Even four atom systems are beyond reach, greatly limiting the chemical systems that can be studied. The move towards the elimination of potential energy surfaces is very recent. As already discussed, several proposals have been made. However, either the theory behind the methods is not well understood or the approximations made are many, and at times confusing. Here I present another method which does not rely on PES. The approximations made are few and simple to understand. The chapter is divided in four parts. The first is the description of the Time-Dependent Variational Principle. The second is the coherent state formalism. In the third part these tools are used to find the equations for electron-nuclear dynamics (the END equations). There is also a description of the approximations made in the first model developed. The fourth part deals with the tools developed to analyze the results of a calculation. The Time-Dependent Variational Principle Three main time-dependent variational principles are used today. They are the McLachlan variational principle [68], the Dirac-Frenkel variational principle [69, 70] and the TimeDependent Variational Principle (TDVP) [71]. All of these have been shown to be equivalent [72] if the trial wavefunction is described with complex parameters and is 24 PAGE 33 25 analytic in these parameters. Because the form of the wavefunction used here satisfies these two conditions, the three variational principles will give the same results. I will describe and use the TDVP. If we assume the existence of a set of electronic and nuclear parameters Â£ which completely define the quantum state of the system, I will label the state with these parameters: \(). Using Hartree atomic units so that h = 1, the following Lagrangian is proposed: = (CIA'IO(CIC)1 where the left-acting time derivative is noted as ^, and the operator K is defined. The action is then A = I* Ldt Jti (3.2) and the TDVP requires that sa = 8 C [K (ci ^ ic> _ ^ ((c|)ic> ) _ (c|//|c> ] (cic>_1 ^ = Â° (33) The TDVP will result in the Schrodinger equation if the trial wavefunction used has no restriction in the Hilbert space of the system. With a partial integration: r< 2 SA = J^[(6C\t^\0-(SC\H\0 (3.4) -W\Q{C\Ql {6C\Q + c.c. ((IQ-'dt i-i. PAGE 34 26 Because the variation is arbitrary 10 (3.5) which is the Schrodinger equation if the right hand side is zero. If the wavefunction is multiplied by global time-dependent phase factor exp(z'7) with 7 satisfying the equation * " IcicT ( } then it is simple to see that the variation of the action results in recovering the Schrodinger equation. The global phase is important in time -correlation functions. In practical applications the trial wavefunction is restricted to a submanifold of the full Hilbert space and an approximation to the time-dependent equation is solved. Introducing the notations (3.7) = exp (t'7)|C) S(C',0 = Â«I0 (3.8) Â£(C,C) = PAGE 35 27 Because all time dependence of the state is through the parameters, Eq. (3.3) is equivalent to dt -f dt -/?[(-?S6-S)* (3.10) dt The second step in Eq. (3.10) involves a partial integration of some terms with respect to t. Since all the 8( and their complex conjugates are independent variations we obtain the dynamical equations (3.11) with the elements of the metric matrix defined as dE d 2 \nS (3.12) The matrix C is clearly Hermitian. The phase factor does not influence the evolution of the other parameters, and can be computed afterwards. In practice, it is convenient to integrate (3.9) along with the equations (3.11) for the other parameters. PAGE 36 28 The equations (3.11) can be written in matrix form as PAGE 37 29 first addressed by Thouless [73] for nuclear systems using the TDHF. The main problem he faced is that a single determinant is unchanged under a unitary transformation of its orbitals. This means that there are redundant parameters in the definition of a determinantal state. The parametrization he found turns out to be a special case of a more general scheme, referred to as coherent states. Schrodinger first proposed the concept of coherent states in 1926 [74] in connection with the classical harmonic oscillator. Coherent states were later used by Glauber [75-77] and Sudarshan [78] in quantum optics in the early 60's. Since then there has been interest in them from both a mathematical [79] and physical [80] point of view. The representations of Lie groups can be studied with coherent states, and the range of physical systems treated with this technique goes from condensed matter and thermodynamics to elementary particle physics and path-integral developments, as well as to study the relationship between quantum and classical dynamics (for a compilation of these developments see Klauder and Skagerstam's book, Ref. 80). Coherent states are also useful in the study of unrestricted Hartree-Fock theory applied to molecules [81]. A coherent state satisfies two properties [80]: 1. A coherent state |^) is a strongly continuous function of the label z. This means that lim I \z) \z') I = 0 (3.17) zÂ— >z' 2. There is a positive measure 6z on the space Q, of labels z such that / = / \z)(z\6z (3.18) PAGE 38 30 The first property indicates that there is a one-to-one mapping between coherent states and points on the label space fi. The second property shows that the coherent states are a complete set, since any state can be decomposed in terms of coherent states. Because of the continuous label, the set cannot be an orthogonal basis, but must be overcomplete. Although there are many forms of coherent states [80], I will restrict myself to ones related to compact Lie groups. Let U(g) be a continuous, irreducible unitary representation of a compact Lie group G and |0) be some normalized reference state, also called the extremal state. Then the states generated by the operation of the group on the reference, satisfy the first property from (g\g') = (0|C/(sr _1 p')|0) and the continuity properties of Lie groups. To show that the second property is also satisfied, take where dg is the group invariant measure. By virtue of this measure, it is clear that for any g', PU(g') = U(g')P. Schur's Lemma then requires that P be proportional to the identity. Renormalizing the invariant measure by a constant leads to P = 7. The states \g) may not all be distinct. To obtain a distinct states let h be an element of a subgroup H of G defined by the property that these group elements only change the reference by a phase factor, i.e. \g) = U{g)\0) (3.19) (3.20) U{h)\0) = exp[ia{h)}\0) (3.21) PAGE 39 31 H is called the stability subgroup and the coset G/H provides a unique decomposition of any element g belonging to G [82]: g = ch (3.22) where c is a coset representative and h belongs to the stability subgroup. Then U(g)\0) = U(c)U(h)\0) = U(c)\0)exp[ia(h)] (3.23) = |c) exp [ic*(/i)] and |c) is the coherent state, defined through the action of the unique coset representative c. For a single determinant, the stability subgroup is the subgroup made up of unitary transformations which only mix the occupied or unoccupied orbitals. These do not change the determinant. By using the coset parameter representation the possibility of parameters changing without changing the determinant is eliminated from the outset. I will return to the single determinant a little further on. First I will illustrate how to apply the above strategy to the simplest coherent state, the harmonic oscillator coherent state, starting from the Heisenberg-Weyl Lie algebra. These coherent states, also called canonical coherent states [80] can be used as a first approximation to treating nuclei quantum mechanically for molecular dynamics. PAGE 40 32 The Heisenberg-Weyl algebra is spanned by the operators /, a, al and n = a^a with the commutation relations: n, a 1 [n, I] = 0, [n,a] = -a, = 0, (3.24) a, a = /, M = o. The carrier space of irreducible representations for this Heisenberg-Weyl group (H4) is spanned by the number eigenstates of n: n\k) = k\k), k = 0,1,2,.... k (3.25) jot)' (*!)' I*) = 77^1Â°) The state |0) will be used as the reference. It is the lowest weight state for the number operator n. The group elements h = exp [i(Sn + PAGE 41 33 where the Baker-Cambell-Hausdorff theorem is used to get the second line and the property of the annihilation operator on the lowest weight state is used to get to the third line. The normalization exp |a| 2 /2^ can be transferred to the metric defining in this way a coherent state that is not normalized, but which has the advantage of being analytic in a. The canonical, analytic coherent state is: \a) = exp (aa f ) |0) . (3.28) In general I will work with analytic coherent states. This choice provides powerful mathematical properties, as can be noted from the fact that the three most common time-dependent variational principles are equivalent [72] with such trial wavefunctions. A general multiconfigurational wavefunction for the electrons has been formulated by Weiner, Deumens and Ohrn [83] using the mathematics of vector coherent states [84, 85]. However, I will not describe that formulation here but restrict the details to the case of the single determinantal approximation for the electronic wavefunction, which is the model that has been developed into a working program. The coherent state description of the single determinant leads to the Thouless parametrization [73]. Assuming an orthonormal spin-orbital basis set made up of K vectors and N particles to be described by a single determinant, this determinant can be found as a unitary transformation acting on the orbital expansion coefficients of some PAGE 42 34 reference single determinant |^o)I will define this reference state as N i*o)=n 6 /i uac > < 3 29 > /=i where b] (6,) are the fermion creation (annihilation) operators of the basis set used. The reference state is a lowest weight state for the irreducible representation [1 N 0 (A: ^] of the group U(K) with generators b]bj because the weight operators 6-6, have eigenvalue 1 for 1=1... JV, and 0 for i=N+l,..X. The stability group of the reference state is U (N) Â®U(KN). Introducing a complex parametrization of