Exchange interaction energy

In the exchange term of Eq. (3), the non-zero range of the interaction probes the non-local space dependence of the density matrix. For short-range interactions, one can expand $\rho(\mbox{{\boldmath {$R$}}},\mbox{{\boldmath {$r$}}})$ to second order with respect to the variable $\mbox{{\boldmath {$r$}}}$, which gives

$$\rho(\bboxr_1,\bboxr_2) = \rho(\mbox{{\boldmath {$R$}}},\mbox{{\boldmath {$r$}}}) = \rho(\mbox{{\boldmath {$R$}}}) + r_a\partial_a\rho(\mbox{{\boldm... ...ial_a\partial_b\rho(\mbox{{\boldmath {$R$}}},\mbox{{\boldmath {$r$}}}) + \ldots$$

$$= \rho(\mbox{{\boldmath {$R$}}}) + i r_aj_a(\mbox{{\boldmath {$R$}... ...\mbox{{\boldmath {$R$}}})-\tau_{ab}(\mbox{{\boldmath {$R$}}})\right] + \ldots ,$$ (14)

where derivatives $\partial_i$ are always calculated at $r_a$=0, and therefore, the result can be expressed in terms of the standard current and kinetic densities [22]:

$$j_a(\mbox{{\boldmath {$R$}}}) ={\textstyle{\frac{1}{i}}}\pa... ...1)}\nabla_b^{(2)}\rho(\bboxr_1,\bboxr_2)_{\bboxr_1=\bboxr_2} .$$ (15)

This parabolic approximation does not ensure that $\rho(\bboxr_1,\bboxr_2)$ $\longrightarrow$0 for large ${r}=\vert\mbox{{\boldmath {$r$}}}\vert=\vert\bboxr_1$$-$$\bboxr_2\vert$. In the spirit of the DME [12], one can improve it by introducing three functions of ${r}$, $\pi_0({r})$, $\pi_1({r})$, and $\pi_2({r})$ [20] that vanish at large ${r}$, i.e., we define the quasi-local approximation of the density matrix by:

$$\rho(\bboxr_1,\bboxr_2) = \pi_0({r})\rho(\mbox{{\boldmath {$R$}}}) + \pi_1({r}) r_a\partia... ...ial_a\partial_b\rho(\mbox{{\boldmath {$R$}}},\mbox{{\boldmath {$r$}}}) + \ldots$$

$$= \pi_0({r})\rho(\mbox{{\boldmath {$R$}}}) + i\pi_1({r}) r_aj_a\rh... ...\mbox{{\boldmath {$R$}}})-\tau_{ab}(\mbox{{\boldmath {$R$}}})\right] + \ldots~.$$ (16)

Such a postulate has to be compatible with the Taylor expansion of Eq. (14), which requires that

$$\pi_0(0)=\pi_1(0)=\pi_2(0)=1 \quad\mbox{and}\quad \pi'_0(0)=\pi'_1(0)=\pi''_0(0)=0.$$ (17)

Of course, for $\pi_0({r})$=$\pi_1({r})$=$\pi_2({r})$=1, one reverts to the parabolic approximation (14).

The product of nonlocal densities in the exchange integral of Eq. (3) to second order reads

$$\rho(\bboxr_1,\bboxr_2)\rho(\bboxr_2,\bboxr_1) = \pi_0^2({r})\rho^2(\mbox{{\boldmath {$R$}}})$$

$$+ \pi_0({r})\pi_2({r})r_a r_b\Big\{ \rho(\mbox{{\boldmath {$R$}}... ...artial_b\rho(\mbox{{\boldmath {$R$}}},\mbox{{\boldmath {$r$}}})]\Big\} + \ldots$$

$$= \pi_0^2({r})\rho^2(\mbox{{\boldmath {$R$}}})$$

$$+ \pi_0({r})\pi_2({r}) r_a r_b \Big\{{\textstyle{\frac{1}{4}}}\rh... ...) +j_a(\mbox{{\boldmath {$R$}}})j_b(\mbox{{\boldmath {$R$}}})\Big\} + \ldots~,$$ (18)

where we have introduced the supplementary condition:

$$\pi_1^2({r})=\pi_0({r})\pi_2({r}).$$ (19)

This condition ensures that the quasi-local approximation of Eq. (16) is compatible with the local gauge invariance [23]. Indeed, the left-hand side of Eq. (18) is manifestly invariant with respect to the local gauge transformation,

$$\rho'(\bboxr_1,\bboxr_2) = e^{i\phi(\bboxr_1)-i\phi(\bboxr_2)} \rho (\bboxr_1,\bboxr_2) ,$$ (20)

and only the difference of terms in the curly brackets in Eq. (18) is invariant with respect to the same transformation [22,19].

Functions $\pi_0({r})$, $\pi_1({r})$, and $\pi_2({r})$ also depend on the parameters defining the approximation (16). In particular, when the infinite matter is used to define functions $\pi_0({r})$, $\pi_1({r})$, and $\pi_2({r})$, like in the DME, they parametrically depend on the Fermi momentum $k_F$. By associating the local density $\rho(\bbox{R})$ with $k_F$, functions $\pi_0({r})$, $\pi_1({r})$, and $\pi_2({r})$ become dependent on $\rho(\bbox{R})$, and hence the damping of the density matrix in the non-local direction $\bbox{r}$ can be different in different local points $\bbox{R}$. However, in order to keep the notation simple, we do not explicitly indicate this possible dependence on density.

