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Isomeric structures and structural transformations in small clusters
Volume41, number3
CHEMICAL
PHYSICS
ISOMERK STRUCTURES AND STRUCTURAL
J.B. KAELBERER, Department
1 August
LET!iXRS
TRAlVSFORMATIONS
1976
lN SMALL CLUSTERS*
R-B. ETTERS and J.C. RAICH
of Physic&Colorado State University, Fort Collins. Colorado 80523, USA
Received 3 May 1976
The temperature dependence of the energies of the isomers of a seven-particle s);stem is studied with a view toward understanding ergodidty probiems in Monte Carlo simulations. It is found that the phase space of particles in a cluster not ergodic at lower temperatures.
is
1. Introduction
2_ Results
Monte Carlo and molecular dynamics studies on small clusters have usually been performed [l--4] with the tacit assumprion +&atthe phase space of the atoms in the cluster is completely ergodic. Even though it is known that a small cluster can exist in severd isomerit forms [S] , the infhence of these isomers on the egodicity condition has not yet been fully realized or investigated. ke have recently performed [6,7] a Monte Carlo calculation on clusters containing varying numbers of atoms, using Metropolis’ algorithm [8] to generate states with a Maxwell-Boltzmanrr distribution. The atoms of the cluster are assumed to interact via a Lennard-Jones pair potential of the form
Of the clusters studied, those containing three and five particles present no ergodicity problems- A threeparticle cluster has only one stable isomer, corresponding to minimum energy, namely, the equilateral triangle (E*=3.0 at T* = 0). The cluster always crystallizes in this form irrespective of the starting configuration. The five-particle cluster has two stable isomers, the square pyramid (E* = -8.48 1 at T* = 0.0) and the trigonal biplramid (E* = - 9.104 at T* = 0.0) [5] . However, even at very low temperatures, a five-particle system never crystallizes in the isomer having higher energy. A Monte Carlo calculation was performed at a temperature of 0.0083, using the square pyramid as the starting configuration_ Within as few as 1000 iterative steps, the system relaxed into the stable trigonal bipyramid. In these two cases, therefore, the average energy and the average configuration are independent of the starting collfiguration at all temperatures. Therefore, phase space is ergodic for three- and five-particle cltisters. In a normal random walk, usually consisting of around IO5 to lo6 steps, ail portions of phase space corresponding to a bound state cluster, will be sampled. This, however, is not true for huger clusters. For example, at lower temperatures, the seven-particle cluster may stablize into one of four different isomeric forms - the pentagonal bipyramid, the incomplete stelIated tetrahedron, the octahedron + 1 and a skewed arrangement having symknetry C, . The energies of these
V = 4E f(*/@
- (&Q6 ] .
The values of the parameters 0 and E for the rare gases are approximately Ar(3.4 a, 12 1 K), Kr(3.6 A, 163 K) and Xe(4.06 A, 231 K) 193. The computations are performed for argoi and extended to the o&er atoms uskg the principle of corresponding states_ Unitless variables (E* = E/E, 19 = P/O, T* = T/e) are used throughout_
* Work suppo-*d 06-002-159.
580
in part by NASA under Grant No. NGL
Volume 41, number3
isomers at 0 K are --L6.50, -15.59, -1593 and -15.53, respectiveiy [S] . At temperatures below about 0.0744, the cluster does not easily move out of one isomeric form into another. At this point, it must be strongly emphasized *&at the clusters are given the freedom to move out of a pa&cular structure. The random walk is performed in such a way that a configuration having &gh energy is accepted as part of the Markov chain, with a probabiiity proportional to the Maxwell-Boltzmaun factor. Certain parameters in the walk are also adjusted such that a trial configuration is accepted 50% of the time, and so there is provision made for a cluster to move into a region of high potential energy, i.e., to cross the energy barrier between two isomeric states. Fig. 1 shows the energies of the seven-particle iso-
N
=
7
SKEWED
STRUCTURE
INCOMPLETE OCTAHEDRON PENTAGONAL
-i 7.01 3
STELLATED +
TETRAHEDRON
1
BIPYRALMID
t 0.1
f/s
I 0.2
Fii. 1. The averagepotentialenergyof the isomersof the seven-p;rrticIe clusteras a functionof temperat,ure.
