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# 13.2.2 Position Intracules

(July 14, 2022)

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. 388 Gill P. M. W., O’Neill D. P., Besley N. A.
Theor. Chem. Acc.
(2003), 109, pp. 241.
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.  684 Lee A. M., Gill P. M. W.
Chem. Phys. Lett.
(1999), 313, pp. 271.
.

Some of the moments of $P(u)$ are physically significant, 392 Gill P. M. W.
Chem. Phys. Lett.
(1997), 270, pp. 193.
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.