Kinetic energy

The kinetic energy of the non-interacting Kohn-Sham system is given by

$$T_{\mathrm s}^{\mathrm J} = - \int {\mathrm d}{\bf r'}~ \lef... ...ight]_{{\bf r}={\bf r'}} = 2 K^{\alpha \beta} T_{\beta \alpha}$$ (7.6)

in which

$$T_{\alpha \beta} = -\frac{1}{2} \int {\mathrm d}{\bf r}~ \phi_{\alpha}({\bf r}) \nabla_{\bf r}^2 \phi_{\beta}({\bf r})$$ (7.7)

are the matrix elements of the kinetic energy operator in the representation of the support functions. Since all of the matrix elements between the spherical-wave basis functions can be calculated analytically,

$$T_{\alpha \beta} = \sum_{n \ell m, n' \ell' m'} c^{n \ell m}... ...pha , n \ell m ;\beta , n' \ell' m'} c^{n' \ell' m'}_{(\beta)}$$ (7.8)

where ${\cal T}$ denotes matrix elements of the kinetic energy operator between spherical-wave basis functions.