1
mers as a function of temperqture. The values at 0 K are ob&ined by extrapolation and compare very weIl [less than 2% difference) v&h the-results of Koare and Pal [Sl , derived by a variationJ method. When X* is greater than O-0744, the structure formerI is &vays a pentagonal bipyramid, irrespective of what starting configuration is introduced. From Of) to 0.0496, the cluster may form in any one of the four isomeric states, depending on the starting configuration. At T* = 0.0496,however, both the stelIated tetrahedrqn and the skewed structure become unstable. A Markov chain initiated on either of these two structures leads invariably to an octahedron f 1 structure. The cluster continues in rhe octahedron f I structure only up to a temperature of T* = O-07$4, after which the ground state pentagonal bipyramid structure is always formed. The energies of the isomers cannot be calculated above temperatures at which they spontaneously deform into the next stable structure. At lower temperatures, the energy can be found by starting the Markov chain in a structure close to the one desired. We are, therefore, able to characterize temperatures of structural transformations. The fast such transitions takes place at T* = 0.0496, where the atoms of the cluster originaliy in either steI.Iated tetrahedral or the skewed structure rearrange themselves to form an octahedron + I, with a decrease of 0.34~ in ener7. The cluster continues in this structure untiI T+ = 0.0744, when the second transition occurs; the structure changing to the pentagonal bipyramid (decrease of 0.63e in potentid energy). Both these transitions are fairly sharp and can be located in a temperature range of 0.0083. This is rather interesting in view of the fact that statistical mechanics predict sharp transitions only for infimitely Iarge systems [lo] _However, the energies reported here have been caIcuIated at temperature intervals of O-0083. If the calculations are performed with greater accuracy, as well as over narrower ranges, some smoothing Of the energy-temperature curve: in the vicinity of the transition may be observed_ It should be mentioned here that several isomaric forms were also observed in larger clusters (N = 9,l I, 13). Although a detailed study has not been made on these clusters, zhe discussion of the isomers and the structural transitions are similar to that presented above for a seven-particle cluster. 581
Voblslle 41,lxzm~_3
CEEX+fICAL PHYSECS LETTERS
123 9: ELLee, J.A. Barker and F.F. Abraham, J. Chem. Phys.
Acknowledgement The authors would like to express their gratitude to Lzr. Gearold ~O%IXXXI and the Colcrado State Uniuersity Computer Center for geenerozsiyhelping to support this work under NSF Grant No. GJ45 1. We aLso acknowledge the considerable help of Mr. Ray Palmer of the NASA Lewis Computer Services Division and Mr. Carl Wade of the Colorado State University Computer Center.
RePerences fiJ E3.3.MCCXnty, J-C&m. Phys. 58 (i97314733; phys. Letters 13 (f971) 525.
1 August 1976
Z&em.
58 11973) 3166. t3f W. Darx%gzarG Kirsensen, E.3. Jensen and R.MJ. Cotteri& J. Chem. m(lvs. 60 (1974) 416%. 141 C.L. Briant and JJ. Burton, J- C&em. Phys. 63 (1975) 2045. @f M.2. Hoare and P. Pal, .4dvan. Phys. 20 (1971) 161; Nature Phys. Sci. 230 (197i) 5; 236 (1972) 35. 161 R.D. Etters and J.B. KaeIberer. Phyr Rev. All. (1975) 1068L t73 J-3. Kaelberer and R.D. Eati, Am. Assoc. CrysztaI Growth, Thisd Am. Co& Proc. July (1975) p- 31; 3. Chem. Phys., submitted for public&ion. PI N. metropolis, A. Rosenbluth, M. Rosenhluth, A. Teller and E. Teller, J. C&em. Phys. 21(1953) 1087. f31 N. Bernzdes, Whys-Rev. I12 (1958) 1534. tw R. Huang,Statistical mechanic-s (Whey, New York, 1963).
