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by: Ms. Juvenal Kertzmann


Ms. Juvenal Kertzmann
GPA 3.76


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This 3 page Class Notes was uploaded by Ms. Juvenal Kertzmann on Saturday September 12, 2015. The Class Notes belongs to Bio Sci 38 at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 68 views. For similar materials see /class/201895/bio-sci-38-university-of-california-irvine in Biological Sciences at University of California - Irvine.

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Date Created: 09/12/15
PRL 96 113401 2006 PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 Structural Transformations and Melting in Neon Clusters Quantum versus Classical Mechanics Pavel A Erantsuzov Dario Meluzzi and Vladimir A Mandelshtam Chemistry Department University of California at Irvine Irvine California 92697 USA Received 17 November 2005 published 20 March 2006 The extraordinary complexity of LennardJones LJ clusters which exhibit numerous structures and phases when eir size or temperature is varied presents a great challenge for accurate numerical simulations even without accounting for quantum effects To study the latter we utilize the variational Gaussian wave packet method in conjunction with the exchange Monte Carlo sampling technique We show that the quantum nature of neon clusters has a substantial effect on their sizetemperature phase diagramsquot particularly the critical parameters of certain structural transformations We also give a numerical con rmation that none of the nonicosahedral structures observed for some classical LJ clusters are favorable in the quantum case DOI 101103PhysRevLett96l13401 Atomic clusters exhibit very rich structural thermody namic and dynamical properties that may vary with size in a nonmonotonic fashion In particular rare gas atomic clusters often modeled using the LennardI ones LI pair potential have been very popular in the last several deca des Typically multifunnel and rough potential energy landscapes in LI clusters make their numerical simulations extremely challenging even when the quantum effects are neglected In the past a number of publications addressed the role of the latter using the path integral Monte Carlo PIMC method A particularly interesting case corre sponds to neon clusters see eg Refs 12 However for a system as small as New a wellconverged heat ca pacity CVT was reported only recently 2 Such a cal culation is apparently a rather dif cult task In Ref 3 the most recent version of a PIMC heat capacity estimator 4 was applied to compute CVT for Ne3g but the conver gence at low temperatures was not quite satisfactory In the same paper the variational Gaussian wave packet VGW method was also applied to Ne3g Although manifestly approximate the latter method had so far demonstrated both numerical ef ciency and surprisingly high accuracy when compared to the exact results for low dimensional systems 5 or to accurate PIMC calculations 6 Unfortunately its application to a system as complex as Ne3g also seemed quite expensive This high cost however was expected as accurate heat capacity calculations for the classical L133 cluster required a numerical effort several orders of magnitude greater than that required for L113 7 It is the relatively good convergence achieved in Ref 3 for Ne3g using both VGW and PIMC methods that may seem rather surprising Unlike its classical counterpart due to strong quantum effects this system has a simple single funnel con guration space at low temperatures and does not undergo structural transformations Otherwise much longer Monte Carlo runs would be needed For example the VGW method in its original formulation 35 is likely to fail for the case of New using any reasonable computa tional resources 0031 9007 06 961111340142300 1134011 PACS numbers 364070 64607i 647071 In this Letter in order to improve the convergence properties of the VGW method we implement a much more ef cient sampling scheme based on the exchange Monte Carlo EMC or parallel tempering procedure 89 rather than on a single Metropolis random walk done at suf ciently high temperature 35 The method is applied to compute the heat capacities for Ne35739 The present result for Ne3g has statistical errors at least an order of magnitude smaller than those in Ref 3 but is achieved with a comparable numerical effort We also perform the VGW analysis of the ground states of Ne for sizes up to n 120 First we brie y review the properties of