Inter-dimensional Degeneracies in van der Waals Clusters and Quantum Monte Carlo Computation of Rovibrational States

Quantum Monte Carlo estimates of the spectrum of rotationally invariant states of noble gas clusters suggest inter-dimensional degeneracy in $N-1$ and $N+1$ spacial dimensions. We derive this property by mapping the Schr\"odinger eigenvalue problem onto an eigenvalue equation in which $D$ appears as a continuous variable. We discuss implications for quantum Monte Carlo and dimensional scaling methods.


I. INTRODUCTION
One of the advantages of Monte Carlo methods is that they scale well with the number of degrees of freedom of a physical system.In this paper we consider van der Waals clusters consisting of N bosonic Lennard-Jones atoms in D spatial dimensions.We treat the atoms as "elementary" particles without internal degrees of freedom, so that in total we deal with clusters with ND quantum mechanical degrees of freedom.We are mainly interested in the energy spectra of these clusters.
Quantum Monte Carlo computations can be made much more efficient by the use of optimized trial wave functions, as is well known.With currently available methods, as a matter of fact, the problem of computing rovibrational spectra with Monte Carlo methods is virtually intractable without good trial functions.One of the questions of interest is the relative importance of the quality of these trial wave functions for n-body correlations with n in excess of the commonly used correlations with n = 2 and n = 3.In this context, the idea of varying the spatial dimensionality of the system quite naturally suggests itself, because particles can be more compact in higher dimensions, which suggests that correlations involving a higher number of particles might become more important as the spatial dimensionality increases.While we have not found clear numerical evidence to support this idea, 1 our computations did produce an interesting by-product, which forms the topic of this paper.
Our computations showed that the energy spectra of N particles in N − 1 and N + 1 spatial dimensions are numerically indistinguishable for states invariant under rotation and translation. 1Indeed, in this paper we show that for these S-states and for D ≥ N − 1, the N particle time-independent Schrödinger equation can be transformed into an eigenvalue equation involving a differential operator with 1 2 (N − 1)N independent variables and an effective potential in which the spatial dimension D appears as a continuously varying parameter.This effective potential turns out to depend quadratically on D and is symmetric about D = N.This implies the aforementioned inter-dimensional degeneracy, in addition to a relationship between the wave functions for D = N − 1 and D = N + 1 dimensions.We note here that, as is also manifest in the frustration of the classical system for D < N − 1 -frustration for example in the sense that not all inter-atomic distances can be equalthe spectrum for values of D < N − 1 cannot be obtained by analytic continuation of the spectrum for D ≥ N − 1; we shall return to this in the discussion at the end of this paper.
Inter-dimensional degeneracy was also derived recently by Gu et al, 2 by a grouptheoretical method.To the best of our knowledge, inter-dimensional degeneracies of S-states of Lennard-Jones clusters have not been observed before, with the exception of the two-body cluster in one and three dimensions which follows from the standard separation of variable solution of the Schrödinger equation.
Even though an extension of our study to dimensions higher than the physical three dimensions is primarily of academic interest, the effect of spatial dimension on quantum systems has been studied since the early days of quantum physics.In fact, Fock 3 as early as 1935 showed that there exists a relationship between the the hydrogen-like wave functions and four-dimensional hyper-spherical harmonics. 4The hyper-spherical coordinate method was used in the late seventies to discover inter-dimensional degeneracies in electron systems.
For the one-electron system a transformation was found that reveals inter-dimensional degeneracy between a system in D dimension and angular momentum l with the same system in D ± 2 dimensions and angular momentum l ∓ 1. 5 Many-electron systems have also been shown to exhibit inter-dimensional degeneracies. 6Further references to other interdimensional studies can be found in Ref. 2.
The structure of this paper is as follows.In Sec.II A we briefly summarize the quantum Monte Carlo technique for excited states developed in Refs.7 and 8 to obtain optimized trial wave functions for van der Waals clusters.In this method, trial wave functions are developed that can be improved systematically.These trial wave functions are linear combinations of elementary basis functions with non-linear variational parameters.The elementary basis functions consist of a prefactor and exponential polynomial that is formulated in terms of all possible N-body correlations.In Sec.II B we present Monte Carlo energy estimates obtained for selected few-body van der Waals clusters in a limited number of dimensions ranging from D = 1 to D = 6.In this study we consider van der Waal clusters composed of atoms of Kr, Ar, Ne and the hypothetical 1 2 -Ne, which has half the (dimensionless) mass of Ne.Kr can be considered as a semi-classical case while that of 1 2 -Ne is more quantum mechanical in nature.Sec.III is devoted to the exact derivation of dimensional degeneracy, with some of the results postponed to the Appendix.In the final Section IV we discuss the relevance of our results, in particular for dimensional scaling methods.

