Expansion coefficient derivatives

The support functions are expanded in spherical-wave basis functions:

$${\phi}_{\alpha}({\bf r}) = \sum_{n \ell m} c^{n \ell m}_{(\alpha)} ~\chi_{\alpha , n \ell m}({\bf r}) .$$ (7.71)

For a functional of the support functions $f[\{ \phi_{\alpha} \}]$, the derivative with respect to the expansion coefficients is

$$\frac{\partial f[\{ \phi_{\alpha} \}]}{\partial c^{n \ell m}_{(\beta)}} = \sum_{\gamma} \int {\mathrm d}{\bf r}~ \frac{\delta f[ \{ \phi_{\... ...bf r})} \frac{\partial \phi_{\gamma}({\bf r})}{\partial c^{n \ell m}_{(\beta)}}$$

$$= \sum_{\gamma} \int {\mathrm d}{\bf r}~ \frac{\delta f[ \{ \phi_{\... ...hi_{\gamma}({\bf r})} \delta_{\gamma}^{\beta} \chi_{\gamma , n \ell m}({\bf r})$$

$$= \int {\mathrm d}{\bf r}~\frac{\delta f[ \{ \phi_{\alpha} \} ]} {\delta \phi_{\beta}({\bf r})} \chi_{\beta , n \ell m}({\bf r}) .$$ (7.72)

For example, for the derivative of the total energy,

$$\frac{\delta E[\rho]}{\delta \phi_{\alpha}({\bf r})} = 4 K^{\alpha \beta} {\hat H} \phi_{\beta}({\bf r}) ,$$ (7.73)

we obtain

$$\frac{\partial E[\rho]}{\partial c^{n \ell m}_{(\beta)}} = 4 K^{\beta \gamma} \int {\mathrm d}{\bf r}~ \chi_{\beta , n \ell m}({\bf r}) {\hat H} \phi_{\gamma}({\bf r})$$

$$= 4 K^{\beta \gamma} \sum_{n' \ell' m'} \langle \chi_{\beta , n \el... ...t {\hat H} \vert \chi_{\gamma , n' \ell' m'} \rangle c^{n' \ell' m'}_{(\gamma)}$$ (7.74)

in which $\langle \chi_{\beta , n \ell m} \vert {\hat H} \vert \chi_{\gamma , n' \ell' m'} \rangle = {\cal H}_{\beta , n \ell m; \gamma , n' \ell' m}$ denotes the matrix element of the Kohn-Sham Hamiltonian with respect to the spherical-wave basis functions.