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tion regarding the correlation structure. However, note that they reflect the underlying (1,4) and (0,5) molecular configurations through the maximum at the origin of the solid line (EXD TI state) or the local minimum at the ori- gin of the dashed line (Moore-Read state), respectively. 

The above analysis provides a caveat against drawing conclusions by relying exclusively on overlaps (as is often the practice in the litterature of fast rotating ultracold bosons [24, 31]). Indeed, we can conclude that the Moore- Read state examined here disagrees in an essential way with the EXD many-body wave functions. 

We stress that the EXD yrast states with the same angular momenta as the MR states exhibit correlations that conform with the molecular structures associated with the magic angular momenta defined in Eq. (30). In particular, L = 8 (corresponding to the MR state for N = 5) is a magic angular momentum associated with a (1,4) configuration (i.e., 8 mod 4 = 0). Another example is L = 12 (corresponding to the MR state for N = 6). In this latter case, the EXD N -body correlation function for the yrast state was studied in Ref. [29], where it was found (see in paricular Figs. 5 and 6 therein) that it conformed to a (0,6) polygonal-ring configuration, in agreement with the fact that 12 mod 6 = 0. 

The many-body Hilbert space corresponding to the translationally invariant part of the LLL spectra of small systems (whether fermions or bosons, and for both low and high angular momenta) is spanned by the RVM trial functions introduced in Eq. (33). The yrast and excited states for both short- and long-range interac- tions can always be expressed as linear superpositions of these RVM functions. Thus the nature of strong correla- tions in the LLL reflects the emergence of intrinsic point- group symmetries associated with rotations and vibra- tions of molecules formed through particle localization. We stress the validity of the molecular theory for low angular momenta, where ”quantum-liquid” physical pic- tures [2, 11, 31] have been thought to apply exclusively. Our analysis suggests that liquid-type pictures, associ- ated with translationally invariant trial functions (e.g., the Jastrow-Laughlin, compact composite-fermion, and Moore-Read functions), are reducible to a description in terms of an excited rotating/vibrating quantal molecule. 

In addition to the above conceptual advances, from a computational point of view, the introduction of the RVM correlated basis is promising concerning future computational developments. Indeed, it has the potential for enabling controlled and stepwise improvements of the variational wave function, in analogy with previous expe- riences from correlated bases in other fields of many-body physics [39–41]. Such developments may enhance com- putational capabilities for systems with a larger number of particles than it is currently possible. 

The main body of the present paper consisted of 

## IV. SUMMARY AND DISCUSSION 

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two parts. The general theoretical background of our methodology was presented in Section II, while Section III was devoted to case studies. In particular: 

The analytic trial functions associated with pure rota- tions of bosonic molecules (i.e., the RBMs) were derived in Section II A, followed by a description of the purely ro- tational electronic molecular functions (i.e., the REMs) in Section II B. Properties of the RBMs and REMs and their association with magic angular momenta were dis- cussed in Section II C. The general ro-vibrational trial functions (i.e., the RVMs) were introduced in Section II D. 

Concerning illustrative examples of the quantal molec- ular description of the LLL spectra, Section III A dis- cussed the case of N = 3 LLL scalar bosons, while Sec- tion III B investigated the case of N = 3 spin-polarized LLL electrons. The case of N = 4 LLL electrons was elaborated 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 molecular theory. Fi- nally, Section III D studied the case of N = 5 LLL bosons, along with an analysis of the Moore-Read state according to the molecular picture. It was shown that the intrinsic correlation structure of the Moore-Read state disagrees strongly with that of the EXD wave function [in spite of having a rather good overlap with the EXD state, calcu- lated by us to be 0.913 (see also [24])]. 

The Appendix discussed in detail the rapid- convergence properties of the RVM basis. 

In the case of bosons in harmonic traps, mean-field vortex solutions of the Gross-Pitaevskii (GP) equation have been proven most useful for the interpretation of experiments in a variety of situations of Bose-Einstein condensates with very large N ; see, e.g., Refs. [62–64]. Consequently, the results presented in this paper, and previously [15, 28, 29], pertaining to crystalline-type cor- relations in finite systems with a small number N of LLL bosons (as well as the earlier prediction of the develop- ment of molecular-crystalline patterns in non-rotating fi- nite boson systems [65]) are unexpected. While these results, are of intrinsic interest for small systems, one may also inquire about the size-dependent evolution of the properties of the system with increasing N . This question, and in particular the possibility of a transition 

