4 that an uncorrelated state is described by a single perma- nent (or Slater determinant) made out from the orbitals in Eq. (4), which are characterized by good single-particle angular momenta l$_{i}$ . A small part (sufficient for our purposes here) of the EXD LLL spectra (as a function of L ) are plotted in Fig. 1 for N = 3 scalar bosons and in Fig. 2 for N = 4 spin-polarized electrons. As is usually done in the LLL, the one-body terms of the Hamiltonian (i.e., confining potential and kinetic-energy) were omitted [11, 15, 30], and the exact diagonalization involved only the two-body interaction [see Eq. (3)]. In Figs. 1 and 2, the yrast bands [36] are denoted by a dashed line. Along the yrast bands there appear spe- cial cusp states denoted by a circle. The cusp states are important because they exhibit enhanced stability when the one-body terms of the Hamiltonian (i.e., external con- finement and kinetic energy) are added [see Eqs. (1) and (2)], and thus they determine [47] the global ground states [11, 15, 24, 29, 48] as a function of the applied magnetic field B or the rotational frequency Ω of the trap (for the correspondence between B and Ω, see Ref. [46]). In all studied cases [11, 15, 24, 28, 29, 31, 48] (including both electrons and bosons up to N = 9 particles), the total an- gular momenta of the global ground states belong to the set of magic angular momenta given by Eq. (30) below. We note that the emergence of these magic angular mo- menta are a direct signature of the molecular nature of the cusp states, a fact that further motivates our present investigations concerning the molecular description of the full LLL spectra beyond the electronic [12, 13, 15] cusp states. In the LLL, all many-body wave functions have the general form[5, 25, 30] where W ( z$_{1}$, z$_{2}$, . . . , z$_{N}$ ) is an homogeneous polynomial of degree L (being antisymmetric for fermions and sym- metric for bosons). In Eq. (5), the symbol | 0 〉 stands for a product of Gaus- sians [see Eq. (4)], i.e., W ( z$_{1}$, z$_{2}$, . . . , z$_{N}$ ) | 0 〉 , (5) | 0 〉 = exp( − N ∑ i =1 z$_{i}$z ∗ $_{i}$/ 2) . (6) D $^{TI}$( L ) = D $^{EXD}$( L ) − D $^{EXD}$( L − 1) . (9) C. Plan of the paper D TI b ( L ) = D TI f ((( L + N ( N − 1) / 2 )) ) , (10) We further note that for N particles (bosons or fermions) where the subscripts b and f stand for bosons and fermions (electrons), respectively. N ( N − 1) / 2 is the smallest value of the total angular momentum for spin polarized fermions in the LLL. is the coordinate of the center of mass [49]. The energies of these spurious states coincide with those appearing at all the other smaller angular momenta [5]. Thus (see TABLES I, II, and IV) z$_{c}$ = 1 N ∑ i =1 z$_{i}$, (8) for any arbitrary constant complex number c . The LLL states belonging to the TI subspaces are de- noted by gray solid dots (marked by an arrow; green online) in Figs. 1 and 2. The dimension D $^{TI}$( L ) of the translational invariant subspace is much smaller than the dimension D $^{EXD}$( L ) of the exact-diagonalization (EXD) [15] space (which is spanned by uncorrelated permanents or Slater determinants as discussed above). The remain- ing D $^{EXD}$( L ) − D $^{TI}$( L ) states are spurious center-of-mass excitations, generated by multiplying the TI states with the operator z m c , m = 0 , 1 , 2 , . . . , where To simplify the notation, this common factor will be omitted henceforth in the algebraic expressions and ma- nipulations [except in Eqs. (18) and (33)], where it is repeated for clarity]. Its contribution, however, is neces- sary when numerical results are calculated. A central property of the LLL spectra is the existence of a translationally invariant (TI) subspace [5, 25, 30] for a given L . This subspace is associated with a special subset of the general wave functions in Eq. (5), i.e., with wave functions having translationally invariant polyno- mials W ( z$_{1}$, z$_{2}$, . . . , z$_{N}$ ). Specifically, the TI polynomials obey the relationship The paper is organized as follows: The analytic trial functions associated with pure ro- tations of bosonic molecules (i.e., the RBMs) are intro- duced in Section II A, followed by a description of the purely rotational electronic molecular functions (i.e., the REMs) in Section II B. Properties of the RBMs and REMs are discussed in Section II C. Section II D introduces the general ro-vibrational trial functions (i.e., the RVMs). Case studies of the quantal molecular description of the LLL spectra are presented in Section III. In par- ticular, Section III A discusses the case of N = 3 LLL scalar bosons, while Section III B discusses the case of N = 3 spin-polarized LLL electrons. The case of N = 4 LLL electrons is presented in Section III C, along with an analysis of the Jastrow-Laughlin state (for fractional filling ν = 1 / 3) from the viewpoint of the present molec- ular theory. Section III D investigates the case of N = 5 LLL bosons, along with an analysis of the Moore-Read state according to the molecular picture. Section IV offers a summary and discussion. Finally, the Appendix discusses the rapid-convergence properties of the RVM basis. W ( z$_{1}$ + c, z$_{2}$ + c, . . . , z$_{N}$ + c ) = W ( z$_{1}$, z$_{2}$, . . . , z$_{N}$ ) , (7)