Essentials of Computational Chemistry

8
Density Functional Theory
8.1 Theoretical Motivation
8.1.1 Philosophy
What a strange and complicated beast a wave function is. This function, depending on one
spin and three spatial coordinates for every electron (assuming fixed nuclear positions), is not,
in and of itself, particularly intuitive for systems of more than one electron. Indeed, one might
approach the HF approximation as not so much a mathematical tool but more a philosophical
one. By allowing the wave function to be expressed as a Slater determinant of one-electron
orbitals, one preserves for the chemist some semblance of clarity by permitting each electron
still to be thought of as being loosely independent. It was noted in Section 7.6.1 that wave
functions can be dramatically improved in quality by removing this rough independence
and including in the wave functional form a dependence on interelectronic distances (R12
methods). The key disadvantage? The wave function itself is essentially uninterpretable – it
is an inscrutable oracle that returns valuably accurate answers when questioned by quantum
mechanical operators, but it offers little by way of sparking intuition.
One may be forgiven for stepping back from the towering edifice of molecular orbital
theory and asking, shouldn’t things be simpler? For instance, rather than having to work with
a wave function, which has rather odd units of probability density to the one-half power,
why can’t we work with some physical observable in determining the energy (and possibly
other properties) of a molecule? That such a thing should be possible would probably not
have surprised physicists before the discovery of quantum mechanics, insofar as such simple
formalisms are widely available in classical systems.
However, we may take advantage of our knowledge of quantum mechanics in asking about
what particular physical observable might be useful. Having gone through the exercise of
constructing the Hamiltonian operator and showing the utility of its eigenfunctions, it would
be sufficient to our task simply to find a physical observable that permitted the apriori
construction of the Hamiltonian operator. What then is needed? The Hamiltonian depends
only on the positions and atomic numbers of the nuclei and the total number of electrons.
The dependence on total number of electrons immediately suggests that a useful physical
observable would be the electron density ρ, since, integrated over all space, it gives the total
Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer
© 2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)