Inherent Structures In The Potential Energy Landscape Of .
Inherent structures in the potential energy landscape of solid 4HeJennifer A. Hodgdon and Frank H. StillingerAT&T Bell Laboratories, Murray Hill, New Jersey 07974 Received 10 September 1993; accepted 27 September 1994!We study the potential energy landscape many-atom potential energy as a function of atomicpositions! of solid hcp 4He in the vicinity of the 0 K crystal structure using an accurate pairpotential. At the melting point, the potential energy of the helium lattice is far above the minimumhcp interatomic potential energy. We confirm previous conclusions based on less accuratepotentials! that all of the classical phonon frequencies at the 0 K melting pressure are imaginary,indicating that the melting-point crystal corresponds to a local maximum in the potential landscape;a pressure of about 1300 bar, however, makes it a local minimum. We find that the atomicarrangements that lie at local minima in the potential landscape ‘‘inherent structures’’! are glassyand porous, and have much lower potential energy than the crystalline form at the same density. Wehave quantitatively characterized the glassy structures by their radial distribution functions andcoordination number distributions; they qualitatively resemble inherent structures for classicalmonatomic liquids, but exhibit differences of detail. A model variational calculation has been carriedout for the melting-density ground state. It utilizes separate basis functions for each of the inherentstructures, predicts a large Lindemann ratio for the crystal, and indicates that the probabilitydistribution is a maximum at the perfect lattice configuration. 1995 American Institute ofPhysics.I. INTRODUCTIONIt has been recognized for many years that of all theelements in their solid forms, helium exhibits the strongestquantum effects. Theoretically, this can be seen by comparing the pair interaction energy, V(a), with the zero-pointenergy, '\2/(ma 2 ! m5atomic mass and a5interatomicspacing!. For most elemental crystals, the interaction energyfar exceeds the above zero-point energy, but for solid heliumthey are comparable, with the zero-point energy slightlylarger.1 This means that while classical mechanics with atoms localized at lattice sites providing a means of distinguishing one atom from another! explains the behavior ofmost crystals under most circumstances, one needs quantummechanics with indistinguishable, intrinsically delocalizedatoms to explain the behavior of solid helium. The presenceof strong quantum effects gives solid helium unusual properties: for instance, 4He remains a liquid at 0 K and onlysolidifies when pressurized to 25 bar;2 also, the large zeropoint motion makes helium a rather soft crystal that has arough, instead of faceted shape at 0 K.3 This and otherunique properties of solid helium have made it a popularsubject for both experimental and theoretical study overmuch of the past century.4To understand the helium crystal fully, one must firstunderstand the multidimensional geometry of the potential inwhich its atoms move. Currently, good empirical pair potentials for helium exist5–7 that fit the experimental gas-phasetransport coefficient data over a wide temperature range; theyappear to be qualitatively similar to Lennard-Jones potentials, but differ in quantitative detail. Further, experimentersrecently observed a very weakly bound helium dimer,8 confirming the prediction of the empirical potentials. However,having a good pair potential is not enough: we lack a comprehensive picture of the landscape of potential energy forJ. Chem. Phys. 102 (1), 1 January 1995the system of a large number of helium atoms; that is, thecollectively determined total potential energy as a function ofthe entire configuration of the N atoms.Detailed study of potential energy landscapes, and inparticular of their ‘‘inherent structures’’ local potential energy minima!, has benefitted the theory of classical liquidsand amorphous solids.9–14 In particular, this approach hasaided in understanding glass transitions,15 clusterdynamics,16 chemical reactivity,17 and molecular conformational equilibria.18 For classical crystals, the inherent structures are simply the crystal structures possibly containingdefects! without phonon motions, and