13.2.2 Position Intracules

The intracule density, $I(𝐮)$, represents the probability for the inter-electronic vector $𝐮={𝐮}_1-{𝐮}_2$:

where $\rho ({𝐫}_1,{𝐫}_2)$ is the two-electron density. A simpler quantity is the spherically averaged intracule density,

where ${\mathrm{\Omega }}_{𝐮}$ is the angular part of $𝐯$, measures the probability that two electrons are separated by a scalar distance $u=|𝐮|$. This intracule is called a position intracule.³⁶⁷ If the molecular orbitals are expanded within a basis set

The quantity $P(u)$ can be expressed as

where ${\mathrm{\Gamma }}_{\mu \nu \lambda \sigma }$ is the two-particle density matrix and ${(\mu \nu \lambda \sigma )}_{\mathrm{P}}$ is the position integral

and ${\varphi }_{\mu }(𝐫)$, ${\varphi }_{\nu }(𝐫)$, ${\varphi }_{\lambda }(𝐫)$ and ${\varphi }_{\sigma }(𝐫)$ are basis functions. For HF wave functions, the position intracule can be decomposed into a Coulomb component,

and an exchange component,

where $D_{\mu \nu }$ etc. are density matrix elements. The evaluation of $P(u)$, $P_{\mathrm{J}}(u)$ and $P_{\mathrm{K}}(u)$ within Q-Chem has been described in detail in Ref. 647.

Some of the moments of $P(u)$ are physically significant,³⁷¹ for example

where $n$ is the number of electrons and, $\mu$ is the electronic dipole moment and $Q$ is the trace of the electronic quadrupole moment tensor. Q-Chem can compute both moments and derivatives of position intracules.