11.9.3 Wigner Intracules

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,⁹³

$$W_2({𝐫}_1,{𝐩}_1,{𝐫}_2,{𝐩}_2)=\frac{1}{{\pi }^6}\int {\rho }_2({𝐫}_1+{𝐪}_1,{𝐫}_1-{𝐪}_1,{𝐫}_2+{𝐪}_2,{𝐫}_2-{𝐪}_2)e^{-2i({𝐩}_1\cdot {𝐪}_1+{𝐩}_2\cdot {𝐪}_2)}𝑑{𝐪}_1𝑑{𝐪}_2$$

(11.42)

can be interpreted as the probability of finding an electron at ${𝐫}_1$ with momentum ${𝐩}_1$ and another electron at ${𝐫}_2$ with momentum ${𝐩}_2$. [The quantity $W_2({𝐫}_1,{𝐫}_2,{𝐩}_1,{𝐩}_2$ 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

$$W(u,v)=\int W_2({𝐫}_1,{𝐩}_1,{𝐫}_2,{𝐩}_2)\delta ({𝐫}_{12}-𝐮)\delta ({𝐩}_{12}-𝐯)𝑑{𝐫}_1𝑑{𝐫}_2𝑑{𝐩}_1𝑑{𝐩}_2𝑑{\mathrm{\Omega }}_{𝐮}𝑑{\mathrm{\Omega }}_{𝐯}$$

(11.43)

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

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

(11.44)

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

$${(\mu \nu \lambda \sigma )}_{\mathrm{W}}=\frac{v^2}{2{\pi }^2}\int \int {\varphi }_{\mu }^{\ast }(𝐫){\varphi }_{\nu }(𝐫+𝐪){\varphi }_{\lambda }^{\ast }(𝐫+𝐪+𝐮){\varphi }_{\sigma }(𝐫+𝐮)j_0(qv)𝑑𝐫𝑑q𝑑{\mathrm{\Omega }}_{𝐮}$$

(11.45)

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,

	${\left[ssss\right]}_{\mathrm{W}}$	$=$	${\displaystyle \frac{u^2{v}^2}{2{\pi }^2}}{\displaystyle \int }{\displaystyle \int }\exp [-\alpha |𝐫-𝐀{|}^2-\beta |𝐫+𝐪-𝐁{|}^2-\gamma |𝐫+𝐪+𝐮-𝐂{|}^2-\delta |𝐫+𝐮-𝐃{|}^2]\times$		(11.46)
			$j_0(qv)d𝐫d𝐪d{\mathrm{\Omega }}_{𝐮}$

can be expressed as

$${\left[ssss\right]}_{\mathrm{W}}=\frac{\pi u^2{v}^2{e}^{-(R+{\lambda }^2{u}^2+{\mu }^2{v}^2)}}{2{(\alpha +\delta )}^{3/2}{(\beta +\gamma )}^{3/2}}\int e^{-𝐏\cdot 𝐮}{j}_0\left(|𝐐+\eta 𝐮|v\right)𝑑{\mathrm{\Omega }}_u$$

(11.47)

or alternatively

$${\left[ssss\right]}_{\mathrm{W}}=\frac{2{\pi }^2{u}^2{v}^2{e}^{-(R+{\lambda }^2{u}^2+{\mu }^2{v}^2)}}{{(\alpha +\delta )}^{3/2}{(\beta +\gamma )}^{3/2}}\sum _{n=0}^{\mathrm{\infty }}(2n+1)i_n(Pu)j_n(\eta uv)j_n(Qv)P_n\left(\frac{𝐏\cdot 𝐐}{PQ}\right)$$

(11.48)

Two approaches for evaluating ${(\mu \nu \lambda \sigma )}_{\mathrm{W}}$ have been implemented in Q-Chem, full details can be found in Ref. 996. The first approach uses the first form of ${\left[ssss\right]}_{\mathrm{W}}$ and used Lebedev quadrature to perform the remaining integrations over ${\mathrm{\Omega }}_{𝐮}$. For high accuracy large Lebedev grids^{542, 543, 540} 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.¹⁷⁷ 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.