ARTICLE pubs.acs.org/JPCA
On the Lindemann Criterion for Quantum Clusters at Very Low Temperature R. Guardiola and J. Navarro* IFIC (CSIC-Universidad de Valencia), Edificio Institutos de Paterna, Apartado Postal 22085, 46071 Valencia, Spain ABSTRACT: The Lindemann criterion to discern the solid-like or liquid-like nature of a quantum cluster at T = 0 is discussed. A critical analysis of current Lindemann parameters is presented and a new parameter is proposed that is appropriate to study quantum clusters made of identical particles. A simple model wave function is introduced to fix the range of variation of these parameters. The model presents two extreme limits that correspond to either a liquid-like or a solid-like system; besides, it fulfills the Bose symmetry and also permits evaluations without symmetrization. Variational and diffusion Monte Carlo calculations are also performed for clusters of spinless bosons interacting through Lennard-Jones potentials. It is shown that the liquid-like or solid-like character of quantum clusters at zero temperature cannot be simply established in terms of a single parameter.
1. INTRODUCTION The Lindemann criterion was introduced a long time ago,1 in the context of the old quantum theory, as an empirical rule to estimate the melting temperature of a crystal. Lindemann assumed that melting occurs when the average displacement of single atoms around the lattice sites Ær2æ1/2 exceeds a certain fraction of the characteristic interparticle distance d. The relevance of Lindemann criterion is that it is related to macroscopic quantities like the Debye temperature or the trends of X-ray scattering, see, for example, ref 2, and in principle it is easy to compute within microscopic theoretical models. The Lindemann parameter is currently defined as pffiffiffiffiffiffiffi Ær 2 æ ð1Þ δ¼ d The critical value is usually taken as δ ≈ 0.15, but in fact it may vary between 0.05 and 0.20, depending on the crystal structure, the specific constituents, and the interaction potential. Early computer simulations3-6 showed that atomic clusters undergo a smooth melting transition. In the context of finite systems, one attributes them to a solid-like nature when a localization of particles, reminiscent of a crystal, appears on a scale comparable to the system size. Conversely, the liquid-like nature of a finite system means that a long-range order is lacking. The Lindemann criterion has been used in finite systems, such as clusters of heavy rare gases,7 polymers,8 or proteins,9 introducing several definitions of the Lindeman parameter. One should keep in mind that the fulfillment of the criterion in the bulk is not a guarantee that it works well in finite clusters. Nevertheless, it has a qualitative validity and can provide some insight, because the r 2011 American Chemical Society
Lindemann parameter increases with temperature with a sizable step that defines the melting temperature, see, for example, ref 10. However, Calvo and Spiegelmann,11 from their analysis of melting in sodium clusters with sizes ranging from 8 to 147 atoms, questioned the interest in focusing on the Lindemann parameter to extract melting temperatures, particularly when premelting effects are strong, that is, when surface melting and surface reconstruction phenomena are clearly manifested. Theoretical studies of small parahydrogen (pH2)N clusters carried out in the last years have raised a very interesting question about their being solid-like or liquid-like in their ground state. Path integral Monte Carlo (PIMC) calculations12 indicated a sizable superfluid fraction below about 2 K in clusters with N = 13 and 18 molecules. Evidence of superfluidity of the (pH2)N cluster with N = 17 molecules was reported by Grebenev et al.13 in an experiment in which clusters of pH2 molecules were assembled around a carbonyl sulfide (OCS) chromophore molecule inside helium droplets in a molecular beam. However, in the production of small (pH2)N clusters by cryogenic jet expansion,14 an enhanced production was observed at N = 13 and possibly also at around N = 33. Both numbers coincide with the first two magic numbers appearing in the classical search of crystal structures for Lennard-Jones clusters.15-17 This solid-superfluid opposition has subsequently stimulated a large number of Special Issue: J. Peter Toennies Festschrift Received: November 22, 2010 Revised: January 27, 2011 Published: February 22, 2011 6843
dx.doi.org/10.1021/jp1111313 | J. Phys. Chem. A 2011, 115, 6843–6850
