7.3 Penalty functional and electron number

The penalty functional $P[\rho]$ is defined by

$$P[\rho] = \int {\mathrm d}{\bf r}~ \left(\rho^2(1-\rho)^2\right)({\bf r},{\bf r})$$

$$= \int {\mathrm d}{\bf r}_1 {\mathrm d}{\bf r}_2 {\mathrm d}{\bf r}... ... \right] \left[ \delta({\bf r}_4,{\bf r}_1) - \rho({\bf r}_4,{\bf r}_1) \right]$$

$$= K^{ij} S_{jk} K^{kl} S_{lm} ( \delta^m_p - K^{mn} S_{np} ) ( \delta^p_i - K^{pq} S_{qi} ) .$$ (7.37)

The derivative with respect to the density-kernel is then

$$\frac{\partial P[\rho]}{\partial K^{\alpha \beta}}= 2 S_{\be... ...^{kl} S_{lm} ) ( \delta^m_{\alpha} - 2 K^{mn} S_{n \alpha} ) .$$ (7.38)

The penalty functional depends implicitly upon the support functions though the overlap matrix:

$$\frac{\delta S_{ij}}{\delta \phi_{\alpha}({\bf r})}= \delta_i^{\alpha} \phi_j({\bf r}) + \delta_j^{\alpha} \phi_i({\bf r})$$ (7.39)

so that

$$\frac{\delta P[\rho]}{\delta \phi_{\alpha}({\bf r})}= 4 K^{\... ...lta_m^{\beta} - 2 S_{mn} K^{n \beta} ) \phi_{\beta}({\bf r}) .$$ (7.40)

For the sake of completeness, we now describe the expressions for the electron number and its derivatives.

$$N = 2 \int {\mathrm d}{\bf r}~ \rho({\bf r},{\bf r}) = 2 K^{ij} S_{ji}$$ (7.41)

$$\frac{\partial N}{\partial K^{\alpha \beta}}= 2 S_{\beta \alpha}$$ (7.42)

$$\frac{\delta N}{\delta \phi_{\alpha}({\bf r})}= 2 K^{\alpha \beta} \phi_{\beta}({\bf r})$$ (7.43)