22.4 Density Functionals

In the following, $\rho_\alpha$ and $\rho_\beta$ are the $\alpha$ and $\beta$ spin densities; the total spin density is $\rho$;

The gradients of the density enter through

$$\sigma_{\alpha\alpha} = \nabla\rho_\alpha \cdot \nabla\rho_\alpha \; , \sigma_{\beta\beta... ...\; , \sigma = \sigma_{\alpha\alpha}+\sigma_{\beta\beta}+2\sigma _{\alpha\beta}.$$ (5)

$$\chi_\alpha = \frac{\sqrt{\sigma_{\alpha\alpha}}}{\rho_\alpha^{4/3}}\;, \chi_\beta = \frac{\sqrt{\sigma_{\beta\beta}}}{\rho_\beta^{4/3}}\;.$$ (6)

$$\upsilon_\alpha = \nabla^2\rho_\alpha \; , \upsilon_\beta=\nabla^2\rho_\beta \; , \upsilon=\upsilon_\alpha+\upsilon_\beta \;.$$ (7)

Additionally, the kinetic energy density for a set of (Kohn-Sham) orbitals generating the density can be introduced through

$$\tau_\alpha = \sum_i^\alpha \left\vert{\bf\nabla}\phi_i\right\vert^2 \; , \tau_... ...ta \left\vert{\bf\nabla}\phi_i\right\vert^2 \;, \tau=\tau_\alpha+\tau_\beta \;.$$ (8)

All of the available functionals are of the general form

$$F\left[\rho_s,\rho_{\bar{s}}, \sigma_{ss},\sigma_{\bar{s}\bar{s}},\sigma_{s\bar{s}}, \tau_s,\tau_{\bar{s}}, \upsilon_s,\upsilon_{\bar{s}} \right] = \int d^3{\bf r} K\left(\rho_s,\rho_{\bar{s}}, \sigma_{ss},\sigma_... ...\sigma_{s\bar{s}}, \tau_s,\tau_{\bar{s}}, \upsilon_s,\upsilon_{\bar{s}} \right)$$ (9)

where $\bar{s}$ is the conjugate spin to $s$.

Below is a list of keywords for the functionals supported by MOLPRO. Additionally there are a list of alias keywords deatailed in the next section for various combinations of the primary functionals listed below.

B86MGC: X $\alpha\beta\gamma$ with Modified Gradient Correction
B86R: X $\alpha\beta\gamma$ Re-optimised
B86: X $\alpha\beta\gamma$
B88C: Becke88 Correlation Functional
B88: Becke88 Exchange Functional
B95: Becke95 Correlation Functional
B97R: Density functional part of B97 Re-parameterized by Hamprecht et al
B97: Density functional part of B97
BR: Becke-Roussel Exchange Functional
BRUEG: Becke-Roussel Exchange Functional -- Uniform Electron Gas Limit
BW: Becke-Wigner Exchange-Correlation Functional
CS1: Colle-Salvetti correlation functional
CS2: Colle-Salvetti correlation functional
DIRAC: Slater-Dirac Exchange Energy
G96: Gill's 1996 Gradient Corrected Exchange Functional
HCTH120: Handy least squares fitted functional
HCTH147: Handy least squares fitted functional
HCTH93: Handy least squares fitted functional
LTA: Local $\tau$ Approximation
LYP: Lee, Yang and Parr Correlation Functional
MK00B: Exchange Functional for Accurate Virtual Orbital Energies
MK00: Exchange Functional for Accurate Virtual Orbital Energies
P86:
PBEC: PBE Correlation Functional
PBEXREV: Revised PBE Exchange Functional
PBEX: PBE Exchange Functional
PW86:
PW91C: Perdew-Wang 1991 GGA Correlation Functional
PW91X: Perdew-Wang 1991 GGA Exchange Functional
PW92C: Perdew-Wang 1992 GGA Correlation Functional
STEST: Test for number of electrons
TH1: Tozer and Handy 1998
TH2:
TH3:
TH4:
THGFCFO:
THGFCO:
THGFC:
THGFL:
VSXC:
VWN3: Vosko-Wilk-Nusair (1980) III local correlation energy
VWN5: Vosko-Wilk-Nusair (1980) V local correlation energy