An Introduction to Computational Quantum Mechanics

Our aim in this chapter will be to establish the basic elements of those quantum mechanical methods that are most widely used in molecular modelling. We shall assume some familiarity with the elementary concepts of quantum mechanics as found in most general physical chemistry textbooks, but little else other than some basic mathematics (see Section 1.10). There are also many excellent introductory texts to quantum mechanics. In Chapter 3 we then build upon this chapter and consider more advanced concepts. Quantum mechanics does, of course, predate the first computers by many years, and it is a tribute to the pioneers in the field that so many of the methods in common use today are based upon their efforts. The early applications were restricted to atomic, diatomic or highly symmetrical systems which could be solved by hand. The development of quantum mechanical techniques that are more generally applicable and that can be implemented on a computer (thereby eliminating the need for much laborious hand calculation) means that quantum mechanics can now be used to perform calculations on molecular systems of real, practical interest. Quantum mechanics explicitly represents the electrons in a calculation, and so it is possible to derive properties that depend upon the electronic distribution and, in particular, to investigate chemical reactions in which bonds are broken and formed. These qualities, which differentiate quantum mechanics from the empirical force field methods described in Qiapter 4, will be emphasised in our discussion of typical applications. 

There are a number of quantum theories for treating molecular systems. The first we shall examine, and the one which has been most widely used, is molecular orbital theory. However, alternative approaches have been developed, some of which we shall also describe, albeit briefly. We will be primarily concerned with the ab initio and semi-empirical approaches to quantum mechanics but will also mention techniques such as Hiickel theory and valence bond theory. An alternative approach to quantum mechanics, density functional theory, is considered in Chapter 3. Density functional theory has always enjoyed significant support from the materials science community but is increasingly used for molecular systems. 

Quantum mechanics is often considered to be a difficult subject, and a cursory glance at the following pages in this chapter may simply serve to reinforce that view However, if followed carefully it is possible to see how models that are developed for very simple 

The starting point for any discussion of quantum mechanics is, of course, the Schrodinger equation. The full, time-dependent form of this equation is 

Equation (2.1) refers to a single particle (e.g. an electron) of mass m which is moving through space (given by a position vector t xi y]+ zk) and time (t) under the influence of an external field y (which might be the electrostatic potential due to the nuclei of a molecule). U is Planck s constant divided by 27t and i is the square root of —1. S is the wavefunction which characterises the particle s motion it is from the wavefunction that we can derive various properties of the particle. When the external potential f is independent of time then the wavefunction can be written as the product of a spatial part and a time part (r, f) = p(r)T(f). We shall only consider situations where the potential is independent of time, which enables the time-dependent Schrodinger equation to be written in the more familiar, time-independent form  

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