The Forms of DFT Functionals

The approximate functionals employed by current DFT methods partition the electronic energy into several terms:

E = E^T + E^V + E^J + E^{XC}

where ET is the kinetic energy term (arising from the motion of the electrons), EV includes terms describing the potential energy of the nuclear-electron attraction and of the repulsion between pairs of nuclei, EJ is the electron-electron repulsion term (it is also described as the Coulomb self-interaction of the electron density), and EXC is the exchange-correlation term and includes the remaining part of the electron-electron interactions.

All terms except the nuclear-nuclear repulsion are functions of ρ, the electron density. EJ is given by the following expression:

E^J = \frac{1}{2}\iint \! \varrho({\vec{r}}_1)({\Delta}r_{12})^{-1}\varrho(\vec{r}_2) \, d{\vec{r}}_1 d{\vec{r}}_2

ET+EV+EJ corresponds to the classical energy of the charge distribution ρ. The EXC term in the preceding equation accounts for the remaining terms in the energy:

  • The exchange energy arising from the antisymmetry of the quantum mechanical wavefunction.
  • Dynamic correlation in the motions of the individual electrons.

Hohenberg and Kohn demonstrated that EXC is determined entirely by the (is a functional of) the electron density. In practice, EXC is usually approximated as an integral involving only the spin densities and possibly their gradients:

E^{XC}(\varrho)= \int \! f(\varrho_{\alpha}(\vec{r}),\varrho_{\beta}(\vec{r}),\nabla\varrho_{\alpha}(\vec{r}),\nabla\varrho_{\beta}(\vec{r})) \, d^3\vec{r}

We use ρα to refer to the alpha spin density, ρβ to refer to the beta spin density, and ρ to refer to the total electon density (ρα + ρβ).

EXC is usually divided into separate parts, referred to as the exchange and correlation parts, but actually corresponding to same-spin and mixed-spin interactions, respectively:

E^{XC}(\varrho) = E^X(\varrho) + E^C(\varrho)

All three terms are again functionals of the electron density, and functionals defining the two components on the right side of the preceding equation are termed exchange functionals and correlation functionals, respectively. Both components can be of two distinct types: local functionals depend on only the electron density ρ, while gradient-corrected functionals depend on both ρ and its gradient, ∇ρ.

We’ll now take a brief look at some sample functionals. The local exchange functional is virtually always defined as follows:

E_{LDA}^X = -\frac{3}{2}(\frac{3}{4\pi})^{\frac{1}{3}} \int \! \varrho^{\frac{4}{3}} \, d^3\vec{r}

where ρ is of course a function of \vec{r}. This form was developed to reproduce the exchange energy of a uniform electron gas. By itself, however, it has weaknesses in describing molecular systems.

Becke formulated the following gradient-corrected exchange functional based on the LDA exchange functional in 1988, which is now in wide use:

E_{Becke88}^X = E_{LDA}^X -\gamma \int \! \frac{\varrho^{\frac{4}{3}}x^2}{(1+6\gamma\sinh^{-1}x)} \, d^3\vec{r}

where x = ρ–4/3|∇ρ|. γ is a parameter chosen to fit the known exchange energies of the inert gas atoms, and Becke defines its value as 0.0042 hartrees. As the preceding equation makes clear, Becke’s functional is defined as a correction to the local LDA exchange functional, and it succeeds in remedying many of the LDA functional’s deficiencies.

Similarly, there are local and gradient-corrected correlation functionals. For example, here is Perdew and Wang’s formulation of the local part of their 1991 correlation functional:

E^C = \int \! \varrho\varepsilon_C(r_s(\varrho(\vec{r})),\zeta) \, d^3\vec{r}, \text{ where } r_s = [\frac{3}{4\pi\varrho}]^{\frac{1}{3}}, \text{ } \zeta = \frac{\varrho_\alpha - \varrho_\beta}{\varrho_\alpha + \varrho_\beta},

\varepsilon_C(r_s,\zeta) = \varepsilon_C(\varrho,0) + a_C(r_s)\frac{f(\zeta)}{f''(0)}(1-\zeta^4) +[\varepsilon_C(\varrho,1)-\varepsilon_C(\varrho,0)]f(\zeta)\zeta^4

f(\zeta) = \frac{[(1+\zeta)^{4/3}+(1-\zeta)^{4/3}-2]}{(2^{4/3}-2)}

rs is termed the density parameter. ζ is the relative spin polarization. ζ=0 corresponds to equal α and β densities, ζ=1 correponds to all α density, and ζ=-1 corresponds to all β density. Note that f(0)=0 and f(±1)=1.

The general expression for εC involves both ρs and ζ. Its final term performs an interpolation for mixed spin cases.

The following function G is used to compute the values of εC(rs,0), εC(rs,1) and –aC(rs):

G(r_s,A,\alpha_1,\beta_1,\beta_2,\beta_3,\beta_4,P) = -2A(1+\alpha_1 r_s) \ln(1+\frac{1}{2A(\beta_1 r_s^{1/2}+\beta_2 r_s + \beta_3 r_s^{3/2} + \beta_4 r_s^{P+1})})

In the preceding equation, all of the arguments to G except rs are parameters chosen by Perdew and Wang to reproduce accurate calculations on uniform electron gases. The parameter sets differ for G when it is used to evaluate each of &epsilonC(rs,0), εC(rs,1) and –aC(rs).

In an analogous way to the exchange functional we examined earlier, a local correlation functional may also be improved by adding a gradient correction.

so-called pure DFT methods are defined by pairing an exchange functional with a correlation functional. For example, the well-known BLYP functional pairs Becke’s gradient-corrected exchange functional with the gradient-corrected correlation functional of Lee, Yang and Parr.

In actual practice, self-consistent Kohn-Sham DFT calculations are performed in an iterative manner that is analogous to an SCF computation. This similarity to the methodology of Hartree-Fock theory was pointed out by Kohn and Sham.

Hartree-Fock theory also includes an exchange term as part of its formulation. Becke was the first to formulate hybrid functionals which include a mixture of Hartree-Fock and DFT exchange along with DFT correlation, conceptually defining EXC as:

E_{hybrid}^{XC} = c_{HF}E_{HF}^X + c_{DFT}E_{DFT}^{XC}

where the c’s are constants. For example, a Becke-style three-parameter functional can be defined via the following expression:

E_{B3LYP}^{XC} = E_{LDA}^X + c_0(E_{HF}^X - E_{LDA}^X) + c_X{\Delta}E^X_{B88} + E_{VWN3}^C + c_C(E^C_{LYP} - E^C_{VWN3})

Here, the parameter c0 allows any admixture of Hartree-Fock and LDA local exchange to be used. In addition, Becke’s gradient correction to LDA exchange is also included, scaled by the parameter cX. Similarly, the VWN3 local correlation functional is used, and it may be optionally corrected by the LYP correlation correction via the parameter cC. In the B3LYP functional, the parameters values are those specified by Becke, which he determined by fitting to the atomization energies, ionization potentials, proton affinities and first-row atomic energies in the G1 molecule set: c0=0.20, cX=0.72 and cC=0.81. Note that Becke used the the Perdew-Wang 1991 correlation functional in his original work rather than VWN3 and LYP. The fact that the same coefficients work well with different functionals reflects the underlying physical justification for using such a mixture of Hartree-Fock and DFT exchange first pointed out by Becke.

Different functionals can be constructed in the same way by varying the component functionals—for example, by substituting the Perdew-Wang 1991 gradient-corrected correlation functional for LYP—and by adjusting the values of the three parameters.