the coset space U (K)/U(N) Â® U(K Â— N) an element g of the group U(k) is written as g = ch as previously discussed. Extending U(K) to GL(K, C) and using the Gauss factorization of c (see notational remarks below) this leads to a parametrization of the coset space in terms of the complex parameters z, with x and y to be determined as functions of z from the condition that g is unitary. In the last expression and in what follows I use a notation intended to make the equations simpler to follow. In the atomic basis, there is a "hole" subspace generated by the first N orbitals which make up the reference single determinant and the "particle" space made up of the rest of the basis spin-orbitals. The field operators of the basis states which belong to the holes will be written with a bullet, i.e. b'\ b', and those associated to the particle states will have an open circle bÂ„, 6Â°. In the same way, when referring to the molecular orbitals made up from the atomic orbitals, the occupied molecular orbitals have a bullet and the virtual orbitals have open circles. So bullets will always be used to describe hole states in a given basis and open circles will represent the (3.30) PAGE 43 35 particle states. Matrices associated with the subspaces of holes or particles will also carry these symbols. Matrix blocks which carry a prime or a double prime indicate the upper or lower off-diagonal block respectively. Matrices associated with the complete one-particle space will not carry any symbols. So, for example, we may write the identity matrix as / = ( 7 Q 7 Â° 0 ) Let T be the unitary irreducible representation of U(K) in Fermi-Fock space. The coherent state is defined as |*,) =T(c fc)|*o) -T(c)T(fc)|Â» 0 ) =r( PAGE 44 36 coefficients, not on the basis functions. This is in accordance with the active point of view of coordinate transformations. I work with the unnormalized coherent state This is the Thouless representation of a determinantal wave function with the elements of the (K Â— N) x N matrix z as time-dependent parameters. Thouless [73] worked in another direction to get to this result. Because his derivation clarifies the meaning of the z parameters I will include it here. If U is the matrix which transforms from the basis spin-orbitals to the occupied and virtual orbitals of the fermionic system, then A' (3.32) t=i (3.33) PAGE 45 37 Following Thouless, m=l[c?\vac) t=i 1=1 \j=l j=N+l = ft ( E K + E E W' I <Â« ) W (3-34) ,=i \/=i \ ;=#+! *=i =Â°n(V + E DjVsr'M t=l \ j=N+lk=l Â«=1 \ jWV+1 jb=l / /=1 where the invariance, up to a constant a, of a determinantal wave function under a linear transformation of its occupied spin orbitals is used. Then the coherent state is recovered as follows i*>=n( l+ e *;vni*o) i=l \ j=N+l ) N K =n n (i+*-v?)i*o) i=l >=JV+1 JV A" =n n exp^-^ivo) t=l j=N+l = ex p(E E 1 1*>, \i=l ]=N+1 (3.35) PAGE 46 38 where X)i=i UflPhi = z jiEq(3.35) we have used the nilpotency of operators The determinantal wavef unction is expressed as det{x(*/)} where K Xi = 1>i+ Â£ (1 < i < AT). (3.36) are nonorthogonal (but linearly independent) dynamical orbitals. The corresponding unoccupied dynamical orbitals may be chosen as N X; = ^-E 2 ;.^ (N+l PAGE 47 39 These relations are used many times when simplifying various expressions in the evolution equations. All of the above assumed a basis of orthonormal spin-orbitals. In practical applications it is preferable to work in the nonorthogonal atomic spin-orbital basis. In this case, since the anticommutation relations of the field operators is not the canonical one but includes the overlap matrix of the orbitals, it is not possible to write the coherent state as an exponential. However, it is still possible to express the molecular orbitals in the same way as in Eq. (3.36) with modified z's: The orbitals of the virtual space are not as simple to express as in the orthonormal basis. Assuming a form K Xi * + E (!<Â»<*). (3.42) j=N+l (3.43) :=1 such that the (K-N)xN matrix v satisfies A* + z tA't + a'z + z \ AÂ°z + A' + z+A'V + z*AÂ° uA* + vA'z + A'* + AÂ°z A 0 + vA' + A'V + uA , u t ) (3.44) PAGE 48 40 i.e. the two subspaces are orthogonal, then vA* + vA'z = -A ,f AÂ°z (3.45) and hence v = (A' t + AÂ°z)(A , + A'z)" 1 (3.46) Â„t = _ (a* + 2 f A't) _1 (A' + *t A 0 ). The Equations of Motion The preceding tools will now be applied to find specific equations. These equations are the ones that are written into the computer code DYNAMO. Here I will derive the general form of the equations and comment on their properties. The details can be found in the appendices. Three approximations are made: 1. In the TDVP the limit of narrow Gaussian wavepackets is taken. The nuclear parameters are then the position and momentum of the nuclei. 2. The electrons are described by means of a single determinant. 3. The spin-orbital electronic basis is truncated to a finite number of orbitals without electron translation factors. The truncated basis limitation is always imposed if an actual calculation is to be performed. The lack of ETF's is something which will be remedied in the near future to allow us to analyze collisions at high speeds. For lower speeds associated with most PAGE 49 41 chemical reactions, the velocities involved are sufficiently small and ETF's are not as critical. The single determinantal wavefunction can be improved upon by including more configurations. The theory to do this for the electronic states can be found in Ref. 83. For nuclei, there is the question of finding a good quantum representation for them. Presently work is being done in this direction. A first approach is to describe them via frozen Gaussians without taking the limit of narrow wavepackets. These are the canonical coherent states discussed in the previous section. However, in work done with Deumens and Ohrn [3], we have shown that a description in which the nuclear wavefunctions are not given the freedom to split, the dynamics becomes nearly identical to that done using classical nuclei. Currently, work is in progress to describe nuclei going beyond the Gaussian wavepacket approximation. Initially I will assume that the electronic basis is complete and is independent of the positions of the nuclei. Then I will show how to derive equations for two different electronic basis sets which are more adequate for calculations. The new equations can be obtained via a symplectic transformation. The basis set that does not depend on the nuclear positions will be referred to as a Non-Following Basis (NFB). Another basis of interest is one with orbitals that are instantaneously centered at the position of the nuclei, but which do not have associated with them the motion of the nuclei, i.e. they do not contain ETF's. This basis set will be referred to as the Static-Following Basis (SFB). A basis set centered at the position of the nuclei with ETF's will be called the Dynamic Following Basis (DFB). For practical calculations with a truncated basis it is desirable to use either the SFB or the DFB. PAGE 50 42 Otherwise, after some time, the nuclei may be far from the vicinity of the basis set location, which would be inadequate for a correct description of the system. There are two equivalent ways to derive the TDVP equations for narrow Gaussians. One is to find the equations of motion and then take the limit of zero width and the other is to take the limit before actually varying the action. I will do the latter. In this case the action becomes A-/[iE(A-*-A-*Hi: Mi + (z\z) dt (3.47) Â— * where H is the molecular Hamiltonian, including the nuclear repulsion term, Rj and Pj are the canonical positions and momenta of the nuclei and \z) is the coherent state description of the electronic single determinant as defined in Eq. (3.32). Defining Z k = R k + iP k (3.48) we can write 1 (3.49) PAGE 51 43 The electronic overlap matrix is given by S(z'\z)=(z'\z) N N Â«=1 ;=1 = det((W+ Â£ Â£ (3.50) = det(^ tJ + Â£ Â«fa-4>)v) V ifc=./V+l / = det(/' + z' t z) and if in the derivation of the TDVP equations the quantity 5 is replaced by S(('\()S d (Z'*,Z) (3.50) then Eq (3.11) can be used to get the equations of motion for the electronic and nuclear parameters: dE dz* dE (3.51) dZi dE dz a 8E dZ k PAGE 52 44 where the unit matrix comes from S c \. In matrix form, these equations become fiC 0 0 0 \ ( i \ / dE/dz* \ 0 il 0 0 z dE/dZ* 0 0 -iC* 0 z* dE/dz ' V o 0 0 -il) Uv \ dE/dZ ) which, with the real form of the nuclear coordinates, can be expressed as \ f z\ (dE/dz*\ z* dE/dz it dE/dR / \ dE/dP J (3.52) (3.53) A transformation that leaves a set of dynamical equations invariant, i.e. does not change the metric at all, is called a symplectic transformation and if the metric contains only ones and zeroes, the transformation is said to be canonical. We are interested in a generalized symplectic transformation that leaves the structure of the equations invariant, but changes the values of the matrix elements of the metric. The invariant is the Poisson bracket. First we consider the transformation from a NFB to a DFB. It has the general form . z = z{z, Z, Z ) (3.54) Z = Z{Z) or with the real form of the nuclear coordinates z = z{z,R, P) R = R(R) (3.55) P = P{P). With this notation I indicate that the z parameters are independent dynamical variables, i.e. they are independent of the "new" nuclear coordinates and momenta. In order to PAGE 53 45 transform the Poisson bracket, we need the matrix of partial derivatives, Jacobian, J (3.56) (d/dz*\ (dz*/dz* 0 0 0\ (d/dz*\ d/dz 0 dz/dz 0 0 d/dz d/dR dz*/dR dz/dR / 0 d/dR V d/dP ) \dz*/dP dz/dP 0 I) \ d/dP J such that J = I ' c* 0 0 0\ 0 c 0 0 r* r I 0 \p* P 0 // (3.57) where c = dz/dz, r = dz/dR, and p = dz/dP. Using the fact that the inverse of a Jacobian is the Jacobian of the inverse transformation on the Poisson bracket results in the transformation {f y g}=difM1 dg = {/, PAGE 54 46 Here we define the matrices Cx k = d 2 \nS(z*,R!,P',z,R,P) dz*dX u R'=R,P'=P and d 2 \ nS(z*,R',P',~z,R,P) dXLdYi R'=R,P'=P (3.61) (3.62) C Xk Y, = -21mwith X and Y standing for R or P. For the case of coherent states in terms of the atomic basis centered on the nuclei the calculation