classical L1 clusters Then we contrast those with the properties of quantum Ne clusters Figure 1 shows the orientational bond order parameter Q6 10 obtained from the global minima of L1 11 as a function of n Here the choice for the order parameter Q6 is motivated by its high sensitivity to cluster symmetry The two dominant structural types that are realized for all but several special cases n 38 75777 98 1027104 are based on either the Mackay icosahedral or antiMackay or polyicosahedral motifs which correspond respectively to an incomplete Mackay or antiMackay overlayer surround ing a Mackay icosahedral core 12715 As seen in Fig 1 at zero temperature the Mackay to antiMackay M gt aM transitions occur at sizes 31 gt 30 and 82 gt 81 Depending on the structure of the global minimum a LI cluster may undergo one or more temperatureinduced structural transformations 716719 according to the fol lowing general rules Clusters with a Mackay overlayer in the global energy minimum will at some nite tempera ture undergo a surface melting transition to an anti Mackay overlayer because the latter is entropically more favorable than the former 1719 A complete melting of the cluster which we loosely de ne as the core meltingquot occurs at higher temperatures 19 but is not always easy to characterize especially for small twolayer clusters for which the 13atom core may be impossible to identify 2006 The American Physical Society PRL 96 113401 2006 PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 x 38 057 C 047 5 a E 7 Q g 03 3 7577 102104 1 quotE 027 o 31X 82 3 017 9 85 0 f 9 98 a 3 55 R9 0 AK 5m 30 Swab 81 l l 100 1 0 0 Cluster size 11 FIG 1 color online Orientational bond order parameter Q as a function of cluster size for global minima of classical LIquot clusters The following symmet types are identi ed Mackay icosahedral icoM antiMackay icosahedral icoaM Trun cated octahedral Oh decahedral dec and Tetrahedral Td uniquely The left panel of Fig 2 compares heat capacities ofLJ3O and L131 The M gt aM transition in L131 gives rise to a small peak in the CVT curve at T N 10 K Here and throughout we assume LJ parameters for the NeNe pair potential The core melting transition results in a broad maximum at T N 118 K TheM gt aM transitions in big ger clusters as in L133 right panel of Fig 2 give rise to l l 71 CVNkB 6 I 7 7 47 L131 7 7 L130 2 l l l 10 15 T K FIG 2 color online Heat capacities for the classical LJ and quantum Ne clusters The broad maxima of CVT at T gt 10 K are presumably due to the cluster core melting A shoulder in the CVT curve at T N 4 K for LJ38 is a result of the structural transformation from the global octahedral minimum toward the Mackay icosahedral local minima LJMV 9 and New at zero temperature have the Mackay overlayer and so does LJ38 at T gt 4 K and as such undergo the temperatureinduced M gt 51M transitions at temperatures where the sharper peaks are situated The thermodynamic properties of LJ30 and New are similar in that they both have antiMackay overlayers at T 0 and do not undergo any lowtemperature transitions below core melting peaks that are stronger and shifted to higher temperatures Clusters with nonicosahedral global minima undergo global structural transformations at some low temperature toward icosahedral local minima because the latter are entropically more favorable than the former 71520 Such a transition in L133 gives a little bump in CVT at T N 4 K and is hard to identify However this and the other structural transitions mentioned above are clearly charac terized by the distributions pQ6 T of the orientational bond order parameter Q6 as a function of temperature see Fig 3 which has distinct values for different symmetries cf Figure 1 These plots recon rm that the structural transitions in nite systems are better described by coex istence regions 21 where eg the distribution pQ6 T for a speci c temperature range may have a bimodal character While much is now understood about classical LJ clus ters less can be said about the effect that the quantum nature of say rare gas clusters can have on their thermo dynamic properties For argon and heavier rare gas clus ters quantum effects can often be treated as small perturbations 22 However neon clusters are expected to exhibit strong quantum effects not only in the low temperature regime but also in the liquid phase 1 By using a procedure based on VGW and similar to that of Ref 3 we were able to estimate the ground state energies and structures of Ne clusters for n S 105 and n 110 