13.2 Intracules 13.2.1 Introduction 13.2.3 Momentum Intracules

13.2.2 Position Intracules

(September 23, 2025)

The intracule density, $I(\mathbf{u})$ , represents the probability for the inter-electronic vector $\mathbf{u}=\mathbf{u}_{1}-\mathbf{u}_{2}$ :

I(\mathbf{u})=\int\rho(\mathbf{r}_{1}\mathbf{r}_{2})\;\delta(\mathbf{r}_{12}-% \mathbf{u})\;d\mathbf{r}_{1}\,\mbox{d}\mathbf{r}_{2}

(13.3)

where $\rho(\mathbf{r}_{1},\mathbf{r}_{2})$ is the two-electron density. A simpler quantity is the spherically averaged intracule density,

P(u)=\int I(\mathbf{u})\mbox{d}\Omega_{\mathbf{u}}\;,

(13.4)

where $\Omega_{\mathbf{u}}$ is the angular part of $\mathbf{v}$ , measures the probability that two electrons are separated by a scalar distance $u=|\mathbf{u}|$ . This intracule is called a position intracule. ⁴³² 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

\psi_{a}(\mathbf{r})=\sum_{\mu}c_{\mu a}\phi_{\mu}(\mathbf{r})

(13.5)

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

P(u)=\sum\limits_{\mu\nu\lambda\sigma}\Gamma_{\mu\nu\lambda\sigma}(\mu\nu% \lambda\sigma)_{\mathrm{P}}

(13.6)

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

(\mu\nu\lambda\sigma)_{\mathrm{P}}=\int\phi_{\mu}^{\ast}(\mathbf{r})\;\phi_{% \nu}(\mathbf{r})\;\phi_{\lambda}^{\ast}(\mathbf{r}+\mathbf{u})\phi_{\sigma}(% \mathbf{r}+\mathbf{u})\;d\mathbf{r}\;d\Omega

(13.7)

and $\phi_{\mu}(\mathbf{r})$ , $\phi_{\nu}(\mathbf{r})$ , $\phi_{\lambda}(\mathbf{r})$ and $\phi_{\sigma}(\mathbf{r})$ 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. 754 Lee A. M., Gill P. M. W.
Chem. Phys. Lett.
(1999), 313, pp. 271. Link .

Some of the moments of $P(u)$ are physically significant, ⁴³⁶ Gill P. M. W.
Chem. Phys. Lett.
(1997), 270, pp. 193. Link for example

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

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13.2.2 Position Intracules