On the Calculation and Accuracy of Theoretical Electron Densities

Abstract

The basic method for the calculation of the electronic structure of atoms and molecules is the Independent Particle Model wherein there is a one-to-one correspondence between electrons and spin-orbitals and therefore each electron moves in the average field of the other N-1 electrons /1/. The wave function in this model is the Slater determinant

$$\Phi(1,2,...,N)=\frac{1}{\sqrt{N!}}\det \left|{{\phi }_{1}}(1){{\phi }_{2}}(2)...{{\phi }_{N}}(N)\right|$$

(1)

where the {⌽_i(J)} are a set of N-spin-orbitals and J is the combined space-spin coordinate of electron j. As we discussed elsewhere /2/, the one-particle density matrix in this model, Г ¹_Φ (1,1’), is called the Fock-Dirac density matrix and may be written in the form

$$\Gamma _\Phi ^1(1,1') = \mathop \Sigma \limits_{i = 1}^N {\phi _i}(1)\phi _i^*(1')$$

(2)

Use of (1) (or equivalently (2)) in the variation principle leads to the Hartree-Fock equations which are a set of integrodifferential equations for the set of orthonormal spin orbitals {φ_i(J)} which are optimum in the sense of the energetic criterion of the variation principle. They may be written in the form

$$\hat{F}(1){{\phi }_{i}}(1)=\sum\limits_{j=1}^{N}{{{\lambda }_{ij}}}{{\phi }_{j}}(1);i=1,...,N,$$

(3)

where $\hat{F}$ (1), called the Fock operator, is a functional of the set {φ_i(J)} and will be defined below. It is a one-electron operator which is the effective Hamiltonian operator for an electron (described by spin-orbital ⌽_i) in the attractive field of the nuclei and the average replusive field of the other electrons.