Kinetic Energy and Local Potentials

Plane waves are eigenfunctions of the kinetic energy operator

$${1 \over 2} \nabla^2 \; e^{i {\bf G} \cdot {\bf r}} = - {1 \over 2} \mid {\bf G} \mid^2 e^{i {\bf G} \cdot {\bf r}} \enspace .$$ (70)

The kinetic energy is therefore easily calculated in Fourier space

$$E_{\rm kin} = \sum_{i} \sum_{\bf G} {1 \over 2} f_{i} \vert{\bf G}\vert^2 \vert c_{i}({\bf G})\vert^2 \enspace ,$$ (71)

and the same is true for the wavefunction forces

$$F_{\rm kin} = {1 \over 2} \vert{\bf G}\vert^2 c_{i}({\bf G}) \enspace .$$ (72)

The plane waves do not depend on the atomic positions, therefore there are no Pulay forces and no contribution of the kinetic energy to the forces on the nuclei.

Local operators act multiplicatively on wavefunctions in real space

$$\int d{\bf r'} \; V({\bf r},{\bf r'}) \Phi({\bf r'}) = V_{\rm loc}({\bf r}) \Phi({\bf r}) \enspace .$$ (73)

The matrix elements of local operators can be calculated from the plane wave expansion of the operator in real space

$$V_{\rm loc}({\bf r}) = \sum_{\bf G} e^{i {\bf G} \cdot {\bf r}}$$ (74)

$$\langle {\bf G}_1 \mid V_{\rm loc}({\bf r}) \mid {\bf G}_2 \rangle = {1 \over \Omega} \sum_{\bf G} V_{\rm loc}({\bf G}) \int d{\bf r} ... ...f G}_1 \cdot {\bf r}} e^{i {\bf G} \cdot {\bf r}} e^{i {\bf G}_2 \cdot {\bf r}}$$ (75)

$$= {1 \over \Omega} \sum_{\bf G} V_{\rm loc}({\bf G}) \int d{\bf r} \; e^{i ({\bf G} - {\bf G}_1 + {\bf G}_2) \cdot {\bf r}}$$ (76)

$$= {1 \over \Omega} V_{\rm loc}({\bf G}_1 - {\bf G}_2) \enspace .$$ (77)

The expectation value only depends on the density

$$E_{\rm loc} = \sum_i f_i \langle \Phi_i \mid V_{\rm loc} \mid \Phi_i \rangle$$ (78)

$$= \int d{\bf r} \; V_{\rm loc}({\bf r}) \left( \sum_i f_i \Phi_i^\star({\bf r}) \Phi_i({\bf r}) \right)$$ (79)

$$= \int d{\bf r} \; V_{\rm loc}({\bf r}) \; n({\bf r})$$ (80)

$$= {1 \over \Omega} \sum_{\bf G} V^\star_{\rm loc}({\bf G}) \; n({\bf G}) \enspace .$$ (81)

Expectation values are calculated in Fourier space as a sum over G-vectors. The local potential is multiplied by the density and therefore only those components of the local potential that are non-zero in the density have to be calculated. Forces are calculated in real space by multiplying the wavefunctions with the potential on the real space grid.