The PBE α Functional

The orbital dependent density functional α is based on the inhomogeneity parame-

ter used in the well known TPSS functional[39] and MGGA_MS2 [40] meta-GGA functionals. These meta-GGA functionals form the third rung in the Jacob’s ladder

metaphor used extensively by Perdew in describing the accuracy of density functionals in comparison to experiment[39]. In the metaphor, the Local Spin Density Approx- imation is the lowest rung and uses only the electron density described in equation 2.1 in the formulation of density functionals. The second rung uses the Generalized Gradient Approximation (GGA), on which PBE is formulated, and makes use of the density and the gradient of the density. The meta-GGA functionals described here use another semi-local ingredient in the formulation of density functionals called the kinetic energy density τ.

τ(r) = occ X i,σ |∇φi,σ(r)|2 2 . (3.1)

It should be noted, in equation 3.1, that the kinetic energy density depends on the gradient of the molecular orbitals and that the sum runs over occupied molecular orbitals of α and beta spin. In practice, the kinetic energy density, like electron density in DFT, is calculated on a spatial grid. However, like the density gradient

∇ρ, kinetic energy density is a semi-local term. In some applications, as in the PKZB functional [42], it is used in conjunction with the Weiczacker kinetic energy density to reduce the self-interaction error inherent in DFT.

τw = |∇ρ| 2

8ρ . (3.2)

However, the kinetic energy density is of interest in this chapter because it is shown to be the main ingredient in an important and sensitive density functional

parameter called the α parameter.

α= τ −τ

w

τunif . (3.3)

In equation 4.4, the τunif _{is the kinetic energy density of the uniform electron}

gas; the fundamental model system upon which all density correlation functionals are

comparisons between these three forms of kinetic energy density. The Wieczacker kinetic energy density is the kinetic energy density of a single orbital, and the kinetic energy density of the uniform electron gas most resembles kinetic energy density resulting from metallic bonding. Thus, if the kinetic energy in the bonding region of a molecule is much less than unity, the bond is covalent. If the kinetic energy density in such regions is close to unity, the bonding is more metallic.

As was shown in the SSG(PBE) method, density functionals, when combined with SSG, have an overall inability to recover correlation effects at intermediate bond lengths when bonds are starting to break. We believe that the sensitivity to changes in bond character that theα functional exhibits may be the key to correcting this error within SSG(DFT). However, we keep the computational simplicity of SSG(DFT)

by converting the α parameter into a density functional to mix with the density

functional chosen in the SSG(DFT) method.

A few changes have been made to the α parameter in order to work within the

SSG formulation. First, we modify the kinetic energy densityτ to correctly calculate the kinetic energy density of SSG as opposed to a single-determinant method. The original definition of the kinetic energy density is applied to only occupied orbitals and assumes whole occupation of said orbitals. In comparison, the SSG wavefunction partially occupies all orbitals and a sum over all orbitals φi is required with an

inclusion of the geminal expansion coefficients Di.

τG=X

i,σ

|∇φi,σ|2D2i

2 . (3.4)

We call this modified version of kinetic energy density the geminal kinetic energy density and signify it as τG_{. In addition, the Wieczacker kinetic energy density is}

modified to make a spin dependent version.

τw =X|∇ρσ| 2

We choose this spin dependent τw

sd in our studies due to a better linear fit of the

spin dependent α functional to atomic correlation results for open shell atoms. The

α parameter is also made dimensionless and multiplied by total density to produce

the resultingα functional.

α(ρ) = τ

G_−τw sd

ρ2/3 . (3.6)

The DFT dynamic correlation calculation is performed outside of self consistent

loops on the resulting SSG wavefunction with the separate PBE and α functionals.

The functionals are both scaled down to reproduce known atomic correlation energies for elements of atomic number 2-17 [35, 36] as was done previously for SSG(PBE) calculations. The scaling is done independently, and the coefficients are basis set specific. The resulting scaled down results are then mixed together using optimized mixing prefactors b and b’ with the sum of these prefactors being equal to unity.

ESSG(P BEα) =bEP BE +b0Eα. (3.7)

The mixing prefactors are optimized empirically with respect to root mean squared deviation (RMSD) from equilibrium bond length for the 28 diatomic molecules in the G2/97 test set[38]. In this way, the createdαfunctional attempts to provide the best possible correction to SSG(PBE) by adding more non-local correlation at intermediate bond lengths.