13.2 Intracules 13.2.3 Momentum Intracules 13.2.5 Intracule Job Control

13.2.4 Wigner Intracules

(November 19, 2024)

The intracules $P(u)$ and $M(v)$ provide a representation of an electron distribution in either position or momentum space but neither alone can provide a complete description. For a combined position and momentum description an intracule in phase space is required. Defining such an intracule is more difficult since there is no phase space second-order reduced density. However, the second-order Wigner distribution, ¹¹⁴ Besley N. A., O’Neill D. P., Gill P. M. W.
J. Chem. Phys.
(2003), 118, pp. 2033. Link

W_{2}(\mathbf{r}_{1},\mathbf{p}_{1},\mathbf{r}_{2},\mathbf{p}_{2})=\frac{1}{% \pi^{6}}\int\rho_{2}(\mathbf{r}_{1}+\mathbf{q}_{1},\mathbf{r}_{1}-\mathbf{q}_{% 1},\mathbf{r}_{2}+\mathbf{q}_{2},\mathbf{r}_{2}-\mathbf{q}_{2})e^{-2i(\mathbf{% p}_{1}\cdot\mathbf{q}_{1}+\mathbf{p}_{2}\cdot\mathbf{q}_{2})}d\mathbf{q}_{1}d% \mathbf{q}_{2}

(13.26)

can be interpreted as the probability of finding an electron at $\mathbf{r}_{1}$ with momentum $\mathbf{p}_{1}$ and another electron at $\mathbf{r}_{2}$ with momentum $\mathbf{p}_{2}$ . [The quantity $W_{2}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{p}_{1},\mathbf{p}_{2}$ is often referred to as “quasi-probability distribution” since it is not positive everywhere.]

The Wigner distribution can be used in an analogous way to the second order reduced densities to define a combined position and momentum intracule. This intracule is called a Wigner intracule, and is formally defined as

W(u,v)=\int W_{2}(\mathbf{r}_{1},\mathbf{p}_{1},\mathbf{r}_{2},\mathbf{p}_{2})% \delta(\mathbf{r}_{12}-\mathbf{u})\delta(\mathbf{p}_{12}-\mathbf{v})d\mathbf{r% }_{1}\,d\mathbf{r}_{2}\,d\mathbf{p}_{1}\,d\mathbf{p}_{2}\,d\Omega_{\mathbf{u}}% \,d\Omega_{\mathbf{v}}

(13.27)

If the orbitals are expanded in a basis set, then $W(u,v)$ can be written as

W(u,v)=\sum\limits_{\mu\nu\lambda\sigma}\Gamma_{\mu\nu\lambda\sigma}\left({\mu% \nu\lambda\sigma}\right)_{\mathrm{W}}

(13.28)

where ( $\mu\nu\lambda\sigma)_{\mathrm{W}}$ is the Wigner integral

(\mu\nu\lambda\sigma)_{\mathrm{W}}=\frac{v^{2}}{2\pi^{2}}\int\int\phi_{\mu}^{% \ast}(\mathbf{r})\phi_{\nu}(\mathbf{r}+\mathbf{q})\phi_{\lambda}^{\ast}(% \mathbf{r}+\mathbf{q}+\mathbf{u})\phi_{\sigma}(\mathbf{r}+\mathbf{u})j_{0}(q\,% v)\;d\mathbf{r}\;d{q}\;d\Omega_{\mathbf{u}}

(13.29)

Wigner integrals are similar to momentum integrals and only have four-fold permutational symmetry. Evaluating Wigner integrals is considerably more difficult that their position or momentum counterparts. The fundamental $\left[ssss\right]_{\mathrm{w}}$ integral,

	$\displaystyle\left[ssss\right]_{\mathrm{W}}$	$\displaystyle=$	$\displaystyle\frac{u^{2}v^{2}}{2\pi^{2}}\;\int\int\exp\left[-\alpha\|\mathbf{r}% \!-\!\mathbf{A}\|^{2}-\!\beta\|\mathbf{r}\!+\!\mathbf{q}\!-\!\mathbf{B}\|^{2}-\!% \gamma\|\mathbf{r}\!+\!\mathbf{q}\!+\!\mathbf{u}\!-\!\mathbf{C}\|^{2}-\!\delta\|% \mathbf{r}\!+\!\mathbf{u}\!-\!\mathbf{D}\|^{2}\right]\times$		(13.30)
			$\displaystyle j_{0}(qv)\;d\mathbf{r}\;d\mathbf{q}\;d\Omega_{\mathbf{u}}$		(13.30)

can be expressed as

\left[ssss\right]_{\mathrm{W}}=\frac{\pi u^{2}v^{2}\;e^{-(R+\lambda^{2}u^{2}+% \mu^{2}v^{2})}}{2(\alpha+\delta)^{3/2}(\beta+\gamma)^{3/2}}\int{e^{-\mathbf{P}% \cdot\mathbf{u}}}j_{0}\left(|\mathbf{Q}+\eta\mathbf{u}|v\right)\;d\Omega_{u}

(13.31)

or alternatively

\left[ssss\right]_{\mathrm{W}}=\frac{2\pi^{2}u^{2}v^{2}e^{-(R+\lambda^{2}u^{2}% +\mu^{2}v^{2})}}{(\alpha+\delta)^{3/2}(\beta+\gamma)^{3/2}}\sum\limits_{n=0}^{% \infty}(2n+1)i_{n}(P\,u)j_{n}(\eta uv)j_{n}(Qv)P_{n}\left({\frac{\mathbf{P}% \cdot\mathbf{Q}}{P\;Q}}\right)

(13.32)

Two approaches for evaluating $(\mu\nu\lambda\sigma)_{\mathrm{W}}$ have been implemented in Q-Chem, full details can be found in Ref. 1365 Wigner E.
Phys. Rev.
(1932), 40, pp. 749. Link . The first approach uses the first form of $\left[ssss\right]_{\mathrm{W}}$ and used Lebedev quadrature to perform the remaining integrations over $\Omega_{\mathbf{u}}$ . For high accuracy large Lebedev grids ⁷³⁴ Lebedev V. I.
Zh. Vychisl. Mat. Mat. Fix.
(1976), 16, pp. 293. Link should be used, grids of up to 5294 points are available in Q-Chem. Alternatively, the second form can be adopted and the integrals evaluated by summation of a series. Currently, both methods have been implemented within Q-Chem for $s$ and $p$ basis functions only.

When computing intracules it is most efficient to locate the loop over $u$ and/or $v$ points within the loop over shell-quartets. ²³⁴ Cioslowski J., Liu G.
J. Chem. Phys.
(1996), 105, pp. 4151. Link However, for $W(u,v)$ this requires a large amount of memory to store all the integrals arising from each $(u,v)$ point. Consequently, an additional scheme, in which the $u$ and $v$ points loop is outside the shell-quartet loop, is available. This scheme is less efficient, but substantially reduces the memory requirements.

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13.2.4 Wigner Intracules