7 C$_{b}$ ( l$_{1}$, l$_{2}$, . . . , l$_{N}$ ) = ( N ∏ i =1 l$_{i}$ ! ) − 1 ( M ∏ k =1 p$_{k}$ ! ) − 1 × ∑ σ { K } cos { [ Kl$_{σ}$$_{1}$ + ( K − 2) l$_{σ}$$_{2}$ + . . . + 2 l$_{σ}$$_{(}$$_{K/}$$_{2)}$ − 2 l$_{σ}$$_{(}$$_{K/}$$_{2+1)}$ − . . . − ( K − 2) l$_{σ}$$_{(}$$_{K}$$_{−}$$_{1)}$ − Kl$_{σ}$$_{K}$ ] π N } , (26) Φ RBM L ( n$_{1}$, n$_{2}$ )[ z ] = l$_{1}$ + l$_{2}$ + ... + l$_{n}$$_{1}$ = L$_{1}$ ∑ 0 ≤ l$_{1}$ ≤ l$_{2}$ ≤ ... ≤ l$_{n}$$_{1}$ l$_{n}$$_{1}$$_{+1}$ + l$_{n}$$_{1}$$_{+2}$ + ... + l$_{N}$ = L$_{2}$ ∑ 0 ≤ l$_{n}$$_{1}$$_{+1}$ ≤ l$_{n}$$_{2}$$_{+2}$ ≤ ... ≤ l$_{N}$ C$_{b}$ ( l$_{1}$, l$_{2}$, . . . , l$_{n}$$_{1}$ ) C$_{b}$ ( l$_{n}$$_{1}$$_{+1}$, l$_{n}$$_{1}$$_{+2}$, . . . , l$_{N}$ )perm[ z l$_{1}$ 1 , z l$_{2}$ 2 , . . . , z l$_{N}$ N ] , (27) 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 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}$$_{q}$ = L$_{q}$ , with ∑ r q $_{=1}$L$_{q}$ = L . B. Purely rotational fermionic trial functions (REMs) The REM expresion for any ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ ) multi-ring configuration (with N = ∑ r q $_{=1}$n$_{q}$ ) was derived earlier in where ∑ σ { K } runs over all permutations of N − 1 indices selected out from the set { 1 , 2 , . . . , N } (of N indices). In both Eqs. (25) and (26), the index M (with 1 ≤ M ≤ N denotes the number of different single-particle angular momenta l$_{j}$ ’s ( j = 1 , 2 , . . . , M ) in the ordered list { l$_{1}$, l$_{2}$, . . . , l$_{N}$ } and the p$_{k}$ ’s are the multiplicities of each one of the different l$_{j}$ ’s [occupations of the corresponding single-particle orbitals ψ$_{l}$$_{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