13.2 Intracules 13.2.2 Position Intracules 13.2.4 Wigner Intracules

13.2.3 Momentum Intracules

(September 23, 2025)

Analogous quantities can be defined in momentum space; $\bar{I}(\mathbf{v})$ , for example, represents the probability density for the relative momentum $\mathbf{v}=\mathbf{p}_{1}-\mathbf{p}_{2}$ :

\bar{I}(\mathbf{v})=\int\pi(\mathbf{p}_{1},\mathbf{p}_{2})\;\delta(\mathbf{p}_% {12}-\mathbf{v})d\mathbf{p}_{1}d\mathbf{p}_{2}

(13.14)

where $\pi(\mathbf{p}_{1},\mathbf{p}_{2})$ momentum two-electron density. Similarly, the spherically averaged intracule

M(v)=\int\bar{I}(\mathbf{v})\mbox{d}\Omega_{\mathbf{v}}

(13.15)

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

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

(13.16)

where $\Gamma_{\mu\nu\lambda\sigma}$ is the two-particle density matrix and $(\mu\nu\lambda\sigma)_{\mathrm{M}}$ is the momentum integral ¹¹⁷ Besley N. A., Lee A. M., Gill P. M. W.
Mol. Phys.
(2002), 100, pp. 1763. Link

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

(13.17)

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

	$\displaystyle(\mu\nu\lambda\sigma)_{\mathrm{M}}=(\nu\mu\lambda\sigma)_{\mathrm% {M}}=(\sigma\lambda\nu\mu)_{\mathrm{M}}=(\lambda\sigma\mu\nu)_{\mathrm{M}}$		(13.18)
	$\displaystyle(\nu\mu\lambda\sigma)_{\mathrm{M}}=(\mu\nu\sigma\lambda)_{\mathrm% {M}}=(\lambda\sigma\nu\mu)_{\mathrm{M}}=(\sigma\lambda\mu\nu)_{\mathrm{M}}$		(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\limits_{\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\limits_{\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: ¹¹⁷ Besley N. A., Lee A. M., Gill P. M. W.
Mol. Phys.
(2002), 100, pp. 1763. Link

\int\limits_{0}^{\infty}v^{0}M(v)dv=\frac{n(n-1)}{2}

(13.22)

\int\limits_{0}^{\infty}u^{0}M_{\mathrm{J}}(v)dv=\frac{n^{2}}{2}

(13.23)

\int\limits_{0}^{\infty}v^{2}P_{\mathrm{J}}(v)dv=2nE_{\mathrm{T}}

(13.24)

\int\limits_{0}^{\infty}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.

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13.2.3 Momentum Intracules