of the metric involves the overlap of two coherent states with different nuclear geometries. If, furthermore, electron translation factors are included in the orbitals, the overlap also depends on the velocity, and hence the momentum, of the nuclei. We find S(z'*, P\ z, R, P) = det (A* + A'z + *' f A" + z'^AÂ°z) (3.63) where the overlap matrix A(R!,P',R, P) (3.64) depends on the nuclear positions and momenta. We have dropped the tilde on the z parameters, something we do from now on, since we will only deal with a following basis. Note that the overlap matrix of two different bases is not Hermitian, and that it becomes the unit matrix when the two nuclear configurations and momenta coincide. In the general derivation it is assumed that the basis set can depend on both the nuclear positions and momenta. This is the case for the DFB. If the dependence of the basis on the nuclear parameters is only through the nuclear positions, the equations become 0 iC R 0 \ -iC* -iC* R 0 / iC 0 tCj -iC T R C RR -I \ 0 0 / \ ( z\ (dE/dz*\ z* dE/dz R dE/dR / {dE/dP/ (3.65) PAGE 55 47 These are the equations that have been coded into DYNAMO, using the results found in Appendix B for a nonorthogonal basis The interpretation of these equations is very interesting. I will indicate in a general form the most salient features. The first point to make is that the equations are timereversible. This comes from the time-reversibility of the TDVP. Energy is conserved. This can be seen most easily using the Poisson brackets E = {E,E} = 0 (3.66) Another property that can be checked directly is conservation of total momentum of the system. The expectation value of the electronic momentum is p e \ = (z\p e i\z) / (z\z) where the operator in brackets is the total electronic momentum operator. The time derivative of this quantity is ft, = Â£ (2ImC^5 C W 4 CrpPi) (3.67) since the time dependence only comes from the electronic and nuclear parameters. By inspecting Eq. (3.60), multiplying the third row of the metric and adding over all the nuclei gives Â£ {2lmC nj + c naA + c *p' p ) -Y.' p i = Y.jw E < 3 68 > *=i k=\ k=\ Â° n i Because the basis set is centered at the position of the nuclei, the energy is invariant to a global translation of the nuclei and so the right hand side vanishes. Using the last two PAGE 56 48 expressions we get: Pel + ^Tot = 0 (3.69) which shows the conservation of total momentum. For a truncated NFB the preceding result is not valid since in that case it can be shown that but in a limited basis, Ehrenfest's theorem is not generally satisfied, which implies that Thus, in a truncated non-following basis total momentum would not be conserved. This property is only satisfied in the basis sets which exhibit following. The next question I consider is the interpretation of the electronic equations of motion. The meaning is identical when the nonorthogonal atomic basis is used, but the more obscure mathematics coming from the nonorthogonality makes interpretation more difficult. Using the orthogonal basis and drawing from the results found in Appendix A to write down the equation of motion for the z's (3.70) Pel ^ ~i (z\WeuH}\z) (3.71) z )) dE (3.72) PAGE 57 49 where the vertical bar after the overlap matrix indicates that we are taking the derivative with respect to the unprimed coordinates in Eq. (3.64). This gives the following expression for the forces in matrix form iz + 7Â°)(Av A A| + ftV A A| (3.73) which can be further reduced to H + i^i-z /^(^V^AI + ^V^AI = (-Â» /Â»)((. + TV(V. Mi r) J ( J 2 *) (3-74) -(-Â« ^(c)The right hand side has a very simple interpretation: the Fock matrix acts on the occupied states and is then projected on the virtual space and only a nonzero projection on the virtual states gives rise to a change in the z coefficients. The Fock matrix in a basis which contains the ETF's of the form exp(iv Â• r) contains two extra terms in the h part of the Fock matrix derived from the operation of the electronic kinetic energy operator on the ETF's. One of them is associated with the kinetic energy v 2 /2, and the other exactly cancels the term coupling with the nuclear velocity in the metric. Thus, only PAGE 58 50 a term associated with the nuclear acceleration is left If we assume, for a moment, that we are treating only a single atom and that the electron is initially in its ground state and that for some reason the nucleus is being accelerated, then the Fock matrix will not be generating any coupling, but the term with the nuclear acceleration generates a dipole coupling between the basis functions which couples to the acceleration of the nucleus. This generates dynamics in the electrons. What happens is that the electrons are being excited through the nuclear acceleration. These are the couplings I mention in the previous chapter which I have not seen implemented in the literature. If a basis without ETF's is used, then the Fock matrix reverts to the standard timeindependent form and the coupling to the nuclear velocity in the metric does not get cancelled. Analyzing this term in a complete basis and assuming that the nucleus moves with a constant velocity shows that this term in the metric is associated with the translation operator. After a time St the nucleus has moved SR = R St. The electronic states are to be described in terms of a basis centered at the new position. The old basis functions must be expressed in terms of the new ones, so the transformation matrix: should be evaluated to apply on the expansion coefficients. For a small translation the electron translation operator is and its matrix elements in a basis which is centered on the nucleus (so that derivatives r _ SR)\ PAGE 59 51 with respect to the electron and nuclear coordinates can be interchanged) gives 6ij + ifiStV A A\ (3.77) Because we use an unnormalized state, the changes in the z's are computed as projected from the occupied to the unoccupied states through the translation operator. Dividing by St leads to a time rate of change of the z's which is precisely the second term found in Eq 3.74. So the action of this coupling is to take care of the fact that the electron coefficients are being continually defined with respect to a basis which has been translated. This coupling, which is sometimes called "unphysical" [6], is just ensuring the correct description of the electronic state in a translating basis. Tools must be developed to understand the results of a time-dependent calculation. I have developed two such tools which will be described here. The first one is used to analyze state-to-state transfer probabilities for the p+H collision. The results found in the static following basis must be projected on eigenstates of the moving hydrogen atom. The eigenstates, which were described in Chapter 2 when discussing ETF's, are Interpretive Tools A m (fR) = exp (iv-ri(v 2 /2 + t nlm )t)4> n i m (r R^j (3.78) where D indicates the dynamic nature of the orbital. The transformation from the static PAGE 60 52 to the dynamic basis is determined from: i*> = E i^> a * k (3.79) = Â£ l^lja! /.it = Â£ itf > a ? /.*r It is easily seen that the part in the dynamic phase which is multiplied by the time t is cancelled in the projection Yl\ ( f > F)( < f > F\ smce * s independent of the electronic coordinates. What is required, therefore, is the transformation matrix defined by (exp(ivf)(f> n i m \ PAGE 61 53 Then, the fundamental matrix elements from which to construct the boost transformation matrix are 3 /2 W*(")lÂ«tp(-Â«?-Â»WÂ«OT) _lu x Â—3Â— Â—3 exp { PAGE 62 54 For a many-electron molecular process in which the final product is made up of several distinct fragments we want to analyze the output of a DYNAMO run and find the probability of having a given number of electrons in each fragment. In general we will also want state-to-state probabilities, even for the case of a single molecule, not many fragments. For a many-electron system these states are timeindependent ones which approximate the eigenstates of the fragments or molecule. I have developed the following method to obtain such information from the single determinantal output from DYNAMO. As discussed in the previous section, our starting point is a single determinant made up of molecular spin-orbitals (MSO's) which are a linear combination of nonorthogonal atomic spin-orbitals (ASO's). This determinant is some final or even intermediate result of a DYNAMO run. I will call the MO's that are obtained in this way dynamic MSO's. The determinant is We can interpret the final result in terms of time-independent multiconfigurational states (I will use MC to denote multiconfigurational, not Monte Carlo) which are an approximation to the exact eigenstates of the system. If there are several fragments then we will want to interpret the result in terms of MC states corresponding to each moiety. The simplest example is that of a collision process: a fragment hits another, a reaction occurs and then fragments separate. Another might be two systems coming A' (3.83) j=N+l PAGE 63 55 in close contact, some reaction occurring and then separating. In all these cases there are two aspects: 1. There are different possible reaction channels, i.e. a different number of electrons associated with each fragment. 