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,^{Besley:2003}

$${W}_{2}({\mathbf{r}}_{1},{\mathbf{p}}_{1},{\mathbf{r}}_{2},{\mathbf{p}}_{2})=\frac{1}{{\pi}^{6}}\int {\rho}_{2}({\mathbf{r}}_{1}+{\mathbf{q}}_{1},{\mathbf{r}}_{1}-{\mathbf{q}}_{1},{\mathbf{r}}_{2}+{\mathbf{q}}_{2},{\mathbf{r}}_{2}-{\mathbf{q}}_{2}){e}^{-2i({\mathbf{p}}_{1}\cdot {\mathbf{q}}_{1}+{\mathbf{p}}_{2}\cdot {\mathbf{q}}_{2})}\mathit{d}{\mathbf{q}}_{1}\mathit{d}{\mathbf{q}}_{2}$$ | (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

$$W(u,v)=\int {W}_{2}({\mathbf{r}}_{1},{\mathbf{p}}_{1},{\mathbf{r}}_{2},{\mathbf{p}}_{2})\delta ({\mathbf{r}}_{12}-\mathbf{u})\delta ({\mathbf{p}}_{12}-\mathbf{v})\mathit{d}{\mathbf{r}}_{1}\mathit{d}{\mathbf{r}}_{2}\mathit{d}{\mathbf{p}}_{1}\mathit{d}{\mathbf{p}}_{2}\mathit{d}{\mathrm{\Omega}}_{\mathbf{u}}\mathit{d}{\mathrm{\Omega}}_{\mathbf{v}}$$ | (13.27) |

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

$$W(u,v)=\sum _{\mu \nu \lambda \sigma}{\mathrm{\Gamma}}_{\mu \nu \lambda \sigma}{\left(\mu \nu \lambda \sigma \right)}_{\mathrm{W}}$$ | (13.28) |

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

$${(\mu \nu \lambda \sigma )}_{\mathrm{W}}=\frac{{v}^{2}}{2{\pi}^{2}}\int \int {\varphi}_{\mu}^{\ast}(\mathbf{r}){\varphi}_{\nu}(\mathbf{r}+\mathbf{q}){\varphi}_{\lambda}^{\ast}(\mathbf{r}+\mathbf{q}+\mathbf{u}){\varphi}_{\sigma}(\mathbf{r}+\mathbf{u}){j}_{0}(qv)\mathit{d}\mathbf{r}\mathit{d}q\mathit{d}{\mathrm{\Omega}}_{\mathbf{u}}$$ | (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,

${\left[ssss\right]}_{\mathrm{W}}$ | $=$ | $\frac{{u}^{2}{v}^{2}}{2{\pi}^{2}}}{\displaystyle \int}{\displaystyle \int}\mathrm{exp}[-\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}]\times $ | (13.30) | ||

${j}_{0}(qv)d\mathbf{r}d\mathbf{q}d{\mathrm{\Omega}}_{\mathbf{u}}$ |

can be expressed as

$${\left[ssss\right]}_{\mathrm{W}}=\frac{\pi {u}^{2}{v}^{2}{e}^{-(R+{\lambda}^{2}{u}^{2}+{\mu}^{2}{v}^{2})}}{2{(\alpha +\delta )}^{3/2}{(\beta +\gamma )}^{3/2}}\int {e}^{-\mathbf{P}\cdot \mathbf{u}}{j}_{0}\left(|\mathbf{Q}+\eta \mathbf{u}|v\right)\mathit{d}{\mathrm{\Omega}}_{u}$$ | (13.31) |

or alternatively

$${\left[ssss\right]}_{\mathrm{W}}=\frac{2{\pi}^{2}{u}^{2}{v}^{2}{e}^{-(R+{\lambda}^{2}{u}^{2}+{\mu}^{2}{v}^{2})}}{{(\alpha +\delta )}^{3/2}{(\beta +\gamma )}^{3/2}}\sum _{n=0}^{\mathrm{\infty}}(2n+1){i}_{n}(Pu){j}_{n}(\eta uv){j}_{n}(Qv){P}_{n}\left(\frac{\mathbf{P}\cdot \mathbf{Q}}{PQ}\right)$$ | (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. Wigner:1932. The
first approach uses the first form of ${\left[ssss\right]}_{\mathrm{W}}$ and used Lebedev
quadrature to perform the remaining integrations over ${\mathrm{\Omega}}_{\mathbf{u}}$. For high
accuracy large Lebedev grids^{Lebedev:1976, Lebedev:1977, Lebedev:1999}
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.^{Cioslowski:1996}
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.