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Advanced Molecular Science: Electronic Structure Theory Abstract Advanced Molecular Science: Electronic Structure Theory Krzysztof Szalewicz et al. Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA (Dated: December 17, 2017) Abstract These Lecture Notes were prepared during a one-semester course at the University of Delaware. Some lectures were given by students and the corresponding notes were also prepared by students. The goal of this course was to cover the material from first principles, assuming only the knowledge of standard quantum mechanics at advanced undergraduate level. Thus, all the concepts are defined and all theorems are proved. There is some amount of material looking ahead which is not proved, it should be obvious from the context. About 95% of the material given in the notes was actually presented in the class, in the traditional blackboard and chalk manner. 1 CONTENTS I. Introduction 5 A. Spinorbitals 6 B. Products of complete basis sets 7 II. Symmetries of many-particle functions 7 A. Symmetric group 8 B. Determinant 10 III. Separation of nuclear and electronic motion 12 A. Hamiltonian in relative coordinates 13 B. Born-Oppenheimer approximation 15 C. Adiabatic approximation and nonadiabatic correction 16 IV. The independent-particle model: the Hartree-Fock method 18 A. Slater determinant and antisymmetrizer 19 B. Slater-Condon rules 21 C. Derivation of Hartree-Fock equations 22 V. Second-quantization formalism 27 A. Annihilation and creation operators 27 B. Products and commutators of operators 28 C. Hamiltonian and number operator 30 D. Normal products and Wick’s theorem 31 1. Normal-Product 31 2. Contractions (Pairings) 32 3. Time-independent Wick’s theorem 32 4. Outline of proof of Wick’s theorem 33 5. Comprehensive proof of Wick’s theorem 34 6. Particle-hole formalism 37 7. Normal products and Wick’s theorem relative to the Fermi vacuum 38 8. Generalized Wick’s theorem 39 9. Normal-product form of operators with respect to Fermi’s vaccum 40 VI. Density-functional theory 41 A. Thomas-Fermi-Dirac method 42 B. Hohenberg-Kohn theorems 47 C. Kohn-Sham method 49 D. Local density approximation 53 E. Generalized gradient approximations (GGA) 55 2 F. Beyond GGA 57 VII. Variational Method 58 A. Configuration Interaction (CI) method 59 1. Size extensivity of CI 61 2. MCSCF, CASSCF, RASSCF, and MRCI 62 B. Basis sets and basis set convergence 64 C. Explicitly-correlated methods 66 1. Coulomb cusp 66 2. Hylleraas function 67 3. Slater geminals 68 4. Explicitly-correlated Gaussian functions 69 VIII. Many-body perturbation theory (MBPT) 70 A. Rayleigh Schrödinger perturbation theory (classical derivation) 70 B. Hylleraas variation principle 74 C. Møller-Plesset perturbation theory 76 D. Diagrammatic expansions for MPPT 82 1. Diagrammatic notation 82 2. One-particle operator 83 3. Two-particle operators 84 4. Hugenholtz diagrams 86 5. Antisymmetrized Goldstone diagrams 87 6. Diagrammatic representation of RSPT 87 E. Time versions 88 1. Time version of the first kind 88 2. Time version of the second kind 88 F. Connected and disconnected diagrams 89 G. Linked and unlinked diagrams 90 H. Factorization lemma (Frantz and Mills) 92 I. Linked-cluster theorem 94 J. Removal of spin 96 IX. Coupled cluster theory 97 A. Exponential ansatz 97 B. Size consistency 98 C. CC method with double excitations 99 D. Equivalence of CC and MBPT theory 108 E. Noniterative triple excitations correction 110 F. Full triple and higher excitations 113 3 X. Linear response theory 114 A. Response function 114 1. Density-density response function 117 2. Calculation of properties from response functions 119 B. Linear response in CC approach 120 1. CC equations 120 2. Hellmann-Feynman theorem 122 3. Linear response CC for static perturbation 124 4. Lambda equations 126 XI. Treatment of excited states 128 A. Excitation energies from TD-DFT 128 B. Limitations of single-reference CC metods 131 C. The equation-of-motion coupled-cluster method 131 D. Multireference coupled-cluster methods 133 XII. Intermolecular interactions 136 A. Symmetry-adapted perturbation theory 137 B. Asymptotic expansion of interaction energy 141 C. Intermolecular interactions in DFT 142 XIII. Diffusion Monte Carlo 143 XIV. Density-matrix approaches 146 A. Reduced density matrices 148 B. Spinless density matrices 150 C. N-representability 151 D. Density matrix functional theory 154 E. Contracted Schrödinger equation 156 XV. Density matrix renormalization group (DMRG) 159 A. Singular value decomposition 159 B. SVD applications 161 C. DMRG wave function 162 D. Expectation values and diagrammatic notation 163 E. Matrix product ansatz 165 F. DMRG algorithm 166 G. DMRG in practice 167 H. Dynamic correlation and excited states 167 I. Applications to atoms and molecules 167 J. Limitations 168 4 I. INTRODUCTION The subject of these lecture notes will be methods of solving Schödinger’s equation