13.2.1 Position Intracules

The intracule density, $I(𝐮)$, represents the probability for the inter-electronic vector $𝐮={𝐮}_1-{𝐮}_2$:

$$I(𝐮)=\int \rho ({𝐫}_1{𝐫}_2)\delta ({𝐫}_{12}-𝐮)𝑑{𝐫}_1\text{d}{𝐫}_2$$

(13.3)

where $\rho ({𝐫}_1,{𝐫}_2)$ is the two-electron density. A simpler quantity is the spherically averaged intracule density,

$$P(u)=\int I(𝐮)\text{d}{\mathrm{\Omega }}_{𝐮},$$

(13.4)

where ${\mathrm{\Omega }}_{𝐮}$ 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

$${\psi }_a(𝐫)=\sum _{\mu }{c}_{\mu a}{\varphi }_{\mu }(𝐫)$$

(13.5)

The quantity $P(u)$ can be expressed as

$$P(u)=\sum _{\mu \nu \lambda \sigma }{\mathrm{\Gamma }}_{\mu \nu \lambda \sigma }{(\mu \nu \lambda \sigma )}_{\mathrm{P}}$$

(13.6)

where ${\mathrm{\Gamma }}_{\mu \nu \lambda \sigma }$ is the two-particle density matrix and ${(\mu \nu \lambda \sigma )}_{\mathrm{P}}$ is the position integral

$${(\mu \nu \lambda \sigma )}_{\mathrm{P}}=\int {\varphi }_{\mu }^{\ast }(𝐫){\varphi }_{\nu }(𝐫){\varphi }_{\lambda }^{\ast }(𝐫+𝐮){\varphi }_{\sigma }(𝐫+𝐮)𝑑𝐫𝑑\mathrm{\Omega }$$

(13.7)

and ${\varphi }_{\mu }(𝐫)$, ${\varphi }_{\nu }(𝐫)$, ${\varphi }_{\lambda }(𝐫)$ and ${\varphi }_{\sigma }(𝐫)$ are basis functions. For HF wave functions, the position intracule can be decomposed into a Coulomb component,

$$P_{\mathrm{J}}(u)=\frac{1}{2}\sum _{\mu \nu \lambda \sigma }{D}_{\mu \nu }{D}_{\lambda \sigma }{(\mu \nu \lambda \sigma )}_{\mathrm{P}}$$

(13.8)

and an exchange component,

$$P_{\mathrm{K}}(u)=-\frac{1}{2}\sum _{\mu \nu \lambda \sigma }\left[D_{\mu \lambda }^{\alpha }{D}_{\nu \sigma }^{\alpha }+D_{\mu \lambda }^{\beta }{D}_{\nu \sigma }^{\beta }\right]{(\mu \nu \lambda \sigma )}_{\mathrm{P}}$$

(13.9)

where $D_{\mu \nu }$ etc. are density matrix elements. The evaluation of $P(u)$, $P_{\mathrm{J}}(u)$ and $P_{\mathrm{K}}(u)$ within Q-Chem has been described in detail in Ref. 553.

Some of the moments of $P(u)$ are physically significant,³⁰² for example

	${\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{u}^{0}P(u)𝑑u$	$=$	${\displaystyle \frac{n(n-1)}{2}}$		(13.10)
	${\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{u}^0{P}_{\mathrm{J}}(u)𝑑u$	$=$	${\displaystyle \frac{n^2}{2}}$		(13.11)
	${\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{u}^2{P}_{\mathrm{J}}(u)𝑑u$	$=$	$nQ-{\mu }^2$		(13.12)
	${\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{u}^0{P}_{\mathrm{K}}(u)𝑑u$	$=$	$-{\displaystyle \frac{n}{2}}$		(13.13)

where $n$ is the number of electrons and, $\mu$ 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.