Wavefunction Gradient

Analytic derivatives of the total energy with respect to the parameters of the calculation are needed for stable molecular dynamics calculations. All derivatives needed are easily accessible in the plane wave pseudopotential approach. In the following Fourier space formulas are presented

$${1 \over f_{i}} {\partial E_{\rm total} \over \partial c_{i}^\star({\bf G})} = {1 \over 2} G^2 c_{i}({\bf G})$$

$$+ \sum_{\bf G'} {V_{\rm loc}^\star({\bf G}-{\bf G'})} c_{i}({\bf G'})$$

$$+ \sum_{I} \sum_{\alpha, \beta} \left(F_{I,i}^{\alpha} \right)^\star h_{\alpha \beta}^{I} P^{I}_{\beta}({\bf G}) S_{I}({\bf G}) \enspace ,$$ (215)

where $V_{\rm loc}$ is the total local potential

$$V_{\rm loc}({\bf G}) = \sum_{I} \Delta V_{\rm local}^{I}({\bf G})... ...}) + V_{\rm xc}({\bf G}) + 4 \pi {n_{\rm tot}({\bf G}) \over G^2} \enspace .$$ (216)

Wavefunction gradients are needed in optimization calculations and in the Car-Parrinello molecular dynamics approach.