CHEMICAL
PHYSICS
ISOMERK STRUCTURES AND STRUCTURAL
J.B. KAELBERER, Department
1 August
LET!iXRS
TRAlVSFORMATIONS
1976
lN SMALL CLUSTERS*
R-B. ETTERS and J.C. RAICH
of Physic&Colorado State University, Fort Collins. Colorado 80523, USA
Received 3 May 1976
The temperature dependence of the energies of the isomers of a seven-particle s);stem is studied with a view toward understanding ergodidty probiems in Monte Carlo simulations. It is found that the phase space of particles in a cluster not ergodic at lower temperatures.
is
1. Introduction
2_ Results
Monte Carlo and molecular dynamics studies on small clusters have usually been performed [l--4] with the tacit assumprion +&atthe phase space of the atoms in the cluster is completely ergodic. Even though it is known that a small cluster can exist in severd isomerit forms [S] , the infhence of these isomers on the egodicity condition has not yet been fully realized or investigated. ke have recently performed [6,7] a Monte Carlo calculation on clusters containing varying numbers of atoms, using Metropolis’ algorithm [8] to generate states with a Maxwell-Boltzmanrr distribution. The atoms of the cluster are assumed to interact via a Lennard-Jones pair potential of the form
Of the clusters studied, those containing three and five particles present no ergodicity problems- A threeparticle cluster has only one stable isomer, corresponding to minimum energy, namely, the equilateral triangle (E*=3.0 at T* = 0). The cluster always crystallizes in this form irrespective of the starting configuration. The five-particle cluster has two stable isomers, the square pyramid (E* = -8.48 1 at T* = 0.0) and the trigonal biplramid (E* = - 9.104 at T* = 0.0) [5] . However, even at very low temperatures, a five-particle system never crystallizes in the isomer having higher energy. A Monte Carlo calculation was performed at a temperature of 0.0083, using the square pyramid as the starting configuration_ Within as few as 1000 iterative steps, the system relaxed into the stable trigonal bipyramid. In these two cases, therefore, the average energy and the average configuration are independent of the starting collfiguration at all temperatures. Therefore, phase space is ergodic for three- and five-particle cltisters. In a normal random walk, usually consisting of around IO5 to lo6 steps, ail portions of phase space corresponding to a bound state cluster, will be sampled. This, however, is not true for huger clusters. For example, at lower temperatures, the seven-particle cluster may stablize into one of four different isomeric forms - the pentagonal bipyramid, the incomplete stelIated tetrahedron, the octahedron + 1 and a skewed arrangement having symknetry C, . The energies of these
V = 4E f(*/@
- (&Q6 ] .
The values of the parameters 0 and E for the rare gases are approximately Ar(3.4 a, 12 1 K), Kr(3.6 A, 163 K) and Xe(4.06 A, 231 K) 193. The computations are performed for argoi and extended to the o&er atoms uskg the principle of corresponding states_ Unitless variables (E* = E/E, 19 = P/O, T* = T/e) are used throughout_
* Work suppo-*d 06-002-159.
580
in part by NASA under Grant No. NGL
Volume 41, number3
isomers at 0 K are --L6.50, -15.59, -1593 and -15.53, respectiveiy [S] . At temperatures below about 0.0744, the cluster does not easily move out of one isomeric form into another. At this point, it must be strongly emphasized *&at the clusters are given the freedom to move out of a pa&cular structure. The random walk is performed in such a way that a configuration having &gh energy is accepted as part of the Markov chain, with a probabiiity proportional to the Maxwell-Boltzmaun factor. Certain parameters in the walk are also adjusted such that a trial configuration is accepted 50% of the time, and so there is provision made for a cluster to move into a region of high potential energy, i.e., to cross the energy barrier between two isomeric states. Fig. 1 shows the energies of the seven-particle iso-
N
=
7
SKEWED
STRUCTURE
INCOMPLETE OCTAHEDRON PENTAGONAL
-i 7.01 3
STELLATED +
TETRAHEDRON
1
BIPYRALMID
t 0.1
f/s
I 0.2
Fii. 1. The averagepotentialenergyof the isomersof the seven-p;rrticIe clusteras a functionof temperat,ure.
