Local densities

Local densities are formed by acting several times on the scalar and vector nonlocal densities with the relative momentum operator $\vec {k}$ and taking the limit of $\vec {r}'=\vec {r}$. Using the spherical representation, the possible coupled $k$-tensors (10) (up to sixth order in derivatives) $K_{nL}$ are those given in Table 1 (replacing $\nabla$ with $k$).

Acting with $K_{nL}$ on the scalar nonlocal density $\rho\left(\vec {r},\vec {r}'\right)$ gives 16 different local densities up to N$^3$LO (one for every term in Table 1). They are listed in Table 3. When acting with $K_{nL}$ on the vector nonlocal densities $s\left(\vec {r},\vec {r}'\right)$, one has to construct all possible ways of coupling the $k$-tensors with the vector density. Obviously, each of the 4 scalar ($L=0$) derivative operators gives one local density, while each of the 12 non-scalar ($L>0$) derivative operators gives three local densities. Altogether, from the vector density one obtains 40 local densities up to N$^3$LO. They are listed in Table 4.

Table 3: Local primary densities (23) up to N$^3$LO built from the scalar nonlocal density $\rho\left(\vec {r},\vec {r}'\right)$ ($v=0$). To simplify the notation the limit of $\vec {r}'=\vec {r}$ is not shown explicitly. Stars ($\star$) mark densities that enter the EDF up to N$^3$LO. Bullets ($\bullet$) mark densities that enter the EDF up to N$^3$LO for conserved spherical, space-inversion, and time-reversal symmetries, see Sec. 4. The last two columns show the $T$ and $P$ parities defined in Eqs. (25) and (26), respectively. In addition, the time-even densities are marked by using the bold-face font.

No.			$\rho_{nLvJ}$ =		density	$n$	$L$	$v$	$J$	$T$	$P$
1	$\star$	$\bullet$	$\bm{\rho_{0000}}$ =		${[}\rho {]}_{0}$	0	0	0	0	1	1
2	$\star$	$$	${}{\rho_{1101}}$ =		${[}k \rho {]}_{1}$	1	1	0	1	$-$1	$-$1
3	$\star$	$\bullet$	$\bm{\rho_{2000}}$ =		${[}{[}kk {]}_{0} \rho{]}_{0}$	2	0	0	0	1	1
4	$\star$	$\bullet$	$\bm{\rho_{2202}}$ =		${[}{[}kk {]}_{2} \rho{]}_{2}$	2	2	0	2	1	1
5	$\star$	$$	${}{\rho_{3101}}$ =		${[}{[}kk {]}_{0} k \rho{]}_{1}$	3	1	0	1	$-$1	$-$1
6	$\star$	$$	${}{\rho_{3303}}$ =		${[}{[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$	3	3	0	3	$-$1	$-$1
7	$\star$	$\bullet$	$\bm{\rho_{4000}}$ =		${[}{[}kk {]}_{0} ^{2} \rho{]}_{0}$	4	0	0	0	1	1
8	$\star$	$\bullet$	$\bm{\rho_{4202}}$ =		${[}{[}kk {]}_{0} {[}kk{]}_{2} \rho{]}_{2}$	4	2	0	2	1	1
9		$$	$\bm{\rho_{4404}}$ =		${[}{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} \rho{]}_{4}$	4	4	0	4	1	1
10	$\star$	$$	${}{\rho_{5101}}$ =		${[}{[}kk {]}_{0}^2 k \rho{]}_{1}$	5	1	0	1	$-$1	$-$1
11		$$	${}{\rho_{5303}}$ =		${[}{[}kk {]}_{0} {[}k{[}kk{]}_{2}{]}_{3} \rho{]}_{3}$	5	3	0	3	$-$1	$-$1
12		$$	${}{\rho_{5505}}$ =		${[}{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5} \rho{]}_{5}$	5	5	0	5	$-$1	$-$1
13	$\star$	$\bullet$	$\bm{\rho_{6000}}$ =		${[}{[}kk {]}_{0} ^3 \rho{]}_{0}$	6	0	0	0	1	1
14		$$	$\bm{\rho_{6202}}$ =		${[}{[}kk {]}_{0}^2 {[}kk{]}_{2} \rho{]}_{2}$	6	2	0	2	1	1
15		$$	$\bm{\rho_{6404}}$ =		${[}{[}kk {]}_{0} {[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} \rho{]}_{4}$	6	4	0	4	1	1
16		$$	$\bm{\rho_{6606}}$ =		${[}{[}k{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5}{]}_{6} \rho{]}_{6}$	6	6	0	6	1	1

Table 4: Same as in Table 3 but for densities built from the vector nonlocal density $\vec {s}\left(\vec {r},\vec {r}'\right)$ ($v=1$).

