19.1 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

$$\sigma_{\alpha\alpha} = \nabla\rho_\alpha \cdot \nabla\rho_\alpha \; , \sigma_{\beta\beta... ...\; , \sigma = \sigma_{\alpha\alpha}+\sigma_{\beta\beta}+2\sigma _{\alpha\beta}.$$ (1)

$$\chi_\alpha = \frac{\sqrt{\sigma_{\alpha\alpha}}}{\rho_\alpha^{4/3}}\;, \chi_\beta = \frac{\sqrt{\sigma_{\beta\beta}}}{\rho_\beta^{4/3}}\;.$$ (2)

$$\upsilon_\alpha = \nabla^2\rho_\alpha \; , \upsilon_\beta=\nabla^2\rho_\beta \; , \upsilon=\upsilon_\alpha+\upsilon_\beta \;.$$ (3)

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

$$\tau_\alpha = \sum_i^\alpha \left\vert{\bf\nabla}\phi_i\right\vert^2 \; , \tau_... ...ta \left\vert{\bf\nabla}\phi_i\right\vert^2 \;, \tau=\tau_\alpha+\tau_\beta \;.$$ (4)

All of the available functionals are of the general form

$$F\left[\rho_s,\rho_{\bar{s}}, \sigma_{ss},\sigma_{\bar{s}\bar{s}},\sigma_{s\bar{s}}, \tau_s,\tau_{\bar{s}}, \upsilon_s,\upsilon_{\bar{s}} \right] = \int d^3{\bf r} K\left(\rho_s,\rho_{\bar{s}}, \sigma_{ss},\sigma_... ...\sigma_{s\bar{s}}, \tau_s,\tau_{\bar{s}}, \upsilon_s,\upsilon_{\bar{s}} \right)$$ (5)

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