Math Needed for Quantum Chemistry

Quantum chemistry uses wavefunctions and operators to calculate electronic energies, bonding, and molecular motion. Its mathematical difficulty is the large number of coupled coordinates. Start with eigenvalue problems and variational approximation, then add the tools needed by the molecular calculation you want to understand.

Molecular orbitals and matrix eigenvalue problems

A useful starting point is an expansion in a finite set of basis functions,

$$|\psi\rangle=\sum_j c_j|\phi_j\rangle.$$

The functions need not be orthogonal. Projecting a linear eigenvalue equation onto them gives

$$Hc=ESc,\qquad H_{ij}=\langle\phi_i|\widehat H|\phi_j\rangle,\qquad S_{ij}=\langle\phi_i|\phi_j\rangle.$$

The overlap matrix $S$ records the inner products of the basis functions. Normalization is $c^\dagger Sc=1$, not generally $c^\dagger c=1$. If the basis is linearly independent, $S$ is positive definite. Nearly dependent functions produce a poorly conditioned overlap matrix and require care even when the Hamiltonian is Hermitian.

Read Bases and Coordinates and Inner Products if these statements are unfamiliar. Then use Rayleigh–Ritz Method to see how the matrix problem arises from varying the coefficients. In Hartree–Fock theory the effective orbital matrix also depends on the occupied orbitals, so the eigenvalue problem is part of a self-consistency iteration.

Choose a physical starting point

Starting problem	Mathematics to review	Continue with
Hydrogen orbitals	Separation of Variables, Spherical Harmonics, and Laguerre Polynomials	Hydrogen Atom
An approximate ground-state energy	Calculus of Variations and normalization constraints	Variational Principle and Helium Variational Estimate
Several electrons	Tensor products and antisymmetric states	Slater Determinants and Spin-Spatial Wavefunctions
Molecular electronic structure	Generalized eigenvalues, variational approximation, and self-consistency	Electronic Structure Overview
Vibrations near equilibrium	Taylor expansion, Hessian matrices, and mass-weighted normal modes	Quantum Harmonic Oscillator
Molecular rotations	Angular momentum and rotation matrices	Spherical Harmonics and Wigner D-Matrices

You can begin with the hydrogen or helium examples without knowing second quantization. Add Fermionic Fock Space and Creation and Annihilation Operators when a many-electron method uses occupation notation.

Fixed nuclei and moving nuclei

For each nuclear configuration $R$, one can solve an electronic eigenvalue problem and plot its energy $E_n(R)$, including nuclear repulsion when defining a molecular potential-energy surface. This is a parameter-dependent spectrum. Using one such surface as an effective potential for nuclear motion makes an additional approximation: transitions and derivative couplings between electronic states must be negligible in the regime considered.

The nuclear wavefunction can still be quantum mechanical. The Born–Oppenheimer approximation does not mean that nuclei follow classical trajectories. Near electronic degeneracies, a single-surface description may fail even when the nuclei are much heavier than the electrons.

What to check in a calculation

A small residual $Hc-ESc$ checks the algebraic solve in the chosen basis. It does not measure the error from an incomplete basis or from a Hartree–Fock ansatz. Change the basis, compare a known limiting case, and keep numerical error separate from the physical approximation. Conditioning and Stability and Convergence Tests explain these numerical checks.

An orbital is a one-electron function, not an electron trajectory. Different occupied orbitals related by a unitary change of basis can represent the same Slater-determinant state up to an overall phase. Physical conclusions should therefore be based on energies, densities, and other well-defined quantities, rather than the appearance of one orbital plot alone.

Check your understanding

1. Two normalized basis functions have real overlap $s$, with $|s|<1$. Write $S$ and find its eigenvalues. Why does $s\to1$ cause numerical difficulty?

Solution

$$S=\begin{pmatrix}1&s\\s&1\end{pmatrix},\qquad \lambda_\pm=1\pm s.$$

Both eigenvalues are positive for $|s|<1$. As $s\to1$, the difference of the two functions has very small norm and the smallest eigenvalue approaches zero. The basis becomes nearly linearly dependent; orthogonalization can amplify numerical errors.

2. A calculation solves the matrix eigenvalue problem very accurately in a small basis. Has it necessarily found the exact molecular ground-state energy?

Solution

No. It has solved the chosen finite problem accurately. Missing basis functions and restrictions on the many-electron trial states can still produce a large error. For a ground-state Rayleigh–Ritz calculation with the same exact Hamiltonian, increasing a nested trial space cannot raise the lowest variational energy, but a small residual alone does not show that this limiting value has been reached.

References

• T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory, Wiley, 2000 — molecular basis sets and many-electron approximations.
• F. Jensen, Introduction to Computational Chemistry, 3rd ed., Wiley, 2017 — numerical and physical sources of error.
• A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, 1996 — matrix formulation and Hartree–Fock theory.