Hartree–Fock Equation

Abstract

In many-electron system having more than two electrons, mathematical method to obtain analytical solution in Schrödinger equation has not been discovered. Instead, we are able to obtain quantum wave function numerically. In quantum computational chemistry, Hartree–Fock equation is used for numerical calculation. Through minimization of total energy in Schrödinger equation, one-electron equation, Hartree–Fock equation is derived. In Schrödinger equation, Hamiltonian operates on total quantum wave function. On the other hand, in Hartree–Fock equation, Fock operator operates on quantum wave function (spin orbital). Fock operator consists of three operators: one-electron operator, Coulomb operator, exchange operator. Orbital energy is given as eigenvalue of Hartree–Fock equation. In closed shell system, two electrons with α and β spins are paired in the same spatial orbital. Total energy in Schrödinger equation is expressed by the use of one-electron operator, Coulomb integral, exchange integral and spatial orbitals, due to orthonormality of spin function. Orbital energy in Hartree–Fock equation is also expressed by the use of one-electron operator, Coulomb integral, exchange integral and spatial orbitals, due to orthonormality of spin function. The derivation processes are explained in details. From the viewpoint of restricted electron-allocation, the Hartree–Fock is called Restricted Hartree–Fock (RHF). In open shell system, different spatial orbitals are prepared for α and β electrons. In the same manner, we consider to express total energy in Schrödinger equation and orbital energy in Hartree–Fock equation. From the viewpoint of unrestricted electron-allocation, the Hartree–Fock is called Unrestricted Hartree–Fock (UHF).