Molecular Orbital Calculation

Abstract

One-electron equation called Hartree–Fock equation is derived by the minimization of total energy given in Schrödinger equation. When basis function is introduced in quantum wave function, the mathematical problem to obtain quantum wave function analytically is converted into calculating expansion coefficient numerically. In closed shell system, α and β electrons can occupy the same spatial orbital. One Hartree–Fock matrix equation is given. On the other hand, in open shell system, different spatial orbitals are prepared for α and β electrons. Two Hartree–Fock matrix equations are given. In this chapter, these Hartree–Fock matrix equations are derived. In atom, spatial orbital is expressed as superposition of basis functions. A set of basis functions for each atom is called basis set. In cluster, spatial orbital is expressed as combination of basis sets. In basis function, Gaussian-type exponential is applied, due to practical calculation with regard to two-electron integral. For correction of radial shape, contraction and split-valence are performed for basis function. In Hartree–Fock approximation, as the Coulomb interaction between electrons is estimated in average procedure, the deviation called electron correlation arises. To incorporate electron correlation, several calculation methods based on Hartree–Fock method have been explored: configuration interaction, coupled cluster etc. In another stream, density functional theory has been developed. By the use of density as variable, Schrödinger equation is converted into Kohn–Sham equation. By solving the equation, molecular orbital under consideration of electron correlation is obtained. When we start molecular orbital calculation, we need to consider three main factors: (1) basis set selection, (2) combination of basis set and calculation method, (3) modelling. The concept of initial atomic orbital (IAO) is useful for molecular orbital analysis. Chemical bonding rule is used to judge chemical bonding character: covalent bonding or ionic bonding. Atomic charge can be estimated based on Mulliken population analysis.