13.2.2 Momentum Intracules

Analogous quantities can be defined in momentum space; $\overline{I}(𝐯)$, for example, represents the probability density for the relative momentum $𝐯={𝐩}_1-{𝐩}_2$:

$$\overline{I}(𝐯)=\int \pi ({𝐩}_1,{𝐩}_2)\delta ({𝐩}_{12}-𝐯)𝑑{𝐩}_1𝑑{𝐩}_2$$

(13.14)

where $\pi ({𝐩}_1,{𝐩}_2)$ momentum two-electron density. Similarly, the spherically averaged intracule

$$M(v)=\int \overline{I}(𝐯)\text{d}{\mathrm{\Omega }}_{𝐯}$$

(13.15)

where ${\mathrm{\Omega }}_{𝐯}$ is the angular part of $𝐯$, is a measure of relative momentum $v=|𝐯|$ and is called the momentum intracule. The quantity $M(v)$ can be written as

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

(13.16)

where ${\mathrm{\Gamma }}_{\mu \nu \lambda \sigma }$ is the two-particle density matrix and ${(\mu \nu \lambda \sigma )}_{\mathrm{M}}$ is the momentum integral⁷⁴

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

(13.17)

The momentum integrals only possess four-fold permutational symmetry, i.e.,

	${(\mu \nu \lambda \sigma )}_{\mathrm{M}}={(\nu \mu \lambda \sigma )}_{\mathrm{M}}={(\sigma \lambda \nu \mu )}_{\mathrm{M}}={(\lambda \sigma \mu \nu )}_{\mathrm{M}}$		(13.18)
	${(\nu \mu \lambda \sigma )}_{\mathrm{M}}={(\mu \nu \sigma \lambda )}_{\mathrm{M}}={(\lambda \sigma \nu \mu )}_{\mathrm{M}}={(\sigma \lambda \mu \nu )}_{\mathrm{M}}$		(13.19)

and therefore generation of $M(v)$ is roughly twice as expensive as $P(u)$. Momentum intracules can also be decomposed into Coulomb $M_{\mathrm{J}}(v)$ and exchange $M_{\mathrm{K}}(v)$ components:

$$M_J(v)=\frac{1}{2}\sum _{\mu \nu \lambda \sigma }{D}_{\mu \nu }{D}_{\lambda \sigma }{(\mu \nu \lambda \sigma )}_{\mathrm{M}}$$

(13.20)

$$M_{\mathrm{K}}(v)=-\frac{1}{2}\sum _{\mu \nu \lambda \sigma }\left[D_{\mu \lambda }^{\alpha }{D}_{\nu \sigma }^{\alpha }+D_{\mu \lambda }^{\beta }{D}_{\nu \sigma }^{\beta }\right]{(\mu \nu \lambda \sigma )}_{\mathrm{M}}$$

(13.21)

Again, the even-order moments are physically significant:⁷⁴

$$\underset{0}{\overset{\mathrm{\infty }}{\int }}{v}^{0}M(v)𝑑v=\frac{n(n-1)}{2}$$

(13.22)

$$\underset{0}{\overset{\mathrm{\infty }}{\int }}{u}^0{M}_{\mathrm{J}}(v)𝑑v=\frac{n^2}{2}$$

(13.23)

$$\underset{0}{\overset{\mathrm{\infty }}{\int }}{v}^2{P}_{\mathrm{J}}(v)𝑑v=2n{E}_{\mathrm{T}}$$

(13.24)

$$\underset{0}{\overset{\mathrm{\infty }}{\int }}{v}^0{M}_{\mathrm{K}}(v)𝑑v=-\frac{n}{2}$$

(13.25)

where $n$ is the number of electrons and $E_{\mathrm{T}}$ is the total electronic kinetic energy. Currently, Q-Chem can compute $M(v)$, $M_{\mathrm{J}}(v)$ and $M_{\mathrm{K}}(v)$ using $s$ and $p$ basis functions only. Moments are generated using quadrature and consequently for accurate results $M(v)$ must be computed over a large and closely spaced $v$ range.