expectation_value:
expectation value w.tr.t. the wave functions (valid only for 1 electron)
$
E_{\rm{px}}[\rho] = -\sum_{\alpha=1}^{M_{p}}\frac{\tilde{\lambda}_{\alpha}^{2}}{2\tilde{\omega}_{\alpha}^{2}}
\langle (\tilde{\mathbf{{\varepsilon}}}_{\alpha}\cdot\hat{\mathbf{J}}_{\rm{p}})\Phi[\rho]
| (\tilde{\mathbf{{\varepsilon}}}_{\alpha}\cdot\hat{\mathbf{J}}_{\rm{p}})\Phi[\rho] \rangle
$
This option only works for the wave function based electron-photon functionals
LDA:
energy from electron density
$
E_{\rm pxLDA}[\rho] = \frac{-2\pi^{2}}{(d+2)({2V_{d}})^{\frac{2}{d}}}
\sum_{\alpha=1}^{M_{p}}\frac{\tilde{\lambda}_{\alpha}^{2}}{\tilde{\omega}_{\alpha}^{2}}
\int d\mathbf{r}\ \rho^{\frac{2+d}{d}}(\mathbf{r})
$
This option only works with LDA electron-photon functionals.