二体演算子の期待値
$A$体系の波動関数
\begin{align} \Phi(\boldsymbol{Z}_{1},\cdots,\boldsymbol{Z}_{A}) & = \frac{1}{\sqrt{A!}}\begin{vmatrix}\langle\boldsymbol{r}_{1}|\varphi_{1}\rangle & \cdots & \langle\boldsymbol{r}_{1}|\varphi_{A}\rangle\\ \vdots & \ddots & \vdots\\ \langle\boldsymbol{r}_{A}|\varphi_{1}\rangle & \cdots & \langle\boldsymbol{r}_{A}|\varphi_{A}\rangle\end{vmatrix}\\ & = \frac{1}{\sqrt{A!}}\det\{\varphi_{1}(\boldsymbol{r}_{1})\cdots\varphi_{A}(\boldsymbol{r}_{A})\} \end{align}
準備
一般に、$n\times n$の$行列B$は以下の性質をもつ。$\Delta_{ij}$は$(i,j)$成分の余因子。 \begin{align} |B| & =\sum_{j}B_{ij}\Delta_{ij}\\ \Delta_{ij} & =B_{ji}^{-1}|B|\label{cofactor} \end{align}
一体演算子の期待値の計算には行列式と余因子の関係(\ref{cofactor})を用いたが、二体演算子の場合は行列式と二重余因子の関係を利用する。 更に余因子を拡張することが想像できるが、Laplace展開と呼ばれる。 \begin{align} |B| & =\sum_{j}B_{ij}\Delta_{ij}\\ & =\sum_{k\ell}B_{ik}B_{j\ell}\Delta_{ij,k\ell}\label{dcofactor} \end{align}
$i$行目に関して余因子展開した後の小行列式の$j$行目に関して再び余因子展開をしている。$j,\ \ell=1,\cdots,n-1$を取り得る。 \begin{align} \Delta_{ij,k\ell} & =|B|\begin{vmatrix}B_{ki}^{-1} & B_{kj}^{-1}\\ B_{\ell i}^{-1} & B_{\ell j}^{-1}\end{vmatrix} \end{align}
計算
二体演算子の期待値を計算する。 \begin{align} \mathcal{O}_{ij} & =\sum_{m_{1}n_{1}m_{2}n_{2}}|\ket{m_{1}(\boldsymbol{r}_{i})m_{2}(\boldsymbol{r}_{j})}\bra{m_{1}(\boldsymbol{r}_{i})m_{2}(\boldsymbol{r}_{j})}|\mathcal{O}_{ij}|\ket{n_{1}(\boldsymbol{r}_{i})n_{2}(\boldsymbol{r}_{j})}\bra{n_{1}(\boldsymbol{r}_{i})n_{2}(\boldsymbol{r}_{j})}|\\ & =\sum_{m_{1}n_{1}m_{2}n_{2}}\bra{m_{1}(\boldsymbol{r}_{i})m_{2}(\boldsymbol{r}_{j})}|\mathcal{O}_{ij}|\ket{n_{1}(\boldsymbol{r}_{i})n_{2}(\boldsymbol{r}_{j})}\underbrace{|\ket{m_{1}(\boldsymbol{r}_{i})}\bra{n_{1}(\boldsymbol{r}_{i})}|}_{o_{m_{1}n_{1}}(\boldsymbol{r}_{i})}\underbrace{|\ket{m_{2}(\boldsymbol{r}_{j})}\bra{n_{2}(\boldsymbol{r}_{j})}|}_{o_{m_{2}n_{2}}(\boldsymbol{r}_{j})} \end{align}
\begin{align} & \bra{\Phi}|\sum_{i\ne j}^{A}\mathcal{O}_{ij}|\ket{\Psi}\\ & =\sum_{i\ne j}^{A}\bra{\varphi_{1}(\boldsymbol{r}_{1})\cdots\varphi_{A}(\boldsymbol{r}_{A})}|\mathcal{O}_{ij}|\ket{\det{\psi_{1}(\boldsymbol{r}_{1})\cdots\psi_{A}(\boldsymbol{r}_{A})}}\\ & =\sum_{m_{1}n_{1}m_{2}n_{2}}\bra{m_{1}(\boldsymbol{r}_{1})m_{2}(\boldsymbol{r}_{2})}|\mathcal{O}|\ket{n_{1}(\boldsymbol{r}_{1})n_{2}(\boldsymbol{r}_{2})}\\ & \ \ \ \ \ \ \ \ \sum_{i\ne j}^{A}\bra{\varphi_{1}(\boldsymbol{r}_{1})\cdots