problems that this equation imposes on ab initio methods. Both the ab initio and the
semiempirical approaches calculate a molecular wavefunction (and molecular
orbital energies), and thus representwavefunction methods. However, a wavefunc-
tion is not a measurable feature of a molecule or atom – it is not what physicists call
an “observable”. In fact there is no general agreement among physicists just what a
wavefunction is – is it “only” a mathematical convenience for calculating observ-
able properties, or is it a real physical entity? [ 1 ].
Density functional theory, DFT, is based not on the wavefunction, but rather on
the electron probability density function or electron density function, commonly
called simply the electron density or the charge density, and designated byr(x,y,z).
This was discussed in Section 5.5.4.5, in connection with atoms-in-molecules
(AIM). This electron densityris the “density” in density functional theory, and
is the basis not only of DFT, but of a whole suite of methods of regarding and
studying atoms and molecules [ 2 ]; unlike the wavefunction, it is measurable, e.g. by
X-ray diffraction or electron diffraction [ 3 ]. Apart from being an experimental
observable and being readily grasped intuitively [ 4 ], the electron density has a
mathematical property particularly suitable for any method with claims to being an
improvement on, or at least a valuable alternative to, wavefunction methods: it is a
function of position only, that is, of justthreevariables (x, y, z), while the
wavefunction of ann-electron molecule is a function of 4nvariables, three spatial
coordinates and one spin coordinate,for each electron. A wavefunction for a ten-
electron molecule will have 40 variables. In contrast, no matter how big the
molecule may be, the electron density remains a function of three variables. The
electron density function, then, trumps the wavefunction in three ways: it is
measurable, it is intuitively comprehensible, and it is mathematically more tractable.
The mathematical term functional, which is akin to function, is explained in
Section7.2.3.1. To the chemist, the main advantage of DFT is that in about the
same time needed for an HF calculation one can often obtain results of about the
same quality as from MP2 calculations (cf. e.g. Sections 5.5.1 and 5.5.2). Chemical
applications of DFT are but one aspect of an ambitious project to recast conven-
tional quantum mechanics, i.e. wave mechanics, in a form in which “the electron
density, and only the electron density, plays the key role” [ 5 ]. It is noteworthy that
the 1998 Nobel Prize in chemistry was awarded to John Pople (Section 5.3.3),
largely for his role in developing practical wavefunction-based methods, and
Walter Kohn,^1 for the development of density functional methods [ 6 ]. The wave-
function is the quantum mechanical analogue of the analytically intractable multi-
body problem (n-body problem) in astronomy [ 7 ], and indeed electron–electron
interaction, electron correlation, is at the heart of the major problems encountered in
(^1) Walter Kohn, born in Vienna 1923. B.A., B.Sc., University of Toronto, 1945, 1946. Ph.D.
Harvard, 1948. Instructor in physics, Harvard, 1948–1950. Assistant, Associate, full Professor,
Carnegie Mellon University, 1950–1960. Professor of physics, University of California at San
Diego, 1960–1979; University of California at Santa Barbara 1979-present. Nobel Prize in
chemistry 1998.
446 7 Density Functional Calculations