# 双电子积分 (量子化学)

## 双电子积分的基本形式

${\displaystyle \int dx_{1}dx_{2}\chi _{1}^{*}(x_{1})\chi _{2}^{*}(x_{2}){\frac {1}{r_{12}}}\chi _{1}(x_{1})\chi _{2}(x_{2})}$

${\displaystyle \langle \chi _{1}(x_{1})\chi _{2}(x_{2})|{\frac {1}{r_{12}}}|\chi _{1}(x_{1})\chi _{2}(x_{2})\rangle }$

## 物理符号与化学符号

### 物理符号

${\displaystyle \int dx_{1}dx_{2}\chi _{i}^{*}(x_{1})\chi _{j}^{*}(x_{2}){\frac {1}{r_{12}}}\chi _{k}(x_{1})\chi _{l}(x_{2})}$

${\displaystyle \langle ij|kl\rangle }$ =${\displaystyle \langle kl|ij\rangle }$

${\displaystyle \langle ij||kl\rangle =\langle ij|kl\rangle -\langle ij|lk\rangle }$

${\displaystyle \int dx_{1}dx_{2}\chi _{i}^{*}(x_{1})\chi _{j}^{*}(x_{2})*{\frac {1-P_{12}}{r_{12}}}\chi _{k}(x_{1})\chi _{l}(x_{2})}$

${\displaystyle \langle ij||kl\rangle =\langle ji||lk\rangle }$

${\displaystyle \langle ij||kl\rangle =-\langle ij||lk\rangle }$

### 化学符号

${\displaystyle \int dx_{1}dx_{2}\chi _{i}^{*}(x_{1})\chi _{j}^{*}(x_{2}){\frac {1}{r_{12}}}\chi _{k}(x_{1})\chi _{l}(x_{2})}$

${\displaystyle [ij|kl]=[ji|kl]=[ij|lk]=[ji|lk]}$  ${\displaystyle =[kl|ij]=[kl|ji]=[lk|ij]=[lk|ji]}$

${\displaystyle [ik||jl]=[ik|jl]-[il|jk]}$

${\displaystyle (ik|jl)}$

### 物理符号与化学符号的关系

${\displaystyle \langle ij|kl\rangle =[ik|jl]}$

${\displaystyle \langle ij||kl\rangle =[ik||jl]}$