11.9 Intracules

11.9.3 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,91

W2(𝐫1,𝐩1,𝐫2,𝐩2)=1π6ρ2(𝐫1+𝐪1,𝐫1-𝐪1,𝐫2+𝐪2,𝐫2-𝐪2)e-2i(𝐩1𝐪1+𝐩2𝐪2)𝑑𝐪1𝑑𝐪2 (11.41)

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 (11.42)

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

W(u,v)=μνλσΓμνλσ(μνλσ)W (11.43)

where (μνλσ)W is the Wigner integral

(μνλσ)W=v22π2ϕμ(𝐫)ϕν(𝐫+𝐪)ϕλ(𝐫+𝐪+𝐮)ϕσ(𝐫+𝐮)j0(qv)𝑑𝐫𝑑q𝑑Ω𝐮 (11.44)

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]× (11.45)
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 (11.46)

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) (11.47)

Two approaches for evaluating (μνλσ)W have been implemented in Q-Chem, full details can be found in Ref. 985. 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 grids538, 539, 536 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.176 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.