the analysis of thesestructures adds little to conventional understanding of thesolid state. However, the inherent structure formalism forstudy of the underlying potential energy landscape can alsobe used in the quantum regime,19 and under proper circumstances has the capacity to illuminate nonobvious aspects ofthe quantum solid state, as results reported below illustrate.In order to examine the potential energy landscape ofhelium, we assume that the additive pair potential from Ref.5 referred to as ‘‘the Aziz et al. potential’’! provides a goodrepresentation of interactions for a large collection of heliumatoms. Using one of the other potentials6,7 does not substantially change the results. Section II below discusses the Azizet al. potential and its implications for helium atoms at, andinfinitesimally displaced from, the regular hcp lattice of solidhelium, over a wide density i.e., pressure! range. The measured lattice spacing a M for solid helium at its 0 K meltingpoint is substantially larger than the value that minimizes thesummed Aziz et al. potential, owing to the dilating effect ofzero-point kinetic energy. In accord with earlier less-accuratecalculations based on Lennard-Jones interactions,20 we findthat at lattice spacing a M all classical phonon modes areunstable, indicating that the observed crystal structure nominally corresponds to a local maximum in the potential land-0021-9606/95/102(1)/457/8/ 6.00 1995 American Institute of Physics457
J. A. Hodgdon and F. H. Stillinger: Potential energy of solid 4He458scape. Upon compression, though, more and more of theclassical phonons become stable, a process that completeswhen the helium crystal is subjected to a pressure of about1300 bar. At this stage the helium crystal corresponds to alocal potential minimum but still not the global minimum!.Section III explores the inherent structures for the lowpressure crystal. Specifically, these are the local minima inthe potential landscape found by gradient descent on the potential energy surface from various starting configurations.The number of distinguishable inherent structures is expected to be exponentially large in N, the number of heliumatoms; we presume only to have generated a few representative examples. Our calculations demonstrate that these inherent structures possess tenuous structures of porous glassycharacter. They have much lower potential energy than theregular hcp lattice and than the configuration from whichthey were generated, and scarcely retain any memory of thelattice periodic order. Pair correlation function and neighbornumber distribution function calculations are also reportedfor the inherent structures in Sec. III.In order to demonstrate the relevance of the inherentstructures to the crystalline 4He ground state, a model variational calculation has been set up and investigated numerically in Sec. IV. Although it is based on a simplified representation of the multidimensional potential energyhypersurface for analytical tractability, it nevertheless conforms to the results obtained for the system’s inherent structures, and utilizes separate basis functions for each inherentstructure. This variational calculation verifies our workingassumption that the perfect lattice configuration is a probability maximum.Finally, in Sec. V, we consider some of the implicationsof our findings. These include the relation of present resultsto inherent structures found previously for classical monatomic systems both solid and liquid!, and to the resonatingvalence bond picture of electronic structure in chemicallybonded systems.II. INTERACTION POTENTIAL AND LATTICEPROPERTIESSolid helium at 0 K forms an hcp crystal with latticespacing a M 53.65 Å at the melting point 25 bar!.2 As a firststep in examining the potential energy landscape of helium,it is natural to ask where in potential energy! this crystalstructure lies in comparison to other possible structures. Toanswer that question, we sum the Aziz et al. pair potential,5V r !5eV*S Dr,rmV * x ! 5A exp 2 a x ! 2H x !5HSexp@ 2 D/x21 ! 2 #1for x DDC 6 C 8 C 1011H x !,x 6 x 8 x 10 2.1!for x,D, where r m 52.9673 Å, e510.8 K, A50.544 850 463106,a513.353 384, C 6 51.373 241 2, C 8 50.425 378 5, C 1050.178 100, and D51.241 314! over all pairs of atoms. Asan example, we have carried out the sums for both hcp andFIG. 