No.			$\rho_{nLvJ}$ =		density	$n$	$L$	$v$	$J$	$T$	$P$
17	$\star$	$$	${}{\rho_{0011}}$ =		${[}s{]}_{1}$	0	0	1	1	$-$1	1
18	$\star$	$$	$\bm{\rho_{1110}}$ =		${[}k s{]}_{0}$	1	1	1	0	1	$-$1
19	$\star$	$\bullet$	$\bm{\rho_{1111}}$ =		${[}k s{]}_{1}$	1	1	1	1	1	$-$1
20	$\star$	$$	$\bm{\rho_{1112}}$ =		${[}k s{]}_{2}$	1	1	1	2	1	$-$1
21	$\star$	$$	${}{\rho_{2011}}$ =		${[}{[}kk{]}_{0} s{]}_{1}$	2	0	1	1	$-$1	1
22	$\star$	$$	${}{\rho_{2211}}$ =		${[}{[}kk{]}_{2} s{]}_{1}$	2	2	1	1	$-$1	1
23	$\star$	$$	${}{\rho_{2212}}$ =		${[}{[}kk{]}_{2} s{]}_{2}$	2	2	1	2	$-$1	1
24	$\star$	$$	${}{\rho_{2213}}$ =		${[}{[}kk{]}_{2} s{]}_{3}$	2	2	1	3	$-$1	1
25	$\star$	$$	$\bm{\rho_{3110}}$ =		${[}{[}kk{]}_{0} k s{]}_{0}$	3	1	1	0	1	$-$1
26	$\star$	$\bullet$	$\bm{\rho_{3111}}$ =		${[}{[}kk{]}_{0} k s{]}_{1}$	3	1	1	1	1	$-$1
27	$\star$	$$	$\bm{\rho_{3112}}$ =		${[}{[}kk{]}_{0} k s{]}_{2}$	3	1	1	2	1	$-$1
28	$\star$	$$	$\bm{\rho_{3312}}$ =		${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$	3	3	1	2	1	$-$1
29	$\star$	$\bullet$	$\bm{\rho_{3313}}$ =		${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$	3	3	1	3	1	$-$1
30	$\star$	$$	$\bm{\rho_{3314}}$ =		${[}{[}k{[}kk{]}_{2}{]}_{3} s{]}_{4}$	3	3	1	4	1	$-$1
31	$\star$	$$	${}{\rho_{4011}}$ =		${[}{[}kk{]}_{0} ^{2} s{]}_{1}$	4	0	1	1	$-$1	1
32	$\star$	$$	${}{\rho_{4211}}$ =		${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{1}$	4	2	1	1	$-$1	1
33	$\star$	$$	${}{\rho_{4212}}$ =		${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{2}$	4	2	1	2	$-$1	1
34	$\star$	$$	${}{\rho_{4213}}$ =		${[}{[}kk{]}_{0} {[}kk{]}_{2} s{]}_{3}$	4	2	1	3	$-$1	1
35	$\star$	$$	${}{\rho_{4413}}$ =		${[}{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{3}$	4	4	1	3	$-$1	1
36		$$	${}{\rho_{4414}}$ =		${[}{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{4}$	4	4	1	4	$-$1	1
37		$$	${}{\rho_{4415}}$ =		${[}{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{5}$	4	4	1	5	$-$1	1

Table 4: continued.

