The intracule density, , represents the probability for the inter-electronic vector :
where is the two-electron density. A simpler quantity is the spherically averaged intracule density,
where is the angular part of , measures the probability that two electrons are separated by a scalar distance . This intracule is called a position intracule.Gill:2003 If the molecular orbitals are expanded within a basis set
The quantity can be expressed as
where is the two-particle density matrix and is the position integral
and , , and are basis functions. For HF wave functions, the position intracule can be decomposed into a Coulomb component,
and an exchange component,
where etc. are density matrix elements. The evaluation of , and within Q-Chem has been described in detail in Ref. Lee:1999.
Some of the moments of are physically significant,Gill:1997 for example
where is the number of electrons and, is the electronic dipole moment and is the trace of the electronic quadrupole moment tensor. Q-Chem can compute both moments and derivatives of position intracules.