1. Summed Aziz et al. Ref. 5! potential energy per particle! for bothhcp and fcc lattices vs lattice spacing. The curve for hcp lies slightly belowthe fcc curve, but the difference is smaller than the thickness of the line.Note that the lattice spacing at the 0 K melting point, 3.65 Å, is far from thepotential energy minimum.fcc lattices for a wide range of lattice spacings. We find thatthe hcp structure always has slightly lower energy than thefcc structure, but the difference of about one part in 104 nearly constant for all lattice spacings! cannot suffice toexplain the observed relative stability of the hcp crystal. Forboth structures the minimum interatomic potential energy perparticle, at a52.90 Å, is 286.6 K, well below the value of235.4 K found at a M see Fig. 1!; also, the interatomic potential energy at reasonable solid-helium densities lies wellbelow the ground-state energy of solid 4He, which has itsminimum value of 26.0 K at the 0 K melting point.21Clearly, the helium crystal found at the 0 K melting pointdoes not correspond to the global minimum in the potentialenergy landscape, and quantum effects e.g., zero-point kinetic energy! must play a large role in stabilizing its crystalline form, besides supplying an energy of about 30 K to thesolid. Qualitatively, these attributes have long been recognized as solid helium properties.20,22This lattice sum calculation does not by itself rule outthe possibility that the observed 0 K melting-point crystalcorresponds to a local minimum in the potential landscape.One probes the local environment in the potential landscapeby calculating the classical phonon spectrum v q! in thestandard way.23 The hcp crystal is a simple hexagonal baselattice with a two-atom basis, so there are six branches in thephonon spectrum. The Aziz et al. potential is short ranged,which makes calculating the six v q! branches to arbitraryprecision possible by cutting off the required sums over thebase lattice at an appropriate reasonable distance.There is no guarantee that the phonon-problem eigenvalues v2 are positive, and in fact, we find that all phononJ. Chem. Phys., Vol. 102, No. 1, 1 January 1995
J. A. Hodgdon and F. H. Stillinger: Potential energy of solid 4He459Fig. 2 b!# and the ideal lattice configuration becomes a localpotential minimum.Experimentally it is known24 that a pressure of about1300 bar is required to attain the lattice spacing a53.2 Åfor hcp crystalline 4He. This is still considerably expandedcompared to the spacing a52.90 Å that minimizes the lattice sum for the Aziz et al. pair interaction, which wouldrequire a pressure of about 6000 bar.24 These attributes contrast vividly with the situation for a classical crystal, which at0 K would always correspond to a local potential minimum,regardless of pressure.III. INHERENT STRUCTURESFIG. 2. Classical harmonic phonon spectra along the @100# direction for hcpcrystals with lattice spacing a! a53.65 Å, b! a53.2 Å, out to the edge ofthe first Brillouin zone. The atoms in the crystal interact with the potential inEq. 1!. We plot the squared frequency v2 in the nonstandard units of K/(m He Å2! vs the wave vector q in Å21. In a!, the crystal found at the 0 Kmelting point, all the phonon modes including those in all other directions!have imaginary frequencies. In b!, the crystal observed under a pressure of1300 bar at 0 K, all phonon modes have real frequencies, and the soundspeed for acoustic modes is about 390 m/s. Solid lines indicate acoustic, anddotted lines optical, phonon modes.modes for the Aziz et al. potential at lattice spacing a M haveimaginary frequencies @see Fig. 2 a! for a typical v2 plotalong one direction in the Brillouin zone#—hardly surprising, since a similar result had been found using the less realistic Lennard-Jones interaction for helium.20 This confirmsthat at the 0 K melting point, the ideal lattice for the hcphelium crystal is unstable in potential energy! to all infinitesimal displacements; i.e., the observed crystalline form corresponds to a local maximum in the potential energy landscape.We have also examined the behavior of the phonons forthe Aziz et al. interaction as the lattice is compressed, corresponding to increasing the pressure on the ground-state helium crystal. When the lattice spacing