content_image/430856.jpg content_image/430857.jpg content_image/430858.jpg content_image/430867.jpg content_image/430868.jpg content_image/430869.jpg content_image/430870.jpg content_image/430848.jpg content_image/430865.jpg
For L = 2, we found Φ exact [1] ∝ Q$_{2}$ , with the index [ i ] indicating the energy ordering in the full EXD spectrum (including both spurious and TI states). Since [see Eq. (34)] 

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this result agrees with the findings of Refs. [25, 59] con- cerning ground states of bosons in the range 0 ≤ L ≤ N . For L = 3, one finds Φ exact [1] ∝ Φ RBM 3 . Since [see Eq. (37)] 

this result agrees again with the findings of Refs. [25, 59]. For L = 5, the single nonspurious state is an excited one, Φ exact [2] ∝ Φ RBM 3 Q$_{2}$ . For L = 6 ( ν = 1 / 2), the ground-state is found to be 

i.e., the bosonic Jastrow-Laughlin function for ν = 1 / 2 is equivalent to an RBM state that incorporates vibrational correlations. 

For L ≥ N ( N − 1) (i.e., ν ≤ 1 / 2), the EXD yrast energies equal zero, and with increasing L the degener- acy of the zero-energy states for a given L increases. It is important that this nontrivial behavior is reproduced faithfully by the present method (see TABLE I). 

Although unrecognized, the solution of the problem of three spin-polarized electrons in the LLL using molecu- lar trial functions was presented by Laughlin in Ref. [60]. Indeed, the main result of Ref. [60] [see Eq. (18) therein] were the following wave functions (we display the poly- nomial part only) 

where the three-particle Jacobi coordinates are 

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## B. Three electrons 

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Expression (42) is precisely of the form Φ REM 3 k Q m 2 , as can be checked after transforming back to Cartesian coordi- nates z$_{1}$ , z$_{2}$ , and z$_{3}$ . Thus the wave functions | k, m 〉 of Ref. [60] describe both pure molecular rotations, as well as vibrational excitations, and they cover the transla- tionally invariant LLL subspace. We note that the pairs of indices { k, m } are universal and independent of the statistics, i.e., the same for both bosons [Eq. (36)] and electrons [Eq. (42)], as can be explicitly seen through a comparison of TABLE I here and TABLE I in Ref. [60]. 

We further note that Laughlin did not present molecu- lar trial functions for electrons with N > 3, or for bosons for any N . This is done in the present paper. 

For N = 4 spin-polarized electrons, one needs to con- sider two distinct molecular configurations, i.e., (0 , 4) and (1 , 3). Vibrations with λ ≥ 2 must also be considered. In this case the RVM states are not always orthogonal, and the Gram-Schmidt orthogonalization is implemented. 

Of particular interest is the L = 18 case ( ν = 1 / 3) which is considered [2] as the prototype of quantum- liquid states. However, in this case we found (see TABLE II) that the exact TI solutions are linear superpositions of the following seven RVM states [involving both the (0,4) and (1,3) configurations]: 

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## C. Four electrons 