2. We want to understand the chemistry in terms of "local" states, where by "local" I mean molecular states written only in terms of the ASO's of the fragment of interest. The general procedure is to do a MC calculation for each of the fragments and for all the channels of interest within a fragment. This will become clearer in a moment. For a given fragment and channel (where by channel I mean a given number of electrons in the fragment) a standard SCF is performed to determine a set of MSO's which are used to perform a MC calculation to a given level of interest. These I will call reference MSO's for the fragment and channel. In principle, it is not necessary to use a HF state as the reference state. A more general reference state can be used if needed. One possibility is to use natural orbitals. What is needed is the transformation between the reference MSO's and the nonorthogonal basis functions. Calling this transformation U, we have PAGE 64 56 Next we need to expand the state we have in terms of the fragments. To make this clearer assume there are only two fragments. It is straightforward to generalize to more fragments. The fragments contain a well defined subset of basis ASO's. The ASO's in each fragment ideally have no overlap with those of the other fragment. If there is some overlap I assume it is small and can be ignored for the calculation. Each dynamic MSO is a sum of the form W = S x > c >Â« + ^ Xl Cli = ^ + ^ (3 ' 86) jcA UB The single determinant is then written, using the fact that a determinant is multilinear in its columns or rows, as (we also normalize the total wavef unction to 1 here) = [{\ PAGE 65 57 terms of the expansion are only those with either L electrons in A or L 1 electrons in A. Otherwise, there will be 2^ determinants in the preceding expansion, which might be necessary in some cases (such as in the a + He collision in which one determinant represents 2-electron transfer, another represents 2-electron excitation and the other two determinants represent only 1 -electron transfer). The scheme just described sets the stage for the MC calculations. According to the channels of interest in the system, a MC calculation is performed to find the ground and excited states for the fragments. Perhaps at most a CI Singles and Doubles (CISD) will be possible due to computational limitations in the size of the system, and even then it might be truncated to only a certain number of excited states. This choice will depend on the system being studied. The next step is to expand the determinants of Eq. (3.87) in terms of the MSO transformation U for each fragment and appropriate channel. Redefining U as Ufo, where C refers to the fragment and M to the number of electrons in the fragment, and defining the reference MSO's of fragment C with M-electrons as <^. t , the dynamic MSO's are written as where Eq. (3.84) is used to write the dynamic MSO in terms of the reference MSO's. Any determinant of Eq. (3.87 ) can be written in terms of determinants of the reference (3.88) PAGE 66 58 MSO's, for example il,S2, Â— ,tjv These determinants are in terms of basis functions that are the same as those used in the MC calculations. Since the MC calculation is done for the fragments, the states we want to compare with are made up with those MC states in the following manner: Ol .Â•Â•Â•>Â£} where the fragment MC states are (3.91) *X;j = Â£ l^i;ir-^f;iil*Oi,..iL}i Â• 0i> -it} Since I assume that the ASO's in A and B are orthogonal, the overlap between the reference MSO's of different fragments are null. When an overlap is taken between these MC states and the dynamic determinant expanded in terms of the reference MSO's, what is obtained is a sum of determinants of overlap matrices. Because of the property of zero overlap between MSO's of different fragments, the overlap determinants can always be brought into a block form, which is just the product of the determinants of each fragment. In this way we come to the final result. The overlap determinants are either (-l) p (to account for any orbital ordering within the determinants) or 0 due to the orthonormality of the MSO's. PAGE 67 59 With this approach, questions which can be asked include: 1. Which MC state is the most important in the final state. 2. What is the probability of finding the system in a given MC state. Some probability will always be lost to higher MC states that are not taken into account, unless a full CI is performed and all possible channels retained. By using common sense and physical and chemical intuition, the size of this loss should be kept reasonably small. It is also easy to check how much probability is lost to eigenstates not considered by adding the calculated probabilities and subtracting the sum from the total of one. This procedure has been coded in a program called PROJECT. It does not incorporate the state-to-state analysis through a MC projection yet. What it does do is calculate the probability of ending up in different channels. It is used in the work presented in the next chapter to calculate the 1and 2-electron transfer probabilities in a + He. PAGE 68 CHAPTER 4 RESULTS The p+H Collision Introduction Proton-hydrogen collisions have been the subject of abundant experimental studies in the last 20 years. Thus they provide an excellent test for new time-dependent theoretical approaches to electron-nuclear dynamics. Most theoretical work on this system is performed at collision energies above 1 keV in the lab frame. Calculations use either straight line or Coulomb trajectories for the nuclei. Such approaches yield electron transfer and excitation cross sections in agreement with experimental results, even if strict conservation laws of energy and total momentum are violated. However, for lower energies the motion of the protons is sufficiently slow that different trajectories significantly alter the results. It is then necessary to treat the electronnuclear and nuclear-nuclear interactions correctly throughout the collision process. Runge et al [60] use the eikonal approximation to treat the nuclei and solve the timedependent electronic density matrices in the linearized TDHF approximation as described in Chapter 2. They analyze proton-hydrogen collisions in the 10 eV-1 keV collisional energy range using only Is states in the basis. They also investigate the effect of using straight line trajectories, Coulomb trajectories and effective potentials. They show the importance of using the correct trajectory for energies below 1 keV. 60 PAGE 69 61 Fritsch and Lin [53] as well as Kimura and Lane [56] have recently published reviews on the semiclassical close-coupling method, and examined the proton-hydrogen collision in some detail, as well as some other collisions. This method, as well as some other methods they mention, all use prescribed trajectories. Fritsch and Lin use the close coupling scheme with an extended basis which they call the AO+ method [54, 55]. It consists of a basis of Is, 2s, 2p (and in Ref. 55, also n=3 states) atomic orbitals corresponding to the free hydrogen atom as well as atomic orbitals corresponding to the united atom (He). The He orbitals are expected to be important for small impact parameters. Their work explores the 1-75 keV energy range. The numerical integration of the time-dependent Schrbdinger equation is realized by Grim et al [87] at 2 keV using a numerical integration on a grid, and avoiding the use of basis sets. Other work includes that by Liidde and Dreizler [88, 89] who solve the time dependent Schrodinger equation using a large set of Hylleraas type basis functions. Few theoretical studies are done for energies below 1 keV. The most recent by Hunter and Kuriyan [49, 50] for collision energies between .0001 eV and 10 eV, uses the PSS method to separate the nuclear from electronic degrees of freedom. Davis and Thorson published a similar study, but for an energy range going from 0 to 0.2 eV [51] and correcting for some errors in the Hunter and Kuriyan work. The results for the two studies in the overlapping energy range are very similar. I will be comparing with Hunter and Kuriyan's work since the information they report is in the form of tables, while Davis and Thorson only publish graphs with a low precision. In both cases the PAGE 70 62 nuclei are treated quantum mechanically, and only the molecular \sa and 2pa states are retained. Twenty years before this work, Dalgarno and Yadav [66] also used a the PSS approach to treat this problem for energies starting as low as 0.25 eV and going up to energies of 100 keV. A little before that Bates and Dalgarno studied this problem using the Born approximation, for the energy range 0-250 keV [90]. The results of this early work shows that a perturbative scheme like the Born approximation works well for energies above 10 keV, but fails for lower energies by several orders of magnitude. An important development at energies above 1 keV is the inclusion of electron translation factors (ETF's) in the basis. In the limit of a complete set, such factors are not needed, but the question always lingers as to how large a basis must be to account correctly for the couplings. A review of these factors can be found in Chapter 2. When using a molecular orbital basis, different ETF's have led to different results, particularly for the 25 excitation and transfer cross section. This can be seen, e.g. in the 10 basis close-coupling calculations of Crothers and Hughes [91] and Kimura and Thorson [52]. I use the computer code DYNAMO to calculate several properties of the protonhydrogen collision as a first test of the general method. No ETF's are used, which limits the collisional energies that can be investigated. I choose an energy range between 0.02-4000 eV which spans the very low energy regime investigated by Hunter and Kuriyan and Davis and Thorson [49-51], as well as the higher energies of other research groups. The transfer or excitation probabilities to different orbitals are calculated by first projecting the results on a moving hydrogenic basis (i.e. one with the factor e tmv T included), as explained in the previous chapter. PAGE 71 63 The momentum conservation law discussed in Chapter 3 allows DYNAMO to be used to show explicitly the interchange of momentum between the three particles of this system. Results Approximate hydrogenic Is, 2s, and 2p states are expressed in terms of 6 Gaussians each. The Is and 2p states are taken from Stewart [92]. The hydrogen 2s was optimized by me. The coefficients and exponents used can be found in Table 4-1. Table 4-1: Contraction coefficients c and exponents a for the basis used in this work. Is orbital* 2s orbital 2p orbital 8 c c a c a 1.30334 6.51095 -3.75318 1.21392 1.01708 1.214225 xlO 1 xlO" 2 xlO 1 xlO" 2 xlO 1 xlO" 2 4.16492 1.58088 -7.61767 2.67784 4.25860 2.64900 xlO 1 xlO 1 xlO 1 xlO" 2 xlO 1 xlO 2 3.70563 4.07099 3.22382 1.60950 4.18036 6.099425 xlO 1 xlO 1 xlO" 2 xlO 1 xlO 1 xlO" 2 1.68538 1.18506 1.77665 4.71398 1.73897 1.585355 xlO 1 xlO 1 xlO 