This sum is over thenoccupied MOscifor a closed-shell molecule, for a total of
2 nelectrons. Equation7.1 applies strictly only to a single-determinant wave-
functionC, but for multideterminant wavefunctions arising from configuration
interaction treatments (Section 5.4) there are similar equations [ 10 ]. A shorthand
forr(x, y, z)dxdydzisr(r)dr, whereris the position vector of the point with
coordinates (x,y,z).
If the electron densityrrather than the wavefunction could be used to calculate
molecular geometries, energies, etc., this might be an improvement over the
wavefunction approach because, as mentioned above, the electron density in an
n-electron molecule is a function of only the three spatial coordinatesx,y,z, but the
wavefunction is a function of 4ncoordinates. Density functional theory seeks to
calculate all the properties of atoms and molecules from the electron density. A
good (and rather “technical”) work on DFT is the magisterial book by Parr and
Yang (1989) [ 11 ]. More recent developments are included in the book by Koch and
Holthausen [ 12 ]. Levine gives a very good yet compact introduction to DFT [ 13 ],
and among the many reviews are those by Friesner et al. [ 14 ], Kohn et al. [ 15 ], and
Parr and Yang [ 16 ]. Quite thorough reviews of DFT are given by Cramer [ 17 ] and
by Jensen [ 18 ].
References oriented toward the development and performance of functionals are
given in Section7.2.3.4.
7.2.2 Forerunners to Current DFT Methods
The idea of calculating atomic and molecular properties from electron density
appears to have arisen from calculations made independently by Enrico Fermi
and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known
as the Fermi-Dirac statistics [ 19 ]. In independent work by Fermi [ 20 ]andThomas
[ 21 ], atoms were modelled as systems withapositivepotential(thenucleus)
located in a uniform (homogeneous) electron gas. Thisobviously unrealistic
idealization, the Thomas-Fermi model [ 22 ], or with embellishments by Dirac
the Thomas-Fermi-Dirac model [ 22 ], gave surprisingly good results for atoms,
but failed completely for molecules: itpredicted all molecules to be unstable
toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi
theory).
The Xa(X¼exchange,ais a parameter in the Xaequation) method gives much
better results [ 23 , 24 ]. It can be regarded as a more accurate version of the Thomas-
Fermi model, and is probably the first chemically useful DFT method. It was
introduced in 1951 [ 25 ] by Slater, who regarded it [ 26 ] as a simplification of the
Hartree–Fock (Section 5.2.3) approach. The Xamethod, which was developed
mainly for atoms and solids, has also been used for molecules, but has been
replaced by the more accurate Kohn–Sham type (Section7.2.3) DFT methods.
448 7 Density Functional Calculations