No.			$\rho_{nLvJ}$ =		density	$n$	$L$	$v$	$J$	$T$	$P$
38	$\star$	$$	$\bm{\rho_{5110}}$ =		${[}{[}kk{]}_{0}^2 k s{]}_{0}$	5	1	1	0	1	$-$1
39	$\star$	$\bullet$	$\bm{\rho_{5111}}$ =		${[}{[}kk{]}_{0}^2 k s{]}_{1}$	5	1	1	1	1	$-$1
40	$\star$	$$	$\bm{\rho_{5112}}$ =		${[}{[}kk{]}_{0}^2 k s{]}_{2}$	5	1	1	2	1	$-$1
41	$\star$	$$	$\bm{\rho_{5312}}$ =		${[}{[}kk{]}_{0} {[}k{[}kk{]}_{2}{]}_{3} s{]}_{2}$	5	3	1	2	1	$-$1
42		$$	$\bm{\rho_{5313}}$ =		${[}{[}kk{]}_{0} {[}k{[}kk{]}_{2}{]}_{3} s{]}_{3}$	5	3	1	3	1	$-$1
43		$$	$\bm{\rho_{5314}}$ =		${[}{[}kk{]}_{0} {[}k{[}kk{]}_{2}{]}_{3} s{]}_{4}$	5	3	1	4	1	$-$1
44		$$	$\bm{\rho_{5514}}$ =		${[}{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5} s{]}_{4}$	5	5	1	4	1	$-$1
45		$$	$\bm{\rho_{5515}}$ =		${[}{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5} s{]}_{5}$	5	5	1	5	1	$-$1
46		$$	$\bm{\rho_{5516}}$ =		${[}{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5} s{]}_{6}$	5	5	1	6	1	$-$1
47	$\star$	$$	${}{\rho_{6011}}$ =		${[}{[}kk{]}_{0} ^3 s{]}_{1}$	6	0	1	1	$-$1	1
48	$\star$	$$	${}{\rho_{6211}}$ =		${[}{[}kk{]}_{0}^2 {[}kk{]}_{2} s{]}_{1}$	6	2	1	1	$-$1	1
49		$$	${}{\rho_{6212}}$ =		${[}{[}kk{]}_{0}^2 {[}kk{]}_{2} s{]}_{2}$	6	2	1	2	$-$1	1
50		$$	${}{\rho_{6213}}$ =		${[}{[}kk{]}_{0}^2 {[}kk{]}_{2} s{]}_{3}$	6	2	1	3	$-$1	1
51		$$	${}{\rho_{6413}}$ =		${[}{[}kk{]}_{0} {[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{3}$	6	4	1	3	$-$1	1
52		$$	${}{\rho_{6414}}$ =		${[}{[}kk{]}_{0} {[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{4}$	6	4	1	4	$-$1	1
53		$$	${}{\rho_{6415}}$ =		${[}{[}kk{]}_{0} {[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4} s{]}_{5}$	6	4	1	5	$-$1	1
54		$$	${}{\rho_{6615}}$ =		${[}{[}k{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5}{]}_{6} s{]}_{5}$	6	6	1	5	$-$1	1
55		$$	${}{\rho_{6616}}$ =		${[}{[}k{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5}{]}_{6} s{]}_{6}$	6	6	1	6	$-$1	1
56		$$	${}{\rho_{6617}}$ =		${[}{[}k{[}k{[}k{[}k{[}kk{]}_{2}{]}_{3}{]}_{4}{]}_{5}{]}_{6} s{]}_{7}$	6	6	1	7	$-$1	1

Finally, all local densities can be denoted by four integers $nLvJ$ as

$$\rho_{nLvJ}(\vec {r})=\left\{[K_{nL}\rho_v(\vec {r},\vec {r}')]_J \right\}_{\vec {r}'=\vec {r}},$$ (23)

where the $n$th-order and rank-$L$ relative derivative operator $K_{nL}$ acts on the scalar ($v=0$) or vector ($v=1$) nonlocal density, and ranks $L$ and $v$ are then vector coupled to $J$. We call these local densities primary densities. The tensor components corresponding to the total rank $J$ are not explicitly shown.

One can also act on each of the local densities with derivative operators $D_{mI}$ of Table 1, and then couple ranks $I$ and $J$ to the total rank $Q$, i.e.,

$$\rho_{mI,nLvJ,Q}(\vec {r})=\left[D_{mI}\rho_{nLvJ}(\vec {r})\right]_Q.$$ (24)

For $m>0$, we call these local densities secondary densities. We do not explicitly list them, because they can be obtained in a straightforward way from the primary densities corresponding to $m=0$, $\rho_{nLvJ}=\rho_{00,InLvJ,J}$, which are listed in Tables 3 and 4.

In Tables 3 and 4, for completeness we also show the time-reversal ($T$) and space-inversion ($P$) parities defined as,

$$T = (-1)^{n+v} ,$$ (25)

$$P = (-1)^{n} .$$ (26)

These definitions are based on the analysis of symmetry properties, which we present in Appendix A. To better visualize the time-reversal properties of the local densities, in Tables 3 and 4 the time-even densities are shown in bold face.

Local densities constructed above are complex. Taking the complex conjugations gives relations derived in Appendix B:

$$\rho_{mI,nLvJ,Q,M}^{*} =\left(-1\right)^{Q-M}\rho_{mI,nLvJ,Q,-M} ,$$ (27)

where the tensor components, denoted $M$, are shown explicitly. These relations allow for expressing positive tensor components through negative ones or vice versa. Therefore, complete information is contained in non-negative or non-positive tensor components only. The $M=0$ components are either real (for even $Q$) or imaginary (for odd $Q$), and hence $2Q+1$ real functions always suffice to describe a given local density of rank $Q$. Moreover, all scalar densities are real, which was the basis of choosing this particular phase convention, as described in Appendix B.