Computational Chemistry

wavefunction calculations (Section 5.4). It may be significant that early in his career
Kohn worked on a many-body problem in atomic physics [ 8 ].
A question sometimes asked is whether DFT should be regarded as a special kind
of ab initio method. The case against this view is that the correct mathematical form
of the DFT functional is not known, in contrast to conventional ab initio theory where
the correct mathematical form of the fundamental equation, the Schr€odinger equa-
tion, is (we think), known. In conventional ab initio theory, the wavefunction can be
improved in a conceptually straightforward way by going to bigger basis sets and
higher correlation levels, which takes us closer and closer to an exact solution of
the Schr€odinger equation, but in DFT there is so far no such straightforward way to
systematically improve the functional (Sections7.2.3.1and7.2.3.2); one must feel
one’s way forward with help from intuition and comparison of the results with
experiment and with high-level conventional ab initio calculations. One might
argue that in this sense current DFT is semiempirical, but the limited use of empirical
parameters(typically from zero to about ten), and the possibility of one day finding
the exact functional, makes it ab initio in spirit. Indeed, DFT using functionals with
no empirical parameters (below) is mathematically as ab initio as wavefunction
methods. Were the exact functional known, DFT might indeed give “chemically
accurate results a priori” (the Dewar quotation at the start of this chapter). The
question of the semiempirical nature of DFT is briefly taken up again after we have
examined the various levels of the method, at the end of Section7.2.3.4g.

7.2 The Basic Principles of Density Functional Theory....................

7.2.1 Preliminaries......................................................

In the Born interpretation (Section 4.2.6) the square of a one-electron wavefunction
cat any pointXis the probability density (with units of volume"^1 ) for the wavefunc-
tion at that point, and |c|^2 dxdydzis the probability (a pure number) at any moment of
finding the electron in an infinitesimal volumedxdydzaround the point (the probabil-
ity of finding the electronata mathematical point is zero). For amultielectron
wavefunctionCthe relationship between the wavefunctionCand the electron
densityris more complicated, being the number of electrons in the molecule times
the sum over all their spins of the integral of the square of the molecular wavefunction
integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM
discussion). It can be shown [ 9 ]thatr(x, y, z)isrelatedtothe“component”one-
electron spatial wavefunctionsci(the molecular orbitals) of a single-determinant
wavefunctionC(recall from Section 5.2.3.1 that the Hartree–FockCcan be
approximated as a Slater determinant of spin orbitalsciaandcib)by

r¼

Xn

i¼ 1

nijcij^2 $(7:1)

7.2 The Basic Principles of Density Functional Theory 447