for atoms, molecules, biomolecular aggregates, and solids. Schrödinger’s equation provides very accurate description of most types of matter under most conditions, where by “most" we will understands the materials and conditions on the Earth. Exceptions include materials that include heavy atoms where relativistic effects have to be accounted for and high-precision measurements where not only the relativistic but also quantum electrodynamics (QED) effects play a role. We will restrict our attention to solutions of Schrödinger’s equation. Incorporation of relativistic and QED effects can be achieved by a fairly straightforward extension of the methods discussed here. We will also restrict our attention to systems built of electrons and nuclei treated as point particles. Thus, we will not consider phenomena which involve nuclear reactions. However, many of the methods discussed here are used in theoretical nuclear physics. For systems with up to 4 electrons and 1 to 3 nuclei, one can now solve Schrödinger’s equation almost to any desired precision, although for the most complicated systems of this type it requires huge amounts of computer resources. Some of the methods used in such calculations, such as the variational method with explicitly correlated functions (i.e., functions depending explicitly on electron-electron distances), will be briefly discussed here, but we will devote most of the time to systems that are larger and for which such methods are not applicable. The difficulty of solving Schrödinger’s equation for systems with 5 and more electrons originates from dimensionality of the problem. For example, the benzene molecule contains 12 nuclei and 42 electrons, so that Schrödinger’s equation for this system is 162-dimensional. Thus, this equation can be solved only by making approximations (although quantum diffusion Monte Carlo methods which will be considered later on do solve such equations “almost" directly). The main approximation applied is many-particle (or many-electron or many-body) expansion. Therefore, most of the material covered here belongs to the branch of physics called many-body physics. The concept of many-particle expansion is based on the observation that in a many-particle system the most important interactions are those involving only two particles. This leads to several method hierarchical in the number of particle interaction considered. The particles that we will consider almost almost exclusively will be electrons. Many- particle theories applied to bound states of such particles are known as electronic structure theory. The reasons for using the word “structure" are uncleared, but probably relates to shell structure of atoms and orbital picture of molecules. 5 A. Spinorbitals 1 Electrons are fermions of spin 2 . The wave function of a single electron depends on the space coordinate r = [x;y;z] with each variable in the range (−∞;1) and on the ± 1 spin coordinate s which takes only values 2 . Therefore, the wave function for a single electron, called spinorbital, can be written in the form of the so-called spinor " # +(r) = $ (r;s) ≡ (x): (r) − where the + component is the amplitude of finding the electron with spin projection, ~ the eigenvalue of the operator Sz, equal to =2 (“spin up") whereas the component − which is the amplitude of finding this particle with spin projection −~=2 (“spin down"). Note that an electron in the state is in general in a mixed spin state. It will be more convenient to use the other form of the wave function shown in the eqution above 1 (r) = (r; ) + 2 1 (r) = (r;− ): − 2 This form is particularly convenient to use in the expectation values of operators (matrix elements) involving spinorbitals. In the one-electron case Z X 3 h jf φi = d r (x)f (r)φ(x); s= 1=2;1=2 − i.e., we sum over spin variable and integrate over the three space variables. Since f does not contain spin operators, the sum over the spin degrees of freedom can be computed immediately. In most cases we will consider pure spin states, i.e., states with the property that either +(r) or (r) is zero. For example, − " # (r) = + 0 is the spinorbital with spin projections ~=2 or “spin alpha (α)" state, whereas the other option is called “spin beta (β)" state. In such cases, the pure-spin spinorbitals can be denoted as +(r) and (r) or (r)α(s) and 0(r)β(s) or α(r) and β(r); − where α(1=2) = β(−1=2) = 1 and α(−1=2) = β(1=2) = 0 and the spatial part is called the orbital. Note the +(r) and (r) are now different spinorbitals, whereas before − they were components of a single spinorbital. We also continue using the symbol (x) 6 assuming that (x) either describes pure spin alpha or is zero. One sometimes uses s=1=2 a somewhat confusing notation where spinorbital and orbital are denoted by the same symbol, so that we have, for example, (x) = (r)α(s). The meaning of this symbol should be obvious from the context.
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