Chemistry Reference
In-Depth Information
[470]. DFT has become since one of the most popular tools in electronic structure theory due to its excellent
performance to-cost ratio as compared with correlated wave function theory (WFT). The accuracy of DFT depends
on the quality of the exchange-correlation functional.
DENSITY FUNCTIONAL THEORY
Kohn and Hohenberg introduced Density functional theory (DFT) in the 1960`s. This theory allows us to
approximately solve the electron correlation problem with resource efficiency similar to that of Hartree-Fock (HF).
The correlation effect in the formal DFT theory is included in the exchange-correlation potential for which the exact
formula is unknown yielding various approaches for the implementation [468-521].
The Hohenberg-Kohn theorem states that for any system of electrons in an external potential the potential is
determined uniquely, except for a constant, by the ground state density whereas the full many-body wavefunction
and all other properties of the system are also completely determined. They indicated in a second theorem that a
universal functional for the energy of the density could be defined for all electron systems whereas the exact ground
state energy is the global minima for a given external potential. The density which minimizes the functional is the
exact ground state density. Using a contradiction argument they proved that in principle one can find all properties
which are functionals of the electron density. For all many body wavefunctions with the same density, we first
minimize for a given density and then minimize to find the density with the lowest energy.
The idea seems simple. The genius of the work was to realize that this provides a new way to approach the many-
body problem. The original many-body problem was replaced with an independent electron problem which can be
solved by requiring the ground state density to be the same as the exact density. The Kohn-Sham equations can be
divided into the exchange-correlation functional, for which we have an exact theory but unknown functionals and
independent particles. The energy can be minimized with constraints once a form is assumed for the exchange
correlation whereas the eigenvalues are approximations to the energies, to add or substract electrons. The practical
DFT does not provide however a guide of how to construct the energy functionals for systems with homogeneous
and non-homogeneous densities. We can see DFT as an effective single-body problem using the Kohn-Sham
equations and we can rewrite the ground state energy functional in terms of functions including a non-interacting
kinetic energy part as well as an exchange-correlation part that depends on the density [470].
Usage by early workers of real spatially inhomogeneous systems led to gradient-expansion approximation (GEA).
The inclusion of more general density functionals and their derivatives in the early eighties, led to the generalized
gradient approximations (GGA) which however does not describe very well weak interactions such as van der
Waals interactions. Incorporation of the kinetic-energy density in addition to GGA led to early Meta GGAs [519].
Addition of the density and its derivatives led to beyond-GGA developments, i.e Meta-GGAs in order to address
issues such as self-interactions, fourth order gradient expansions and finite exchange potential at the nucleus. Time-
dependent DFT (TD-DFT) can be used to study excited states and spin-DFT (SDFT) employs one density for each
spin. Indirect orbital approaches to minimize the energies include optimized-potential model (OPM) and the self-
interaction correction (SIC) [484].
Summarizing, the last two decades have seen important progress in the development and validation of density
functionals. The LSDA which yields good predictions for solid-state physics is not as useful for chemistry due to
overbinding of chemical bonds and underestimation of barrier heights. The GGA functional yields more accurate
predictions for thermochemistry than LSDA although they still underestimate barrier heights. In the third generation
functionals the spin kinetic energy densities are included yielding the meta-GGAs whereas all these are local
functionals.
Mixing with nonlocal HF exchange yields the hybrid functionals which are more accurate than local functionals for
main-group thermochemistry. Much of the more recent focus is now on including noncovalent interactions.
Extensive applications of DFT methods in chemistry started with the introduction by Becke in 1993 of the B3LYP
functional which largely replaced MP2 in theoretical energy evaluations [470,482]. The B3LYP functional combines
