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Mathematics for Nerds

This chapter presents the mathematical framework underlying a Single Point Energy calculation. Unlike the previous chapter, which described the computational workflow conceptually, the present discussion focuses on the quantum mechanical equations solved during the calculation.

Although Gaussian automates these procedures, every electronic structure method ultimately seeks an approximate solution to the electronic Schrödinger equation.

For readers interested primarily in running calculations, this chapter may be skipped without affecting the remaining tutorials. However, for those wishing to understand how Gaussian computes molecular energies, the following sections provide the necessary theoretical background.

Roadmap

The mathematical development follows the sequence

Many-Particle Schrödinger Equation
                │
                ▼
Born–Oppenheimer Approximation
                │
                ▼
Electronic Hamiltonian
                │
                ▼
Hartree–Fock Approximation
                │
                ▼
Basis Set Expansion
                │
                ▼
Roothaan–Hall Equations
                │
                ▼
Density Functional Theory
                │
                ▼
Kohn–Sham Equations
                │
                ▼
SCF Iterations
                │
                ▼
Final Electronic Energy

Each of these topics builds upon the previous one and collectively describes the theoretical foundation of modern quantum chemistry calculations.

Topics Covered

This chapter includes

  • Born–Oppenheimer approximation
  • Electronic Schrödinger equation
  • Molecular Hamiltonian
  • Hartree–Fock theory
  • Slater determinants
  • Basis functions
  • Linear Combination of Atomic Orbitals (LCAO)
  • Roothaan–Hall equations
  • Density Functional Theory (DFT)
  • Kohn–Sham formalism
  • Exchange–correlation functionals
  • The B3LYP functional
  • Self-Consistent Field (SCF) procedure
  • Electronic energy evaluation

Each section introduces one important component of the mathematical framework used by Gaussian.

Prerequisites

Readers are expected to have a basic familiarity with

  • Linear algebra
  • Multivariable calculus
  • Quantum mechanics
  • Matrix notation
  • Eigenvalue problems

No prior knowledge of Density Functional Theory is assumed. The required concepts will be introduced gradually throughout this chapter.

Notation

Throughout this chapter,

Symbol Meaning
r Electronic coordinates
R Nuclear coordinates
Ψ Many-electron wavefunction
φ Molecular orbital
χ Basis function
ρ Electron density
H Hamiltonian operator
F Fock or Kohn–Sham matrix
S Overlap matrix
C Molecular orbital coefficients
ε Orbital energies

These symbols will be used consistently in all subsequent mathematical derivations.

Learning Objectives

After completing this chapter, readers should understand

  • why the electronic Schrödinger equation cannot be solved exactly,
  • how Hartree–Fock approximates the many-electron problem,
  • how basis sets convert differential equations into matrix equations,
  • how Density Functional Theory differs from Hartree–Fock,
  • why the Self-Consistent Field procedure is iterative,
  • and how Gaussian ultimately computes the electronic energy reported in the output file.

The next section begins with the Born–Oppenheimer approximation, which forms the foundation of nearly all electronic structure calculations.