13.2 Intracules

13.2.1 Position Intracules

(June 30, 2021)

The intracule density, I(𝐮), represents the probability for the inter-electronic vector 𝐮=𝐮1-𝐮2:

I(𝐮)=ρ(𝐫1𝐫2)δ(𝐫12-𝐮)𝑑𝐫1d𝐫2 (13.3)

where ρ(𝐫1,𝐫2) is the two-electron density. A simpler quantity is the spherically averaged intracule density,

P(u)=I(𝐮)dΩ𝐮, (13.4)

where Ω𝐮 is the angular part of 𝐯, measures the probability that two electrons are separated by a scalar distance u=|𝐮|. This intracule is called a position intracule. 358 Gill P. M. W., O’Neill D. P., Besley N. A.
Theor. Chem. Acc.
(2003), 109, pp. 241.
Link
If the molecular orbitals are expanded within a basis set

ψa(𝐫)=μcμaϕμ(𝐫) (13.5)

The quantity P(u) can be expressed as

P(u)=μνλσΓμνλσ(μνλσ)P (13.6)

where Γμνλσ is the two-particle density matrix and (μνλσ)P is the position integral

(μνλσ)P=ϕμ(𝐫)ϕν(𝐫)ϕλ(𝐫+𝐮)ϕσ(𝐫+𝐮)𝑑𝐫𝑑Ω (13.7)

and ϕμ(𝐫), ϕν(𝐫), ϕλ(𝐫) and ϕσ(𝐫) are basis functions. For HF wave functions, the position intracule can be decomposed into a Coulomb component,

PJ(u)=12μνλσDμνDλσ(μνλσ)P (13.8)

and an exchange component,

PK(u)=-12μνλσ[DμλαDνσα+DμλβDνσβ](μνλσ)P (13.9)

where Dμν etc. are density matrix elements. The evaluation of P(u), PJ(u) and PK(u) within Q-Chem has been described in detail in Ref. 629.

Some of the moments of P(u) are physically significant, 362 Gill P. M. W.
Chem. Phys. Lett.
(1997), 270, pp. 193.
Link
for example

0u0P(u)𝑑u = n(n-1)2 (13.10)
0u0PJ(u)𝑑u = n22 (13.11)
0u2PJ(u)𝑑u = nQ-μ2 (13.12)
0u0PK(u)𝑑u = -n2 (13.13)

where n is the number of electrons and, μ is the electronic dipole moment and Q is the trace of the electronic quadrupole moment tensor. Q-Chem can compute both moments and derivatives of position intracules.