8 Φ REM L ( n$_{1}$, n$_{2}$ )[ z ] = l$_{1}$ + l$_{2}$ + ... + l$_{n}$$_{1}$ = L$_{1}$,l$_{n}$$_{1}$$_{+1}$ + l$_{n}$$_{1}$$_{+2}$ + ... + l$_{N}$ = L$_{2}$ ∑ 0 ≤ l$_{1}$<l$_{2}$<...<l$_{n}$$_{1}$<l$_{n}$$_{1}$$_{+1}$<...<l$_{N}$ C$_{f}$ ( l$_{1}$, l$_{2}$, . . . , l$_{n}$$_{1}$ ) C$_{f}$ ( l$_{n}$$_{1}$$_{+1}$, l$_{n}$$_{1}$$_{+2}$, . . . , l$_{N}$ )det[ z l$_{1}$ 1 , z l$_{2}$ 2 , . . . , z l$_{N}$ N ] , (28) where the fermionic coefficients C$_{f}$ ( l$_{1}$, l$_{2}$, . . . , l$_{n}$$_{1}$ ) and C$_{f}$ ( l$_{n}$$_{1}$$_{+1}$, l$_{n}$$_{1}$$_{+2}$, . . . , l$_{N}$ ) are calculated by applying to each one of them the single-ring [(0 , N )] expression configurations. Indeed under condition (32) the C$_{b}$ ( . . . ) and C$_{f}$ ( . . . ) coefficients are identically zero, as can be eas- ily checked using MATHEMATICA [57]. In other words, purely rotational states are allowed only for certain an- gular momenta that do not conflict with the intrinsic molecular point-group symmetries. The yrast states corresponding to magic L ’s [Eq. (30)] are associated with the special cusp states described pre- viously in Figs. 1 and 2 and in Section I B. Furthermore, the enhanced stability associated with the cusp states (see Section I B) is obviously due to the selection rule described by Eqs. (31) and (32). An important property of the REM and RBM trial functions is their translational invariance (in the sense described in Section I B). D. General ro-vibrational trial functions (RVMs) The RVM functions that account for the general exci- tations of the rotating molecules have the form (within a normalization constant): C$_{f}$ ( l$_{1}$, l$_{2}$, . . . , l$_{N}$ ) = ( N ∏ i =1 l$_{i}$ ! ) − 1   ∏ 1 ≤ i<j ≤ N sin [ π N ( l$_{i}$ − l$_{j}$ ) ]   . (29) . . . + l$_{n}$$_{q}$$_{−}$$_{1}$$_{+}$$_{n}$$_{q}$ = L$_{q}$ , with ∑ r q $_{=1}$L$_{q}$ = L . Apart from the permanent being replaced by a deter- minant, we note two other main differences between the REM and RBM expressions. That is, for electrons (1) M = N in all instances (single occupancy) and (2) a product of sine terms in the coefficients C$_{f}$ ( . . . ) replaces the sum of cosine terms in the coefficients C$_{b}$ ( . . . ). It is straighforward to generalize the two-ring REM expression in Eq. (28) to more complicated or simpler [i.e., (0 , N ) and (1 , N − 1)] configurations by (I) consid- ering a separate factor C$_{f}$ ( l$_{n}$$_{q}$$_{−}$$_{1}$$_{+1}$, l$_{n}$$_{q}$$_{−}$$_{1}$$_{+2}$, . . . , l$_{n}$$_{q}$$_{−}$$_{1}$$_{+}$$_{n}$$_{q}$ ) for each q th ring; (II) restricting the summation of the associated n$_{q}$ angular momenta, i.e., l$_{n}$$_{q}$$_{−}$$_{1}$$_{+1}$ + l$_{n}$$_{q}$$_{−}$$_{1}$$_{+2}$ + C. Properties of RBMs and REMs The index RXM stands for either REM, i.e., a ro- tating electron molecule, or RBM, i.e., a rotating boson molecule. The purely rotational functions Φ RXM L ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ ) have been described in detail in earlier sections. The product in Eq. (33) combines ro- tations with vibrational excitations, the latter being de- noted by Q m λ , with λ being an angular momentum; the superscript denotes raising to a power m . Both Φ RXM L and Q m λ are homogeneous polynomials of the complex particle coordinates z$_{1}$, z$_{2}$, . . . , z$_{N}$ , of order L and λm , respectively. The total angular momentum = L + λm . Q m λ is always symmetric in these variables; Φ RXM L is anti- symmetric (symmetric) for fermions (bosons). | 0 〉 is the product of Gaussians defined in Eq. (6); this product of Gaussians is usually omitted. The vibrational excitations Q$_{λ}$ are given by the same expression for both bosons and electrons, namely, by the symmetric polynomials: Φ RXM L ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ ) Q m λ | 0 〉 , (33) L = L 0 + r ∑ q =1 n$_{q}$k$_{q}$, (30) Φ RXM L ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ )[ z ] = 0 (31) The analytic expressions for Φ RXM L ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ )[ z ] (the index RXM standing for either RBM or REM) de- scribe pure molecular rotations associated with magic an- gular momenta with k$_{q}$ , q = 1 , . . . , r being nonnegative integers; L 0 = N ( N − 1) / 2 for electrons and L 0 = 0 for bosons. A central property of these trial functions is that iden- tically for both bosons and electrons when L ̸ = L 0 + r ∑ q =1 n$_{q}$k$_{q}$ (32) Q$_{λ}$ = N ∑ i =1 ( z$_{i}$ − z$_{c}$ ) $^{λ}$, (34) where z$_{c}$ is the coordinate of the center of mass defined This selection rule follows directly from the point group symmetries of the ( n$_{1}$, n$_{2}$, . . . , n$_{r}$ ) multi-ring polygonal