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Notes on Modern Quantum Chemistry

Started: 08 Jun 2025
Updated: 08 Jun 2025

Title: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory
Author: Attila Szabo and Neil S. Ostlund (1982)

Table of Content

Table of Content

Computational Notes

see: https://github.com/antble/modernquantumchemistry

First Part: Quantum Mechanical Framework for Molecular Systems

chapter 1: Essential Mathematical Tools and Physical Postulates

before tackling many-electron systems, first 2 chapters establishes the fundamental mathematical tools, physical postules, and notational conventions that form the bebdrock of modern quantum chemistry
- essential mathematical tools and physical postulates
  - toolkit:
    - linear algebra (matrices, determinants, n-dimensional complex vector spaces, eigenvalue problem)
    - variation method: the energy calculated from any approximation trial wavefunction will always be an upper bound to the true ground state energy
      - the cornerstone of the Hartree-Fock method and Configuration Interactions
  - central problem: solving the time-independent Schrodinger equation for a molecule
    - full molecular Hamiltonian operator, contains terms for the kinetic energy of the electrons, and nuclei as well as the potential energy from electron-nuclear attraction, electron-electron repulsion, and nucleus-nucleus repulsion.
    - solving this is intractable for all but the simplest systems
  - solution (partial): Born-Oppenheimer approximation
    - based on the vast difference in mass between electrons and nuclei, their motions can be effectively decoupled
    - assumption: the nuclei are fixed in space (“clamped”), allowing for the solution of a purely electronic Schrodinger equation
    - repeating this calculation for many different nuclear arrangements generates a potential energy surface (PES), which in turn governs the motion of the nuclei (vibration and rotation)

chapter 2: Constructing Many-electron wavefunctions

addresses the challenge of how to write down a mathematically acceptable wavefunction for a system with many electrons
- starting point: Pauli exclusion principle
  - fermions like electrons, manifests as the antisymmetry principle: a many-electron wavefunction, must change sign upon the exchange of the spatial and spin coordinates of any two electrons
  - problem: a simple product of one-electron orbitals, known as Hartree product, fails to satisfy this condition
  - solution: construct the wavefunction as a Slater determinant
    - a mathematical construct, built from a set of one-electron spin-orbitals, that inherently possess the required antisymmetry and is the fundamental building block for nearly all wavefunction-based methods
    - the book is widely praised for developing a strong intuition for the properties and manipulation of Slater determinants
- second quantization formalism: a framework that recasts operator and wavefunctions in terms of creation ($a^\dagger$) and annihilation ($a$) operators
  - it acts on an abstract Fock space to add or remove an electron from a specific spin-orbital, respectively
  - operators for fermions obey specific anticommutation relations
  - this formalism provides a compact and powerful language that greatly simplifies the derivation of complex expressions for matrix elements between determinants
  - it becomes the natural language for describing many-body perturbation theory and coupled-cluster theory

chapter 3: Hartree-Fock approximation: a foundational mean-field model

computational notes:
- two-electron SCF fortran code for $HeH^+$ from Appendix B: twoeSCF.f
- python attempts for chapter 3: chapter 3.5.1, 3.5.2
the intellectual core of the book’s first half and is widely regarded as one of the most lucid and thorough treatments of the Hartree-Fock (HF) method
it is the cornerston of ab initio quantum chemistry
- serve as the conceptual starting point and computational reference for virtually all more advanced methods
- derivation and interpretation of the HF equations
  - HF method seeks an approximate solution to the electronic Schrodinger equation by making a pivotal simplifications: it assumes that exact N-electron wavefunction can be well-represented by a single Slater determinant
    - mean-field approximation: each electron is treated as moving independently in an average electrostatic field generated by the other N-1 electrons, rather. than accounting for the instantaneous repulsions between them
  - canonical Hartree-Fock equations: obtained by applying the variational principle—systematically varying the spin-orbitals within the Slater determinant to find the set that minimizes the total energy—one
    - a set of coupled, one-electron eigenvalue problem: \begin{equation} f\ket{\chi_a} = \varepsilon_a\ket{\chi_a} \end{equation}
    - $f$ is the Fock operator, which acts as the effective one-electron Hamiltonian for electron $a$.
    - $f(1) = h(1) + v^{HF}(1)$,
      - $v^{HF}(1)$ one-electron potential operator (Hatree-Fock potential)
        
