theory_underconstruction.md

Theory

Green's function

$$ G^\sigma(\mathbf{r},\mathbf{r}',\omega) = \sum_i \frac{ \varphi_i^\sigma(\mathbf{r}) \varphi_i^\sigma(\mathbf{r'})} { \omega - \epsilon_i^\sigma - \mathrm{i} \eta} + \sum_a \frac{ \varphi_a^\sigma(\mathbf{r}) \varphi_a^\sigma(\mathbf{r'})} { \omega - \epsilon_a^\sigma + \mathrm{i} \eta} $$

Linear combination of atomic orbitals (LCAO)

$$ \varphi_n^\sigma(\mathbf{r}) = \sum_\mu C_{\mu n}^\sigma \phi_\mu(\mathbf{r}) $$

where $\phi_\mu(\mathbf{r})$ are Gaussian-Type Orbital centered on atom $\mathbf{R_A}$

$$ \phi_\mu(\mathbf{r}) = \mathcal{Y}{lm}(\widehat{ \mathbf{r - R_A}}) \left| \mathbf{r-R_A}\right|^l \sum_b c{ b} e^{ -\alpha_{b} \left| \mathbf{r-R_A}\right|^2 } $$

AO to MO transform

This scales formally as $N^4$:

$$ ( P | p q ) = \sum_\alpha C_{\alpha p} \left[ \sum_\beta C_{\beta q} ( \alpha \beta | P ) \right] $$

Polarizability for imaginary frequencies

$$ (v^{1/2} \chi_0 v^{1/2}){PQ}(\mathrm{i}\omega) = \sum{ia} ( P | i a ) ( Q | i a ) \left[ \frac{1}{\mathrm{i} \omega - \epsilon_a + \epsilon_i} - \frac{1}{\mathrm{i} \omega - \epsilon_i + \epsilon_a} \right] $$

$$ (v^{1/2} \chi v^{1/2}){PQ}(\mathrm{i}\omega) = (v^{1/2} \chi_0 v^{1/2}){PQ}(\mathrm{i}\omega)

• \sum_{R} (v^{1/2} \chi_0 v^{1/2}){PR}(\mathrm{i}\omega) (v^{1/2} \chi_0 v^{1/2}){RQ}(\mathrm{i}\omega) $$

$$ (v^{1/2} \chi v^{1/2})_{PQ}(\mathrm{i}\omega)

\sum_R (v^{1/2} \chi_0 v^{1/2}){PR}(\mathrm{i}\omega) \left[ I - v^{1/2} \chi_0 v^{1/2}(\mathrm{i}\omega) \right]^{-1}{RQ} $$