Fundamentals of Computational Quantum Chemistry

Abstract

The chief aim of this course is to describe practicable methods of solving the eigen-value problem:

$$\hat H\Psi \quad = \quad E\Psi$$

((1.1))

and the realisation of these methods on electronic digital computers. Here E is one of a set of possible energies {E_n} and Ψ one of a set of associated state functions {Ψ_n}. The Hamiltonian Operator Ĥ we shall take to be

$$\widehat {\text{H}}\left( {1,2 - - N} \right) = \sum\limits_{1 = 1}^{\text{N}} {\hat h} \left( i \right) + \frac{1}{2}\sum\limits_{1j = 1}^{{{\text{N}}_e}2} {{/_{4\pi { \in _o}{r_{ij}}}}}$$

((1.2a))

$$\hat h\left(i\right)= -{h^2}{/_{2m}}{\nabla ^2}\left(i\right) - {\sum\limits_{n = 1}^{{{\rm N}_n}} {{{\rm Z}_n}} ^{{}_e{2_/}}}{}_{4\pi { \in _o}{r_{ni}}}$$

((1.2b))

which describes (in conventional notation), the motion of N electrons, moving in the field provided by N_n nuclei each with charge Z_n, fixed in space, assuming only electrostatic interactions. We shall generally quote this Hamiltonian in atomic units, by quoting all distances as multiples of the fundamental length a = 4 πβ_oħ²/me².