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17.1 Density Functionals

In the following, $\tilde\rho, \bar\rho$ are the $\alpha,\beta$ spin densities; the total and spin densities are
$\displaystyle \rho$ $\textstyle =$ $\displaystyle \tilde\rho+\bar\rho\; , \qquad
\hat \rho=\tilde\rho-\bar\rho
\; ;$ (1)

the gradients of the density enter through
$\displaystyle \sigma$ $\textstyle =$ $\displaystyle \nabla\rho \cdot \nabla\rho \; , \qquad
\hat \sigma = \nabla\rho ...
...\rho\; , \qquad
\hat {\hat \sigma} = \nabla\hat \rho \cdot \nabla\hat \rho \; .$ (2)
$\displaystyle \upsilon$ $\textstyle =$ $\displaystyle \nabla^2\rho \; , \qquad
\hat\upsilon=\nabla^2\hat\rho \; .$ (3)

Additionally, the kinetic energy density for a set of (Kohn-Sham) orbitals generating the density can be introduced through
$\displaystyle \tau$ $\textstyle =$ $\displaystyle \left(\sum_i^\alpha+\sum_i^\beta\right)
\left\vert{\bf\nabla}\phi...
...\sum_i^\alpha-\sum_i^\beta\right)
\left\vert{\bf\nabla}\phi_i\right\vert^2
\; .$ (4)

All of the available functionals are of the general form

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



Subsections

P.J. Knowles and H.-J. Werner
molpro-support@tc.bham.ac.uk
Mar 8, 2000