Variational and Perturbation Methods

Because molecular Schrodinger equations cannot be solved exactly, quantum chemistry relies on the variational principle and perturbation theory to obtain accurate approximate energies and wavefunctions.

Definition

Variational and perturbation methods are the principal approximation techniques of quantum chemistry: the variational method minimizes the energy of a trial wavefunction, while perturbation theory corrects a solvable reference problem by small successive terms.

Scope

This topic covers the systematic approximation methods of quantum chemistry: the variational principle, which guarantees that any trial wavefunction gives an energy above the true ground state and provides the basis for the secular equations and the Hartree-Fock self-consistent field method; and perturbation theory, including Rayleigh-Schrodinger and Moller-Plesset treatments of electron correlation. It also introduces density functional theory as an alternative route based on the electron density. The qualitative molecular orbital picture and the heavily computational implementations are treated in neighbouring topics.

Core questions

• Why does the variational principle guarantee an upper bound to the ground-state energy?
• How does the Hartree-Fock method use the variational principle to obtain molecular orbitals?
• How does perturbation theory recover electron correlation missing from Hartree-Fock?
• How does density functional theory reformulate the problem in terms of the electron density?

Key concepts

• Variational principle and trial wavefunctions
• Secular equations and the Hartree-Fock method
• Electron correlation
• Rayleigh-Schrodinger and Moller-Plesset perturbation theory
• Density functional theory

Key theories

Variational principle

The expectation value of the energy for any normalized trial wavefunction is never below the true ground-state energy, so minimizing it over adjustable parameters yields the best approximation within the chosen functional form.

Density functional theory

The Hohenberg-Kohn theorems establish that the ground-state energy is a functional of the electron density alone, and the Kohn-Sham equations recast the problem as non-interacting electrons in an effective potential, making accurate calculations on large systems practical.

Clinical relevance

These methods make quantitative electronic-structure calculation possible, supplying the energies, geometries, and reaction barriers used in computational chemistry, catalyst and materials design, and structure-based drug discovery, with density functional theory now the workhorse of the field.

History

The variational self-consistent field method was developed by Hartree and Fock in the late 1920s and 1930s; Moller-Plesset perturbation theory followed in 1934, and the density functional theory of Hohenberg, Kohn, and Sham in the 1960s, recognized with the 1998 Nobel Prize, transformed the practical scope of quantum chemistry.

Key figures

• Douglas Hartree
• Vladimir Fock
• Walter Kohn

Seminal works

• szabo1996
• hohenberg1964
• kohn1965

Frequently asked questions

Why can the variational method never give an energy that is too low?

Any trial wavefunction is a mixture of the true energy eigenstates, and since the ground state is the lowest, the weighted average energy of the mixture is always at least the ground-state energy; equality holds only when the trial function is exact.

Why is density functional theory so widely used?

It captures much of the electron correlation that Hartree-Fock omits while working with the three-dimensional electron density rather than the full many-electron wavefunction, giving a favourable balance of accuracy and computational cost that scales to large molecules and solids.