Two-electron integrals
The notation of the two-electron integrals is often a source of confusion, because we have three different notations for the same quantity. Mulliken’s notation \(g_{PQRS}\) is favored by Helgaker (T. Helgaker, Jørgensen, and Olsen 2000), and in this notation, the two-electron integrals are written as: \[\begin{equation} \require{physics} g_{PQRS} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_P^\dagger(\vb{r}_1) \phi_Q(\vb{r}_1) \thinspace \frac{1}{ \norm{\vb{r}_1 - \vb{r}_2} } \thinspace \phi_R^\dagger(\vb{r}_2) \phi_S(\vb{r}_2) \thinspace . \end{equation}\] The other two notations are chemist’s notation \((PQ|RS)\) and physicist’s notation \(\braket{PQ}{RS}\) and they are linked by the following relation: \[\begin{equation} g_{PQRS} = (PQ|RS) = \braket{PR}{QS} \thinspace . \end{equation}\]
The two-electron Coulomb repulsion integrals obey the following permutational symmetries: \[\begin{equation} g_{PQRS}^* = g_{QPSR} \end{equation}\] and \[\begin{equation} g_{RSPQ} = g_{PQRS} \thinspace , \end{equation}\] which are completely analogous to the permutational symmetries of the two-electron excitation operators. If we are employing a real spinor basis, we additionally have the following four-fold permutational symmetry: \[\begin{equation} g_{PQRS} = g_{PQSR} = g_{QPRS} = g_{QPSR} \thinspace . \end{equation}\]
Furthermore, specifically in some physics literature and in diagrammatic coupled-cluster theory, we sometimes find another type of two-electron integrals. These are the antisymmetrized integrals: \[\begin{equation} \tilde{g}_{PQRS} = g_{PQRS} - g_{PSRQ} \thinspace , \end{equation}\] and are most often used in physicist’s notation: \[\begin{equation} \braket{PQ}{|RS} = \braket{PQ}{RS} - \braket{PQ}{SR} \thinspace . \end{equation}\] Both notations are analogously linked as the non-antisymmetrized versions: \[\begin{equation} \tilde{g}_{PQRS} = \braket{PR}{|QS} \thinspace . \end{equation}\] The main use of this representation of the two-electron integrals is that they now possess the same permutational symmetries as the two-electron excitation operator: \[\begin{equation} \tilde{g}_{PQRS} = -\tilde{g}_{RQPS} = \tilde{g}_{RSPQ} = -\tilde{g}_{PSRQ} \end{equation}\] and \[\begin{equation} \tilde{g}_{PQRS}^* = \tilde{g}_{QPSR} \thinspace . \end{equation}\]
We often find expressions using the Coulomb integrals \[\begin{equation} J_{PQ} = g_{PPQQ} \end{equation}\] and the exchange integrals \[\begin{equation} K_{PQ} = g_{PQQP} \thinspace . \end{equation}\]
Again, but now for spatial orbitals, we can introduce a chemist’s notation and a physicist’s notation: \[\begin{equation} g_{pqrs} = (pq|rs) = \braket{pr}{qs} \thinspace , \end{equation}\] which are symmetric in a switching of the pair indices: \[\begin{equation} g_{rspq} = g_{pqrs} \end{equation}\] and whose complex conjugates are related in the following way: \[\begin{equation} g_{qpsr} = g_{pqrs}^* \thinspace . \end{equation}\]