# Deriving the equations for two-electron integrals in Hartree-Fock theory

In Szabo and Ostlund's Modern Quantum Chemistry, the matrix elements for the hamiltonian of a two-electron system, with an operator $$\hat O$$, are written on pages 64-66 as

$$\langle \Psi_0 | \hat O | \Psi_0 \rangle=\int dx_1 dx_2 [2^{-\frac{1}{2}} (\chi_1(x_1)\chi_2(x_2)-\chi_2(x_1)\chi_1(x_2))]^* \cdot \hat O [2^{-\frac{1}{2}} (\chi_1(x_1)\chi_2(x_2)-\chi_2(x_1)\chi_1(x_2))]$$

for a spin orbital $$\chi$$ of electron x defined as the product of a spacial orbital $$\psi(r)$$ and spin $$\alpha(\omega)$$ or $$\beta(\omega)$$, e.g., $$\chi(x)=\psi(r)\alpha(\omega)$$.

This is multiplied out into

$$\langle \Psi_0 | \hat O | \Psi_0 \rangle=\frac{1}{2} \int dx_1 dx_2 \{\chi_1^*(x_1)\chi_2^*(x_2)\hat O \chi_1(x_1)\chi_2(x_2)+\chi_2^*(x_1)\chi_1^*(x_2)\hat O \chi_2(x_1)\chi_1(x_2)-\chi_1^*(x_1)\chi_2^*(x_2)\hat O \chi_1(x_1)\chi_2(x_2)-\chi_2^*(x_1)\chi_1^*(x_2)\hat O \chi_2(x_1)\chi_1(x_2)\}$$

This expanded equation is what I will base my question on.

For the core-hamiltonian (one-electron integrals),

$$\hat O=h(r_1)+h(r_2)$$

and for the electron repulsion (two-electron integrals),

$$\hat O=\frac{1}{r_{12}}$$

For the core hamiltonian, you can integrate out the electron which is not being acted upon by the operator to reduce the expanded equation to a one-electron integral.

For the electron repulsion, on page 66 the two-electron integral is simplified by stating the first two terms of the expanded equation are equal. This means:

$$\int \chi_1^*(x_1)\chi_2^*(x_2)\frac{1}{r_{12}} \chi_1(x_1)\chi_2(x_2)= \int \chi_2^*(x_1)\chi_1^*(x_2)\frac{1}{r_{12}} \chi_2(x_1)\chi_1(x_2)$$

This is what I do not follow. How is this derived? I can permute the electrons to achieve the same form, but that would introduce a negative sign.

Furthermore, for the equality to be true, does it not imply one of two scenarios: that either the spatial functions are degenerate, i.e.,

$$\chi_1^*(x_1)=\chi_2^*(x_1)$$ $$\chi_2^*(x_2)=\chi_1^*(x_2)$$ $$\chi_1(x_1)=\chi_2(x_1)$$ $$\chi_2(x_2)=\chi_1(x_2)$$

or otherwise that the electrons have the same spin and coordinate such that $$x_1=x_2$$

$$\chi_1^*(x_1)=\chi_1^*(x_2)$$ $$\chi_2^*(x_1)=\chi_2^*(x_2)$$ $$\chi_1(x_1)=\chi_1(x_2)$$ $$\chi_2(x_1)=\chi_2(x_2)$$

neither of which would make sense.

Since $$r_{12} = r_{21}$$, we can interchange the dummy variables of integration in the second term [...].
• I wasn't sure on what they meant by the dummy variables of integration, or why $r_{12}=r_{21}$ is a relevant property for this interchange. What precisely is being permuted here? I only have the one pair here, $x_1$ and $x_2$, to interchange, correct? – Blaise Jul 21 '19 at 19:19
• Basically, one is interested in the distance between the electrons. (That is a directionless quantity, thus $r_{12} = r_{21}$.) For the integral, it is then without consequence how the integration variables are chosen or ordered. (That's why Szabo and Ostlund call them dummy variables.) – TAR86 Jul 22 '19 at 4:42