Gauss-Hermit Integration

The matrix element of the non-local pseudopotential

$$V^{\rm nl}({\bf G},{\bf G^\prime}) = \sum_{L} {1 \over \Omega} \int d{\bf r} e^{-i {\bf G} \cdot {\bf r}} \Delta V^{L}({\bf r}) e^{i {\bf G^\prime} \cdot {\bf r}}$$ (178)

$$= \sum_{L} \int_0^{\infty} dr \langle {\bf G} \mid Y_{\rm L} \rangl... ...elta V^{L}(r) \langle Y_{\rm L} \mid {\bf G^\prime} \rangle_{\omega} \enspace ,$$ (179)

where $\langle \cdot \mid \cdot \rangle_{\omega}$ stands for an integration over the unit sphere. These integrals still depend on $r$ . The integration over the radial coordinate is replaced by a numerical approximation

$$\int_0^{\infty} r^2 f(r) dr \approx \sum_i w_i f(r_i) \enspace .$$ (180)

The integration weights $w_i$ and integration points $r_i$ are calculated using the Gauss-Hermit scheme. The non-local pseudopotential is in this approximation

$$V^{\rm nl}({\bf G},{\bf G^\prime}) = \sum_{L} {1 \over \Omega} \sum_i w_i \Delta V^{L}(r_i) \langle {\... ...gle_{\omega}^{r_i} \langle Y_{\rm L} \mid {\bf G^\prime} \rangle_{\omega}^{r_i}$$ (181)

$$= \sum_{L} {1 \over \Omega} \sum_i w_i \Delta V^{L}(r_i) P_i^{L\star}({\bf G}) P_i^{L}({\bf G^\prime}) \enspace ,$$ (182)

where the definition for the projectors $P$

$$P_i^{L}({\bf G}) = \langle Y_{\rm L} \mid {\bf G} \rangle_{\omega}^{r_i}$$ (183)

has been introduced. The number of projectors per atom is the number of integration points (5 - 20 for low to high accuracy) multiplied by the number of angular momenta. For the case of s and p non-local components and 15 integration points this accounts to 60 projectors per atom.

The integration of the projectors can be done analytically

$$P_i^{L}({\bf G}) = \int_{\omega} Y_L^\star(\omega) e^{i G r_i} d\omega$$ (184)

$$= \int_{\omega} Y_L^\star(\omega) 4 \pi \sum_{l=0}^{\infty} i^l j_l(G r_i) \sum_{m\prime=-l}^l Y_{lm\prime}^\star(\omega) Y_{lm\prime}(\bf G) d\omega$$ (185)

$$= 4 \pi i^l j_l(G r_i) Y_L (\hat G) \enspace ,$$ (186)

where the expansion of a plane wave in spherical harmonics has been used. $j_l$ are the spherical Bessel functions and $\hat G$ the angular components of the Fourier vector ${\bf G}$ .