        encapsulates the average electron-electron repulsion
        
        coulomb operator : $\mathscr{J}$, represents the classical electrostatic repulsion between an electron in spin orbital $\chi_a$ and the average charge distribution of all other electrons
        
        exchange operator : $\mathcal{H}$, a purely quantum mechanical term with no classical analog, it arises directly from the antisymmetry of the slater determinant and lowers the energy for electrons of parallel spin
      - $h(1)$ is the core-Hamiltonian operator
    - solution to these equations provide a set of optimal spin-orbitals $\chi_a$ and their corresponding orbital energies $\varepsilon_a$
  - Physical significant of these results is illuminated by two key theorems
    - Koopman’s theorem: states that the negative of a canonical HF orbital energy is a reasonable appoximation to the ionization potential for removing an electron from that orbital
    - Brillouin’s theorem: shows that the matrix element between the HF ground-state determinant and any singly excited determinant is zero, which has important consequences for Configuration Interaction theory.
Roothaan-Hall equations and the SCF procedure
- Hartree-Fock equations are integro-differential equations, which are computationally challenging to solve for general molecules
  - solution: introduction of a finite set of known, atom-centered basis function ${\phi_\mu}$ to represent the unknown molecular orbitals ${\psi_i}$, this is the linear combination of atomic orbitals (LCAO) approximation: \begin{equation} \psi_i = \sum_{\mu=1}^{K} C_{\mu i} \phi_\mu \quad i=1,2,\cdots, K \end{equation}
  - it transforms the difficult equations into a more tractable matrix algebra problem resulting to the Roothan-Hall equations in matrix form:
    \begin{equation} \mathbf{FC} = \mathbf{SC}\mathbf{\varepsilon}\label{eq:roothanhall} \end{equation}
    - this is a generalized eigenvalue equation
      - $\mathbf{F}$: Fock matrix, the Fock operator in the chosen basis
      - $\mathbf{C}$: the matrix of the molecular orbital coefficients $C_{\mu i}$ that one seeks to find
      - $\mathbf{S}$: the overlap matrix of the non-orthogonal basis functions
      - $\mathbf{\varepsilon}$: the diagonal matrix of orbital energies
    - critical feature of this equation: non-linearity
      - the Fock matrix depends on the electron density, which in turn depends on the orbital coeffient matrix, the very quantity being sought
  - this necessitates an iterative solution known as Self-Consistent Field (SCF) procedure
    - initial guess of the $\mathbf{C}$ -> a Fock matrix $\mathbf{F}$ is constructed -> Roothan-Hall (equation \ref{eq:roothanhall}) is solve to obtain a new coefficient $\mathbf{C}$ -> a new Fock matrix -> …
      - the cycle repeats until the coefficients, electron density, and energy no longer changes between iterations, at which point the solution is deemed “self-consistent”
Basis Sets and Open-Shell systems
- quality of an HF calculation is critically dependent on the set of basis functions used
- 2 primary types introduced in the book:
  - Slater-Type orbitals (STOs): have the correct physical form (e.g. cusp at the nucleus) but lead to computationally expensive two-electron integrals
  - Gaussian-Type orbitals (GTOs): less physically accurate but are far more efficient due to the Gaussian Product Theorem - the product of two Gaussian on different centers is another Gaussian
- Modern quantum chemistry almost universally employs contracted basis sets, where a fixed linear combination of several GTOs is used to approximate the shape of a single STO
  - hierarchy of Pople-style basis sets:
    - STO-3G : each STO is mimicked by 3 GTOs
    - 4-31G: split-valence sets
    - 6-31G: polarized sets, add functions of higher angular momentum to allow for greater flexibility in describing chemical bonds
- restricted HF method: forces paired electrons to occupy the same spatial orbital
- unrestricted HF method
  - electrons with $\alpha(\uparrow)$ and $\beta(\downarrow)$ are allowed to have different spatial orbitals
    - often inadequate (?)
  - provides a better description for processes like bond dissociation
    - can lead to a wavefunction that is not a pure spin state, a problem known as spin contamination

