12 of the following 19 [= D $^{TI}$( L = 30)] RVM functions: D. Five bosons Φ REM 6+4 $_{k}$(0 , 4) Q 12 − 2 k 2 , with k = 0 , 1 , 2 , 3 , 4 , 5 , 6; Φ REM 6+3 $_{k}$(1 , 3) Q 12 − 3 k/ 2 2 , with k = 2 , 4 , 6; Φ REM 6+4 $_{k}$(0 , 4) Q 8 − 4 k/ 3 3 , with k = 0 , 3; Φ REM 6+3 $_{k}$(1 , 3) Q 8 − k 3 , with k = 2 , 3 , 4 , 5 , 6 , 7 , 8 . (49) Φ $^{MR}$[ z ] = S ∏ i<j ∈ A ( z$_{i}$ − z$_{j}$ ) 2 ∏ k<l ∈ B ( z$_{k}$ − z$_{l}$ ) $^{2}$, (50) For N = 5 spinless bosons, one needs to consider rovi- brational states [see Eq. (33)] for three distinct molecular configurations, i.e., Φ RBM 5 k (0 , 5) Q m λ , Φ REM 4 k (1 , 4) Q m λ , and Φ REM 3 k (2 , 3) Q m λ . Vibrations with ≥ 2 must also be con- sidered; since the RVM states are not always orthogonal, a Gram-Schmidt orthogonalization is implemented. TA- BLE IV summarizes the quantal molecular description in the start of the LLL spectrum (0 ≤ L ≤ 12). Of particular interest is the L = 8 case, since it corre- sponds to the bosonic Moore-Read state, given [24, 31] by the analytic expression: Diagonalization of the Coulomb interaction in the above TI subspace yielded an energy 0.25084902 e $^{2}$/ ( κl$_{B}$ ) per electron for the yrast state; this value agrees again, within machine precision, with the EXD result. The most sophisticated variants of the composite-fermion the- ory [including composite-fermion diagonalization (CFD), composite-fermion crystal (CFC), and mixed liquid-CFC states [11, 17–19]] fall short in this respect. Indeed the following higher energies were found [17, 61]: 0.250863(6) (CFD), 0.25094(4) (mixed), 0.25101(4) (CFC). The CFD basis is not translationally invariant [18, 19]. Conse- quently, to achieve machine-precision accuracy, the CFD will have to be performed in the larger space of dimension D $^{EXD}$( L = 30) = 169. Based on sizeable overlaps with the EXD wave func- tions [24, 31], the bosonic Moore-Read states [7] are thought to represent the yrast states in the lowest Lan- dau level with the same angular momentum, i.e., L = where sets A and B contain ( N − 1) / 2 and ( N + 1) / 2 particles, respectively, if N is odd. (For N even, both sets contain N/ 2 particles.) S symmetrizes over all possible ways of carrying such a division of N particles into the two sets. L D $^{EXD}$( D $^{TI}$) [( n 1 , n 2 ) { k, λ, m } ] Energy eigenvalues (RVM diag. or EXD $^{TI}$) 0 1(1) [(0,5) { 0, λ ,0 } ] 5.00000000[1] 2 2(1) [(0,5) { 0,2,1 } ] 3.75000000[1] 3 3(1) [(0,5) { 0,3,1 } ] 3.12500000[1] 4 5(2) [(1,4) { 1, λ ,0 } ] [(0,5) { 0,2,2 } ] 2.50000000[1] 3.18750000[3] 5 7(2) [(0,5) { 1, λ ,0 } ] [(0,5) { 0,5,1 } ] 1.87500000[1] 3.03125000[3] 6 10(3) [(1,4) { 1,2,1 } ] [(0,5) { 0,2,3 } ] [(0,5) { 0,3,2 } ] 1.90664171[2] 2.42925914[3] 3.02347415[5] 7 13(3) [(1,4) { 1,3,1 } ] [(0,5) { 1,2,1 } ] [(2,3) { 1,2,2 } ] 1.77354877[1] 2.07258062[4] 3.00543311[7] 8 18(5) [(1,4) { 2, λ ,0 } ] [(1,4) { 1,2,2 } ] [(0,5) { 1,3,1 } ] 1.18821986[1] 1.60795253[2] 1.91688009[6] [(0,5) { 0,2,4 } ] [(2,3) { 2,2,1 } ] 2.24905854[8] 3.00273273[11] 9 23(5) [(1,4) { 1,5,1 } ] [(0,5) { 1,2,2 } ] [(2,3) { 3, λ ,0 } ] 1.12895814[1] 1.52195553[3] 1.89102917[7] [(2,3) { 2,3,1 } ] [(2,3) { 1,3,2 } ] 2.06074601[10] 3.00082677[15] 10 30(7) [(0,5) { 2, λ ,0 } ] [(0,5) { 1,5,1 } ] [(0,5) { 0,5,2 } ] 0.90026059[1] 1.29362646[4] 1.51398054[5] [(1,4) { 2,2,1 } ] [(1,4) { 1,2,3 } ] [(1,4) { 1,3,2 } ] 1.66194766[8] 1.95923264[14] 2.12274862[17] [(2,3) { 2,2,2 } ] 3.00035191[21] 11 37(7) [(0,5) { 1,3,2 } ] [(0,5) { 1,2,3 } ] [(1,4) { 2,3,1 } ] 1.03755324[2] 1.07552423[3] 1.50429489[7] [(2,3) { 3,2,1 } ] [(2,3) { 2,5,1 } ] [(2,3) { 1,2,4 } ] 1.58737738[10] 1.94750687[18] 2.03365831[20] [(2,3) { 1,4,2 } ] 3.00012024[27] 12 47(10) [(0,5) { 2,2,1 } ] [(1,4) { 3, λ ,0 } ] [(1,4) { 2,2,2 } ] 0.61480761[1] 0.93028069[3] 1.05066256[5] [(1,4) { 2,4,1 } ] [(1,4) { 1,4,2 } ] [(1,4) { 1,2,4 } ] 1.34022509[10] 1.50000444[11] 1.50755634[13] [(2,3) { 4, λ ,0 } ] [(2,3) { 3,3,1 } ] [(2,3) { 1,3,3 } ] 1.64279523[18] 1.98164620[27] 2.05477689[29] [(2,3) { 0,4,3 } ] 3.00004768[36] TABLE IV: LLL spectra of five spinless bosons interacting via a repulsive contact interaction gδ ( z i − z j ) . Second column: Dimensions of the full EXD and the nonspurious TI (in parenthesis) spaces (the EXD space is spanned by uncorrelated permanents of Darwin-Fock orbitals). Last three columns: Energy eigenvalues [in units of g/ ( π Λ $^{2}$)] from the diagonalization of the contact interaction in the TI subspace spanned by the trial functions Φ RBM 5 k (0 , 5) Q m λ , Φ RBM 4 k (1 , 4) Q m λ , and Φ RBM 3 k (2 , 3) Q m λ (RVM digonalization). Third to sixth columns: the molecular configurations ( n 1 , n 2 ) and the quantum numbers k , λ and m are indicated within brackets. There is no nonspurious state with L = 1. The full EXD spectrum at a given L is constructed by including, in addition to the listed TI energy eigenvalues [ D $^{TI}$( L ) in number], all the energies associated with angular momenta smaller than L . An integer in square brackets indicates the energy ordering in the full EXD spectrum [including both spurious and TI states (EXD $^{TI}$)], with [1] denoting an yrast state. Eight decimal digits are displayed, but the energy eigenvalues from the RVM diagonalization agree with the corresponding EXD TI ones within machine precision.