content_image/381084.jpg content_image/381083.jpg content_image/381081.jpg content_image/381071.jpg content_image/381076.jpg content_image/381078.jpg content_image/381072.jpg content_image/381073.jpg
We note that, going from N = 3 to N = 5 particles, the molecular description requires consideration of suc- cessively larger numbers of isomeric molecular structures as elaborated in Section III. In particular, for N = 3 only one molecular isomer is needed, while three different molecular isomers are needed for N = 5. It is remark- able that these isomers are independent of the statistics (bosons or fermions). 

## II. MOLECULAR TRIAL FUNCTIONS 

The molecular trial functions introduced in this paper are derived with the help of a first-principles methodol- ogy of hierarchical successive approximations which con- verge to the exact solution of the many-body Schr¨ odinger equation [15]. Specifically, this methodology is based on the theory of symmetry breaking at the mean-field level and of subsequent symmetry restoration via projection techniques [15]. In this Section, we present (and/or re- view where appropriate) this derivation in some detail. 

## A. Purely rotational bosonic trial functions (RBMs) 

RBM analytical wave functions in the LLL for N bosons in two-dimensional rotating traps can be derived following earlier analogous derivations for the case of elec- trons [12]. Our approach consists of two steps: 

(I) At the first step , one constructs a permanent (Slater determinant for fermions) Ψ $^{N}$( z$_{1}$, . . . , z$_{N}$ ) out of dis- placed single-particle states u ( z$_{j}$, Z$_{j}$ ), j = 1 , . . . , N that represent scalar bosons localized at the positions Z$_{j}$ , with (omitting the particle indices) z = x + iy = and Z = X + iY = Re $^{iφ}$. 

with the matrix M $^{N}$[ z ] being 

<img src='content_image/381084.jpg'>

For the permanent of a matrix, we follow here the def- inition in Ref. [51], that is, the permanent is an analog of a determinant where all the signs in the expansion by minors are taken as positive. This definition provides an unnormalized expression for the permanent. If a nor- malized expression is needed, one has to multiply with a normalization constant 

where { p$_{1}$, p$_{2}$, . . . , p$_{M}$ } denote the occupations (multi- plicities) of the orbitals, assuming that there are M dis- tinct orbitals in a given permanent ( M ≤ N ) [52]. 

<img src='content_image/381083.jpg'>

<img src='content_image/381081.jpg'>

In the LLL, one can specifically consider the limit when the confining potential can be neglected compared to the effect induced by the gauge field. The localized u ( z, Z ) single-particle states (referred to also as orbitals) are then taken to be displaced zero-node Darwin-Fock states with appropriate Peierls phases due to the presence of a per- pendicular magnetic field [see Eq. (1) in Ref. 12], or due to the rotation of the trap with angular frequency Ω. Then, assuming a symmetric gauge, the orbitals can be represented [12, 13, 53] by displaced Gaussian analytic functions, centered at different positions Z$_{j}$ ≡ X$_{j}$ + Y$_{j}$ according to the equilibrium configuration of N classical point charges[54, 55] arranged at the vertices of nested regular polygons (each Gaussian representing a localized particle). Such displaced Gaussians are written as 

<img src='content_image/381071.jpg'>

where the phase factor is due to the gauge invariance. z ≡ x + iy , and all lengths are in units of Λ in the case of a rotating trap or $_{l$_{B}$}$√$_{2 in the case of an applied magnetic}$ field; see Section I B. 

The localized orbital u ( z, Z ) can be expanded in a series over the complete set of zero-node single-particle wave functions in Eq. (4). One gets [see Appendix A in Ref. [56]] 

with 

for Z ̸ = 0. Naturally, C$_{0}$ (0) = 1 and C$_{l>}$$_{0}$ (0) = 0. For an N -particle system, the bosons are situated at the apexes of r concentric regular polygons. The ensu- ing multi-ring structure is denoted by ( n$_{1}$, n$_{2}$, ..., n$_{r}$ ) with ∑ r q $_{=1}$n$_{q}$ = N . The position of the j -th electron on the q -th ring is given by 

<img src='content_image/381076.jpg'>

The single permanent Ψ molecule. Using Eq. (15), one finds the following expan- sion (within a proportionality constant): 

<img src='content_image/381078.jpg'>

where P ( l$_{1}$, l$_{2}$, ..., l$_{N}$ ) ≡ perm[ z l$_{1}$ 1 , z l$_{2}$ 2 , . . . , z l$_{N}$ N ]; the el- ements of the permanent are the functions z l$_{j}$ i , with z l$_{1}$ 1 , z l$_{2}$ 2 , . . . , z l$_{N}$ N being the diagonal elements. The Z$_{k}$ ’s (with 1 ≤ k ≤ N ) in Eq. (18) are the Z q $_{j}$’s of Eq. (17), but relabeled. 

<img src='content_image/381072.jpg'>

<img src='content_image/381073.jpg'>

