A novel statistical mechanical methodology is applied to clusters of N ≤ 7 atoms. Exact statistical analogs for any energy derivative of entropy mS/∂Em are used in rigorous microcanonical Monte Carlo simulations to vastly enlarge the pool of measurable thermodynamic properties relative to previous work. All analogs are given for two alternative partition functions of the microcanonical ensemble. Coarse grained energy distributions are used to establish the existence of melting transitions. LJ7, LJ5, and LJ4 are found to exhibit trimodal distributions, a feature not being reported before. Varieties of combinations of entropy derivatives are tested for a direct detection of the melting region. It is shown that for such a purpose, derivatives of at least fourth order are necessary.

1.
R.
Lustig
, “
Direct molecular NVT simulation of the isobaric heat capacity, speed of sound and Joule–Thomson coefficient
,”
Mol. Simul.
37
,
457
465
(
2011
).
2.
R.
Lustig
, “
Statistical analogues for fundamental equation of state derivatives
,”
Mol. Phys.
110
,
3041
3052
(
2012
).
3.
H. W.
Graben
and
J. R.
Ray
, “
Eight physical systems of thermodynamics, statistical mechanics, and computer simulations
,”
Mol. Phys.
80
,
1183
1193
(
1993
).
4.
P.
Ströker
,
R.
Hellmann
, and
K.
Meier
, “
Systematic formulation of thermodynamic properties in the NpT ensemble
,”
Phys. Rev. E
103
,
023305
(
2021
).
5.
P.
Ströker
and
K.
Meier
, “
Classical statistical mechanics in the grand canonical ensemble
,”
Phys. Rev. E
104
,
014117
(
2021
).
6.
P.
Ströker
and
K.
Meier
, “
Rigorous expressions for thermodynamic properties in the NpH ensemble
,”
Phys. Rev. E
105
,
035301
(
2022
).
7.
P.
Ströker
and
K.
Meier
, “
Classical statistical mechanics in the μVL and μpR ensembles
,”
Phys. Rev. E
107
,
064112
(
2023
).
8.
M.
Thol
,
G.
Rutkai
,
A.
Köster
,
R.
Lustig
,
R.
Span
, and
J.
Vrabec
, “
Equation of state for the Lennard-Jones fluid
,”
J. Phys. Chem. Ref. Data
45
,
023101
(
2016
).
9.
R.
Lustig
, “
On the Lennard-Jones and Devonshire theory for solid state thermodynamics
,”
Mol. Phys.
115
,
1362
1377
(
2017
).
10.
R.
Lustig
,
G.
Rutkai
, and
J.
Vrabec
, “
Thermodynamic correlation of molecular simulation data
,”
Mol. Phys.
113
,
910
931
(
2015
).
11.
M.
Thol
,
G.
Rutkai
,
A.
Köster
,
M.
Kortmann
,
R.
Span
, and
J.
Vrabec
, “
Fundamental equation of state for ethylene oxide based on a hybrid dataset
,”
Chem. Eng. Sci.
121
,
87
99
(
2015
).
12.
M.
Thol
,
F. H.
Dubberke
,
G.
Rutkai
,
T.
Windmann
,
A.
Köster
,
R.
Span
, and
J.
Vrabec
, “
Fundamental equation of state correlation for hexamethyldisiloxane based on experimental and molecular simulation data
,”
Fluid Phase Equilib.
418
,
133
151
(
2016
).
13.
M.
Thol
,
G.
Rutkai
,
A.
Köster
,
S.
Miroshnichenko
,
W.
Wagner
,
J.
Vrabec
, and
R.
Span
, “
Equation of state for 1,2-dichloroethane based on a hybrid data set
,”
Mol. Phys.
115
,
1166
1185
(
2017
).
14.
M.
Thol
,
S. M.
Pohl
,
D.
Saric
,
R.
Span
, and
J.
Vrabec
, “
Fundamental equation of state for mixtures of nitrogen, oxygen, and argon based on molecular simulation data
,”
J. Chem. Phys.
160
,
174102
(
2024
).
15.
P.
Ströker
,
R.
Hellmann
, and
K.
Meier
, “
Thermodynamic properties of argon from Monte Carlo simulations using ab initio potentials
,”
Phys. Rev. E
105
,
064129
(
2022
).
16.
P.
Ströker
,
R.
Hellmann
, and
K.
Meier
, “
Thermodynamic properties of krypton from Monte Carlo simulations using ab initio potentials
,”
J. Chem. Phys.
157
,
114504
(
2022
).
17.
C. W.
Glass
,
S.
