# 斯萊特行列式

## 形式

### 基本形式

${\displaystyle \Psi _{(x_{1},x_{2},\cdots ,x_{n})}={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}\chi _{i(x_{1})}&\chi _{j(x_{1})}&\cdots &\chi _{k(x_{1})}\\\chi _{i(x_{2})}&\chi _{j(x_{2})}&\cdots &\chi _{k(x_{2})}\\\vdots &\vdots &\ddots &\vdots \\\chi _{i(x_{n})}&\chi _{j(x_{n})}&\cdots &\chi _{k(x_{n})}\end{vmatrix}}}$

${\displaystyle {\begin{vmatrix}\chi _{i(x_{1})}&\chi _{j(x_{1})}&\cdots &\chi _{k(x_{1})}\\\chi _{i(x_{2})}&\chi _{j(x_{2})}&\cdots &\chi _{k(x_{2})}\\\vdots &\vdots &\ddots &\vdots \\\chi _{i(x_{n})}&\chi _{j(x_{n})}&\cdots &\chi _{k(x_{n})}\end{vmatrix}}=-{\begin{vmatrix}\chi _{i(x_{2})}&\chi _{j(x_{2})}&\cdots &\chi _{k(x_{2})}\\\chi _{i(x_{1})}&\chi _{j(x_{1})}&\cdots &\chi _{k(x_{1})}\\\vdots &\vdots &\ddots &\vdots \\\chi _{i(x_{n})}&\chi _{j(x_{n})}&\cdots &\chi _{k(x_{n})}\end{vmatrix}}}$

### 其他形式

• 考慮到行列式在書寫過程中的不便，通常人們用右矢的形式代表斯萊特行列式：

${\displaystyle \Psi _{(x_{1},x_{2},\cdots ,x_{n})}=\mid \chi _{i},\chi _{j},\cdots ,\chi _{k}\rangle }$

• 將行列式展開後，可以用置換算子形式來表示斯萊特行列式:

${\displaystyle \Psi _{(x_{1},x_{2},\cdots ,x_{n})}={\frac {1}{\sqrt {N!}}}\sum _{n=1}^{N!}(-1)^{p_{n}}P_{n}\left[\chi _{i(x_{1})}\chi _{j(x_{2})}\cdots \chi _{k(x_{n})}\right]}$

• 對Slater行列式的置換算子形式進一步簡化可以用反對稱化算子形式來表示：

${\displaystyle \Psi _{(x_{1},x_{2},\cdots ,x_{n})}=A\left[\chi _{i(x_{1})}\chi _{j(x_{2})}\cdots \chi _{k(x_{n})}\right]}$

## 應用

${\displaystyle \Phi =\sum _{i}C_{i}\Psi _{i}}$