1 xlO 1 xlO 1 4.23592 4.93615 5.42463 1.68965 3.76794 5.10090 xlO" 2 xlO 2 xlO" 2 xlO 1 9.16360 2.31030 1.01853 9.22099 3.75970 2.577175 xlO" 3 xlO 1 xlO 2 xlO" 3 a: Ref. [92]. The hydrogen atom in its ground state is initially placed at rest at the origin of the laboratory coordinate system and the proton at a distance of 50 a.u. from the origin and given different impact parameters and an initial momentum corresponding to the energy PAGE 72 64 of the collision. The system is allowed to evolve until the nuclei are again separated by 50 a.u. For collision energies from 10 eV to 4 keV, the approximate n=l and n=2 hydrogenic basis is used. A comparison between this larger basis and a smaller basis of a single Is state per center is made at 10 eV and the difference in total cross-section is found to be less than 2%. Thus for energies from 0.02 eV to 10 eV only the Is functions are used. Impact parameters are chosen differently for various energy ranges: from 0.1 to 7.9 a.u in steps of 0.2 a.u.'s for energies from 100 eV to 4 keV; from 0.1 to 9.9 a.u.'s in steps of 0.2 aoi.'s for energies of 10 eV to 40 eV, and 0.0 to 13.9 a.u.'s in steps of 0.1 a.u.'s for energies of 0.02 eV and 0.18 eV. When only the Is basis is used, the initial and final separations were taken to be 30 a.u.'s. The transition probabilities are computed after projecting on the basis with ETF's included. Then transfer and excitation crosssections are computed from: where n,l define the state, and P(E, b) is the probability at the given collision energy E and impact parameter, b. As a first step to determine if DYNAMO is capable of reproducing good results for this system I plot in Fig. 4-1 the probability of electron transfer times the impact parameter as a function of the impact parameter and compare it to an exact integration of the time-dependent Schrodinger equation where the wavefunction is represented on a grid. This last work is done by Griin et al [87] at 2 keV. Also shown are results of work done by Liidde and Dreizler who use a close coupling method without ETF's and a basis (4.1) PAGE 73 65 of 64 Hylleraas-type basis functions [88]. The agreement between our results and those of Griin et al is excellent, while Ludde and Dreizler's results are not that good. Work by Fritsch and Lin using the close-coupling with ETF's and their AO+ method also shows very good agreement with the results by Griin et al [53]. 4.00 3.00 oT 2 00 1.00 0.00 , , , , 1 , 1 1 . Â• 1 1 1 1 /Â• \ \ ' \ * i \ \ / ' \ v Â• / Â• \ /> /< \ \ V A 1 , \V', 1 , 1 , L_*. i ' : Â• -i 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Impact Parameter b (a.u.) Figure 4-1: Weighted transition probabilities for total electron transfer at 2 keV as a function of impact parameter. All data in atomic units. Dots are the results by Griin et al. The full line are the results of calculations by Ludde and Dreizler. The dash dotted line are the results from DYNAMO. (See text for references). Table 4-2 lists the total transfer cross-sections for the p+H collisions for the energies studied and compares them with the experimental results. Fig 4-2 plots the experimental total transfer cross-sections and our calculations. From the previous table and this plot, it is clear that our results reproduce experiments over five orders of magnitude in the kinetic energy of the colliding proton. PAGE 74 66 Table 4-2: Total transfer cross-sections for proton colliding with a hydrogen atom (x 10~ 16 cm 2 ). Collision Energy (eV). Total Transfer Experiment (Energy, eV) Cross-section. 0.02 69.55 U. 10 j / 59.4 .9/7(0. lo)^ in n ^7 37.0 3 _4.7(9.7) a ' a 20 33.22 32.8Â±4.8 (22.2)^ 40 30.25 30.0Â±4.9 (40.4) a d 100 25.60 23.7Â±3.5 (109.6) 8 500 19.44 18.9Â±3.2 (500) b 1000 16.78 16.3Â±2.9 (1000) b 2000 14.07 13.9Â±3.5 (2000) b 3000 12.43 12.1+0.61 (3040) c 4000 11.33 11.1Â±0.55 (3820) c a: Newman et ai, Ref. [93]. b: Gealy and Van Zyl, Ref. [94]. c: McClure, Ref. [95]. d: At energies below 100 ev, collisions are between protons and deuteron atoms. Newman et al. [93] measure total transfer cross-sections for energies between 0.18 eV and 300eV. For energies below 100 eV, they use protons colliding on deuterium atoms. Hunter and Kuriyan [49, 50] use a partial wave expansion for the nuclei moving on a Born-Oppenheimer lsa, 2pa electronic potential energy surfaces at low energies (below 10 eV). Their results indicate that below 1 eV appreciable differences appear between the total transfer cross-sections of a proton on deuterium and that of the proton on hydrogen. As the former process is slightly endothermic ( AE = 0.0037 eV), this result is to be expected at some point. We study collisions between protons and deuterium and PAGE 75 67 D + -hydrogen collisions. Our results, as well as Hunter and Kuriyan's, for energies below 1 eV are shown in Table 4-3. The calculations with the deuterium atom and ion are done using the same basis as the proton on hydrogen, except that the exponents in the basis are changed to reflect the change in the reduced mass of the deuteron-electron system. Energy (eV) Full circles: Ref. 93. Full squares: Ref 94. Full triangles: Ref. 95. Figure 4-2: Total transfer cross sections from 0.02 eV to 4000 eV. Comparison of experiment to theory. DYNAMO computations are the larger open circles joined by solid line. Comparing the results from DYNAMO with those of Hunter and Kuriyan [49, 50] and noting that the experimental value of the cross-section at 0.18 Â± 0.06 eV for the p+D->H+D + is 59.4 IJ^ 6 x 10" 16 cm 2 [93], their result, at 0.2 eV, is just within the error bar and considerably lower than the one reported here. The present calculations show only a slight difference between the proton colliding against a hydrogen atom and against PAGE 76 68 a deuterium atom. Even at 0.02 eV the difference is not significant. Since at 0.18 eV DYNAMO agrees more closely with the experimental value, it would be interesting if experiments could be repeated at these and perhaps even lower energies to verify when and if the isotopic effect becomes important. Due to obvious problems in colliding a proton to a hydrogen atom, it would be easier and more interesting to measure collisions of a proton on deuterium and collisions of D + on hydrogen. Hunter and Kuriyan report significant differences in the transfer cross-sections of these two processes for energies below 0.2 eV. Table 4-3: Total transfer cross-sections for collisions of a proton on hydrogen and deuterium atoms, as well as that of a D + ion on hydrogen, for energies below 1 eV, compared to results by Hunter and Kuriyan (H&K) (Cross-section units xlO _16 cm 2 ). Energy (eV) p+H^H+p p+D-+H+D + D + +H->D+p H&K a 0.02 72.63 60.78 72.17 Present work 0.02 69.55 69.23 69.25 H&K a 0.2 55.88 50.64 51.31 Present work 0.18 57.32 56.89 a: Refs. 49, 50. For transfer and excitation cross-sections to n=2 states at energies of 10-40 eV, the z coefficients to the n=l states are below the accuracy requested from the integrator. Thus transfer and excitation crosssections to these states are reported here only for energies of 100 eV and above. The results are shown in Table 4-4. Fig. 4-3 and Fig. 4-4 compare total 2p and 25 cross-sections with experiment for excitation and transfer processes respectively. PAGE 77 69 Table 4-4: Excitation and transfer cross-sections for 2s, 2p\, 2pz states and total 2p cross-sections (x 10 16 cm 2 ). Enerev Excitation Transfer 2s 2px 2pz 2p 2s 2px 2pz 2p lOOeV 0.01 0.42 0.72 1.14 0.19 0.56 0.68 1.24 500eV 0.98 12.45 2.64 15.09 2.16 11.35 2.57 13.92 lOOOeV 1.59 23.76 3.54 27.30 4.61 11.28 9.38 20.66 2000eV 6.78 30.42 3.71 34.13 10.37 10.65 18.98 29.63 3000eV 10.54 31.77 1.85 33.62 12.99 10.78 32.66 43.44 4000eV 11.51 33.13 1.90 35.02 13.64 10.48 45.22 55.70 Fig. 4-3 and Fig. 4-4 compare total 2p and 2s cross-sections with experiment for excitation and transfer processes respectively. The results from DYNAMO are very close to experimental values for total 2p excitations. They are slightly larger than experiments after 2 keV. These results and other recent theoretical work by others coincide with the more recent measurements rather than with the older Stebbings et al [96] results, which, for excitations, appear to be too low and for transfer seem to be too high. The calculated 2s excitation cross-sections fall right between the values of the two experiments performed at these energies. Other theoretical treatments [54, 55, 89, 52] locate the 2s excitation cross-sections below these experimental values. The lack of ETF's for excitations is not as important as in the case of electron transfer since little momentum is transferred to the target for the important impact parameters. The only problem is that the projectile states are not well reproduced, since they lack PAGE 78 70 important ETF's, an omission which limits the accuracy of the excitation crosssections for higher energies. 