Dual-Space Gaussian (Goedecker-Teter-Hutter) Pseudopotentials

Pseudopotentials in the Kleinman-Bylander form have the advantage of requiring minimal amount of work in a plane wave calculation by still keeping most of the transferability and general accuracy of the underlying semilocal pseudopotential. However, one wonders if it would not be possible to generate directly pseudopotentials in the separable form fulfilling the Hamann-Schlüter-Chiang conditions. It was found [102] that indeed it is possible to optimize a small set of parameters defining an analytical form for the local and non-local form of a pseudopotential that fulfills those conditions and reproduces even additional properties leading to highly transferable pseudopotentials.

The local part of the pseudopotential is given by

$$V_{\rm loc}(r) = {- Z_{\rm ion} \over r} \left[\bar{r}/\sqrt{2} \right] + \exp\left[-{1 \over 2} \bar{r}^2... ...s \left[ C_1 + C_2 \bar{r}^2 + C_3 \bar{r}^4 + C_6 \bar{r}^6 \right] \enspace ,$$ (195)

where erf denotes the error function and $\bar{r}=r/r_{\rm loc}$ . $Z_{\rm ion}$ is the ionic charge of the atomic core, i.e. the total charge minus the charge of the valence electrons. The non-local contribution to the pseudopotential is a sum of separable terms

$$V_l({\bf r},{\bf r}^\prime) = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{m=-... ...at{r}) p^l_i(r) \; h^l_{ij} \; p^l_j(r) Y^\star_{lm}(\hat{r}^\prime) \enspace ,$$ (196)

where the projectors $p^l_i(r)$ are Gaussians of the form

$$p^l_i(r) = { \sqrt{2} r^{l+2(i-1)} \exp\left[-{r^2 \over 2 r_l^2}... ... \over r_l^{l+(4i-1)/2} \sqrt{\Gamma\left[l+{4i-1 \over 2}\right]} } \enspace ,$$ (197)

where $\Gamma$ is the gamma function. The projectors are normalized

$$\int_0^{\infty} r^2 p^l_i(r) p^l_i(r) dr = 1 \enspace .$$ (198)

This pseudopotential also has an analytical form in Fourier space. In both real and Fourier space, the projectors have the form of a Gaussian multiplied by a polynomial. Due to this property the dual-space Gaussian pseudopotential is the optimal compromise between good convergence properties in real and Fourier space. The multiplication of the wavefunction with the non-local pseudopotential arising from an atom can be limited to a small region around the atom as the radial projectors asymptotically tend to zero outside the covalent radius of the atom. In addition, a very dense integration grid is not required, as the projector is reasonably smooth because of its good decay properties in Fourier space.

The parameters of the pseudopotential are found by minimizing a target function. This function is build up as the sum of the differences of properties calculated from the all-electron atom and the pseudo-atom. Properties included are the integrated charge and the eigenvalues of occupied and the lowest unoccupied states.