13.2 Intracules

13.2.3 Wigner Intracules

The intracules P(u) and M(v) 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

W2(𝐫1,𝐩1,𝐫2,𝐩2)=1π6ρ2(𝐫1+𝐪1,𝐫1-𝐪1,𝐫2+𝐪2,𝐫2-𝐪2)e-2i(𝐩1𝐪1+𝐩2𝐪2)𝑑𝐪1𝑑𝐪2 (13.26)

can be interpreted as the probability of finding an electron at 𝐫1 with momentum 𝐩1 and another electron at 𝐫2 with momentum 𝐩2. [The quantity W2(𝐫1,𝐫2,𝐩1,𝐩2 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

W(u,v)=W2(𝐫1,𝐩1,𝐫2,𝐩2)δ(𝐫12-𝐮)δ(𝐩12-𝐯)𝑑𝐫1𝑑𝐫2𝑑𝐩1𝑑𝐩2𝑑Ω𝐮𝑑Ω𝐯 (13.27)

If the orbitals are expanded in a basis set, then W(u,v) can be written as

W(u,v)=μνλσΓμνλσ(μνλσ)W (13.28)

where (μνλσ)W is the Wigner integral

(μνλσ)W=v22π2ϕμ(𝐫)ϕν(𝐫+𝐪)ϕλ(𝐫+𝐪+𝐮)ϕσ(𝐫+𝐮)j0(qv)𝑑𝐫𝑑q𝑑Ω𝐮 (13.29)

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 [ssss]w integral,

[ssss]W = u2v22π2exp[-α|𝐫-𝐀|2-β|𝐫+𝐪-𝐁|2-γ|𝐫+𝐪+𝐮-𝐂|2-δ|𝐫+𝐮-𝐃|2]× (13.30)
j0(qv)d𝐫d𝐪dΩ𝐮

can be expressed as

[ssss]W=πu2v2e-(R+λ2u2+μ2v2)2(α+δ)3/2(β+γ)3/2e-𝐏𝐮j0(|𝐐+η𝐮|v)𝑑Ωu (13.31)

or alternatively

[ssss]W=2π2u2v2e-(R+λ2u2+μ2v2)(α+δ)3/2(β+γ)3/2n=0(2n+1)in(Pu)jn(ηuv)jn(Qv)Pn(𝐏𝐐PQ) (13.32)

Two approaches for evaluating (μνλσ)W have been implemented in Q-Chem, full details can be found in Ref. Wigner:1932. The first approach uses the first form of [ssss]W and used Lebedev quadrature to perform the remaining integrations over Ω𝐮. 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 s and p basis functions only.

When computing intracules it is most efficient to locate the loop over u and/or v points within the loop over shell-quartets.Cioslowski:1996 However, for W(u,v) this requires a large amount of memory to store all the integrals arising from each (u,v) point. Consequently, an additional scheme, in which the u and v points loop is outside the shell-quartet loop, is available. This scheme is less efficient, but substantially reduces the memory requirements.