A. Optimization of ground and excited state wave functions
We consider clusters in D dimensions consisting of N atoms with positions specified by the D × N matrix of Cartesian coordinates R = (r 1 r 2 . . .r N ), with We shall use the following definitions for difference vectors and their lengths.
For a system of N bosonic van der Waals atoms with atomic mass µ and interacting via a pair potential, the dimensionless Hamiltonian is with and where V is the dimensionless Lennard-Jones potential The inverse dimensionless mass is given by m −1 = 2 /2 1 3 µσ 2 ǫ, which is proportional to the square of the de Boer parameter, 9 where −ǫ is the minimum of the Lennard-Jones potential and 2 1 6 σ the corresponding inter-particle distance.A preliminary step in our optimization procedure is to generate a sample of configurations R σ , with σ = 1, . . ., s, which are sampled from a relative probability density function ψ g (R σ ) 2 .The guiding function ψ g used for the computations reported in this paper is defined in terms of a trial function ψ, which approximates the ground state.In simple cases, we used ψ 2 g = ψ2/ρ , where the parameter ρ is chosen in the range 2 ρ 3.Where necessary, we used a more sophisticated guiding function 10 so as to generate a sample with substantial overlap with all the excited states under consideration.
The trial wave functions are linear combinations of elementary basis functions β i , each of which implicitly depends on non-linear variational parameters, and we use different procedures to optimize the linear and non-linear parameters.For reasons explained in detail below, we define the re-weighted functions βi For a complete set of elementary basis functions β i the Schrödinger equation can be written in In practical applications, the set of functions β i is, of course, far from complete, but the n×n matrix E may still be determined by solving Eq. ( 6) for E in a least-squares sense given the re-weighting just introduced.Note that Eq. ( 6) is exactly satisfied if the functions β i span an invariant subspace of the Hamiltonian H, even if they do not form a complete set; this provides an important zero-variance principle for the corresponding part energy spectrum.
It is convenient to rewrite Eq. ( 6) in matrix form where B σi = βi (R σ ) and B ′ σi = β′ i (R σ ).Multiplying Eq. ( 7) from the left by the transpose of B, one obtains by inversion with Nij = σ βi (R σ ) βj (R σ ) and Ĥij = σ βi (R σ ) β′ j (R σ ).As can be easily verified, Eq. ( 8) is indeed the least-squares solution of Eq. ( 6).Note that for an infinite sample the hermiticity of the Hamiltonian guarantees that Ĥ is a symmetric matrix, but this is not the case for a finite Monte Carlo sample.If Ĥ is symmetrized in Eq. ( 8), the resulting E no longer satisfies the least-squares property nor the aforementioned zero-variance principle.
The optimal linear combinations of the basis functions β i are computed by constructing the spectral decomposition of E: where dk j and d k i are the components of the left and right eigenvectors of E with eigenvalues Ẽk .This yields the trial functions Before we continue this review of our optimization procedure, some comments should be made.First of all, the matrix N frequently is ill-conditioned.This problem can be dealt with by using a singular value decomposition of the matrix B to obtain a numerically regularized inverse of N. 7,8 Secondly, we mention that Eq. ( 10) can also be derived from the condition that the quantum mechanical expectation value of the Hamiltonian in the states ψk is stationary with respect to variation of the coefficients d k i .This condition yields a generalized eigenvalue equation involving matrices N and H, the Monte Carlo estimators of which are the matrices N and Ĥ introduced previously.The re-weighting defined before Eq. ( 6) was introduced so that these estimators are unbiased.
As mentioned, the basis function β i depend implicitly on non-linear variational parameters.These are optimized iteratively and it should be kept in mind that for each choice of the non-linear parameters, new optimized linear parameters have to be computed.The full optimization of all parameters therefore entails a linear optimization nested in a non-linear one.The linear optimization is a standard linear algebra problem; the optimization of the non-linear parameters is performed by minimizing the variance of the local energy of the wave function: where ψk = ψ −1 g ψk and ψk′ = ψ −1 g H ψk .The trial wave functions produced by this method yield estimates of the energy levels that are upper bounds to the exact energies, if statistical errors are negligible.2][13] This reduces the variational bias in the eigenvalue estimates, but it usually increases the statistical errors in the estimates.In a formal sense, the reduction of variational errors obtained in correlation function Monte Carlo is accomplished by introducing a new and improved basis by means of the substitution For increasing projection time t the spectral weight of more highly excited states in the new basis is reduced, and with it the variational error.In the limit t → ∞ all states of the new basis collapse onto the ground state, which implies that as t increases, the overlap matrix of the t-dependent basis states becomes more nearly singular, which increases the statistical errors.In principle, the errors increase exponentially; in practice, the method as we currently use it, breaks down once the Monte Carlo estimate of the overlap matrix develops negative eigenvalues.
We use elementary basis functions of the following general form 14 where the term involving A imposes short-and long-range asymptotics; the s i and s j are bosonically symmetrized monomials.The exact structure of these basis functions is of no concern in this paper.A detailed description of the above mentioned method and the structure of the basis functions can be found in Refs.7,8.