has been reduced froma M 53.65 Å to a53.3 Å, for example, some but not all ofthe phonons have converted from imaginary to real frequencies; at this stage the ideal lattice configuration has become ahigh-order saddle point on the multidimensional potential energy surface. Further compression to a53.2 Å completesthe conversion—all phonons then have real frequencies @seeThe information presented in Sec. II concerned the perfect lattice and its infinitesimal distortions. Our main objective in this paper, however, involves construction and characterization of inherent structures for the low-pressurehelium crystal, which are appreciably displaced from the perfect lattice. These inherent structures owe their existence tothe presence of imaginary classical phonon frequencies forthe undistorted lattice, and can be found by following gradient paths on the potential surface starting in the immediatevicinity of the lattice configuration, or from configurationswhich differ appreciably from the perfect lattice, as we willsee below.Inherent structures in the quantum regime19 are the localpotential energy minima that contain configurations sampledby the quantum wave function C in their basins of attraction.These structures can be found by gradient descent on thepotential energy surface from configurations supplied by thedensity matrix. In low-density 0 K solid helium, observationtells us that C2 has a maximum at each of the N! equivalentperfect hcp lattice arrangements of the N helium atoms, although these arrangements are local potential energymaxima. We would expect C2 to spread out approximatelyisotropically around these configurations into the descent basins of the various inherent structures that have the perfecthcp lattice structure as a common boundary point, and this is,in fact, what we find in the model variational calculationpresented below in Sec. IV. For that reason, one way tosample the density matrix is to add very small displacementsto the hcp crystal configuration; each choice of random displacement will select a descent basin for an inherent structure. This method may not, however, result in finding all ofthe inherent structures that could be sampled by the heliumwave function—conceivably there might be other inherentstructures that do not contain the hcp lattice in their basins ofattraction, but that are nonetheless sufficiently close to theperfect crystal configuration to have appreciable wave function amplitude. These inherent structures could be found bygradient descent from configurations which have large random displacements from the perfect crystal.We model the infinite crystal undergoing local rearrangements as a small cell, typically 83838 atoms, with periodicboundary conditions that preserve the cell volume. We searchfor nearby local minima in interatomic potential energy byadding random perturbations to the perfect hcp lattice, andthen minimizing the total interatomic potential energy usingthe conjugate gradient method we sum over all pairs of at-J. Chem. Phys., Vol. 102, No. 1, 1 January 1995
J. A. Hodgdon and F. H. Stillinger: Potential energy of solid 4He460FIG. 3. Structures of solid helium, all shown from the same perspective. a!An hcp helium crystal with a small amount of random displacement, thestarting point for a typical conjugate-gradient minimization of the latticepotential energy. b! The inherent structure that results from the minimization. c! An atypical cracked inherent structure, obtained by adding a biastowards forming a crack to the initial configuration and then minimizing.oms in the cell, using the pair potential of the closest imageof each pair in the periodic boundary conditions!. In some ofour searches, we use a small random perturbation of about5% of a lattice constant, to sample the inherent structuresclosest to the perfect lattice configuration, which should havethe highest access probability. For other searches, we use alarge random perturbation of about 1/3 of a lattice constant,to sample inherent structures with lower access probability.We find that the inherent structures for these two differentstarting configurations were very similar, as described below.We also conducted one search from a biased starting configuration that led to the formation of a crack. We expect thatsuch inherent structures