4.00 0.00 0.00 1.00 2.00 3.00 Energy (keV) 4.00 5.00 Open diamonds: Ref. 97. Open boxes: Ref. 98. Open triangles: Ref. 96. Solid boxes: Ref. 99. Solid triangles: Ref. 100. Figure 4-3: n=2 excitation cross sections. Comparison of DYNAMO results with experiments. Total 2p excitation calculations are shown by open circles joined by a solid line. 2s excitation calculations are shown by solid circles joined by a dotted line. DYNAMO 2p transfer cross-sections agree with experiments up to about 2keV. Above that they are too large. For 2s transfer the results are well above the existing experimental values. Other theoretical studies, in particular by Kimura and Thorson [52] and by Fritsch and Lin [54, 55] do a better job of reproducing experimental results. Our omission of ETF's in these calculations is probably the cause of the discrepancies at these collisional energies. PAGE 79 71 7.00 0.00 1.00 2.00 3.00 Energy (keV) 4.00 5.00 Figure 4-4: n-2 transfer cross sections. The symbols and lines have the same meaning as in the previous figure. The integral alignment provides a measure of the relative excitation to the different 2p states of the target atom. The available experimental results are from Hippler and collaborators [101, 102]. The measurements they perform start from 1 keV and go up to energies of 25 keV. The integral alignment is computed as: A 2 o = PAGE 80 72 Fig. 4-5 compares my results to experiment. In Refs. 101, 102 the experiments are compared to theoretical work by Liidde and Dreizler [89] and Fritsch and Lin [54, 55]. As the energy drops to 1 keV, all these theoretical results converge to an alignment of 46%, just above the experimental error bars, and none predict the decrease in the alignment below 2 keV that experiments show. The straight line approximation used in these other works appears to cause errors in the alignment at these lower energies which are not present when using a full electronic-nuclear dynamics treatment as in DYNAMO. My results reproduce the decrease in the alignment and predict a sharp fall in the alignment as energy continues to drop. This warrants further experimental work. xlO 1 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Energy (keV) Solid triangles: Ref. 101. Solid boxes: Ref 102. Figure 4-5: Computed integral alignment and experimental results, as a function of collision energy. PAGE 81 73 Finally, Fig. 4-6 shows the x components of the nuclear and electronic momenta which are initially zero for a collision at 40 eV and an impact parameter of 3.0 a.u. The basis set used for this calculation is a Is orbital on each nucleus. The x component of the total momentum remains zero throughout the calculation. Momentum (a.u.) 4.0 3.0 2.0 1.0 0.0 Â•1J0 -2.0 Â•3.0 -4.0 350. 400. 450. 500. 550. 600. 650. Tim* (a.u.) Figure 4-6: Expectation value of x component of electron momentum and nuclear momentum as a function of time. Middle curve is the electron expectation value, the upper curve is the momentum corresponding to the incoming proton and the lower curve is that of the proton originally at rest at the origin. PAGE 82 74 The a+He Collision Introduction After finding the excellent agreement with experiment that DYNAMO can achieve for a one-electron system, it is important to determine how it can handle a system with more than one electron. One possibility is to test it on the p+He collision. However this system is essentially a one-electron transfer process and it seems more important to test our method on a system for which different channels are possible. The simplest system for which there is enough experimental work is the a+He collision. Experiments to determine the oneand two-electron transfer on the a+He collisions have been performed by several groups [103-108]. The most comprehensive work has been done by Afrosimov et al in the middle 70's [103, 104]. They used a coincidence measurement technique to draw information about different excited states in the products. They used 3 He in their experiments. The results by this group and results by Bayfield and Khayrallah [105] and by Berkner et al [106] all agree within the error bars of about 20% for each of these groups. Except for the results by Afrosimov et al and Berkner et al, all of the experiments are done at collision energies above 10 keV. Berkner et al provide results at energies above 7 keV, and Afrosimov et al present results at energies above 2.5 keV. Experiments at lower energy have been performed by Hertel and Koski [108] for energies above 0.5 keV and by Latypov et al [107] for energies above 0.1 keV. These latter results have such large uncertainties that is is fruidess to compare calculations with them. PAGE 83 75 On the theoretical side, work on this system has been done using the PSS approach [109], different close-coupling methods [110, 111] and using the TDHF [112, 113]. An investigation of variational improvement of the TDHF is carried out by Gazdy and Micha [114-1 17] for the a+He collision. However, this work is formal, carried out for the oneelectron transfer at energies between 30-100 keV and using only a minimal basis set of Is orbitals. The PSS work covered an energy range of 10-100 keV [109] obtaining close agreement with experiment. The basis used is an expansion in 7 single determinants each determinant formed with one-electron diatomic orbitals (i.e. orbitals found for the diatom containing only a single electron) and not using ETF's. The close-coupling work by Kimura [110] explores an energy range from 2-400 keV using 15 determinants from STO basis functions. He uses an MO approach with MO-ETF's. His results also agree well with experiment, using straight line or Coulomb trajectories depending on the energy. Gramlich ex al [111] use a Gaussian orbital expansion with ETF's to look at the energy range of 8-^400 keV. They only use straight line trajectories. This affects the quality of their results at lower energies. They use a total of 29 determinants for their calculations. At higher energies, where curved trajectories are not as important, their results are closer to experiments than those of Kimura. The TDHF work is done in the energy range of 30-150 keV by Devi and Garcia [112] and 20-160 keV by Stich et al [113]. The first used a prescribed Coulomb trajectory and several simplifying assumptions on the TDHF equations. In the second case straight lines were prescribed for the nuclear motion. The electronic basis functions were expanded PAGE 84 76 on a large basis of Hylleraas type functions ( the actual number varied according to the distance between nuclei, but was on the order of 150). The results of these two calculations are good. However for energies around 30 keV and under the one-electron transfer is overestimated in both cases. This has been explained as a lack of electronic correlation, i.e. the use of a single determinant. It has also been stated by Stich et al that to lift the single determinantal description is a very difficult problem. Here I use the single determinant approximation for the electrons at a much lower energy range than the TDHF results cited above. The purpose was to determine the importance of different basis sets as well as to test a single determinantal description at lower energies. I look at the energy range of 4-10 keV and find the total oneand two-electron cross-sections. Results Four different basis sets are used. The first one (Basis I) is identical to the one used for the p+H collision, except that the exponential coefficients are multiplied by a factor of 3.9967 to account for the charge of +2 for the He nucleus and also for the effect of the mass of the He atom on the Bohr radius. The second basis (Basis II) is made up of two contracted s functions, the first made up of 3 Gaussians and the second of 1 Gaussian and a p set made from a single contracted Gaussian [118]. This basis is a basis optimized for hydrogen. The third basis (Basis III) is similar to Basis II, but optimized for helium. The fourth basis (Basis IV) has 4 s basis functions from the contraction of 7 s-type Gaussians and 2 p basis functions made from the contraction of 3 p-type Gaussians [119]. This last basis set is also optimized for the He atom. PAGE 85 77 The calculations are started with the nuclei separated by 50 a.u. with the He atom at the origin of coordinates and in the ground state as determined by the optimization of the He atom in each basis. The ion is given an impact parameter ranging from 0.0 to 2.9 a.u. in steps of 0.1 a.u. for collision energies of 2.5, 4, 6, 8 and 10 keV, although all these energies are not used for all basis sets. At the end of a run the program PROJECT, described in Chapter 3, is used to determine the probability for oneand two-electron transfer. The transfer cross section is calculated using for A being one of the two transfer processes. The range of 0.0-2.9 a.u. is used because at 2.9 a.u. the one-electron transfer probability is of the order of 10" 5 , while for the 2-electron transfer it is 10" 10 . The results of the calculations are tabulated in Table 4-5 for one-electron transfer and Table 4-6 for two-electron transfer. Experimental results of Afrosimov [103, 104] are also shown. Table 4-5: One-electron transfer cross sections using three different basis sets (see text for description of the basis sets) are compared with experiments by Afrosimov. (Cross-section units x 10" 17 cm 2 .) Energy Basis I Basis II Basis UJ Basis IV Exp. (error) (4.4) 2.5 keV 4.54 3.49 4.0 (0.6) 4.0 (0.6) 3.9 (0.6) 3.9 (0.6) 3.9 (0.6) 4 kev 4.97 6.32 6.77 6keV 7.56 7.58 7.76 8.36 8keV 14.99 lOkeV 10.60 10.34 PAGE 86 78 Table 4-6: Two-electron transfer cross sections using three different basis sets (see text for description of the basis sets) are compared with experiments by Afrosimov. (Cross-section units x 10" 17 cm 2 .) Energy Basis I Basis II Basis III Basis IV Exp. (error) 2.5 keV 18.4 15.3 31.0 (4.7) 4keV 15.0 16.1 14.8 25.0 (3.7) 6keV 15.2 17.2 16.9 15.6 23.0 (3.5) 8keV 15.0 21.0 (3.1) lOkeV 16.0 17.9 20.0 (3.0) The results for the different basis sets used are very similar, with slightly better results for the two-electron transfer using the basis set Basis IL The nearly constant experimental crosssection for the two-electron transfer is qualitatively reproduced by DYNAMO. The one-electron transfer is not reproduced very well. At the low end (2.5 keV) the agreement is very close. Otherwise, the cross-sections are off by a factor of 2 or 3. This agrees with overestimation of one-electron transfer probabilities found using the TDHF by Devi and Garcia [112] and Stich et al [113], although they never got as low as this in energy. The two-electron transfer process is essentially a resonant transfer phenomenon [103]. Electrons tend to transfer from ground state to ground state of the He atom. On the other hand, the one-electron process tends to be a transfer to an excited n=2 or higher state of He + [104]. By using a single determinant, we force it to describe both these processes in a single configuration. This could be a possible explanation for the results not being completely satisfactory at these energies. PAGE 87 79 As the energy goes down, the single determinant seems to provide a better picture for the one-electron transfer. This could be indicating another possible reason for a poor description of one-electron transfer: the lack of ETF's in the basis