The Journal of Physical Chemistry A theoretical studies of the structures of small pH2 clusters (see, e.g., refs 18-20 and references therein). The Lindemann criterion has been used to discern the solid-like or liquid-like nature of pH2 puddles.21 Modified definitions of the Lindeman parameter have been used in the case of pH2 clusters22-26 at very low temperatures, down to T = 0. Taking into account the small mass of H2 molecules, their quantum vibrations around their equilibrium positions give the clue to explain the structure of pH2 clusters. Motivated by these studies on parahydrogen clusters, we present in this paper a critical analysis of currently used Lindemann parameters, and introduce a new one, which we consider more appropriate to study quantum clusters. The main objective is to check the utility of using a Lindemann parameter to discern the solid-like or liquid-like nature of a quantum cluster at T = 0. We shall study the Lindeman parameter as a function of the cluster size and the pairwise interaction. Along the paper we shall refer to different Monte Carlo techniques. The reader not familiar with them may find refs 27-30 useful. The paper is organized as follows. In section 2 we introduce a simple model of a cluster made of bosons, based on harmonic oscillations around some fixed cluster positions. The model depends on a single parameter, the ratio of the harmonic oscillator constant, and the nearest neighbor distance whose extreme values, namely, zero and infinity, correspond to a liquid-like or solid-like system, respectively. The model fulfills the Bose symmetry and, at the same time, permits evaluations with labeled particles, that is, without the symmetrization. Within that simple model, current definitions of the Lindemann parameter are analyzed in section 3, and a new parameter is proposed and quantified in section 4. The validity of the new parameter is assessed in section 5, using optimized variational trial functions as well as diffusion Monte Carlo estimates (VMC and DMC, respectively) for clusters of spinless bosons interacting through Lennard-Jones potentials. Finally, the main conclusions of this work are presented in section 6.
2. SIMPLE MODEL FOR CLUSTER WAVE FUNCTIONS We present a simple model for a cluster consisting of N spinless bosons, based on a noncorrelated wave function. We start from the classical minimum for a given cluster of N constituents, interacting through the pairwise Lennard-Jones (LJ) potential ! σ 12 σ 6 V ðrÞ ¼ 4ε 12 - 6 ð2Þ r r A complete review of classical LJ clusters may be found in The Cambridge Cluster Database,31 where in particular the static coordinates corresponding to the minimum of LJ potential are given for a large number of clusters. Let us denote by {ri }N i=1 the constituent coordinates, whose values at the minimum are specified by the set of coordinates {ci }N i=1 given in ref 31. These values have been obtained by minimizing the potential energy of static constituents interacting through a dimensionless LJ potential, that is, with energies and lengths given in units of parameters ε and σ, respectively. It is reasonable to assume that if a quantum cluster becomes solid, the average positions of its
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constituents will be close to the mentioned cluster coordinates. The typical nearest neighbor distance is equal to the position of the minimum of LJ potential, namely, d = 2 1/6 σ ≈ 1.12σ. Our simplified model assumes that the motion of each particle around its corresponding equilibrium position is described by a harmonic oscillator well of inverse length constant R. In the ground state, the normalized single-particle wave functions are given by R3=2 1 2 2 ð3Þ φk ðrÞ ¼ 3=4 exp - R ðr - ck Þ 2 π and the following product of single-particle wave functions Ψ¼
N Y i¼1
φi ðri Þ
ð4Þ
provides an unsymmetrized wave function of the cluster ground state. To obtain the symmetrized wave function required by Bose symmetry one must impose the permutation of labels, either of particles or cluster positions, dealing with products of the form N Y i¼1
φP i ðri Þ
ð5Þ
where the symbol P stands for a permutation. The fully symmetrized wave function is thus ΨB ¼
∑
N Y
P i¼1
φP i ðri Þ
ð6Þ
This function, which is not normalized, corresponds to the socalled permanent of the matrix φj(ri). Working with permanents is not common in quantum calculations. We refer the reader to the book of Nijenhuis and Wilf32 for a brief but precise description of permanents. The model has the nice property of switching from a liquid to a solid according to the value of its single parameter R. For very small values of R there is a very large overlap of single-particle wave functions, so that all ci may be neglected, and the wave function reduces to a simple Gaussian form !N " # R3 2 2 2 exp -R ri jΨB, R f 0 j ð7Þ π3=2 i
∑
which does not require any explicit symmetrization. In the opposite limit, for a very large value of R, each particle is narrowly tied to a cluster position, thus, representing a solid. The squared wave function may be written as Y δðri -cP i Þ ð8Þ jΨB, R f ¥ j2