have extremely low access probability, but they are still of interest since they are a part of theoverall landscape of inherent structures for helium.Both the cracked and uncracked inherent structures thatwe find have irregular forms, with compacted regions andrarefied regions see Fig. 3!. These structures do not look atall crystalline, and in the course of the computation, the atoms move a distance on the order of the original lattice spacing a M , subtracting the center of mass motion see Table I!.The interaction energies of the inherent structures are muchlower than the energy of the hcp crystal at the 0 K meltingpoint, but not as low as the 286.6 K per particle found forthe minimum-energy hcp crystal see Table I!; the structurewith the crack has even lower energy than the uncrackedstructures. We also note that both the hcp crystal and theinherent structures we find at this density have much lowerenergy than the ‘‘lattice gas’’ formed by removing particlesat random from the minimum-energy hcp lattice to reach thedesired density: the melting-point 0 K crystal is less densethan the minimum-energy hcp crystal by a factor of 0.502,which gives the lattice gas an energy of 0.502!2 286.6K!5221.8 K.We use several methods to characterize the structuresTABLE I. Characteristics of solid helium inherent structures and the perfect crystal. The first line of the tableshows the characteristics of the hcp solid at the 0 K melting point, which corresponds to a local maximum inthe potential landscape. The rest of the lines in the table correspond to inherent structures obtained by descending into potential energy minima from various starting points: small random displacements from the perfectcrystal, larger random displacements, and slightly cracked. We show the lattice energy of the structures, in unitsof K per particle; the ratio between the number of nearest-neighbor bonds in the structure and in the perfectcrystal; the Debye–Waller factors corresponding to Bragg reflection off the 001! and 100! planes; and the rootmean square distance moved by an atom in a structure from its starting point in the perfect crystal, subtractingthe center-of-mass motion, in Å. 001! DWF 100! DWFStarting pointEnergyBond frac.Perfect hcp235.41.001.001.000.00Small dispSmall dispSmall disp263.7265.5263.30.8520.8860.8510.000 340.000 240.000 880.000 470.001 60.000 134.134.193.79Large dispLarge dispLarge disp264.0263.6265.80.8680.8730.8880.000 800.000 700.000 500.002 00.000 040.001 14.063.884.58Crack268.50.9280.003 50.000 143.57J. Chem. Phys., Vol. 102, No. 1, 1 January 1995Dist. moved
J. A. Hodgdon and F. H. Stillinger: Potential energy of solid 4He461FIG. 4. Pair distribution function g(r) dimensionless! vs r Å!, averagedover the three typical inherent structures; g(r) for the cracked inherentstructure is very similar, and both differ markedly from the crystalline pairdistribution function.geometrically. First, we look at the pair distributionfunction231g r !5rNN( d r1xa 2xb ! & , 3.1!FIG. 5. Coordination number distributions for helium inherent structures.Fraction of atoms p(C) with coordination number C open symbols! for a!the three typical inherent structures @the filled symbols are the averagep(C)# and b! the cracked inherent structure. Lines are guides to the eye.aÞbwhere r5number density, N5number of atoms, d r! is theDirac delta function, and the brackets denote an average overthe direction of r. For the uncracked inherent structures, g(r)has a strong peak at 2.90 Å the lattice spacing of theminimum-energy hcp crystal!, with several other peaks atlarger distances, eventually settling to g(r)51 see Fig. 4!;the cracked inherent structure has a very similar g(r). Thisform for g(r) is very different from a crystalline g(r), whichhas large, sharp peaks at the first, second, third, etc., nearestneighbor distances, out to r5 . However, the inherent structure g(r) is qualitatively similar to that found for LennardJones glasses,25,26 except that the peaks at small r are largerand narrower in the present inherent structure g(r). We believe the difference is due to the pores in the helium inherentstructures: atoms bordering on an open space have the freedom to reorder into a more crystalline structure, which bothsharpens and increases the height of the peaks in the heliumpair