B. Numerical results in various dimensions
In this section we present numerical results that show that the energy spectrum as a function of dimensionality for D ≥ N − 1 is symmetric about D = N.We discuss results for Kr, Ar, Ne and the hypothetical 1 2 -Ne, which are defined respectively by the following inverse masses: 1.9128 × 10 −4 (Kr) 6.9635 × 10 −4 (Ar) 7.0920 × 10 −3 (Ne) 1.4184 × 10 −2 ( 1 2 -Ne).

The three-body case
Table I shows the ground state energies E 1 for Kr 3 , Ar 3 and 1 2 -Ne 3 in dimensions ranging from D = 1 to D = 6.We fitted the computed values for D ≥ 2 to a parabola with its  minimum at D = 2.The difference between the computed and fitted results ∆E 1 is also shown in the Table I.As is the case with the classical minimum of the energy, which equals −2.03 for D = 1 and −3 for D ≥ 2, the quantum mechanical ground state energy at D = 1 is nowhere near the curve.

The four-body case
Table II shows the ground state energies E 1 for a four-body cluster, Ar 4 , in various dimensions.From these results it is evident that an inter-dimensional degeneracy exists The results in TableV illustrate the loss of accuracy that occurs for five particle clusters.
The differences between the estimates of the energies of corresponding levels for four and six dimensions are due to the failure to converge of the correlation function Monte Carlo.This reflects the fact that our trial wave functions can in principle be systematically improved only for cluster sizes N ≤ 4, because they contain fully adjustable n-body correlations with n ≤ 4 only.

A. Clusters in arbitrary number of dimension
The Schrödinger equation for an N particle cluster in D spatial dimensions is a differential equation in ND variables.For a S state the wave function is invariant under rotations and translations.Therefore, one can write the wave function as a function of fewer than ND variables.To accomplish this we proceed as follows.
Consider the N − 1 difference vectors r 21 , r 31 , . . ., r N 1 as defined by Eq. (2a).Note that these vectors cannot be linearly independent unless D ≥ N − 1, in which case they define a parallelepiped P .Precisely 1 2 (N − 1)N independent variables are required to define P up to a congruence transformation.One possible choice of such variables consists of: (1) the angles θ ij between the the vectors r i1 and r j1 or their cosines with 1 < i < j ≤ N; and (2) the lengths of the vectors r i1 with 1 < i ≤ N.Alternatively, as independent variables one may choose the lengths of all distinct inter-particle distances r ij = r ji with i = j.These are the variables we shall use in this paper with the assumption, required for linear independence, that D ≥ N − 1.

B. Generalized Schrödinger equation
We consider a D-dimensional Schrödinger equation of the form with a Hamiltonian slightly more general than the one defined in Eq. ( 3) with a potential that is rotationally and translationally invariant, but not necessarily a sum of two-body contributions.Furthermore, the mass of each particle may be different.
We restrict ourselves to S states and to cases in which D ≥ N − 1 so that, as discussed in the previous subsection, the wave functions can be considered to be a function of independent inter-particle distances r ij with 1 < i < j ≤ N.
By straightforward application of the differential operator identity one obtains where and With the inter-particle distances as independent variables, the Schrödinger equation assumes a form that involves: (1) a linear differential operator that explicitly depends on the spatial dimensionality D; and (2) a second-order differential and a potential energy operator that are independent of D, as is clear from Eqs. ( 17) and (19).
Next, we transform the Schrödinger equation into an equation in which the second-order operator is unchanged, the linear operator is absent, and in which the potential is modified by an additional term. 15This is accomplished as follows: and The action of the operator on the right-hand side of an arbitrary function is to be evaluated from right to left, so that multiplying by χ takes precedence over operating by H.
This yields a special case of Eq. (3.8) of Ref. 16 with We define square matrices of order for i = 1, . . ., N. The matrix Ĝi is the Grammian associated with the N − 1 vectors r ij with j = 1, . . ., i − 1, i + 1, . . ., N. Its determinant is the square of the volume of the parallelepiped defined by the vectors pointing from particle i to all other particles.This volume is equal to N! times the volume of the (N − 1)-simplex of which the N particles are the vertices, which explains why ω does not depend on i, as our notation indicates.
In the Appendix we show that T i vanishes for the choice while Furthermore, using the fact that g i;jk , defined in Eq. ( 19), is an inner product, one can rewrite the sum in the Eq.(30b) as a sum of squares.This implies that to linear order in perturbation theory the energy eigenvalues depend quadratically on D with a minimum at D = N, in agreement with our numerical estimates presented in Section II B.

