Phương pháp mới giải phương trình Schrödinger - Luận án tiến sĩ

Novel Approaches to Solving the Electronic Schrödinger Equation for Molecules by Anthony Dean Dutoi B. (Saint Louis University) 1999 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Martin P. Head-Gordon, Chair Professor David Chandler Professor Michael F. Crommie Spring 2006 UMI Number: 3228314 Copyright 2006 by Dutoi, Anthony Dean All rights reserved.

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Box 1346 Ann Arbor, MI 48106-1346 Novel Approaches to Solving the Electronic Schrodinger Equation for Molecules Copyright 2006 by Anthony Dean Dutoi Abstract Novel Approaches to Solving the Electronic Schrédinger Equation for Molecules by Anthony Dean Dutoi Doctor of Philosophy in Chemistry University of California, Berkeley Professor Martin P. Head-Gordon, Chair In the study of the electronic structure of matter, the Hamiltonian can generally be divided into two parts, a mean-field portion and a fluctuation potential. The mean- field approximation involves solving for the states of single particles in the averaged presence of others, self-consistently defining an effective one-particle Hamiltonian. A correlated calculation allows the state of the system to be a function of the coordinates of all the particles in an inseparable manner, but an exact such eigenstate of the full Hamiltonian can generally only be approximated.

Electron correlation is usually defined as the deviation of a state from a mean-field solution. This can loosely be divided into either static and dynamic or short- and long-range components. This work is a study of the correlation problem from four distinct angles. In the first part of this work, we attempt to make a well-defined computational distinction between short and long range by dividing the Coulomb operator itself into two pieces and deriving the resultant non-trivial molecular integrals.

We then proceed to look into present Kohn-Sham density-functional approaches, which attempt to cap- ture correlation effects in a one-particle model but suffer from electron self-interaction. Terms which can correct for the most severe errors are borrowed from the mean-field Hamiltonian, using our divided Coulomb operator. Although Kohn-Sham methods have shown an ability to make useful estimates of the dynamic correlation energy, the physics of static correlation are out of reach of any single-electron theory. Static correlation itself merits study, as it is an important part of the descriptions of species like low-spin multiradicals.

We propose a specific, model-independent definition of general multiradical character, and we find that it identifies the state-space coordi- nates which best describe strong correlations in qualitatively correct wavefunctions. Finally, molecular Hamiltonians define a class of problems whose mapping to exact algorithms is ideal in terms of quantum computing resources, and we conclude by developing the details of such a quantum algorithm. To my parents, who taught me the most important things I could ever learn il Acknowledgments I would like to begin by thanking my advisor, Martin Head-Gordon, for his patient guidance while I defined my interests as somewhere between esoteric and practical. I am grateful to Alex Sodt for his expert programming advice and administration of my favorite computer cluster.

I had many interesting discussions about latest ideas with my office-mate Greg Beran, and he is also thanked for system administration. Joe Subotnik was an invaluable resource on all sorts of mathematical topics. I remember productive discussions with Andreas Dreuw concerning TD-DFT charge transfer and how to fix it. I thank Alán Aspuru-Guzik for efforts equal to my own on the quantum computing project; he and Peter Love were excellent people to work collaboratively with.

Garnet Chan inspired me a lot in the early days, and John Herbert was a great source of knowledge on many topics. I am grateful to Geoff Galitz and Leslie Silvers for their battles won against UNIX and bureaucracy, respectively. Finally, I thank everyone who was in the Head-Gordon group during my tenure in 45 Gilman. The group environment was always interesting and friendly.

The whole Pitzer Center has a great academic atmosphere, and I made many friends there. I thank all the inspiring teachers I have had, particularly, Mrs. Durham and Prof. I am grateful to past research advisors, Chris Marshall, Peter Botschwina, and especially Ronald See, my undergraduate mentor, who inspired me to earn my doctorate.

I would also like to thank the people who provided the majority of my escapes from science here. I could not have done without Josh’s enthusiasm, Alex’s wit, Matt’s hospitality, Kim’s stories, Telly’s grin, Jeff’s sense of humor or Harsha organizing us. These people can tell you what the words olfactory, pot and starfish have in common. I don’t think I could enjoy living with three other people more than I did with Andrew, Katie and Kateri.

I’m grateful for what all of my many friends brought to my life here. I especially want to thank Lianne for making me smile more. Finally, I have a close family to thank for who Jam. My parents, Dean and Carol, made a large effort to nurture my interests, and they always reminded me, along with my siblings, Karen and Brian, that life is happiest when centered around people.

iii Contents List of Figures V List of Tables 1x 1 Introduction 1 1.1 Notation and Basic Concepts .1 Three Formal Approaches to the Electronic Schrédinger Equation 2 1.2 The Hartree-Fock Model .3 The Kohn-Sham Model.4 General Self-consistent Field Methods.5 Static and Dynamic Correlation .2 Outline of Chapters. 00000 0 và va 14 2 A Two-parameter Coulomb Attenuator: Introducing an Explicit Cut-off Radius into Two-electron Repulsion Integrals 17 2. c ee gà và kg va 17 2.1 Repulsion Integrals in General.2 Specific Attenuated Fundamental Integrals .3 Fundamental Integral Evaluation. 30 3 Self-interaction Error in Local Density Functionals: Ground-state and Time-dependent 31 3.2 Ground-state DFT: Alkali-halide Dissociations .2 Results and Discussion.3 Time-dependent DFT: Inclusion of Long-range Exchange .8 Results and Discussion.ee 59 An Orbital-based Definition of Radical and Multiradical Character 62 4.