Reiser
,
G.
Rutkai
,
S.
Deublein
,
A.
Köster
,
G.
Guevara-Carrion
,
A.
Wafai
,
M.
Horsch
,
M.
Bernreuther
,
T.
Windmann
,
H.
Hasse
, and
J.
Vrabec
, “
ms2: A molecular simulation tool for thermodynamic properties, new version release
,”
Comput. Phys. Commun.
185
,
3302
3306
(
2014
).
18.
I.
Nitzke
and
J.
Vrabec
, “
Numerical discrimination of thermodynamic Monte Carlo simulations in all eight statistical ensembles
,”
J. Chem. Theory Comput.
19
,
3460
3468
(
2023
).
19.
I.
Nitzke
,
S.
Stephan
, and
J.
Vrabec
, “
Topology of thermodynamic potentials using physical models: Helmholtz, Gibbs, Grand, and Null
,”
J. Chem. Phys.
160
,
214104
(
2024
).
20.
R.
Lustig
, “
Microcanonical thermodynamics of three and four atoms
,”
J. Chem. Phys.
150
,
074303
(
2019
).
21.
J. E.
Jones
, “
On the determination of molecular fields. I. From the variation of the viscosity of a gas with temperature
,”
Proc. R. Soc. A
106
,
441
462
(
1924
).
22.
J. E.
Jones
, “
On the determination of molecular fields. II. From the equation of state of a gas
,”
Proc. R. Soc. A
106
,
463
477
(
1924
).
23.
P.
Schwerdtfeger
and
D. J.
Wales
, “
100 years of the Lennard-Jones potential
,”
J. Chem. Theory Comput.
20
,
3379
3405
(
2024
).
24.
G.
Rutkai
,
M.
Thol
,
R.
Span
, and
J.
Vrabec
, “
How well does the Lennard-Jones potential represent the thermodynamic properties of noble gases?
,”
Mol. Phys.
115
,
1104
1121
(
2017
).
25.
R. S.
Berry
,
T. L.
Beck
,
H. L.
Davis
, and
J.
Jellinek
, “
Solid-liquid phase behavior in microclusters
,” in
Evolution of Size Effects in Chemical Dynamics, Part 2
,
Advances in Chemical Physics
, edited by
I.
Prigogine
and
S. A.
Rice
(
Wiley
,
New York
,
1988
), Vol.
LXX
, pp.
75
138
.
26.
D. J.
Wales
and
J. P. K.
Doye
, “
Theoretical predictions of structure and thermodynamics in the large cluster regime
,” in
Large Clusters of Atoms and Molecules
,
NATO ASI Series E: Applied Sciences
, edited by
T. P.
Martin
(
Kluwer
,
Dordrecht
,
1996
), Vol.
313
, pp.
241
279
.
27.
R. S.
Berry
, “
Phases and phase changes of clusters
,” in
Large Clusters of Atoms and Molecules
,
NATO ASI Series E: Applied Sciences
, edited by
T. P.
Martin
(
Kluwer
,
Dordrecht
,
1996
), Vol.
313
, pp.
281
297
.
28.
D. L.
Freeman
and
J. D.
Doll
, “
Computational studies of clusters: Methods and results
,”
Annu. Rev. Phys. Chem.
47
,
43
80
(
1996
).
29.
D. J.
Wales
,
Energy Landscapes: With Applications to Clusters, Biomolecules and Glasses
(
University of Cambridge
,
Cambridge
,
2003
).
30.
F.
Baletto
and
R.
Ferrando
, “
Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects
,”
Rev. Mod. Phys.
77
,
371
423
(
2005
).
31.
P.
Labastie
and
F.
Calvo
, “
Thermodynamics and solid–liquid transitions
,” in
Nanomaterials and Nanochemistry
, edited by
C.
Bréchignac
,
P.
Houdy
, and
M.
Lahmani
(
Springer
,
Berlin
,
2007
), pp.
55
87
.
32.
A.
Aguado
and
M. F.
Jarrold
, “
Melting and freezing of metal clusters
,”
Annu. Rev. Phys. Chem.
62
,
151
172
(
2011
).
33.

The value of a random variable X at a maximum is called a mode of the distribution p(X). In this work, distributions are measured along random walks as histograms with sufficiently small bin widths. The identifiable peaks then provide accurate measures of modes.

34.
J.
Jellinek
,
T. L.
Beck
, and
R. S.
Berry
, “
Solid–liquid phase changes in simulated isoenergetic Ar13
,”
J. Chem. Phys.
84
,
2783
2794
(
1986
).
35.