distribution function; these effects decrease as r exceedscharacteristic pore dimensions. Note also that the asymmetric triplet of second, third, and fourth peaks is better resolvedin the present case than for Lennard-Jones glasses.25,26The first minimum in the inherent structure g(r) lies at3.8 Å, and this provides a convenient cutoff for the definitionof nearest-neighbor atoms. With this definition, we can makea second geometrical characterization of the structures bycomputing the atoms’ coordination numbers distributionsshown in Fig. 5! and the number of nearest-neighbor bondsin the structure as a fraction of the number in the perfectlattice shown in Table I!. We find that the most probablecoordination number in the inherent structures is 12, thevalue in the hcp crystal, but that the distributions of coordination numbers are rather wide and asymmetric. The distributions have shoulders at a coordination number of 9 or 10,which we believe largely correspond to atoms on the edgesof open spaces pores or cracks! in the structures. The inherent structures have about 85% of the number of nearestneighbor bonds in the perfect hcp crystal, and not surprisingly! there is a strong negative correlation between thenumber of nearest-neighbor bonds and the energy see TableI!. The qualitative nature of the coordination number distribution and the correspondence of total bond number withinteratomic potential energy do not change when we useother reasonable cutoffs for the nearest-neighbor definition.Finally, we also characterize the inherent structures bycomputing their Debye–Waller factors. The Debye–Wallerfactor for a structure is defined to be the ratio of the scattering intensity I for a wave undergoing momentum change Gdivided by its scattering intensity I 0 in the perfect crystallattice27J. Chem. Phys., Vol. 102, No. 1, 1 January 1995
J. A. Hodgdon and F. H. Stillinger: Potential energy of solid 4He462I15 2I0 NU(UN2 3.2!exp iG x j ! .jand all with the same depthWe find that for both the cracked and uncracked inherentstructures the Debye–Waller factor is near or below thevalue 1/N50.0020 expected for a completely random placement of the 512 atoms in the cell, for two different choices ofG see Table I!; this implies that virtually none of the longrange order of the hcp crystal remains in the structures. Theone slight exception to this is for G perpendicular to thecrack surface in the atypical cracked inherent structure,where the Debye–Waller factor of 0.0035 indicates that ashadow of the original crystal lattice structure remains forplanes parallel to the crack surface.IV. VARIATIONAL WAVE FUNCTION2 3N 5exp N ln 8 ! 4.1!exhibits the requisite exponential rise with N the ln 8 is anapproximation to the correct coefficient, at present not accurately known!.In order to represent this array of minima surroundingthe central maximum, it is necessary to invoke at least quartic anharmonicity in the expansion of the potential energyhypersurface. Without needing to specify the hypercube orientation, we can introduce collective configuration variablesu 1 ,.,u 3N by coordinate axis rotation so that the potentialenergy locally has the simple approximate formF52A(S( D3Nu 2i 2 B/3N !i51u 2i23N1Ci51(u 4i , 4.5!Data in Table I can be used to fix R and Fmin . Equivalently this provides two constraints on the model potential’sparameters A, B, and C. Specifically we findA53.432 498 K/Å2 , 4.6!C5B10.305 551 0 K/Å4 .The remaining parameter B must eventually be chosen sothat the ground state energy occurs at 4.7! 4.2! j51,.,3N ! ,all at a common distance from the origin equal toto agree with experiment.The F minima can be denoted by a set of Ising spinsm1 ,.,m 3N 561, where the sign of m j is the same as that ofthe collective variable u j . Then for each minimum we introduce a Gaussian basis functionS 4.3!D3Nf uu m! 5exp 2 a( u i 2s m i ! 2 .i51 4.8!Here a and s are nonlinear variational parameters. Althoughthe center of this Gaussian is displaced from the origin toward its corresponding potential minimum, the two need notbe coincident.The nodeless ground state C will be approximated by asymmetric sum over all basis functionsC u! 5(mf uu m! . 4.9!Three quantities are required, the normalization integralI a ,s ! 5EC 2 du, 4.10!the kinetic energy matrix elementK a ,s ! 