10 -4  0  4 -4  0  4 -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4 -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4  6 N=4e L=18 |1> |2> |5> [1] [4] [2] R VMs [Eq.  (1)] EXD -TI -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4  6 -6 -4 -2  0  2  4 pure (0,4) REMGLYPH<13> ( =18) GLYPH<j76> pure (1,3) REMGLYPH<13> ( =18) GLYPH<j76> (0,4) REM ( =14) +GLYPH<13> ( =2, m=2) GLYPH<j76> GLYPH<j108> FIG. 3: (Color online) CPDs for N = 4 LLL electrons with L = 18 ( ν = 1 / 3). Top row: The three lowest-in-energy EXD TI states (see TABLE II). Bottom row: The RVM trial functions associated with the largest expansion coefficients (underlined, see TABLE III) of these three EXD TI states in the correlated RVM basis. See the text for details. The solid dot denotes the fixed point r 0 . Distances in nm. 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. C. Four electrons B. Three electrons 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)] 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)] Q$_{2}$ ∝ ( z$_{1}$ − z$_{c}$ )( z$_{2}$ − z$_{c}$ )+( z$_{1}$ − z$_{c}$ )( z$_{3}$ − z$_{c}$ )+( z$_{2}$ − z$_{c}$ )( z$_{3}$ − z$_{c}$ ) , (39) Φ RBM 3 ∝ ( z$_{1}$ − z$_{c}$ )( z$_{2}$ − z$_{c}$ )( z$_{3}$ − z$_{c}$ ) , (40) Φ exact [1] ∝ − 160 9 Φ RBM 6 + 1 4 Q 3 2 = ( z$_{1}$ − z$_{2}$ ) $^{2}$( z$_{1}$ − z$_{3}$ ) $^{2}$( z$_{2}$ − z$_{3}$ ) $^{2}$, (41) 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) 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]: | 1 〉 = Φ REM 18 (0 , 4) , | 2 〉 = Φ REM 14 (0 , 4) Q 2 $_{2}$, | 3 〉 = Φ REM 10 (0 , 4) Q 4 $_{2}$, 4 = Φ REM 6 (0 , 4) Q 6 $_{2}$, | 5 〉 = Φ REM 18 (1 , 3) , | 6 〉 = Φ REM 12 (1 , 3) Q 3 $_{2}$, | 7 〉 = Φ REM 15 (1 , 3) Q$_{3}$. (46) where the three-particle Jacobi coordinates are | k, m 〉 ∝ [ ( z$_{a}$ + iz$_{b}$ ) 3 k − ( z$_{a}$ − iz$_{b}$ ) 3 k 2 i ] ( z 2 a + z 2 $_{b}$)$^{m}$, (42) z$_{c}$ = z$_{1}$ + z$_{2}$ + z$_{3}$ 3 , (43) z$_{a}$ = ( 2 3 ) 1 / 2 [ z$_{1}$ + z$_{2}$ 2 − z$_{3}$ ] , (44) z$_{b}$ = 1 √$_{2}$ ( z$_{1}$ − z$_{2}$ ) . (45)