is skewing couplings and degrading the results. The method yields acceptable results. Being a factor of 2 or 3 off from experiment is not bad! The results have the right orders of magnitude and certain trends are reproduced. Having tested four different basis sets shows that the results have approximately converged within the approximations of the method. The differences with experiment do not come from too small a basis set. These results as well as the ones for the p+H collision indicate that ETF's should be included if DYNAMO is to be used to predict results of collisions in the keV range. Another conclusion is that the single determinant may not be able to predict quantitatively accurate results when singlet and triplet states both play significant roles in a chemical process, such as for the a+He collision. This requires an extension, which in our case, contrary to the problems faced by Stich et al, is quite simple to implement. PAGE 88 CHAPTER 5 CONCLUSIONS The single determinantal Hartree-Fock approximation in electronic structure calculations is not always very accurate. Yet it is usually an excellent starting place for methods which include electronic correlation and it is quite capable of generating qualitatively and even quantitative results which are in good agreement with experiment [81]. For large systems the Hartree-Fock approach is the starting point for even further simplifications leading to semiempirical methods. In the same way, a single determinantal approach to time-dependent methods seems to be a good starting point. Although this assertion needs to be investigated further, it is clear from the results presented here and those we have already published [4] that the single determinantal approach does yield quite good results. The alternatives at present are to either do dynamics on potential energy surfaces using quantum mechanical methods, or using a full CI approach to time dependent methods. The first approach can produce excellent results for three atom systems when only two or three potential energy surfaces are important in the dynamics. Not many chemical systems fall in this category, however. The full CI approach is embodied in the closecoupling methods. These methods, as implemented today, ignore nuclear dynamics and so are limited to collisions in the keV ranges. By doing this they violate conservation of energy, momentum and angular momentum. They also face the same problem that CI does in electronic structure calculations: computer time and memory needed grow 80 PAGE 89 81 factorially with the number of electrons and basis functions. For this reason closecoupling calculations have been limited to systems of no more than 2 electrons. An intermediate position, i.e. one where a systematic multiconfigurational approach is used without going to full CI has not been formulated in the literature. The field of time-dependent dynamics with a single determinant is not new in physics, although its application to chemical systems is fairly recent. The advantage of the END equations presented here as compared to similar methods lies in the transparency of its approximations and the ease with which they can be improved upon. This method has already been formulated in such a way as to use a multiconfigurational electronic description [83], although a computer code has not been implemented to work with such a state yet. The conservation properties of the equations are also important. No other single determinantal method has been developed to the extent of the one presented here. For the two systems studied here I have shown that the lack of ETF's can hinder accurate results for energies above a few keV. If collisions or reactions are to be studied at lower energies than this then the method yields good results. We also cover a range of energies that methods based on PES or the close-coupling approach can handle, i.e. between 10 eV and 1 keV. The single determinant approach must be tested further to learn more about it's strengths and weaknesses. A first upgrade is to write a new integral code which is more adequate for the kind of operations that a time-dependent method requires. For example it should include electron translation factors, and it should be vectorizable to make it efficient on modern computers. PAGE 90 82 Another extension is in two very different directions; on one hand to include more determinantal states for chemical processes where electron correlation is important, and on the other hand to develop a semiempirical approximation starting from a single determinant to permit the study of large systems. Another line of investigation is to include a quantum description of the nuclei in such a way as to permit wavepacket splitting for the nuclei. The approach to take is one in which not all nuclei have a quantum description since heavier nuclei are less likely to experience significant quantum effects from tunneling or zero point energy fluctuations. PAGE 91 APPENDIX A DETAILS OF DERIVATION IN ORTHONORMAL BASIS The derivations in this appendix and the next can be found in the technical report "Coherent State Approach to Time Evolution with a Hartree-Fock State" by Deumens, Diz and Ohm [120]. In Chapter 3 the overlap kernel for the electrons is found to be: S{z'\ z) = (z'\z) = det(7" + z' f z) (A.l) from which we can derive expressions for its derivatives d dzi -ln(zV)|^ =2 =(.|z)1 ^^M v Minor(M),^| 2 ^ (A.2) ((/Â• + A)-V jk In the first step the determinant is expanded in the j'-th row with M denoting the matrix M = r + z*z. (A.3) The next step uses the expression for the inverse of a matrix in terms of its minors Minor(M)v >J % det(M) V ' 83 PAGE 92 84 and the fact that none of the elements z/y do occur in the minor ij because it does not contain column / of M. Similarly we find for the derivative with respect to the complex conjugate parameter Hi (A.5) -(.(r+A)-") This can be written in matrix form as 9 Tln(z'| Z )| z =z = (/' + A)" 1 ^ dz T ^ln(z'|z)| 2 . = * = *(/' + A)"*. (A.6) dz Eq. (3.63) shows S{z'\B!,P\z,R,P) = (z'\B!,P'\z,R,P) (A.7) = det(A* + A'z + z'^A" + z'^AÂ°z) from which the derivative with respect to the nuclear position can be evaluated. It is the purely imaginary quantity V* \n(z',R',P'\z,R,P) I R " x ' ' 1 1 \z'=z,R'=R,P'=P (A.8) = Tr(/+ ^^(v^A-l + V R A'\z + z^ & A"\ + z^ n AÂ°\z), where we used the following property dlndet (M) . ddet (M) dM tJ dx =d6t(M) ?"lj^T" Â«i = TrM" 5t ' PAGE 93 85 We also introduced the abbreviation of a vertical bar to indicate that the derivative of the overlap matrix is to be taken with respect to the R dependence of one side only. The derivatives with respect to nuclear momentum are given by an identical expression with the gradients V^. Therefore we do not show them in the following. For the second derivatives of the overlap, we need the derivatives of the inverse of M. From the relation XX~ l = I (A.10) for any matrix X depending on a variable x, we find ^= -X-^X-\ (AM) Ox ox Therefore the derivatives of the inverse of M are and The second derivatives of the overlap are then given by (A.14) -((r + .t.)-.t) (( r+A )-.t) PAGE 94 86 and ^ln(z|z) dz as well as <^ln(z|z) _ ^ dz (A.15) The derivatives with respect to nuclear position follow in the same way _d dz where we used the relation -i / (A.16) -V n \n(z',#\z,R) l, =z , R , =R =(0 DVj A|( / ;)(/+ ,t 2 )1 -,(r + ,t 2 )I (/ . 2 t )Vj? A|( / 2 , )(/+ 2 t ,)" 1 (A.17) z^' + A) = (7Â°W) z, (A.18) which follows immediately from (l 0 + zz^z = z(l* + z*zy (A.19) PAGE 95 87 The derivatives with respect to the nuclear coordinates can be expressed as = Tr[(/+ ^)"V ,t )V ^A|( 7 2 ') which is a Hermitian matrix. The kernel for the one-particle density matrix or 1-matrix is defined as Y ix {z\z) = {z\z)-\z\b\b ] \z) The 1-matrix has the familiar block form r For the occupied block we find the following expression 1-1 / P T'\ r; t ={z\z)~ l (vae\ N / K . /=1 \ m=N+l N / K \ [ b k + E h Â°^mk)\vac) k=l V m=N+l / = (z\z)1 (-) ,+3 det((S lk+ Â£ *ml*mk)â€¢i,ht ) ) \ m=N+l ) = det (M) -1 Minor(M) tJ =((^n. (A.21) (A.22) (A.23) PAGE 96 88 For the unoccupied block we find r jt = {z\z)l (vac\\(z tl -(b? + Â£ A V m=N+l bÂ°Jz ml )b: N K II K + E h Â°m^l 1=2 \ m=N+l AW A\ II K + E ^mtlbac). k=2 \ m=A+l / After moving the annihilators further through, we obtain N N /=1 *=1 det ( (6 pq + 2J z* mp z mq ) p ^i^ k V m=l > A = det (M)" 1 ^ z^MinortM)^ Jt=i so that (A.24) (A.25) (A.26) The off-diagonal block is obtained with the help of the first derivatives of the logarithm PAGE 97 89 of overlap kernel from T' }l =(z\z)-\z\b?b)\z) =^J ln <*l*> (A.27) Therefore the 1-matrix is given by r(z\z)=( r z ^(r + z< z y\r *t) (A.28) r(z'*,/2\P\z,i2,P) = (^(a' + AV' + AW^aVJ'V * f ) Note that this is the projector onto the (non-orthonormal) occupied orbitals [121]. The two-particle density matrix or 2-matrix for a single determinantal wave function is a simple function of the l-matrix[122] {zlz^iz^bihlz) = T ki T h T kj T u . (A.29) With the total molecular Hamiltonian written in second quantization in the orthonormal molecular basis as (A.30) K K PAGE 98 90 we find, with the above kernels (A.23-A.27) and (A.29) for the densities, the following expression for the energy of the coherent state E(z*,z) =Â£(Â°) + Tr^ 1 )) + Â±Tt(vtW) =Â£(Â°) + Tv(hT) + iTr(Tr(V at;o6 r) a r). jTr(Tr(K aMa r) a r) t (A.31) =E(Â°) + Tv(hT) + l -Tr(Tv(V ab . >ab T) T). 