Slater-Condon rule for a two-electron operator

The Slater-Condon rule for a two-electron operator is an expression of the expectation value of the two-electron operator involving Slater determinants.

Let’s consider the expectation value of the two-electron Hamiltonian operator: $\langle\Psi_a\vert\hat h\vert\Psi_a\rangle$ , where $\hat h=\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\frac{1}{r_{ij}}$ and $\Psi_a$ is given by eq66.

Substituting $\hat h=\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\frac{1}{r_{ij}}$ in eq85,

$\langle\Psi_a\vert\hat h\vert\Psi_a\rangle=\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\int\cdots\int\Psi_s^*\frac{1}{r_{ij}}\sum_{t=1}^{n!}(-1)^t\hat P_t\Psi_s dx_1\cdots dx_n$

Substituting $\Psi_s=\phi_1(x_1)\phi_2(x_2)\cdots\phi_x(x_n)$ in the above equation and simplifying, we have

$\langle\Psi_a\vert\hat h\vert\Psi_a\rangle=\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\biggr[\int\phi_1^*(x_1)\phi_1(x_1)dx_1\cdots\int\int\phi_i^*(x_i)\phi_j^*(x_j)\frac{1}{r_{ij}}\phi_i(x_i)\phi_j(x_j)dx_idx_j$ $\cdots\int\phi_n^*(x_n)\phi_n(x_n)dx_n-\int\phi_1^*(x_1)\phi_1(x_1)dx_1$ $\cdots\int\int\phi_i^*(x_i)\phi_j^*(x_j)\frac{1}{r_{ij}}\phi_j(x_i)\phi_i(x_j)dx_idx_j\cdots\int\phi_n^*(x_n)\phi_n(x_n)dx_n\biggr]$

$\langle\Psi_a\vert\hat h\vert\Psi_a\rangle=\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\left(J_{ij}-K_{ij}\right)=\frac{1}{2}\sum_{j=i}^n\sum_{i=1}^n\left(J_{ij}-K_{ij}\right)\;\;\;\;\;\;\;\;(89)$

where $J_{ij}=\int\int\phi_i^*(x_i)\phi_j^*(x_j)\frac{1}{r_{ij}}\phi_i(x_i)\phi_j(x_j)dx_idx_j$ and $K_{ij}=\int\int\phi_i^*(x_i)\phi_j^*(x_j)\frac{1}{r_{ij}}\phi_j(x_i)\phi_i(x_j)dx_idx_j$ .

$J_{ij}$ and $K_{ij}$ are known as the Coulomb integral and the exchange integral, respectively.

Question

Show that $\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\left(J_{ij}-K_{ij}\right)=\frac{1}{2}\sum_{j=i}^n\sum_{i=1}^n\left(J_{ij}-K_{ij}\right)$ .

Answer

Since $J_{kk}-K_{kk}=0$ , we can add $n$ terms of $J_{kk}-K_{kk}$ , where $k=1,\cdots,n$ to $\frac{1}{2}\sum_{j \neq i}\sum_{i=1}^n\left(J_{ij}-K_{ij}\right)$ , resulting in $\frac{1}{2}\sum_{j=i}^n\sum_{i=1}^n\left(J_{ij}-K_{ij}\right)$ , i.e.

Given that any expectation value of a physical property of an electron is invariant to a unitary transformation of the electron’s wavefunction expressed as a Slater determinant, we substitute eq74, eq75, eq76 and eq77 in $\sum_{j=i}^n\sum_{i=1}^nJ_{ij}$ of eq89 to give

$\sum_{j=i}^n\sum_{i=1}^nJ_{ij}=\int\int\sum_{r,t,s,j,q,i=1}^nu_{iq}^*u_{ir}\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_q'(x)\phi_s'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}u_{js}^*u_{jt}dxdx'$

where we have changed the dummy variables from $x_i$ and $x_j$ to $x$ and $x'$ respectively.

Substituting eq82 in the above equation and simplifying

$\sum_{j=i}^n\sum_{i=1}^nJ_{ij}=\int\int\sum_{r,t,s,j,q=1}^n\delta_{qr}\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_q'(x)\phi_s'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}u_{js}^*u_{jt}dxdx'$ $=\int\int\sum_{r,t,s,j=1}^nu_{js}^*u_{jt}\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_r'(x)\phi_s'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}dxdx'$
Similarly, substituting eq83 in the above equation and simplifying

$\sum_{j=i}^n\sum_{i=1}^nJ_{ij}=\int\int\sum_{r,t}^n\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_r'(x)\phi_t'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}dxdx'\;\;\;\;\;\;\;\;(90)$

Substituting eq74, eq75, eq76 and eq77 in $\sum_{j=i}^n\sum_{i=1}^nK_{ij}$ of eq89 and repeating the above logic, we have

$\sum_{j=i}^n\sum_{i=1}^nK_{ij}=\int\int\sum_{r,t}^n\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_t'(x)\phi_r'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}dxdx'\;\;\;\;\;\;\;\;(91)$

Substitute eq90 and eq91 back in eq89

$\langle\Psi_a\vert\hat h\vert\Psi_a\rangle=\frac{1}{2}\sum_{r=i}^n\sum_{t=1}^n\int\int\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_r'(x)\phi_t'(x')-\phi_t'(x)\phi_r'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}dxdx'$ $=\frac{1}{2}\sum_{t\neq r}\sum_{t=1}^n\int\int\phi_r^{'*}(x)\phi_t^{'*}(x')\frac{\phi_r'(x)\phi_t'(x')-\phi_t'(x)\phi_r'(x')}{\vert\boldsymbol{\mathit{r}}-\boldsymbol{\mathit{r}}'\vert}dxdx'\;\;\;\;\;\;\;\;(92)$

Eq92 is used in the derivation of the Hartree-Fock equations.

Slater-Condon rule for a two-electron operator

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