13.2 Intracules

13.2.1 Position Intracules

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.298 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. 553.

Some of the moments of P(u) are physically significant,302 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.