User:楊肇峰/DFT

$\left[-{\frac {1}{2}}\nabla ^{2}+V(\mathbf {r} )\right]\varphi _{i}(\mathbf {r} )+\sum _{j\neq i}\int {\frac {|\varphi _{j}(r')|^{2}}{|r-r'|}}\varphi _{i}(r)dr'-\sum _{j\neq i,||}\int {\frac {\varphi _{j}^{*}(r')\varphi _{i}(r')}{|r-r'|}}\varphi _{j}(r)dr'=E_{i}\varphi _{i}(\mathbf {r} )$

${\hat {H}}=-{\frac {\hbar ^{2}}{2{m_{e}}}}\sum _{i}\nabla _{\mathbf {r} _{i}}^{2}-{\frac {\hbar ^{2}}{2{M_{i}}}}\sum _{i}\nabla _{\mathbf {R} _{i}}^{2}-\sum _{i,j}{\frac {Z_{i}e^{2}}{|\mathbf {R} _{i}-\mathbf {r} _{j}|}}+{\frac {1}{2}}\sum _{i\neq j}{\frac {e^{2}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}+{\frac {1}{2}}\sum _{i\neq j}{\frac {e^{2}Z_{i}Z_{j}}{|\mathbf {R} _{i}-\mathbf {R} _{j}|}}$

$n(r)=\sum _{\sigma }n(r,\sigma )=\sum _{\sigma }\sum _{i=1}^{N^{\sigma }}|\psi _{i}^{\sigma }(r)|^{2}$

$T_{s}=-{\frac {1}{2}}\sum _{\sigma }\sum _{i=1}^{N^{\sigma }}\langle \psi _{i}^{\sigma }|\nabla ^{2}|\psi _{i}^{\sigma }\rangle ={\frac {1}{2}}\sum _{\sigma }\sum _{i=1}^{N^{\sigma }}d^{3}r|\nabla \psi _{i}^{\sigma }(r)|^{2}$

$E_{Hartree}={\frac {1}{2}}\int d^{3}rd^{3}r'{\frac {n(r)n(r')}{|r-r'|}}$

$E_{KS}=T_{s}[n]+\int drV_{ext}(r)n(r)+E_{Hartree}[n]+E_{11}+E_{xc}[n]$

$V_{ks}=V_{ext}(r)+{\frac {\delta E_{Hartree}}{\delta n(r)}}+{\frac {\delta E_{xc}}{\delta n(r)}}=V_{ext}(r)+V_{Hartree}(r)+V_{xc}(r)$

$E_{tot}[n_{0}]=\sum \varepsilon _{i}-\int n_{0}(r)[V_{Hartree}(r)+V_{xc}(r)]d^{3}r+E_{Hartree}[n_{0}]+E_{xc}[n_{0}]$

$E_{xc}^{LDA}[n]=\int n(r)\varepsilon _{xc}^{LDA}(n(r))d^{3}r$

$E_{xc}^{GGA}[n]=\int n(r)\varepsilon _{xc}^{GGA}(n(r),\nabla n(r))d^{3}r$

$E_{xc}=\int _{0}^{1}d\lambda U_{xc}$