today, DFT is not variational – it can give an energy below the true energy. In the
Kohn–Sham approach the energy of a system is formulated as a deviation from
the energy of an idealized system with noninteracting electrons. The energy of the
idealized system can be calculated exactly since its wavefunction (in the Kohn–
Sham approach wavefunctions and orbitals were introduced as a mathematical
convenience to get at the electron density) can be represented exactly by a Slater
determinant. The relatively small difference between the real energy and the energy
of the idealized system contains the exchange-correlation functional, the only
unknown term in the expression for the DFT energy; the approximation of this
functional is the main problem in DFT. From the energy equation, by minimizing
the energy with respect to the Kohn–Sham orbitals the Kohn–Sham equations can
be derived, analogously to the Hartree–Fock equations. The molecular orbitals of
the KS equations are expanded with basis functions and matrix methods are used to
iteratively refine the energy, and to get a set of molecular orbitals, the KS orbitals,
which are qualitatively similar to the orbitals of wavefunction theory.
The simplest version of DFT, the local density approximation (LDA), which treats
the electron density as corresponding in a restricted way (Section 7.2.3.4a) to that of a
uniform electron gas, and also pairs two electrons of opposite spin in each KS orbital,
is little used nowadays. It has been largely replaced by methods which use gradient-
corrected (“nonlocal”) functionals and which assign one set of spatial orbitals to
a-spin electrons, and another set of orbitals tob-electrons; this latter “unrestricted”
assignment of electrons constitutes the local-spin-density approximation (LSDA).
The best results appear to come from so-called hybrid functionals, which include
some contribution from Hartree–Fock type exchange. The most popular current DFT
method is the LSDA gradient-corrected hybrid method which uses the B3LYP
(Becke three-parameter Lee-Yang-Parr) functional. However, this may soon be
largely replaced by new functionals, like those of the M06 family.
Gradient-corrected and, especially, hybrid functionals, give good to excellent
geometries. Gradient-corrected and hybrid functionals usually give fairly good
reaction energies, but, especially for isodesmic-type reactions, the improvement
over HF/3-21G or HF/6-31G calculations does not seem to be dramatic (as far as
the relative energies of normal, ground-state organic molecules goes; for energies
and geometries of transition metal compounds, DFT is the method of choice). For
homolytic dissociation, correlated methods (e.g. B3LYP and MP2) are vastly better
than Hartree–Fock-level calculations; these methods also give tend to give fairly
good activation barriers.
DFT gives reasonable IR frequencies and intensities, comparable to those from
MP2 calculations. Dipole moments from DFT appear to be more accurate than
those from MP2, and B3LYP/6-31G moments on AM1 geometries are good.
Time-dependent DFT (TDDFT) is the best method (with the possible exception
of semiempirical methods parameterized for the type of molecule of interest) for
calculating UV spectra reasonably quickly. DFT is said to be better than Hartree–
Fock (but not as good as MP2) for calculating NMR spectra. Good first ionization
energies are obtained from B3LYP/6-31+G//B3LYP/3-21G()energy differences
(using AM1 geometries makes little difference, at least with normal molecules).
7.5 Summary 511