17.1 Density Functionals

In the following, $\tilde\rho, \bar\rho$ are the $\alpha,\beta$ spin densities; the total and spin densities are

$$\rho = \tilde\rho+\bar\rho\; , \qquad \hat \rho=\tilde\rho-\bar\rho \; ;$$ (1)

the gradients of the density enter through

$$\sigma = \nabla\rho \cdot \nabla\rho \; , \qquad \hat \sigma = \nabla\rho ... ...\rho\; , \qquad \hat {\hat \sigma} = \nabla\hat \rho \cdot \nabla\hat \rho \; .$$ (2)

$$\upsilon = \nabla^2\rho \; , \qquad \hat\upsilon=\nabla^2\hat\rho \; .$$ (3)

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

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

All of the available functionals are of the general form

$$F\left[\rho,\hat\rho, \sigma,\hat\sigma,\hat{\hat\sigma}, \tau,\hat\tau, \upsilon,\hat\upsilon \right] = \int d^3{\bf r} K\left(\rho,\hat\rho, \sigma,\hat\sigma,\hat{\hat\sigma}, \tau,\hat\tau, \upsilon,\hat\upsilon \right)$$ (5)