IV. DISCUSSION
We transformed the Schrödinger equation for the rotationally and translationally invariant states of an N-particle cluster in D ≥ N −1 spatial dimensions into a differential equation in 1 2 (N − 1)N independent variables in which the dependence on D is fully contained in an effective potential energy.Here D can be interpreted as a continuously varying parameter, as is commonly done in dimensional scaling studies. 17In agreement with work by Gu et al., 2 we observed that there exists an inter-dimensional degeneracy, of an N-body cluster in D = N − 1 and D = N + 1 dimensions.This degeneracy exists for all S states, i.e., for both the ground and excited states.Furthermore, the minimum energy was observed to be at the dimension D = N.
We stumbled upon this inter-dimensional degeneracy numerically by Monte Carlo methods, for which the generalization to arbitrary discrete dimensions is simple.In retrospect, knowing that this degeneracy is an exact property of the Schrödinger equation is useful because it provides a powerful check of the validity of our computer code and of our estimates of systematic and statistical errors in our numerical results.to infinite order and re-summed to yield a convergent expression valid for all D, the result would agree with this the analytic continuation discussed above, but not with the physics of clusters with more than four particles in three dimensions.

in D = 3
and D = 5 dimensions.Again, ∆E 1 represents the difference of the computed energies and the results obtained from a parabolic fit with its minimum at D = N = 4, this time excluding D = 1 and D = 2.The quantum mechanical estimates can be compared with the classical minima, viz., −3.07 for D = 1, −5.07 for D = 2, and −6 for D ≥ 3.
Comparison of the excited state energies E k (with errors in the least significant digit) of Ar 4 in D = 3 and 5 dimensions.Comparison of ground and excited state energies E k (with uncontrolled errors) of Ne 5 in D = 4 and 6 dimensions.
) Clearly, U i depends on the spatial dimensionality via its D dependent amplitude, which is symmetric in D about D = N. Recalling that this derivation is valid only for values for D ≥ N − 1, we find that for S states the Schrödinger equation has the same energy eigenvalues in D = N − 1 and D = N + 1 -and for those values of D only-while the eigenstates are related via ψ(D = N − 1) = √ ω ψ(D = N + 1).
Our results have another interesting implication.Our transformation of the Schrödinger equation to a differential equation in 1 2 (N − 1)N independent variables is valid only for D ≥ N − 1.However, D in the resulting equation can be interpreted as a continuous variable, and the equation has an analytical continuation for D < N − 1 and is symmetric about D = N.As a consequence, the energy spectrum for D < N − 1 of the transformed equation is analytic in D and symmetric about D = N.Whatever might be the meaning of this spectrum obtained by analytic continuation of the spectrum for large values of the D spectrum, it cannot have anything to do with the physical spectrum of N particle clusters for D < N − 1.This in turn implies that there is a fundamental problem with the work by Gonzalez and Leal, 18 who have used 1/D expansion to estimate energy levels of Lennard-Jones clusters in D = 3 with N = 3, 4, . . ., 7 and 13.If such calculations could be done

TABLE I :
Ground state energies E 1 (with errors in the least significant digit) and deviations from quadratic fits ∆E 1 for Kr 3 , Ar 3 and 1 2 -Ne 3 in dimensions D = 1 through D = 6.

TABLE II
Table III and IV strongly suggest that the same holds for excited states.The first table shows the four lowest excited state energies obtained for Ar 3 in D = 2, 3 and 4 dimensions.Once again, as observed for the ground state energies, these degeneracies exist in this three-body cluster for the D = 2 and D = 4 case.The D = 3 case is included in this table to indicate that the energies obtained here are different and lower than the other two cases.The second table shows the four lowest excited state energies obtained for Ar 4 in D = 3 and 5 dimensions; we denote energy levels by : Ground state energies (with errors in the least significant digit) and deviations from quadratic fits ∆E 1 for Ar 4 in dimensions D = 1 through D = 6.3.Excited statesThus far we have only numerically verified that inter-dimensional degeneracies exist for ground state energies.

TABLE III :
Comparison of the excited state energies E k (with errors in the least significant digit) of Ar 3 in D = 2, 3 and 4 dimensions.