∑ P
In both limiting cases, the calculation of expectation values is very simple. For example, the expectation values of rij are pffiffiffi 2 2 ð9Þ ÆΨB jrij jΨB æR f 0 pffiffiffi R π 6844
dx.doi.org/10.1021/jp1111313 |J. Phys. Chem. A 2011, 115, 6843–6850
The Journal of Physical Chemistry A
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and ÆΨB jrij jΨB æR f ¥
∑
2 jci -cj j NðN - 1Þ i < j
ð10Þ
An important question is to determine the critical value of R, or more precisely the critical range of values separating liquid-like from solid-like systems. To this end we compute the volume around a given cluster position at a distance d/2, i.e. the probability of finding the particle in a sphere of radius d/2, which is given by Z Rd jφðrÞj2 dr ¼ erf ðRd=2Þ-pffiffiffi exp½-R2 d2 =4 ð11Þ π r < d=2 where erf(z) is the error function. One can reasonably accept that a probability greater than 0.9 corresponds to a solid and a probability lesser than 0.7 corresponds to a liquid. In terms of the dimensionless harmonic oscillator parameter, this defines two values: RS = 3.2, and RL = 2.4. The precise values of the mentioned probabilities are not crucial for the present purposes, as the model does not provide a sharp liquid/solid frontier. What matters is that the above criterion means a solid-like cluster if R g RS, a liquid-like one if R e RL, and a coexistence regime in between these two reasonably estimated values.
3. CURRENT DEFINITIONS OF LINDEMANN PARAMETER FOR CLUSTERS The original Lindemann parameter requires an external input for the lattice distance or the nearest neighbor distance d, and thus, it is not always adequate to characterize melting of finite clusters. Several authors have given other definitions of the Lindemann parameter appropriate for computer simulations, keeping a qualitative resemblance with the original one but without external input. Etters and Kaelberer4 introduced the concept of bond fluctuations in the study of clusters of rare gases and defined a Lindemann parameter based on the width of the peaks of the pair distribution function, with the lattice distance being obtained along the simulation. Actually, the parameter used by these authors is the half-width at half of the maximum (HWHM) of the pair distribution peaks divided by 2σ, where σ is the length parameter of the LJ potential of each species. This is a very reasonable criterion, specially adapted to study the melting process as a function of the temperature, but it is not adequate to answer the question of whether the ground state (T = 0) of a cluster is a liquid or a solid. Beck et al.6,33 introduced an alternative parameter based on the relative mean bond length fluctuations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ærij2 æ - Ærij æ2 2 ð12Þ δB ¼ NðN - 1Þ i < j Ærij æ
∑
Brackets in the previous definition stand for the average along a Monte Carlo path integral evolution, which in the quantum case should coincide with the corresponding expectation value. This definition is thus well adapted both for classical and quantum calculations. Zhou et al.34 have performed molecular dynamics simulations for clusters composed of 64 particles interacting pairwise through square-well and Morse potentials. They have computed the previous δB and a new parameter, which differs from δB in that the averaged sum is restricted to neighboring
pairs, where members of each pair are separated by less than a nearest-neighbor cutoff distance. Their results show that both parameters lead to similar critical values for both interactions. The parameter δB has been used in various papers dealing with small para-hydrogen clusters.22-25 However, it should be properly modified when dealing with quantum systems of identical particles. Indeed the Hamiltonian H is invariant under the permutation of particle coordinates and, as it is well-known, among all solutions of the eigenvalue equation HΨ = EΨ, only those that are fully symmetric or fully antisymmetric appear in nature. In both cases, the probability density |Ψ|2 is a symmetric function, and when computing the expectation value of any operator it acts as a projector, extracting from this operator exclusively the symmetric part. In other words, relevant quantum mechanical operators in a system of identical particles must be symmetric. At the same time, the use of particle labels has no sense, apart from being a way of labeling the 3N dummy coordinates of the wave function. Now, if the brackets in eq 12 are interpreted as quantum mechanical expectation values and the wave function is Bose symmetric, the following equality holds Ærij æ ¼
∑
2 Æ rkl æ NðN - 1Þ k < l
ð13Þ
for any pair (i,j), and the parameter δB should be equivalent to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Æ rij2 =Np æ - Æ rij =Np æ2 δG ¼
∑
∑
i
