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# 13.2.4 Wigner Intracules

(December 20, 2021)

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, 100 Besley N. A., O’Neill D. P., Gill P. M. W.
J. Chem. Phys.
(2003), 118, pp. 2033.

 $W_{2}(\mathbf{r}_{1},\mathbf{p}_{1},\mathbf{r}_{2},\mathbf{p}_{2})=\frac{1}{% \pi^{6}}\int\rho_{2}(\mathbf{r}_{1}+\mathbf{q}_{1},\mathbf{r}_{1}-\mathbf{q}_{% 1},\mathbf{r}_{2}+\mathbf{q}_{2},\mathbf{r}_{2}-\mathbf{q}_{2})e^{-2i(\mathbf{% p}_{1}\cdot\mathbf{q}_{1}+\mathbf{p}_{2}\cdot\mathbf{q}_{2})}d\mathbf{q}_{1}d% \mathbf{q}_{2}$ (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) $\displaystyle j_{0}(qv)\;d\mathbf{r}\;d\mathbf{q}\;d\Omega_{\mathbf{u}}$

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.  1215 Wigner E.
Phys. Rev.
(1932), 40, pp. 749.
. 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 645 Lebedev V. I.
Zh. Vychisl. Mat. Mat. Fix.
(1976), 16, pp. 293.
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. 208 Cioslowski J., Liu G.
J. Chem. Phys.
(1996), 105, pp. 4151.
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.