1
mers as a function of temperqture. The values at 0 K are ob&ined by extrapolation and compare very weIl [less than 2% difference) v&h the-results of Koare and Pal [Sl , derived by a variationJ method. When X* is greater than O-0744, the structure formerI is &vays a pentagonal bipyramid, irrespective of what starting configuration is introduced. From Of) to 0.0496, the cluster may form in any one of the four isomeric states, depending on the starting configuration. At T* = 0.0496,however, both the stelIated tetrahedrqn and the skewed structure become unstable. A Markov chain initiated on either of these two structures leads invariably to an octahedron f 1 structure. The cluster continues in rhe octahedron f I structure only up to a temperature of T* = O-07$4, after which the ground state pentagonal bipyramid structure is always formed. The energies of the isomers cannot be calculated above temperatures at which they spontaneously deform into the next stable structure. At lower temperatures, the energy can be found by starting the Markov chain in a structure close to the one desired. We are, therefore, able to characterize temperatures of structural transformations. The fast such transitions takes place at T* = 0.0496, where the atoms of the cluster originaliy in either steI.Iated tetrahedral or the skewed structure rearrange themselves to form an octahedron + I, with a decrease of 0.34~ in ener7. The cluster continues in this structure untiI T+ = 0.0744, when the second transition occurs; the structure changing to the pentagonal bipyramid (decrease of 0.63e in potentid energy). Both these transitions are fairly sharp and can be located in a temperature range of 0.0083. This is rather interesting in view of the fact that statistical mechanics predict sharp transitions only for infimitely Iarge systems [lo] _However, the energies reported here have been caIcuIated at temperature intervals of O-0083. If the calculations are performed with greater accuracy, as well as over narrower ranges, some smoothing Of the energy-temperature curve: in the vicinity of the transition may be observed_ It should be mentioned here that several isomaric forms were also observed in larger clusters (N = 9,l I, 13). Although a detailed study has not been made on these clusters, zhe discussion of the isomers and the structural transitions are similar to that presented above for a seven-particle cluster. 581
Voblslle 41,lxzm~_3
CEEX+fICAL PHYSECS LETTERS
123 9: ELLee, J.A. Barker and F.F. Abraham, J. Chem. Phys.
Acknowledgement The authors would like to express their gratitude to Lzr. Gearold ~O%IXXXI and the Colcrado State Uniuersity Computer Center for geenerozsiyhelping to support this work under NSF Grant No. GJ45 1. We aLso acknowledge the considerable help of Mr. Ray Palmer of the NASA Lewis Computer Services Division and Mr. Carl Wade of the Colorado State University Computer Center.
RePerences fiJ E3.3.MCCXnty, J-C&m. Phys. 58 (i97314733; phys. Letters 13 (f971) 525.
1 August 1976
Z&em.
58 11973) 3166. t3f W. Darx%gzarG Kirsensen, E.3. Jensen and R.MJ. Cotteri& J. Chem. m(lvs. 60 (1974) 416%. 141 C.L. Briant and JJ. Burton, J- C&em. Phys. 63 (1975) 2045. @f M.2. Hoare and P. Pal, .4dvan. Phys. 20 (1971) 161; Nature Phys. Sci. 230 (197i) 5; 236 (1972) 35. 161 R.D. Etters and J.B. KaeIberer. Phyr Rev. All. (1975) 1068L t73 J-3. Kaelberer and R.D. Eati, Am. Assoc. CrysztaI Growth, Thisd Am. Co& Proc. July (1975) p- 31; 3. Chem. Phys., submitted for public&ion. PI N. metropolis, A. Rosenbluth, M. Rosenhluth, A. Teller and E. Teller, J. C&em. Phys. 21(1953) 1087. f31 N. Bernzdes, Whys-Rev. I12 (1958) 1534. tw R. Huang,Statistical mechanic-s (Whey, New York, 1963).