User:Ivan Shmakov/QC Algorithm 2013

(Some formulae for a certain quantum chemistry model I was asked to render. Hopefully these would make it to either here or the Russian-language section.)

$$\begin {align} \bar r' _i &= r _i - \alpha \bar \gamma _i,\quad & i &= 1, \ldots, N;\\ \alpha &= 1.5 N k T      \frac { \sqrt { \sum _{i = 1} ^N \gamma _i ^2 m _i ^{-2}}}{ \sum _{i = 1} ^N \gamma _i ^2 m _i ^{-1}} \end {align}$$

$$\begin {align} \bar \gamma _i &= \sum _{j = 1} ^N B _{ij} \frac {E _{ij}} {\rho _{ij}},\quad & i &= 1, \ldots, N;\\ E _{ij} &= -2 \beta _{ij} \kappa _{ij} \left[\lambda _{ij} \kappa _{ij} - \sqrt {D _{ij} \lambda _{ij}}\right];\\ \kappa _{ij} &= \exp [- \beta _{ij} \rho _{ij}];\\ \lambda _{ij} &= D _{ij} \exp [2 \beta _{ij} R _{ij}];\\ \rho _{ij} &= \sqrt {(\bar r _i - \bar r _j) ^2};\\ \end {align}$$