Within the quasi-local approximation, one obtains the exchange interaction energy,

$${\cal E}^{\text{int}}_{\text{exc}} = \int{\rm d}^3\mbox{{\... ...{\cal H}^{\text{int}}_{\text{exc}}(\mbox{{\boldmath {$R$}}}),$$ (21)

where up to second order,

$${\cal H}^{\text{int}}_{\text{exc}}(\mbox{{\boldmath {$R$}}}... ...rho\Delta\rho -(\rho\tau-\bbox{j}^{\,2})\Big)\Big] + \ldots,$$ (22)

and where $\tau=\tau_{aa}$. The coupling constants, $V^{00}_{\pi0}$ and $V^{02}_{\pi2}$, are given by the following moments of the interaction,

$$V^{ij}_{\pi n} = \int{\rm d}^3\mbox{{\boldmath {$r$}}}\, r^... ...) = 4\pi\int{\rm d}r\, r^{n+2} \pi_i({r}) \pi_j({r}) V(r) .$$ (23)

Unlike the coupling constants defining the direct term (11), these in Eq. (23) have to be understood as the running coupling constants, which in the DME depend on the scale of the Fermi momentum $k_F$ or density $\rho(\bbox{R})$.

Again, the separation of scales between the range of interaction and the rate of change of the density matrix in the non-local direction results in the dependence of the local energy density on two coupling constants only, and not on the details of the interaction. For the parabolic approximation of Eq. (14), the coupling constants that define the direct and exchange energies are identical, i.e., $V^{00}_{\pi0}$=$V_0$ and $V^{02}_{\pi2}$=$V_2$; however, for the quasi-local approximation of Eq. (16) they are different. This important observation is discussed in Sec. 3.2 in more detail.

In nuclei, the separation of scales discussed above is not very well pronounced. The characteristic parameter, which defines these relative scales, is equal to $k_Fa$, where $a$ stands for the range of the interaction. For example, within the Gogny interaction, which we analyze in detail in Sec. 4, there are two components with the ranges of $a=0.7$ and 1.2fm, whereas $k_F=1.35$fm$^{-1}$, which gives values of $k_Fa=0.95$ and 1.62 that are dangerously close to 1. Therefore, in the expansion of the local energy density (22), one cannot really count on the moments $V^{ij}_{\pi n}$ decreasing with the increasing order $n$.

Instead, as demonstrated by Negele and Vautherin [12], one can hope for tremendously improving the convergence by using at each order the proper counter-terms, which make each order vanish in the infinite matter. Without repeating the original NV construction, here we only note that the net result consists of adding and subtracting in Eq. (16) the infinite-matter term, that is,

$$\rho(\bboxr_1,\bboxr_2) = \nu_0({r})\rho(\mbox{{\boldmath {$R$}}}) + i\nu_1({r}) r_aj_a\rho(\mbox{{\boldmath {$R$}}})$$

$$+ {\textstyle{\frac{1}{2}}} \nu_2({r}) r_a r_b \left[{\textstyle{\f... ...e{\frac{1}{5}}}\delta_{ab}k_F^2\rho(\mbox{{\boldmath {$R$}}})\right] + \ldots~,$$ (24)

where we have defined functions $\nu_i(r)$ such that:

$$\nu_0(r) = \pi_0(r) - {\textstyle{\frac{1}{10}}}(k_F r)^2\p... ...quad \nu_1(r) = \pi_1(r) , \quad\quad \nu_2(r) = \pi_2(r) .$$ (25)

By neglecting the term quadratic in $\nu_2$, we can now use the approximation (24) to calculate the product of densities in Eq. (18), which gives

$$\rho(\bboxr_1,\bboxr_2)\rho(\bboxr_2,\bboxr_1) = \nu_0^2({r})\rho^2(\mbox{{\boldmath {$R$}}}) + \nu_0({r})\nu_2({... ...math {$R$}}})-\rho(\mbox{{\boldmath {$R$}}})\tau_{ab}(\mbox{{\boldmath {$R$}}})$$

$$~~~~~~~~~~~~~~~~~~~~ +{\textstyle{\frac{1}{5}}}\delta_{ab}k_F^2\... ...) +j_a(\mbox{{\boldmath {$R$}}})j_b(\mbox{{\boldmath {$R$}}})\Big\} + \ldots~,$$ (26)

where again the gauge invariance requires that

$$\nu_1^2({r})=\nu_0({r})\nu_2({r}).$$ (27)

We note that the gauge-invariance conditions for the functions $\pi _i(r)$ and $\nu_i(r)$, Eqs. (19) and (27), are compatible with one another only up to the term $\pi_2^2({r})$, which was shifted to higher orders. Finally, approximation (26) gives the energy density analogous to (22),

$${\cal H}^{\text{int}}_{\text{exc}}(\mbox{{\boldmath {$R$}}}... ...) +{\textstyle{\frac{3}{5}}}k_F^2\rho^2 \Big)\Big] + \ldots,$$ (28)

where the coupling constants, $V^{00}_{\nu0}$ and $V^{02}_{\nu2}$, are given by the moments of the interaction calculated for functions $\nu_i(r)$, namely,

$$V^{ij}_{\nu n} = \int{\rm d}^3\mbox{{\boldmath {$r$}}}\, r^... ...) = 4\pi\int{\rm d}r\, r^{n+2} \nu_i({r}) \nu_j({r}) V(r) .$$ (29)

We see that the two sets of auxiliary functions, $\pi _i(r)$ and $\nu_i(r)$, are suitable for discussing the approximate forms of the nonlocal density $\rho(\bboxr_1,\bboxr_2)$ and exchange energy density ${\cal H}^{\text{int}}_{\text{exc}}(\mbox{{\boldmath {$R$}}})$, respectively. Although for the nonlocal density they correspond to a simple reshuffling of terms, which gives relations (25) between $\pi _i(r)$ and $\nu_i(r)$, for the exchange energy density they constitute entirely different approximations, given in Eqs. (22) and (28), with expansion (28) having a larger potential for faster convergence. Only this latter expansion is further discussed.