2 / a >b' Here h andV are the one-electron and and symmetrized two-electron integrals in the spinorbital molecular basis Viy,kl =(ij\\kl) =(ij\kl) (ij\lk) (A.32) =(ik\jl) (il\jk), with (til*/) = J ^*(r 1 )0*(r 2 )V'Jt(n)V'/(''2) J 3 , 3 ^ : d nd r 2 . (A.33) H -r 2 When straight, i.e. not antisymmetrized, two-electron integrals are used the factor of one quarter needs to be replaced by one half. And eÂ«Â» = rA + r 4*Â£ii_ (a.34> 2M fc & Ri\ is the classical nuclear kinetic and nuclear repulsion energy. When the nuclei are treated quantum mechanically this term has to be replaced with the expectation value in the PAGE 99 91 appropriate nuclear coherent state and the oneand two-electron integrals have to be replaced similarly. The one-electron integrals form a Hermitian matrix. The two-electron integrals have the following symmetries: (ij\kl) =(ji\lk) (A.35) =(ki\ijy = (ik\jiy. When real atomic orbitals are used, there are additional symmetries so that we have the following: (tj|*Q =(ji\lk) =(kl\ij) = (lk\ji) (A.36) =(kj\il) = (jk\li) =(il\kj) = (li\jk). In order to obtain an explicit expression for the energy in terms of the coherent state parameters z we write =EW + kijTji KWO (iWWkiTij (A.37) =EW + EW(z*,z) + EM{z*,z). PAGE 100 92 The one-electron energy is then & l \z\ z) =Tv(hT) = Tr (h( ) (/* + ~\ I' z< )j =Tr(AT) + 2Re(Tr(/iT' t )) + Tr{hÂ°TÂ°) =TtU*(r + Â« t Â«)" 1 ) + 2Re^Tr^'z(/' + z^)"* (A.38) For the two-electron energy we find *V,*) =^Tr(Tr(v; 6;a6 r) a r) fc = ^WÂ«rÂ« =Wiv(v (lM i(^(/' + *t,)" 1 (iÂ« ,t)^ (A . 39) (^)(r + .t,)> .t,) ; Worked out in blocks this becomes Â£ ( V, z) JE Viwnn + E VwVN + \ E KwHrirj + E + Re (E ( A 4 Â°) + 2Re(Â£ V^li,.rj.) + 2Re(Â£ V^r^), where we have used the symmetries (A.35) of the two-electron integrals to reduce the PAGE 101 93 16 terms to 7. We conclude E(z*,z) + Tr(h(^ r z ^(r + z*z) \p z*)) + W < &(v -g^Vr + ***)" 1 (i* z*)) a (A.41) ( 7 ;)( r+A ) _1(7, Using the derivatives (A. 12 and A. 13) of the inverse of M, we can compute the derivatives with respect to i of the one-electron energy (A.38) ^r=l4('(';)(-*-)-Â»Â•Â•>). K + ' ( ( o i-^^JCr + .t,)" 1 ) (A.42) + Using the same relation as in Eq. (A. 17), we find dE^(z\z) ((,.,Â«.)-(-* n ^)(r + ^-') t . (A.43, Applying the same calculation to the double trace and using the symmetries (A.35 and A.36) of the antisymmetrized two-electron integrals, we obtain the derivatives with PAGE 102 94 respect to z of the two-electron term (A.39). With the one-electron term this gives (A + Tr(v^(^)(/ , + * t *)" 1 (/ ) (A.44) C)e +A )1We recognize the Fock operator F = /t + Tr(V aM6 r) a (A.45) = ft + Tr([(oa|66) (afc|6a)]r) in this expression. The derivatives with respect to the nuclear coordinates R and P are obtained by taking the derivatives of the oneand two-electron integrals h and V and of the overlap matrix A. The derivatives with respect to z are the same whether we consider orbitals that depend on nuclear positions or not, and for that reason we have not shown the dependence of E on the nuclear coordinates and momenta. The expressions for the derivatives are Eq. (A.37-A.41) with the integrals replaced by the corresponding derivative-integrals, plus the reorthonormalization terms involving the derivatives of the overlap matrix. All derivatives with respect to nuclear coordinates are understood to be derivatives with PAGE 103 (A.46) 95 respect to R', F which , afterwards are set equal to R and P, respectively, i.e. ! Z k Z t e 2 (R k -Mi) V* E(z*,R,P,z,R,P) = --> v , ' + Tr(v^ftr)-Tr(v A ArAr) + 5 Tr ( Tr ( v *. v '^ r )/) J -Tr^VjArV,,,,^^. When the nuclei are treated classically, both the integrals and their derivatives with respect to R are given by available programs. The derivatives with respect to P are similarly Â— V p E(z\R,P,z,R,P) =^+ Tv{y p hT)-Tv(y p AThT) + iTr^Tr(v A V a6;afc r) a r^ (A.47) -Tr( Tr (v A Arv; 6;a6 r) a r) 6 . The terms after the first are only present when travelling molecular orbitals are used. Currently no programs are available to give such integrals or their derivatives. For more advanced nuclear descriptions, the integral evaluation requires a more detailed treatment and the derivatives get correspondingly more involved. PAGE 104 APPENDIX B DETAILS OF DERIVATION IN ATOMIC BASIS From Eq. (B.l) for the coefficients v of the dynamic orbitals orthogonal to the occupied ones, we define the following matrices AÂ»(*) = (JÂ« *t)^* ^VQ=A* + ^At + A'Â« + Â«tAÂ°z (B.l) A being the atomic basis overlap matrix and \Â°(v) = (v P)(Â£ ^j^j=AÂ° + ,A' + AV + V AV (B.2) which naturally occur in many expressions. From the expression of the parameters v as a function of z the following derivatives are calculated Cz mn V / nj + ((a'* + AÂ°z) (A" + A'*)" 1 A') + A'z)1 ) (B.3) = -(AÂ° + ^) im ((A' + A',)1 )^ ^=((A. + ^)-) j> (At(A. + ^)I (A' + ^)) my -((a' + ^A*)" 1 ) A^ (B.4) = -((a+ ^)-') (a'+aV)^. 96 PAGE 105 97 As in the previous appendix, the derivatives of the logarithm of the overlap 5 = det{(7* f )A('*)} = det(A* + A'z + 2 f A' f + z^AÂ°z) det(A*(z)) have to be computed. Repeating the calculations in (A.2-A.8), we obtain ~-ln(zV>U'=Â« = E ( A ' + zlA Â°) tk ( A * + A ' 2 + ztA ' f + z ^Â° z ) -i -(A-'(A' + ,tA-)) (B.5) (B.6) ^In^'l^l^, = T ( A'* + AÂ°z) (A* + A'z + zW + ztA 0 z) = ((A'. + AÂ°,)A-) ti and Valn^i^P'I^P) z'=z,R'=R,P'=P = TV[(A' + A'z + 2 f A" + 2 f A 0 z) -1 (B.7) (B.8) K A 'l + *A A 'l* + **Vj A "l + * f V A A Â°l*)] M and an identical expression for the derivative with respect to nuclear momentum. PAGE 106 98 The derivatives of the inverse of overlap matrix of the occupied dynamic orbitals are similar to (A. 12 and A. 13) d dz kj and A* + A'z + A' f + z^AÂ°z\ J / mn --(A^r^A' + jftA )) (B.9) ^-((A' + A'^ + ^A't + ztA^) = -fA*(^)1 ) (B.10) V / mi ((At + A-Â»)AW -') The second derivatives are then (A'(.)->(A' + .tAÂ«)). and = (A 0 (A't + A^A'^)" 1 ^' + jJA-JMA-M1 ) (B.ll) = Â£ j^A'W^A' + .tA-^i + (A^)" 1 )^ , =1 ^/m (B 12 ) PAGE 107 99 Using the virtual dynamic orbitals v given by (3.46) and the relation / P occ = P virt expressed in terms of the atomic orbitals, then J Â— P occ is (A 9 A'\ ~\A* A 0 J' ( (A* + A'z)A*{z)1 (A' + ^ A 't) (A* + A'*)A # (*) _1 (A' + z^AÂ°) \ ' V (A't + AÂ°z)A'(z)1 (A* + z f A't) (A* + AÂ°z)A*(z)~ 1 (A' + A 0 ) J (B.13) and the virtual projector is (fvtÂ£o)AÂ»-VA+ A't + / (AV + A')AÂ°(u) _1 (vA* + A' f ) (AV + A')A 0 (u) _1 (uA' + A 0 ) = V (A^v* + A 0 )A 0 (v) _1 (Â«A* + A't) (A'V + A^A^v)" 1 ^' + A 0 ) (B.14) Then, from the equality of these two expressions we can simplify Eq. (B.12) to and g=((.V + i ')AÂ»-'K + A.)) n g8--((^ + A-.)A^) ta (B.15) ((A^+A'ijA'W" 1 ) . (B.16) PAGE 108 100 The mixed derivatives with the nuclear coordinates are (A" + tfz)\\z)-\r z^V^Al^y^z)1 The double derivative with respect to the nuclear coordinates is given by V^V^ln^i^,/?)^^ = Tr A*(z)-\r **)V A V 4 A|I (B.17) (B.18) The 1-matrix (Eqs. (A.23)-(A.28)) becomes in the atomic basis r at =A^ / 2 *^(A , + A'z + 2 t A' t + z t AÂ°z) \l* z f )A =ArA. (B.19) Notice the overlap matrices on both sides as is typical for projection operators in a nonorthogonal basis. T is the operator form of the 1-matrix, whereas T at is the matrix element form It is typical in nonorthogonal bases that every linear operator has two matrices representing it. These are the covariant and contravariant forms. PAGE 109 101 Then the energy (A.41) of the coherent state becomes (B.20) E{z\ z) =f?(Â°) + Tv(hT) + ^Tr(Tr(K 6;ai r) a r) 6 + iTr(Tr(v a6;a6 ( / ;)A-(z)1 (/^ ( 7 ;)a-( 2 )-(/. .t^ which is only slightly more complex than in the orthonormal basis! For the derivatives of the energy, only the derivative of the one-electron part is shown in detail, the two-electron part is similar. Using (B.9 and B. 10), the derivatives of (B.20) give dz ii dEi y >*) ( A t + AÂ«z) (A+ A'z + zt A 't + .tAO,)1 (A' + zU't ti + 2 ihÂ°) + {h't hÂ°)^j (B.21) (^(a' + A's + z^ + ^z) ^ . Using the Eqs. (B.13) and (B.14) with the rightmost A replaced by the one-electron PAGE 110 102 integrals h hÂ° (A't + AÂ°z) A'(z)1 (h' + ( P z ) A'(z)1 ^ = (((A'V + A 0 ) A'ivy 1 (vh' + A't) (AV + A 0 ) A'iv)1 (vti + >> 0 )) ( ^ ) A*(^) -1 ) = ^(A'V + A 0 )AÂ°( l ;)1 (t; P^'^A'iz)1 ^ . With a similar derivation for the two-electron part we find (B.22) ^f)=((AV + A-)AÂ»-'(Â» /Â•) c>>-), with v given by Eq. (3.46). 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Today all his friends call him Augie, helping to distinguish him from his son, also named Agustfn. At the age of one, Augie went to live in Tucuman, Argentina. He later returned briefly to Chicago. He started kindergarten and elementary school at Burning Tree Elementary in Maryland. His family then moved to Geneve, Switzerland, where he finished elementary and started middle school at Ecolint, or International School of Geneva. His family then moved to Mexico City for two and a half years, where he finished middle school and started high school at the American School. Then they returned to Buenos Aires, Argentina, where Augie finished high school at St. Andrews Scots School. A few weeks before he began his first class at the Universidad de Buenos Aires, he met the love of his life, Marcela, at a disguise party where one was expected to go as a "punk" (not knowing this he dressed up as a Mexican, and was saved from humiliation by a friend just before leaving for the party). He went on to obtain his Licenciatura en Ciencias Ffsicas at the Universidad de Buenos Aires graduating in 1985. He married Marcela a week after his Licenciatura defense. Two years later he came to the University of Florida with Marcela and their baby son Agustfn. Now they also have a daughter, Ximena. Augie's fascination with physics started early and was well stoked by his father's interest in mathematics and the sciences. He initially thought of working in cosmology. Ill PAGE 120 112 Finally he did an about face and decided to work on the very small, but not the tiny. It is of small consequence, but interesting to note, that at the age of 7 he asked his mother what jobs there were which did NOT use mathematics, since he simply hated the subject. That was to change only after he was introduced to algebra. PAGE 121 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. M ]nm^haijman N. Yngve Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosopl Charles F. Hooper Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. HendrilfJ. Monkhorst Professor of Physics and Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Samuel B. Trickey Professor of Physics and Chemistry PAGE 122 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. William Weltner, Jr Professor of Chemistry This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1992 Dean, Graduate School xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E9JI2FD9S_5YIHV2 INGEST_TIME 2015-04-22T21:13:41Z PACKAGE AA00029965_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |