The Pauli Exclusion Principle — Question 9

Question 9

The exchange term changes two-electron interaction energies

For two orthonormal spatial orbitals u,v define normalized symmetric and antisymmetric states $\psi_\pm=[u(1)v(2)\pm v(1)u(2)]/\sqrt2$. A spin-independent symmetric interaction V has direct integral J and real exchange integral K: $J=\langle uv|V|uv\rangle=\langle vu|V|vu\rangle$, $K=\langle uv|V|vu\rangle$.

1. Derive each interaction expectation.

2. For $K>0$, compare singlet and triplet energies when their one-electron energies are equal.

Question 9 - Solution

The exchange term changes two-electron interaction energies

(a) Method and calculation

$$\langle V\rangle_\pm=\tfrac12[J+J\pm K\pm K].$$$$\boxed{\langle V\rangle_+=J+K,\quad\langle V\rangle_-=J-K}.$$

(b) Interpretation and further calculation

$$\text{Singlet: symmetric space};\qquad\text{triplet: antisymmetric space}.$$$$\boxed{E_s-E_t=(J+K)-(J-K)=2K>0}.$$

Physical check

The energy ordering here follows the supplied interaction integrals, not exclusion alone. Different orbital occupancies add one-electron energy costs that must also be considered.