The many dimensions of electronic wave functions makes them difficult to analyze and interpret. It is often convenient to reduce this large number of dimensions, yielding simpler functions that can more readily provide chemical insight. The most familiar of these is the one-electron density $\rho (\mathbf{r})$, which gives the probability of an electron being found at the point $\mathbf{r}$. Analogously, the one-electron momentum density $\pi (\mathbf{p})$ gives the probability that an electron will have a momentum of $\mathbf{p}$. However, the wave function is reduced to the one-electron density much information is lost. In particular, it is often desirable to retain explicit two-electron information. Intracules are two-electron distribution functions and provide information about the relative position and momentum of electrons. A detailed account of the different type of intracules can be found in Ref. Gill:2003. Q-Chem’s intracule package was developed by Aaron Lee and Nick Besley, and can compute the following intracules for or HF wave functions:
Position intracules, $P(u)$: describes the probability of finding two electrons separated by a distance $u$.
Momentum intracules, $M(v)$: describes the probability of finding two electrons with relative momentum $v$.
Wigner intracule, $W(u,v)$: describes the combined probability of finding two electrons separated by $u$ and with relative momentum $v$.