H. L.
Davis
,
J.
Jellinek
, and
R. S.
Berry
, “
Melting and freezing in isothermal Ar13 clusters
,”
J. Chem. Phys.
86
,
6456
6464
(
1987
).
36.
J. D.
Honeycutt
and
H. C.
Andersen
, “
Molecular dynamics study of melting and freezing of small Lennard-Jones clusters
,”
J. Phys. Chem.
91
,
4950
4963
(
1987
).
37.
D. J.
Wales
, “
Coexistence in small inert gas clusters
,”
Mol. Phys.
78
,
151
171
(
1993
).
38.
M.
Schmidt
and
H.
Haberland
, “
Phase transitions in clusters
,”
C. R. Phys.
3
,
327
340
(
2002
).
39.
D. D.
Frantz
, “
Magic numbers for classical Lennard-Jones cluster heat capacities
,”
J. Chem. Phys.
102
,
3747
3768
(
1995
).
40.
R. D.
Etters
and
J.
Kaelberer
, “
Thermodynamic properties of small aggregates of rare-gas atoms
,”
Phys. Rev. A
11
,
1068
1079
(
1975
).
41.
J. B.
Kaelberer
and
R. D.
Etters
, “
Phase transitions in small clusters of atoms
,”
J. Chem. Phys.
66
,
3233
3239
(
1977
).
42.
R. D.
Etters
and
J.
Kaelberer
, “
On the character of the melting transition in small atomic aggregates
,”
J. Chem. Phys.
66
,
5112
5116
(
1977
).
43.
N.
Quirke
and
P.
Sheng
, “
The melting behavior of small clusters of atoms
,”
Chem. Phys. Lett.
110
,
63
66
(
1984
).
44.
E. M.
Pearson
,
T.
Halicioglu
, and
W. A.
Tiller
, “
Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble
,”
Phys. Rev. A
32
,
3030
3039
(
1985
).
45.
J. R.
Ray
, “
Microcanonical ensemble Monte Carlo method
,”
Phys. Rev. A
44
,
4061
4064
(
1991
).
46.
R.
Lustig
, “
Microcanonical Monte Carlo simulation of thermodynamic properties
,”
J. Chem. Phys.
109
,
8816
8828
(
1998
).
47.
M.
Creutz
, “
Microcanonical Monte Carlo simulation
,”
Phys. Rev. Lett.
50
,
1411
1414
(
1983
).
48.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation
(
Academic
,
San Diego
,
1996
).
49.
M. J.
Grimson
, “
Microcanonical Monte Carlo simulation of the melting behaviour of small clusters
,”
Chem. Phys. Lett.
195
,
92
96
(
1992
).
50.
M. A.
Miller
and
D. J.
Wales
, “
Isomerization dynamics and ergodicity in Ar7
,”
J. Chem. Phys.
107
,
8568
8574
(
1997
).
51.
F.
Calvo
and
P.
Labastie
, “
Monte-Carlo simulations of rotating clusters
,”
Eur. Phys. J. D: At, Mol. Opt. Phys.
3
,
229
236
(
1998
).
52.

Reference 29, Sec. 8.2, Appendix B.

53.
F.
Calvo
,
J. P.
Neirotti
,
D. L.
Freeman
, and
J. D.
Doll
, “
Phase changes in 38-atom Lennard-Jones clusters. II. A parallel tempering study of equilibrium and dynamic properties in the molecular dynamics and microcanonical ensembles
,”
J. Chem. Phys.
112
,
10350
10357
(
2000
).
54.
J.
Jellinek
and
A.
Goldberg
, “
On the temperature, equipartition, degrees of freedom, and finite size effects: Application to aluminum clusters
,”
J. Chem. Phys.
113
,
2570
2582
(
2000
).
55.
M. A.
Carignano
, “
Monte Carlo simulations of small water clusters: Microcanonical vs canonical ensemble
,”
Chem. Phys. Lett.
361
,
291
297
(
2002
).
56.
R.
Becker
,
Theory of Heat
, 2nd ed. (
Springer
,
New York
,
1967
);
R.
Becker
,
Theorie der Wärme
, 3rd ed. (
Springer
,
Berlin
,
1985
).
57.
J.
Spanier
and
K. B.
Oldham
,
An Atlas of Functions
(
Hemisphere
,
Washington
,
1987
).
58.
H.
Flyvbjerg
and
H. G.
Petersen
, “
Error estimates on averages of correlated data
,”
J. Chem. Phys.
91
,
461
466
(
1989
).
59.
R.
