Nonlocal nuclear EDF

The interaction energy, which is the potential part of the EDF, is derived by averaging the pseudopotential $\hat{V}$ (2) over the uncorrelated nuclear wave function expressed as Slater determinant. The functional obtained in this way is in general nonlocal, meaning that it contains terms depending on one-body densities non-diagonal with respect to the spatial coordinates.

If we leave out the exchange operator from the pseudopotential (2), we can define it as $\hat{V}=\mathcal{V} \left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right)$. This allows us to express the EDF in the following way,

$$\hspace*{-2cm} {\cal E} = \frac{1}{2} \int {\rm d}{\bm{r}'_1}{\rm d}{\bm{r}'_2}{\rm d}{\bm{... ...'_{2} \sigma'_2 \tau'_2, \bm{r}_{1} \sigma_1 \tau_1 \bm{r}_{2} \sigma_2 \tau_2)$$

$$(\rho(\bm{r}_{1} \sigma_1 \tau_1,\bm{r}'_{1} \sigma'_1 \tau'_1)\r... ...gma'_1 \tau'_1)\rho(\bm{r}_{1} \sigma_1 \tau_1,\bm{r}'_{2} \sigma'_2 \tau'_2)),$$ (7)

where the two-body spin-isospin (non-antisymmetrized) matrix elements of $\mathcal{V}$ are defined as

$$\mathcal{V} (\bm{r}'_{1} s'_1 t'_1 \bm{r}'_{2} s'_2 t'_2, \bm{r}_... ...bm{r}'_{1},\bm{r}'_{2},\bm{r}_{1}, \bm{r}_{2}) \vert s_1 t_1 , s_2 t_2\rangle ,$$ (8)

and $\rho(\bm{r}_{1} s_1 t_1,\bm{r}'_{1} s'_1 t'_1)$ and $\rho(\bm{r}_{2} s_2 t_2,\bm{r}'_{2} s'_2 t'_2)$, are the one-body density matrices in spin-isospin channels. In Eq. (7), the two terms that are bilinear in densities lead to the standard direct and exchange terms, which are respectively, local and nonlocal in space coordinates.

We performed derivations of average energies (7) separately for all terms of the pseudopotential (2). The final result of this derivation is given by linear combinations of terms of the EDF appearing on the rhs of the following expression,

$$\langle C_{\tilde{n} \tilde{L},v_{12} \tilde{S}}^{\tilde{n}... ...\mathcal{L}} T_{a, \alpha,Q}^{a',\alpha', t,\mathcal{L}}.$$ (9)

In this expression, $C_{a,\alpha,Q}^{a',\alpha',t,\mathcal{L}}$ and $T_{a, \alpha,Q}^{a',\alpha', t,\mathcal{L}}$ denote, respectively, the coupling constants and terms of the EDF according to the compact notation introduced in Ref. [25], where the Greek indices $\alpha=\left\{ n_{\alpha}S_{\alpha}v_{\alpha}J_{\alpha}\right\}$ and Roman indices $a=\left\{m_a I_a\right\}$ combine all the quantum numbers of the local densities $\rho_{\alpha}(\bm{r})$ and derivative operators $D_{a}$, as defined below. The spherical-tensor formalism for the higher-order EDF has been developed in Ref. [9]. Because here we treat the isospin degree of freedom explicitly, and because we deal with a nonlocal functional, we enriched the notation by adding superscripts $t$, which denote the isoscalar ($t=0$) or isovector ($t=1$) channels, and by adding labels $\mathcal{L}$ that distinguish between local ($\mathcal{L}=L$) and nonlocal ($\mathcal{L}=N$) terms of the functional. Index $Q$ corresponds to the total rank of densities, which are coupled to a scalar.

The formalism developed in Ref. [9], along with the straightforward extension to the isospace introduced in Ref. [12], which originally pertained to quasilocal higher-order EDF and zero-range pseudopotential, can easily be accommodated to express the EDF discussed in this paper. Then, direct (local) terms of the functional, bilinear in local densities, read

$$T^{a',\alpha', t,L}_{a,\alpha,Q}=\int {\rm d}{\bm{r}_1}{\rm d}{\b... ...^t(\bm{r}_1)\right]_Q [D_{a}\rho_{\alpha}^t(\bm{r}_2)]_{Q} \right]^0 \right]_0.$$ (10)

They have been obtained using the integration by parts to transfer all derivatives onto the density matrices, and then employing the locality deltas to perform integrations over two out of four space coordinates. In Eq. (10), subscripts and superscripts denote the standard coupling of the angular momentum and isospin, respectively, and the higher-order derivative operators $D_{a}$ of order $m_a$ and rank $I_a$, are built from the order-one, rank-one derivative operators,

$$D_{11}= 2\boldsymbol\nabla _{\!1} \quad\mbox{or}\quad D_{11}= 2\boldsymbol\nabla _{\!2},$$ (11)

depending on whether they act on variables $\bm{r}_1$ or $\bm{r}_2$.

