11.9 Intracules

11.9.2 Momentum Intracules

Analogous quantities can be defined in momentum space; I¯(𝐯), for example, represents the probability density for the relative momentum 𝐯=𝐩1-𝐩2:

I¯(𝐯)=π(𝐩1,𝐩2)δ(𝐩12-𝐯)𝑑𝐩1𝑑𝐩2 (11.29)

where π(𝐩1,𝐩2) momentum two-electron density. Similarly, the spherically averaged intracule

M(v)=I¯(𝐯)dΩ𝐯 (11.30)

where Ω𝐯 is the angular part of 𝐯, is a measure of relative momentum v=|𝐯| and is called the momentum intracule. The quantity M(v) can be written as

M(v)=μνλσΓμνλσ(μνλσ)M (11.31)

where Γμνλσ is the two-particle density matrix and (μνλσ)M is the momentum integral89

(μνλσ)M=v22π2ϕμ(𝐫)ϕν(𝐫+𝐪)ϕλ(𝐮+𝐪)ϕσ(𝐮)j0(qv)𝑑𝐫𝑑𝐪𝑑𝐮 (11.32)

The momentum integrals only possess four-fold permutational symmetry, i.e.,

(μνλσ)M=(νμλσ)M=(σλνμ)M=(λσμν)M (11.33)
(νμλσ)M=(μνσλ)M=(λσνμ)M=(σλμν)M (11.34)

and therefore generation of M(v) is roughly twice as expensive as P(u). Momentum intracules can also be decomposed into Coulomb MJ(v) and exchange MK(v) components:

MJ(v)=12μνλσDμνDλσ(μνλσ)M (11.35)
MK(v)=-12μνλσ[DμλαDνσα+DμλβDνσβ](μνλσ)M (11.36)

Again, the even-order moments are physically significant:89

0v0M(v)𝑑v=n(n-1)2 (11.37)
0u0MJ(v)𝑑v=n22 (11.38)
0v2PJ(v)𝑑v=2nET (11.39)
0v0MK(v)𝑑v=-n2 (11.40)

where n is the number of electrons and ET is the total electronic kinetic energy. Currently, Q-Chem can compute M(v), MJ(v) and MK(v) using s and p basis functions only. Moments are generated using quadrature and consequently for accurate results M(v) must be computed over a large and closely spaced v range.