content_image/426312.jpg content_image/426313.jpg
<img src='content_image/426312.jpg'>

where ∑ σ { K } runs over all permutations of N − 1 indices selected out from the set { 1 , 2 , . . . , N } (of N indices). In both Eqs. (25) and (26), the index M (with 1 ≤ M ≤ N denotes the number of different single-particle angular momenta l$_{j}$ ’s ( j = 1 , 2 , . . . , M ) in the ordered list { l$_{1}$, l$_{2}$, . . . , l$_{N}$ } and the p$_{k}$ ’s are the multiplicities of each one of the different l$_{j}$ ’s [occupations of the corresponding single-particle orbitals ψ$_{l}$$_{j}$ ( z )]. For example, for N = 4 and { l$_{1}$ = 2 , l$_{2}$ = 2 , l$_{3}$ = 2 , l$_{4}$ = 5 } , one has M = 2 and p$_{1}$ = 3, p$_{2}$ = 1; for { l$_{1}$ = 0 , l$_{2}$ = 0 , l$_{3}$ = 0 , l$_{4}$ = 0 } , one has M = 1 and p$_{1}$ = 4; for { l$_{1}$ = 1 , l$_{2}$ = 2 , l$_{3}$ = 3 , l$_{4}$ = 9 } , 

one has M = 4 and p$_{1}$ = p$_{2}$ = p$_{3}$ = p$_{4}$ = 1. 

We further note that for both Eqs. (25) and (26) the total number of distinct cosine terms is N ! / 2 [58], with the division by 2 following from the symmetry properties of cosine, i.e., from cos( − x ) = cos( x ). 

The RBM expresion for an ( n$_{1}$, n$_{2}$ ) two-ring configura- tion (with N = n$_{1}$ + n$_{2}$ ) can be derived following similar steps as in the derivation of the expressions for the multi- ring REMs [12]. If L$_{1}$ + L$_{2}$ = L , the final two-ring RBM expression is 

<img src='content_image/426313.jpg'>

where C$_{b}$ ( l$_{1}$, l$_{2}$, . . . , l$_{n}$$_{1}$ ) and C$_{b}$ ( l$_{n}$$_{1}$$_{+1}$, l$_{n}$$_{1}$$_{+2}$, . . . , l$_{N}$ ) are calculated by applying the single-ring expressions of Eqs. (25) and (26). 

Generalizations of expression (27) to structures with a larger number r of concentric rings involve for each q -th ring (1 ≤ q ≤ r ): (I) Consideration of a separate factor C$_{b}$ ( l$_{n}$$_{q}$$_{−}$$_{1}$$_{+1}$, l$_{n}$$_{q}$$_{−}$$_{1}$$_{+2}$, . . . , l$_{n}$$_{q}$$_{−}$$_{1}$$_{+}$$_{n}$$_{q}$ ); (II) A restric- tion on the summation of the associated n$_{q}$ angular mo- menta, i.e., l$_{n}$$_{q}$$_{−}$$_{1}$$_{+1}$ + l$_{n}$$_{q}$$_{−}$$_{1}$$_{+2}$ + . . . + l$_{n}$$_{q}$$_{−}$$_{1}$$_{+}$$_{n}$$_{q}$ = L$_{q}$ , with ∑ r q $_{=1}$L$_{q}$ = L . 

## B. Purely rotational fermionic trial functions (REMs) 

The REM expresion for any ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ ) multi-ring configuration (with N = ∑ r q $_{=1}$n$_{q}$ ) was derived earlier in 

Refs. [12, 13] following similar steps as those in Section II A. A determinant needs to be used, however, instead of a permanent, to conform with the antisymmetrization properties of the electronic (fermionic) many-body wave function. If L$_{1}$ + L$_{2}$ = L , the final two-ring REM expres- sion is 