52 \ 2 /2m !EC¹ 2 C du, 4.11!and the potential energy matrix elementV a ,s ! 5i51where A,B, and C.B are suitable positive constants. Here,for convenience, we set the potential equal to zero atthe perfect lattice configuration at a53.65 Å u 1 5 5u 3N 50!. It is trivial to show that the minima of Fare located at the 2 3N positionsu j 56 @ A/2 C2B !# 1/2F min523NA 2 /4 C2B ! .E 0 /N529.4 K,Inherent structure results contained in Table I for the 0 Kcrystal at its melting point display a striking consistency withrespect to their energy, mean atomic displacement from theregular lattice, and overall geometric pattern. Although rareexceptions exist e.g., last line of Table I!, the conclusion isthat nominally distinct but very similar inherent structuresare almost always reached by steepest descent, starting alongindependent directions from the unstable lattice configuration. These observations suggest that a simple model calculation, in variational format, can be carried out for the crystalground state to illustrate how the inherent structures combineto yield the system’s wave function.To be specific, we assume that the potential energyminima are all at a common distance in the 3N-dimensionalconfiguration space from the perfect-lattice configuration,and are uniformly distributed in direction. The vertices of a3N-dimensional hypercube provide a specific realization ofthis geometry; the number of its vertices,3N 4.4!R5 @ 3NA/2 C2B !# 1/2 ,EC 2 F du. 4.12!Then the ground state is inferred from the variational minimum with respect to a and s ofE 0 /N5 min K1V ! /I. a ,s ! 4.13!Upon inserting Eq. 4.8! into Eq. 4.9!, it is straightforward to expand C about the origin the perfect lattice!. Oneeasily obtainsJ. Chem. Phys., Vol. 102, No. 1, 1 January 1995
J. A. Hodgdon and F. H. Stillinger: Potential energy of solid 4HeFG3NC52 3N exp 2 a s 2 ! 11 a 2 a s 2 21 !(u 2i 10 u 4 ! .i51 4.14!Consequently it is the sign of 2 a s 2 21 that determineswhether C and probability distribution C2! possesses a localmaximum or a local minimum at the origin.Evaluation of I( a ,s) is greatly simplified by the fact thatoverlap integrals for distinct basis functions depend only onthe number of Ising spin discrepancies between the two, andnot on which spins differ. One readily obtains the simpleexact resultI a ,s ! 5 2 p / a ! 3N/2 @ 11exp 22 a s 2 !# 3N . 4.15!Obtaining the K and V expressions is somewhat more tedious, but reasonably straightforward. We findSK a ,s ! 5 \ 2 /2m ! 3NI ! a 2V a ,s ! 523NIA23NIB13NICSSSD4 a 2 s 2 exp 22 a s 2 !;11exp 22 a s 2 !21s14 a 11exp 22 a s 2 !s2114 a 11exp 22 a s 2 !DD 4.16!2D3s 4 1 3s 2 /2 a !110 1 ! . 4.17!16 a 2 11exp 22 a s 2 !Our interest concerns primarily the large-N limit, so the 0 1!portion of V( a ,s) has been suppressed.Variational minimization leads to the following numerical results when condition 4.7! is imposed:a 52.2293 Å 22 ,s50.4238 Å, 4.18!B542.92 K Å 24 .Since this implies thata 2 a s 2 21 ! 520.4441, 4.19!the probability density is at a maximum at the origin.One of the contributions included in V( a ,s) is equivalent to the mean-square particle displacement, due to zeropoint notion, in the ground state. Specifically,3NN21(i51 u 2i & 53S1s214 a 11exp 22 a s 2 !5 0.8416 Å ! 2D 4.20!upon substituting the values found for a and s. The nominallattice spacing is 3.65 Å at the melting point as noted before.Consequently the Lindemann ratio of rms particle displacement to lattice spacing! is found to be 4.21!0.8416/3.65 0.23.28This is in close agreement with results from extensivevariational Monte Carlo calculations for crystalline 4He, andsignificantly exceeds the corresponding melting point ratiosfound for classical many body systems 0.15!.463V. DISCUSSIONThe construction and analysis of inherent structures forsolid helium at 0 K and 25 bar provides an illuminatingdemonstration of the profound influence of quantum effects,and generates instructive contrasts with the correspondinginherent structures obta
We study the potential energy landscape many-atom potential energy as a function of atomic positions! of solid hcp 4He in the vicinity of the 0 K crystal structure using an accurate pair potential. At the melting point, the potential energy of the helium lattice is far above the m
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