Lecture: ab initio Methods Introduction

This is a short description of the comonly used calculational procedure to solve Schrodinger's equation, known as the Hartree-Fock approximation. This calculation is fundamental to an understanding of molecules. Within the framework of the approximations used in the Hartree-Fock approach, the distribution of electric charge is determined. In addition, a molecular wavefunction is calculated and various molecular properties are found.

ab initio , is a latin phrase meaning from first principles. Common usage in CCM is to compute the electronic charge distribution in a molecule using quantum mechanics. The principle type of calculation is called Hartee-Fock using a set of approximations to reduce the complexity of the 4N dimensional problem, where N is the number of electrons and nuclei. The factor of 4 arises from spatial coordinates plus the spin.

The basic set of approximations are:

Born-Oppenheimer Approximation

Decouples the electronic and nuclear degrees of freedom. Assumes the nuclear centers of mass are fixed for a given calculation. I.e., the wave function is parameterized with respect to the nuclear coordinates.

Hartree-Fock Approximation

The many electron problem is approximated by a sequential calculation of the response of the i^th electron in the average potential of the rest of the electrons. This one-electron operator is called the Fock operator.

Anti-symmetric wave function.

Wave functions describing electrons obey Fermi Dirac statistics, that is, they must be anti-symmetric with respect to an interchange of coordinates. This is conveniently expressed in terms of a determinant called the Slater Determinant.

Roothan Equations

The Hartree-Fock equations are a set of coupled integro-differential equations. Using basis vectors of Hilbert space, this can be transformed into a set of algebraic equations This gives a generalized eigenvalue problem to solve

HF-SCF Hartree-Fock Self Consistent Field

Name of the procedure to calculate the charge distribution. The problem is non-linear (the Fock matrix is dependent on the variational coefficients that are being calculated) thus an iterative approach is used (self consistency)

Post Hartree-Fock Calculations

The Hartree-Fock calculation has a built in deficiency; no electron-electron correlation. Thus, further work needs to be performed. Lumped as post Hartree-Fock calculations. Most of these schemes involve the diagonalization of large matrices.

Author: Ken Flurchick