115 120 For the case of N633 the method proved to be reliable when veri ed by extensive PlMC calcula tions Here for each n several long random walks with classical canonical distributions are generated at a series of temperatures between 5 and 9 K using a standard EMC procedure similar to that utilized in Ref 19 Once in every 1000 MC steps per temperature the cluster con gu ration is selected as the initial condition for propagating the VGW in imaginary time to a high value low temperature in order to obtain a stationary Gaussian state We also construct the stationary states from all the con gurations given in Ref 1 l The state with the lowest energy is then accepted as a putative ground state Since this particular calculation uses a more accurate version of the VGW method ie fully correlated Gaussians that scales numeri cally as n it becomes prohibitively expensive for large clusters Figure 4 displays the orientational bond order parameter Q6 computed for the ground states as a function of n This diagram con rms the conclusion of Ref 22 based on the harmonic approximation HA that none of the nonicosahedral con gurations cf Figure l survives for the quantum case of New However we note that in the Ne case the HA gives grossly incorrect energy estimates due to the excessive value of the quantum delocalization parameter A Wax 0095 which can even lead to delocalization of the ground state over several local poten tial energy minima 3 In particular most of the ground state structures reported in Ref 1 1 that are different from the classical global minima do not actually give stationary Gaussians with the lowest energy 1134012 week ending PRL96113401 2006 PHYSICAL REVIEW LETTERS 24MARCH2006 I I 06 06 06 LJ3O quot2 LJ38 Negg Oh 04 04 04 02 icoM icoaM 02 02 0 39 39 39 0 0 5 10 15 0 06 06 06 LJ39 04 04 04 06 icoM 02 ico M 02 02 3 icoaM l 39COaM ICON IcoaM 0 0 FIG 3 color online and quantum Ne Lennard Jones clusters The quantum effects make the antiMackay symmetry more favorable than the Mackay symmetry shifting the corresponding transition to higher cluster sizes In particu lar it is only starting at n 39 that the ground states of all the doublelayer Ne clusters have the Mackay symmetry and are localized over the global minimum of the potential energy Figure 5 displays two and threelayer neon clus ters that are representative of the two symmetry types The antiMackay overlayers are less compact and seem to have more liquidlike character By analogy to the classical case 19 we expect to observe the temperatureinduced M gt aM transition in clusters with Mackay overlayer while for antiMackay clusters the core melting at T 10 K should be the only structural transformation 02 o ico aM o ico M 015 0 h o o of 0 A 0 89 92 a o 39 I 3 o0 S 01 39 55 9 0 2 E o O 38 ta 005 o 93 6 0 o 00 o 2 0 95 Oo 0 Q o M9 3 0 33 0 I I 100 0 Cluster size 11 FIG 4 color online Orientational bond order parameter Q6 for the putative ground states of Ne clusters 10 15 0 5 10 15 T K Contourplots of the distributions pQ6 T of the orientational bond order parameter Q6 for the classical LI In order to support our conjecture we performed the heat capacity calculations for N63539 in the temperature range 1 K lt T lt 16 K A con ning radius of RC 30 was used While for n S 38 the convergence was relatively easy to achieve due to the lack of structural transformations at low temperatures the results for N639 were particularly slow to converge For this case 14 replica temperatures Tk distributed in the temperature range of interest gave rise to 14 random walks which were executed in a parallel fash ion on a 14processor computer cluster During the equili bration run the temperature grid was adjusted to make the exchange rate between replicas at adjacent temperatures equal to approximately 50 The temperaturedependent observables for each interval between the two adjacent replicas Tk lt T lt Tk were evaluated by the method of Ref 3 utilizing the contribution from the k 1th rep lica The convergence was monitored by comparing aver ages over independent successive runs The total number of MC steps needed to obtain reasonably wellconverged results for N639 was NMC 4 X 106 where one MC step corresponds to one accepted Gaussian wave packet in the N638 icoaM Ne39 icoM N688 icoaM FIG 5 Some Ne ground state con gurations estimated by the VGW method and representative of the two possible overlayer symmetry types The core atoms are shown in dark color 1134013


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