3 Theoretical background
3.2 Density functional theory
3.2.2 Exchange-correlation functionals
With the introduction of the Kohn-Sham formalism most contributions to the total energy can be calculated exactly. The remaining unknown parts are assembled in the exchange-correlation functional. Good approximations to EXC[ρ] are crucial to obtain reliable results in a DFT calculation. The first attempt to find an expression for EXC[ρ] was based on the homogeneous electron gas for which the exact exchange-correlation energy is known. In this local density approximation (LDA) the exchange-correlation energy for the homogeneous electron gas is used for the non-homogeneous system. The basic assumption is that exchange and correlation depend only on the local value of the density. One approximates the real inhomogeneous electron density as a sum of small cells each of which has a homogeneous electron density. It is assumed that EXC[ρ(r)] at position r is identical to EXCLDA[ρ(r)] of the homogeneous electron gas of the same density. The exchange-correlation functional is then given by
[
ρ( )
r]
=∫
rρ( )
r εXC(
ρ( )
r)
LDA
XC d
E , (3.26)
where εXC is the exchange-correlation energy per particle of the homogeneous electron gas. EXCLDA
[ρ(r)] can be split into an exchange and a correlation contribution EXLDA and ECLDA. The exchange part can be given analytically:
[ ]
4 33
3 1
4
3 ρ
ρ π
∫
−
= dr
EXLDA , (3.27)
while the correlation energy is only known numerically from quantum Monte Carlo calculations from Ceperley and Alder (1980). The correlation part was parameterized by Vosko et al. (1980) and by Perdew and Wang (1992). Both parameterizations give usually very similar results.
Although the local density approximation is a rather unrealistic model for real systems, it gives for slowly varying electron densities as in simple crystalline metals very accurate results. Even for other systems it is, due to a fortunate error cancellation, often comparable to the Hartree-Fock method. However, the LDA approximation typically
overestimates binding energies and underestimates bond lengths. An improvement on LDA can be achieved by including the first derivative of the electron density, ∇ρ, in the exchange-correlation functional. The exchange-correlation functional can then be written in this generalized gradient approximation (GGA) as:
[
ρ( )
r]
=∫
dr f(
ρ( )
r ,∇ρ( )
r)
EGGAXC . (3.28)
One generalized gradient exchange functional used in this work was presented by Becke (1988), two popular generalized gradient correlation functionals, also used, are the LYP correlation functional of Lee, Yang and Parr (1988) and Perdew’s 1986 correlation functional. Various other generalized gradient exchange-correlation functionals have been developed and can be found, e.g., in the book from Jensen (1999). In most cases GGA functional energies are more reliable than LDA results, but due to their ambiguous definition there is a certain variation in the energies obtained from different GGA functionals. In addition, most GGA calculations are strictly spoken not an ab initio calculation, as some experimental information is used in there construction. Both LDA and GGA calculations can fail badly if the Kohn-Sham non-interacting wave function is not a single Slater determinant, or when the non-interacting energies are nearly degenerate (Perdew and Kurth, 2003).
It should be noted that even if the real wave function cannot be described with a single Slater determinant, DFT is still often able to calculate the energy of such a system provided that the electron density can be fitted to a single determinant ansatz for the Kohn-Sham orbitals and that an exact exchange-correlation functional is known (Gritsenko and Baerends, 1997). Such cases are termed non-interacting pure state Vs
representable. On the other hand, Schipper et al. (1998) demonstrated that for some systems (e.g., for the 1
∑
+g state of the C2 molecule) it is not possible to fit the electron density to a single Slater determinant. For these systems the non-interacting ground state density cannot be represented by a single Kohn-Sham determinant even if the exact exchange-correlation functional is known.Many systems of interest possess an odd number of electrons or have to be treated even with an even number of electrons as radicals. If the exact exchange-correlation functional is available, the Kohn-Sham formalism is in principle suitable for any kind of atom or molecule, regardless of closed-shell or open-shell character. Even if a system with an odd number of electrons is considered, the basic variable remains the total density which can be constructed from the individual α and β spin densities, ρ = ρα + ρβ. In practice, the current exchange-correlation functionals cannot account for open-shell problems in a realistic manner, and an unrestricted ansatz is commonly used.
Spin-density functionals for exchange and correlation are employed that explicitly depend on the α and β spin densities. This unrestricted Kohn-Sham (UKS) approach can capture more of the essential physics in open-shell systems than the spin-independent functionals. However, the use of UKS has some drawbacks. The spatial symmetry of the total wave function (charge density) is often lower than the symmetry of the system. This unphysical phenomenon is termed spatial symmetry breaking and the resulting wave functions are often termed broken symmetry solutions. Additionally, spin unrestricted wave functions are not eigenfunctions of the total spin angular momentum operator Sˆ2 anymore. The deviation of the expectation value <Sˆ2 > from the correct value S(S + 1) is taken as a measure for the “contamination” of the wave function with wave functions of other spin multiplicities. From a fundamental point of view, it is to be noted that the Kohn-Sham determinant in DFT does not represent the true wave function of the real system, but only the wave function of the fictitious non-interacting reference system. The true wave function is not available with DFT. Thus, the deviation of <Sˆ2> is not necessarily a probe for the quality of the UKS energies.
This is in contrast to the unrestricted Hartree-Fock formalism, where the unrestricted Slater determinant is actually meant to represent the true wave function. In practice, however, the performance of the UKS approach depends on the system under consideration, and some severe failures have been noted, see e.g. Bauschlicher et al.
(1999) and Alcami et al. (2000). In summary, the broken symmetry solutions obtained by the UKS scheme provide in many cases with strong non-dynamcial correlation effects surprisingly good results, at the price of unphysical spin densities. Nevertheless the performance of UKS should be evaluated against reliable reference data, and its applicability remains a case-by-case decision.
Using an atomic orbital (AO) basis for the Kohn-Sham orbitals in Eqn. (3.21), just as in the Hartree-Fock scheme, similar equations to the Roothaan matrix equations (Eqn.
(3.17)) for closed shell molecules and the Pople–Nesbet equations (Eqn. (3.18)) for open shell molecules can be derived.