The intracules and provide a representation of an electron distribution in either position or momentum space but neither alone can provide a complete description. For a combined position and momentum description an intracule in phase space is required. Defining such an intracule is more difficult since there is no phase space second-order reduced density. However, the second-order Wigner distribution,Besley:2003
can be interpreted as the probability of finding an electron at with momentum and another electron at with momentum . [The quantity is often referred to as “quasi-probability distribution” since it is not positive everywhere.]
The Wigner distribution can be used in an analogous way to the second order reduced densities to define a combined position and momentum intracule. This intracule is called a Wigner intracule, and is formally defined as
If the orbitals are expanded in a basis set, then can be written as
where ( is the Wigner integral
Wigner integrals are similar to momentum integrals and only have four-fold permutational symmetry. Evaluating Wigner integrals is considerably more difficult that their position or momentum counterparts. The fundamental integral,
can be expressed as
Two approaches for evaluating have been implemented in Q-Chem, full details can be found in Ref. Wigner:1932. The first approach uses the first form of and used Lebedev quadrature to perform the remaining integrations over . For high accuracy large Lebedev gridsLebedev:1976, Lebedev:1977, Lebedev:1999 should be used, grids of up to 5294 points are available in Q-Chem. Alternatively, the second form can be adopted and the integrals evaluated by summation of a series. Currently, both methods have been implemented within Q-Chem for and basis functions only.
When computing intracules it is most efficient to locate the loop over and/or points within the loop over shell-quartets.Cioslowski:1996 However, for this requires a large amount of memory to store all the integrals arising from each point. Consequently, an additional scheme, in which the and points loop is outside the shell-quartet loop, is available. This scheme is less efficient, but substantially reduces the memory requirements.