gà lv TT v v v v va 66 43 Method. cu ng ee 69 4.4 Results and DiscusSion.2 Diradical Character: General Discussion .3 Diradical Character: Results.4 Diradical Character: Discussion of Results. eee ee ee 83 4.6 Relationship of Multiradical Character to Monoradical Characters 84 4.7 Algebraic Connection to the “Odd-electron” Distribution.8 Behavior in the Limiting Case of the Number of “Odd Electrons” 89 4. v2 TT v và 90 Simulated Quantum Computation of Molecular Energies 91 5.

và cv v và v. cv gà Nà v v v v.4 Discussion of Scaling ConcernS.2 On the Future of Local Correlation Methods. 103 Bibliography 107 List of Figures 2.1 Plots of attenuators and attenuated Coulomb potentials. The func- tions terf and terfc are plotted, and the shape of erf is shown for refer- ence.

A division of the Coulomb potential into short- and long-range components by terfc and terf, respectively (w=5/ro).1 The UKS dissociation curves of LiF with Ms = 0, +1 for three different functionals: a. The Ms = 0 dissocia- tion asymptote is closer to the physically correct result with increased inclusion of HF-like exchange.2 The UHF dissociation curves of LiF with Ms = 0,+1. Both curves dissociate to the correct size-consistent asymptote.3 Atomic partial charges verses the difference in magnitude between IP“ and EA* for all four alkali-halide homologues with BLYP. Closer com- petition between A* and X for the “last” electron gives larger erroneous charges at dissociation, ©.4 Plots of the functions erf, terf and sterf.

The specific forms of these functions were chosen for technical reasons involving molecular integral evaluation; most notably terf(mr, mro)/r and sterf(m’r, m/rạ) are both symmetric with respect to r, and neither is ever singular. 44 Comparison of the erf and terf attenuated Coulomb potentials. If the parameters in the two types of potential are related by m” = 0.7336m when mrp = 1/ V⁄2, as shown above, then they intersect at the inflection point of the terf attenuated one. Now the short-range behaviors can be compared for two functions which become Coulombic at comparable distances.

Note that the terf attenuated function is very flat out to approximately rọ =1/mV2.6 Sample kernels V'® from Equation 3.14 for different values of rp and Eé. In the above kernels, ky = 0, kg = 1, m = 1/rov⁄2, rạ = 2.079/rạ and Ey = —E§T— ksv/2/e#jro (Es = Ef/erf(m'r))). The total vertical drop of the kernel between the origin and the asymptote is controlled by both rọ and Bg.7 Deviation of the TD-KS ionization potential from A-KS for BLYP CH¿. In the cross-hatched parameter region, the TD-KS ionization potential is within +0.

of the target unmodified A-KS value (0. The ionization potential was computed at the intersections of the cross- hatched grid and the circles are centered where cubic splines along the grid connecting lines indicate that the TD-KS deviation is zero. The dark curve is the best fit of the form #4(ro) = w + œeT”9 — +/rọ through these points; the parameter values for this fit are given in Table 3.4 for each molecule in our data set. The existence of this E&(ro) relationship reduces our parameter choice to a one-dimensional optimization for each molecule.8 The optimal parameter region.

The dark cross-hatched region is where the root-mean-squared deviation of the energies for the reactions in Table 3.2 from their parent BLYP values is outside chemical accu- racy. Outside the light cross-hatched region, all of the reaction ener- gies match the unmodified values to within chemical accuracy. Each dashed curve indicates, for a given molecule, the locus of parameters for which the BLYP TD-KS ionization energy is exactly that of un- modified A-KS, as illustrated for CH, in Figure 3.7; these curves are described by the data in Table 3. From top to bottom, the curves are for HF, F2, HạO, No, four together (CO, NH3, HOOH and HCN), two together (CH, and CHạO), CHạOH, HCCH, NHạNH;, CHạCH; and CH3CHs, respectively.

In general, we would like to choose parameters outside of the cross-hatched regions, but with ro as small as possible.9 Ground- and excited-state spectra of Be---F2 as a function of inter- molecular separation R for varied rp (“T” geometry with an F-F sepa- ration of 1. The solid line is 0. is the asymptotic A-KS CT energy using unmodified BLYP. The spectra are all relative to the ground-state energy of the separated molecules with the same parameters.

Eg was chosen by the prescrip- tion in Equation 3.26 for each rp: a 1. Decreased rp improves the curvature of the CT state but degrades its asymptote.10 Ground- and excited-state spectra of Be---Es as a function of inter- molecular separation R at the parent BLYP level (“T” geometry with an F-F separation of 1. The solid line is 0. is the asymptotic A-KS CT energy using unmodi- fied BLYP.

The spectra are all relative to the ground-state energy of the separated molecules with this functional. The correct Coulombic behavior of the CT state is clearly absent.1 Radical character as a function of orbital for the FCI/6-31G Li atom (0 = black < gray < purple < dark blue < light blue < green < yellow < orange % 1). The |1s) and |3s) orbitals have nearly zero probability of being singly occupied, as is true for the |2p) and |3p) functions that are not shown here.