A diffusion Monte Carlo study of small para-Hydrogen clusters

Theoretical studies of parahydrogen clusters have attracted a growing interest in the past years, partly motivated by a recent experiment [1] in which Raman scattering was used in cryogenic free jets of the pure gas. Small changes in the frequency near the Q 1 (0) line of the monomer, were observed and interpreted as intermolecular effects on the intramolecular potential. (p-H 2 ) N clusters with N = 2 -8 were clearly identified through frequency shifts ranging from ∆ν = -0.40cm -1 for N = 2 to ∆ν = -2.35cm -1 for N = 8. The experiment also showed a bump at N = 13, N = 33 and N = 55, which were interpreted as a signal of magical clusters. However, it is worth mentioning that these three values are actually extrapolations from smaller clusters and presumably are approximate.

Magical numbers appear in classical Lennard-Jones clusters, related to geometrical shapes [2]. Several papers have appeared in the last year with the main objective of checking the magical numbers found in Ref. [1], and/or studying possible superfluidity effects in parahydrogen clusters. Indeed, Path Integral Monte Carlo (PIMC) calculations [3] have found a large superfluid fraction in clusters with N=13 and 18 at temperatures T ≤ 2 K. A superfluid response has been observed in small clusters consisting of a carbonyl sulfide cromophore surrounded by 15-17 p-H 2 molecules, all within a large helium droplet [4]. This has been confirmed by several MC simulations [5,6,7,8,9] of doped p-H 2 clusters.

Systematic studies of (p-H 2 ) N clusters, covering the range from N = 3 to N = 50 molecules, have been done based on powerful many-body techniques, as difusion Monte Carlo (DMC) [10], PIMC [12,13,14], and PIMC adapted to the ground state (PIGS) [11]. Whereas up to N ≃ 22 all these calculations are substantially in agreement, for heavier clusters there are noticeable differences between DMC and PIMC results, particularly for N ≥ 26. PIMC chemical potentials show very prominent peaks at N=26, 29, 34 and 39, in contrast with the smooth behavior obtained with DMC.

In this work we present new DMC calculations, improving our previous ones [10] so as to get very precise results within our computational capacity. Specifically, we consider three aspects: the importance sampling func-tion, the time step adjustement and the estimation of the statistical errors.

The DMC procedure is significantly improved when using a good importance sampling wave function, the main effect being the reduction of the variance of the stochastic procedure. We have used a Jastrow function with two-and three-body correlations:

with

and

with

Indices i, j, l run over the number of molecules in the cluster. This function is described in terms of five variational parameters, p 5 , p 1 , s T , ω T , and λ T . In our previous calculations [10] we used the standard two-body Jastrow function Φ T = exp(u 2 ). The present trial function includes an enlarged the two-body variational space and also three-body correlations in the form suggested in Ref. [19], which still has O(N 2 ) computational complexity. DMC is based in a short-time approximation of the Green’s function related to the imaginary time Schrödinger equation. In this way, an initial wave function Φ T (t = 0) evolves to the exact ground state wave function Ψ at large t after many short-time steps τ . We have used the O(τ 3 ) approximation to the Green’s function as described in Refs. [17,18], which provides energies O(τ 2 ). The time step adjustment is the following: from calculations at the relative large steps 0.001 and 0.0005K -1 , we obtain the Richardson extrapolated value

based on the τ expansion E(τ ) = E(0) + Cτ 2 + • • • . This value turns out to be very close to the calculations with much smaller time steps, as it may be checked in the last three rows of Table I. This checks that the algorithm behaves as O(τ 2 ), as expected, and suggests to use the value τ = 0.0001K -1 for massive calculations with a negligible bias. A further improvement of the calculation regards the estimate of the statistical error. Because of the sequential Markov chain nature of Monte Carlo algorithms, successive samples are strongly correlated, and the typical way of estimating the variance, σ 2 = H 2 -H 2 , may be too optimistic. To avoid these correlations we computed a number of times (10, typically), the binding energies, with independent and randomized runs, and estimate the variance from these results. This requires a considerable increasing in computational time, but the obtained standard deviations are very precisely computed. Specifically, we have used 1000 walkers with 10 5 steps plus 20000 stabilization steps in each walker.

Hydrogen molecules interact through weak van der Waals forces that, nevertheless, are sufficiently strong to bound clusters with any number of molecules. Several forms have been derived to describe the p-H 2 -p-H 2 interaction. Two of them are of particular interest because they combine ab initio prope

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