Second Part: Capturing Electron Correlation

a post-Hartree-Fock methods
its about sovling the “problem of correlation”
- the HF method neglects the instantaneous, correlated motions of electrons
- electron correlation energy = (energy of the exact non-relativistic energy) - (energy at the Hartree-Fock limit)
  - recovering this energy is essential for achieving quantitative accuracy
- it presents 3 distinct philosophical and mathematical approaches to rectify the central flaw of the HF model:
  - Configuration Interaction
  - Coupled-Cluster Theory
  - Many-Body perturbation

chapter 4: Configuration Interaction (CI): The Variational Path to Correlation

essence of CI: express the true wavefunction as a linear combination of many Slater determinants to improve upon the single-determinant HF wavefunction
- a straightforward method for including electron correlation
- additional determinants: $\{\Phi_i\}$, generated by “exciting” one or more electrons from occupied HF orbitals to virtual (unoccupied) orbitals
  - CI wavefunction: \begin{equation} |\Psi_{\text{CI}}\rangle = c_0 |\Psi_0\rangle + \sum_{i,a} c_i^a |\Psi_i^a\rangle + \sum_{i < j, a < b} c_{ij}^{ab} |\Psi_{ij}^{ab}\rangle + \dots \end{equation}
    - $\Psi_0$ is the HF reference determinant
    - $\Psi_i^a$ is a singly excited determinant
    - $\Psi_{ij}^{ab}$ is a doubly excited determinant
  - The coefficients $c$ are determined variationally by diagonalizing the Hamiltonian matrix in the basis of these determinants.
Full CI calculation includes all possible excitations within a given basis set, provides the exact solution to the electronic Schrodinger equation for that basis
- problem: the number of determinants grows factorially with the system size, making Full CI computationally prohibitive for all but the smallest molecules
- solution: expansion must be truncated
CISD, includes all single and double excitations
- problem: truncated CI are not size-extensive (or size-consistent)
  - the CISD energy of two non-interacting hydrogen molecules is not equal to twice the CISD energy of a single hydrogen molecule
- solution: alternative methods (e.g. CC)

chapter 5: Coupled-Cluster (CC) Theory: The Gold Standard of Single-Reference Methods

it solves elegantly the size-extensivity problem of truncated CI
coupled-cluster approximation (CCA)
- CCA with only double excitations is called CCD (coupled-clusters doubles)
- CCSD include single and higher excitations
CCSD(T) methd, adds a perturbative correction for triple excitations to the CCSD result, is often called the “golden standard” of single-reference quantum chemistry for its exceptional accuracy

chapter 6: Many-Body Perturbation Theory (MBPT): A Perturbative Correction

instead of a variational expansion, Many-Body Perturbation Theory (MBPT) treats electron correlation as a small perturbation to the HF solution.
- employs Rayleigh-Schrödinger perturbation theory (RS-PT) to systematically calculate corrections to the HF energy and wavefunction
- total Hamiltonian is partitioned such that the unperturbed zeroth-order Hamiltonian is the sum of the Fock operators, whose exact eigenfunction is the HF determinant
  - perturbation is then the difference between the true Coulomb repulsion and the mean-field HF potential.
Møller-Plesset theorem: the sum of the zeroth- and first-order energies in this partitioning is exactly the Hartree-Fock energy ($E_{\text{HF}} = E^{(0)} + E^{(1)}$)
- the first meaningful correction that introduces electron correlation is the second-order energy, $E^{(2)}$
- MP2 method is one of the most popular and cost-effective ways to account for electron correlation
Diagrammatic perturbation theory:
- using Goldstone diagrams, the complex algebraic terms in the perturbation expansion can be represented visually, providing powerful physical insight into the interactions being described and greatly simplifying their derivation

chapter 7: Many-body Green’s function

One-Particle Many-Body Green’s Function method

a powerful and advanced formalism based on propagators that is particularly well-suited for directly calculating properties like ionization potentials and electron affinities.