User:Partialoff/sandbox

Density functional perturbation theory

Overview
Density-functional perturbation theory (DFPT) arose as an extension of density functional theory (DFT) to calculate properties that arise from perturbations from the electronic ground state of a system.

Formalism
The Hohenberg-Kohn theorem shows that the expectation values of quantum operators can be written in terms of the density


 * $$ \hat H \Psi = \left[\hat T + \hat V + \hat U\right]\Psi = \left[\sum_{i=1}^N \left(-\frac{\hbar^2}{2m_i} \nabla_i^2\right) + \sum_{i=1}^N V(\mathbf r_i) + \sum_{i<j}^N U\left(\mathbf r_i, \mathbf r_j\right)\right] \Psi = E \Psi, $$

Applications
Calculating the Cutoff Energy:

$$ \phi_{j,\vec{k}}(\vec{r}) = \sum_{j=1}^\infty C_{j,\vec{k}}e^{i(\vec{k}+\vec{G}_{j}) \cdot \vec{r}} \approx \sum_{j=1}^{||\vec{k}+\vec{G}||^2/2 \leq E_{cut}} C_{j,\vec{k}}e^{i(\vec{k}+\vec{G}_{j}) \cdot \vec{r}} $$

where,

$$ \vec{G} = \frac{2\pi}{\vec{a_1}\cdot(\vec{a_2} \times \vec{a_3})}[\nu_1(\vec{a_1}\times \vec{a_3}) + \nu_2(\vec{a_2}\times \vec{a_3}) + \nu_3(\vec{a_3}\times\vec{a_1})] $$

where,

$$ \nu_1,\nu_2,\nu_3 \in \mathbb{Z} $$

The KPoint mesh is calculated as follows:

$$ n(\vec{r}) = \int_{v_{BZ}}d^3 \vec{k} \sum_{j=1}^{N_{KS}} \left| \phi_{j,\vec{k}}(\vec{r}) \right|^2 \approx \sum_{k\in\{\vec{k}\}}\sum_{j=1}^{N_{KS}} \left|\phi_{j,\vec{k}}(\vec{r})\right|^2 $$

The energy convergence threshold is as follows:

$$ {^m}E[n] - {^{m-1}}E[n] < E_{threshold}, m = \text{SCF iteration}, $$

$$ {^m}E[n] = \sum_{j=1}^{N_{KS,occ}} {^m}\varepsilon_j + \int d^3 \vec{r} n \epsilon_{XC}(n, \vec{\nabla}n, ...) $$