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.367 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. 647.
Some of the moments of are physically significant,371 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.