Determination of functions $\pi _i(r)$

For a given nonlocal density $\rho(\bboxr_1,\bboxr_2)$, the auxiliary functions $\pi _i(r)$ or $\nu_i(r)$, which define its quasi-local approximation, can be calculated as their best possible approximations in terms of local densities. However, the usefulness of the expansion relies on the assumption that generic forms of these functions can be estimated and then applied to all many-body systems of a given kind.

The standard Slater approximation [24,25], which is routinely used to evaluate the Coulomb exchange energy (cf. Refs. [26,27]), corresponds to

$$\pi_0(r) = \nu_0(r) = \frac{3j_1(k_Fr)}{k_Fr} \quad\mbox{and}\quad \pi_2(r) = 0 .$$ (30)

The NV expansion [12] gives a second-order estimate by making the momentum expansion around the Fermi momentum $k_F$ of an infinite system. This gives:

$$\pi_0(r) = \frac{6j_1(k_Fr)+21j_3(k_Fr)}{2k_Fr} \simeq 1 - \frac{(k_Fr)^4}{504},$$ (31)

$$\pi_2(r) = \frac{105j_3(k_Fr)}{(k_Fr)^3} \simeq 1 - \frac{(k_Fr)^2}{18} + \frac{(k_Fr)^4}{792},$$ (32)

$$\nu_0(r) = \frac{3j_1(k_Fr)}{k_Fr} \simeq 1 - \frac{(k_Fr)^2}{10} + \frac{(k_Fr)^4}{280},$$ (33)

where $j_n(k_Fr)$ are the spherical Bessel functions.

The NV functions $\pi _0(r)$, $\pi _2(r)$, and $\nu _0(r)$ are plotted in Fig. 1 with solid, dashed, and dotted lines, respectively. One can see that for large $k_Fr$, functions $\pi _0(r)$ and $\pi _2(r)$ have zeros close to one another and the same signs. Indeed, asymptotically both behave like $\cos(k_Fr)/k_Fr$. Therefore, the gauge-invariance condition (19) can be satisfied almost everywhere. On the other hand, functions $\nu _0(r)$ and $\nu_2(r)=\pi_2(r)$, have asymptotically opposite signs, and the corresponding gauge-invariance condition (27) can be almost nowhere satisfied. Nevertheless, as discussed in Sec. 2.2, what really matters are the moments of interaction (23) and (29), where functions $\pi _i(r)$ and $\nu_i(r)$ are probed only within the range of the interaction, that is, up to $k_Fr\,\simeq\,$1-2. Therefore, for the NV expansion, one can safely use approximations:

$$\pi_1(r) = +\sqrt{\vert\pi_0(r)\pi_2(r)\vert} ,$$ (34)

$$\nu_1(r) = +\sqrt{\vert\nu_0(r)\nu_2(r)\vert} ,$$ (35)

which are valid up to the first zero of $j_3$ or $j_1$, respectively, that is, up to $k_Fr\,\simeq\,$7.0 and 4.5.

By the same token, we can replace in Eqs. (31)-(33) the Bessel functions by Gaussians having the same leading-order dependence on $k_Fr$, namely,

$$\pi_0(r) = \exp\left(- \frac{(k_Fr)^4}{504}\right),$$ (36)

$$\pi_2(r) = \exp\left(-\frac{(k_Fr)^2}{18} - \frac{(k_Fr)^4}{1100}\right),$$ (37)

$$\nu_0(r) = \exp\left(- \frac{(k_Fr)^4}{504}\right) - \frac{(k_Fr)^2}{10} \exp\left(-\frac{(k_Fr)^2}{18} - \frac{(k_Fr)^4}{1100}\right).$$ (38)

As seen in the bottom panel of Fig. 1, in this way, in the region of small $k_Fr$, one obtains a very good reproduction of the NV functions $\pi _i(r)$ and $\nu_i(r)$.

Figure: Dependence of the functions $\pi _0(r)$ (solid lines), $\pi _2(r)$ (dashed lines), and $\nu _0(r)$ (dotted lines) on $k_Fr$. The top and middle panels show the NV functions of Eqs. (31)-(33), with the top panel plotted in expanded scale to better show the details at large $k_Fr$. The bottom panel shows the functions $\pi _0(r)$, $\pi _2(r)$, and $\nu _0(r)$ approximated by Gaussians as in Eqs. (36)-(38).