In an analogous way, exchange (nonlocal) terms of the functional read

$$T^{a',\alpha', t,N}_{a,\alpha,Q} = \int\!\! {\rm d}{\bm{r}_1}{\rm d}{\bm{r}_2}\, g_a(\bm{r}) \left[ ... ...right]_Q [D_{a}\rho_{\alpha}^t(\bm{r}_{2},\bm{r}_{1})]_{Q} \right]^0 \right]_0,$$

where the nonlocal densities are defined as,

$$\rho_{\alpha}^{t}(\bm{r}_{1},\bm{r}_{2})=[K_{nS}\rho_v^t(\bm{r}_{1},\bm{r}_{2})]_J ,$$ (12)

with the order-$n$ and rank-$S$ relative derivative operators $K_{nS}$ acting on nonlocal densities and built from the relative-momentum operators,

$$K_{11}= \frac{1}{2 {\mathrm i}} (\boldsymbol\nabla _{\!1}-\boldsymbol\nabla _{\!2}).$$ (13)

Derivative operators $D_{a}$ of order $m_a$ and rank $I_a$, are built from the building blocks,

$$D_{11}= \boldsymbol\nabla _{\!1}+\boldsymbol\nabla _{\!2}.$$ (14)

We do not use different notations for operators (11) and (15) - which one of them is used is clear from the context.

The coupling constants of the functional that we consider in the present study do not depend on densities. Therefore, in principle, in expressions (10) and (12) one could perform integrations by parts and thus, in an attempt to achieve the same form as that for the quasilocal EDF considered in Ref. [9], transfer the derivative operators $D_{a'}$ onto the second density. However, this transformation would have created a series of extra terms produced by the action of the derivative operators onto the regularized delta, in such a way that the functional would have had a mixed form composed of derivatives of densities and an expansion in the parameter $a$ of (1). For this reason, we keep the form of Eqs. (10) and (12) with the derivatives operators acting on both densities.

Explicit calculations of linear combinations in Eq. (9) are formally identical to those performed for the zero-range pseudopotential (see Sec. III of Ref. [11] for details). In this Section, we present general results for the finite-range pseudopotential (2), which we also supplement by those pertaining to the symmetric spin-saturated nuclear matter. Table 2 lists the numbers of independent terms (10) and (12) of the functional obtained from the finite-range pseudopotential. Each allowed combination of indexes ( $a',\alpha', t,a,\alpha,Q$) gives a pair of EDF terms, one local and another one nonlocal; therefore, the numbers shown are twice the numbers of such allowed combinations. The fact that the isospin does not couple with operators belonging to spin and position-coordinate space, along with the requirement that the EDF is isoscalar, implies that isoscalar and isovector densities, respectively $t$=0 and $t$=1, give rise to two isospin channels in the functional having the same structure. Therefore only one isospin channel is accounted for in Table 2. For the symmetric nuclear matter, the isovector terms do not contribute.

Table 2 also displays numbers of EDF independent terms obtained separately from the central ($\tilde{S}$=0), SO ($\tilde{S}$=1), and tensor ($\tilde{S}$=2) terms of the finite-range pseudopotential. Strictly speaking, the EDF cannot be divided into central and tensor contributions, even though the SO part of the EDF is decoupled from the other two. This is at the origin of the functional terms that mix scalar and vector densities in spin spaces. However there are terms of the EDF that can be produced by both central and tensor terms of the pseudopotential. This explains why the sums of terms in the second, third, and fourth columns do not equal to the corresponding values in the fifth column.

Table 2: Numbers of terms defined in Eqs. (10) and (12) of different orders in the EDF up to N$^3$LO, given for one isospin channel. In the second, third, and fourth columns, numbers terms stemming from central ($\tilde{S}$=0), SO ($\tilde{S}$=1), and tensor ($\tilde{S}$=2) finite-range pseudopotential are given, respectively. The last column gives the numbers of terms of the EDF when it is applied to the symmetric spin-saturated nuclear matter.

	Order	$\tilde{S}$=0	$\tilde{S}$=1	$\tilde{S}$=2	Total	Nuclear matter
	0	4	0	0	4	2
	2	24	8	16	36	3
	4	144	64	114	222	8
	6	640	336	564	1010	12
	N$^3$LO	776	408	694	1272	25