o_{m_{1}n_{1}}^{\dagger}(\boldsymbol{r}_{i})\varphi_{i}(\boldsymbol{r}_{i})\cdots o_{m_{2}n_{2}}^{\dagger}(\boldsymbol{r}_{j})\varphi_{j}(\boldsymbol{r}_{j})\cdots\varphi_{A}(\boldsymbol{r}_{A})}|\ket{\det{\psi_{1}(\boldsymbol{r}_{1})\cdots\psi_{A}(\boldsymbol{r}_{A})}}\\ & =\sum_{m_{1}n_{1}m_{2}n_{2}}\bra{m_{1}(\boldsymbol{r}_{1})m_{2}(\boldsymbol{r}_{2})}|\mathcal{O}|\ket{n_{1}(\boldsymbol{r}_{1})n_{2}(\boldsymbol{r}_{2})}\sum_{i\ne j}^{A}\begin{vmatrix}\bra{\varphi_{1}}|\ket{\psi_{1}} & \cdots & \bra{\varphi_{1}}|\ket{\psi_{A}}\\ \vdots & & \vdots\\ \bra{\varphi_{i}}|o_{m_{1}n_{1}}|\ket{\psi_{1}} & \cdots & \bra{\varphi_{i}}|o_{m_{1}n_{1}}|\ket{\psi_{A}}\\ \vdots & & \vdots\\ \bra{\varphi_{j}}|o_{m_{2}n_{2}}|\ket{\psi_{1}} & \cdots & \bra{\varphi_{j}}|o_{m_{2}n_{2}}|\ket{\psi_{A}}\\ \vdots & & \vdots\\ \bra{\varphi_{A}}|\ket{\psi_{1}} & \cdots & \bra{\varphi_{A}}|\ket{\psi_{A}}\end{vmatrix}\\ & =\sum_{m_{1}n_{1}m_{2}n_{2}}\bra{m_{1}(\boldsymbol{r}_{1})m_{2}(\boldsymbol{r}_{2})}|\mathcal{O}|\ket{n_{1}(\boldsymbol{r}_{1})n_{2}(\boldsymbol{r}_{2})}\sum_{i\ne j}^{A}\sum_{k\ell=1}\bra{\varphi_{i}}|o_{m_{1}n_{1}}|\ket{\psi_{k}}\bra{\varphi_{j}}|o_{m_{2}n_{2}}|\ket{\psi_{\ell}}\Delta_{ij,k\ell}\\ & =\sum_{ijk\ell}\bra{\varphi_{i}\varphi_{j}}|\mathcal{O}|\ket{\psi_{k}\psi_{\ell}}\Delta_{ij,k\ell}\\ & =\sum_{ijk\ell}\bra{\varphi_{i}\varphi_{j}}|\mathcal{O}|\ket{\psi_{k}\psi_{\ell}}|B|\begin{vmatrix}B_{ki}^{-1} & B_{kj}^{-1}\\ B_{\ell i}^{-1} & B_{\ell j}^{-1}\end{vmatrix}\\ & =\bra{\Phi}|\ket{\Psi}\sum_{ijk\ell}\bra{\varphi_{i}\varphi_{j}}|\mathcal{O}|\ket{\psi_{k}\psi_{\ell}}\begin{vmatrix}B_{ki}^{-1} & B_{kj}^{-1}\\ B_{\ell i}^{-1} & B_{\ell j}^{-1}\end{vmatrix} \end{align}
\begin{align} \bra{\Phi}|\frac{1}{2}\sum_{i\ne j}^{A}\mathcal{O}_{ij}|\ket{\Psi} = \boxed{ \frac{1}{2}\bra{\Phi}|\ket{\Psi}\sum_{ijk\ell}\bra{\varphi_{i}\varphi_{j}}|\mathcal{O}|\ket{\psi_{k}\psi_{\ell}}\begin{vmatrix}B_{ki}^{-1} & B_{kj}^{-1}\\ B_{\ell i}^{-1} & B_{\ell j}^{-1}\end{vmatrix} } \end{align}
参考文献
- D.$\ $M. Brink, in Proceedings of the International School of Physics Enrico Fermi, Course 36, Varenna, edited by C. Bloch (AcademicPress, New York, 1966)
- Chapter III. Kernels of GCM, RGM and OCM and Their Calculation Methods | Progress of Theoretical Physics Supplement | Oxford Academic