Lustig
, “
Statistical thermodynamics in the classical molecular dynamics ensemble. III. Numerical results
,”
J. Chem. Phys.
100
,
3068
3078
(
1994
).
60.
F. G.
Amar
and
R. S.
Berry
, “
The onset of nonrigid dynamics and the melting transition in Ar7
,”
J. Chem. Phys.
85
,
5943
5954
(
1986
).
61.
T. L.
Beck
,
J.
Jellinek
, and
R. S.
Berry
, “
Rare gas clusters: Solids, liquids, slush, and magic numbers
,”
J. Chem. Phys.
87
,
545
554
(
1987
).
62.
D. J.
Wales
and
R. S.
Berry
, “
Coexistence in finite systems
,”
Phys. Rev. Lett.
73
,
2875
2878
(
1994
).
63.
R. S.
Berry
and
B. M.
Smirnov
, “
Heat capacity of isolated clusters
,”
J. Exp. Theor. Phys.
98
,
366
373
(
2004
).
64.
R. S.
Berry
and
B. M.
Smirnov
, “
Observability of coexisting phases of clusters
,”
Int. J. Mass Spectrom.
280
,
204
208
(
2009
).
65.
R. S.
Berry
and
B. M.
Smirnov
, “
Phase transitions in various kinds of clusters
,”
Phys.-Usp.
52
,
137
164
(
2009
).
66.
R. S.
Berry
and
B. M.
Smirnov
, “
Phase transitions in clusters
,”
Low Temp. Phys.
35
,
256
264
(
2009
).
67.
R. S.
Berry
and
B. M.
Smirnov
, “
Entropy and phase coexistence in clusters: Metals vs. nonmetals
,”
Entropy
12
,
1303
1324
(
2010
).
68.
A.
Proykova
,
R.
Radev
,
F.-Y.
Li
, and
R.
Stephen Berry
, “
Structural transitions in small molecular clusters
,”
J. Chem. Phys.
110
,
3887
3896
(
1999
).
69.
A.
Proykova
,
S.
Pisov
, and
R. S.
Berry
, “
Dynamical coexistence of phases in molecular clusters
,”
J. Chem. Phys.
115
,
8583
8591
(
2001
).
70.
A.
Proykova
and
R. S.
Berry
, “
Insights into phase transitions from phase changes of clusters
,”
J. Phys. B: At., Mol. Opt. Phys.
39
,
R167
R202
(
2006
).
71.
J. L.
Lebowitz
,
J. K.
Percus
, and
L.
Verlet
, “
Ensemble dependence of fluctuations with application to machine computations
,”
Phys. Rev.
153
,
250
254
(
1967
).
72.
C. J.
Tsai
and
K. D.
Jordan
, “
Use of the histogram and jump-walking methods for overcoming slow barrier crossing behavior in Monte Carlo simulations: Applications to the phase transitions in the (Ar)13 and (H2O)8 clusters
,”
J. Chem. Phys.
99
,
6957
6970
(
1993
).
73.
M.
Rapacioli
,
N.
Tarrat
, and
F.
Spiegelman
, “
Melting of the Au20 gold cluster: Does charge matter?
,”
J. Phys. Chem. A
122
,
4092
4098
(
2018
).
74.
T.
Çaǧin
and
J. R.
Ray
, “
Fundamental treatment of molecular-dynamics ensembles
,”
Phys. Rev. A
37
,
247
251
(
1988
).
75.
R.
Lustig
, “
Statistical thermodynamics in the classical molecular dynamics ensemble. I. Fundamentals
,”
J. Chem. Phys.
100
,
3048
3059
(
1994
).
76.
R.
Lustig
, “
Statistical thermodynamics in the classical molecular dynamics ensemble. II. Application to computer simulation
,”
J. Chem. Phys.
100
,
3060
3067
(
1994
).
77.
J. R.
Ray
and
H.
Zhang
, “
Correct microcanonical ensemble in molecular dynamics
,”
Phys. Rev. E
59
,
4781
4785
(
1999
).
78.
K.
Meier
and
S.
Kabelac
, “
Pressure derivatives in the classical molecular-dynamics ensemble
,”
J. Chem. Phys.
124
(
6
),
06414
(
2006
).
79.
M. R.
Hoare
and
P.
Pal
, “
Physical cluster mechanics: Statics and energy surfaces for monatomic systems
,”
Adv. Phys.
20
,
161
196
(
1971
).
80.
R. S.
Berry
, “
Remarks on negative heat capacities of clusters
,”
Isr. J. Chem.
44
,
211
214
(
2004
).
You do not currently have access to this content.