^{1,a)}, M. Basire

^{1}, J.-M. Mestdagh

^{1}and C. Angelié

^{1}

### Abstract

A novel Monte Carlo flat histogram algorithm is proposed to get the classical density of states in terms of the potential energy, *g*(*E* _{ p }), for systems with continuous variables such as atomic clusters. It aims at avoiding the long iterative process of the Wang-Landau method and controlling carefully the convergence, but keeping the ability to overcome energy barriers. Our algorithm is based on a preliminary mapping in a series of points (called a σ-mapping), obtained by a two-parameter local probing of *g*(*E* _{ p }), and it converges in only two subsequent reweighting iterations on large intervals. The method is illustrated on the model system of a 432 atom cluster bound by a Rydberg type potential. Convergence properties are first examined in detail, particularly in the phase transition zone. We get *g*(*E* _{ p }) varying by a factor 10^{3700} over the energy range [0.01 < *E* _{ p } < 6000 eV], covered by only eight overlapping intervals. Canonical quantities are derived, such as the internal energy *U*(*T*) and the heat capacity C_{V}(*T*). This reveals the solid to liquidphase transition, lying in our conditions at the triple point. This phase transition is further studied by computing a Lindemann-Berry index, the atomic cluster density n(*r*), and the pressure, demonstrating the progressive surface melting at this triple point. Some limited results are also given for 1224 and 4044 atom clusters.

I. INTRODUCTION

II. METHOD TO GET THE DENSITY OF STATES

A. Relevance of to express the DOS

B. Algorithm

1. Preliminary step

2. First step: Linear bias

3. Second step: Metropolis exploration

4. Final remarks

III. ILLUSTRATION OF THE METHOD AND DISCUSSION OF THE CONVERGENCE

A. Computational details

B. A Debye crystal

C. A Rydberg crystal

IV. CANONICAL QUANTITIES AND PHASE TRANSITION CHARACTERIZATION

A. Caloric curve and heat capacity

B. Lindemann-Berry analysis

C. Volumic density analysis

D. Pressure study

V. DISCUSSION RELATIVE TO AND

A. Discussion of the variation

B. Comparison of the -linear bias method with the Wang-Landau algorithm

VI. CONCLUSION AND PERSPECTIVES

### Key Topics

- Monte Carlo methods
- 51.0
- Phase transitions
- 43.0
- Solid liquid phase transitions
- 17.0
- Electron densities of states
- 15.0
- Atomic and molecular clusters
- 14.0

## Figures

Histogram obtained with a σ-linear bias (, *B* = 40 eV^{−1}) giving σ(349.93 eV) = 1103.20 for a 432 atom system.

Histogram obtained with a σ-linear bias (, *B* = 40 eV^{−1}) giving σ(349.93 eV) = 1103.20 for a 432 atom system.

(a) Histograms obtained with the bias b^{(1)} (respectively b^{(2)}) after the first (respectively second) iteration, for the range [230 − 1530 eV] above the phase transition. (b) Histograms obtained in the phase transition zone [150 − 220 eV] after 7.5 × 10^{9} MC steps (1), 11.7 × 10^{9} MC steps (2), 50 × 10^{9} MC steps (3). All histograms are obtained for a 432 atom system.

(a) Histograms obtained with the bias b^{(1)} (respectively b^{(2)}) after the first (respectively second) iteration, for the range [230 − 1530 eV] above the phase transition. (b) Histograms obtained in the phase transition zone [150 − 220 eV] after 7.5 × 10^{9} MC steps (1), 11.7 × 10^{9} MC steps (2), 50 × 10^{9} MC steps (3). All histograms are obtained for a 432 atom system.

Functions σ(*E* _{ p }) (red curve) and β(*E* _{ p }) (green curve, eV^{−1}) for a 432 atom system, with two different energy scales: (a) from 0 to 400 eV and (b) from 400 to 6000 eV.

Functions σ(*E* _{ p }) (red curve) and β(*E* _{ p }) (green curve, eV^{−1}) for a 432 atom system, with two different energy scales: (a) from 0 to 400 eV and (b) from 400 to 6000 eV.

Density of states *g*(*E* _{ p }) plotted for a 432 atom system from 0.01 to 6000 eV in Log_{10} scale.

Density of states *g*(*E* _{ p }) plotted for a 432 atom system from 0.01 to 6000 eV in Log_{10} scale.

Normalized curves: σ × 1000/*N* _{ at } as a function of *E* _{ p } × 1000/*N* _{ at } (eV) for 432, 1224, and 4044 atoms.

Normalized curves: σ × 1000/*N* _{ at } as a function of *E* _{ p } × 1000/*N* _{ at } (eV) for 432, 1224, and 4044 atoms.

Caloric curve *U*(*T*) (red line, *eV*), and heat capacity C_{V}(*T*) (green line, in *k* _{ B } units) for a 432 atom system.

Caloric curve *U*(*T*) (red line, *eV*), and heat capacity C_{V}(*T*) (green line, in *k* _{ B } units) for a 432 atom system.

Phase transition temperature *T* _{ ph } (K), in terms of .

Phase transition temperature *T* _{ ph } (K), in terms of .

Lindemann index L_{B} of a 432 atom cluster, for the core C (0 to 9 Å) and the surface shell S (9 to 12 Å), at 2 × 10^{9} and 9 × 10^{9} MC steps (indicated by: C2, C9, S2, and S9).

Lindemann index L_{B} of a 432 atom cluster, for the core C (0 to 9 Å) and the surface shell S (9 to 12 Å), at 2 × 10^{9} and 9 × 10^{9} MC steps (indicated by: C2, C9, S2, and S9).

Volumic density n(*r*) of a 432 atom cluster at different energies: 80, 135, 235, 650, 800, 950, 1100, and 1250 eV.

Volumic density n(*r*) of a 432 atom cluster at different energies: 80, 135, 235, 650, 800, 950, 1100, and 1250 eV.

Pressure curve *P*(*T*) of a 432 atom cluster in semi-logarithmic scale. The left part depicts the zone of the triple point from 1000 to 3500 K. The right part is expanded up from 3500 to 120 000 K and shows the critical point.

Pressure curve *P*(*T*) of a 432 atom cluster in semi-logarithmic scale. The left part depicts the zone of the triple point from 1000 to 3500 K. The right part is expanded up from 3500 to 120 000 K and shows the critical point.

Histograms obtained in the phase transition zone [150 − 210 eV] for a 432 atom system. “**This work**” refers to the curve already presented in Fig. 2(b) with 50 × 10^{9} MC steps. “**WL** ^{+}” refers to the result of a Metropolis calculation (100 × 10^{9} MC steps) following a Wang-Landau process (27 × 10^{9} MC steps).

Histograms obtained in the phase transition zone [150 − 210 eV] for a 432 atom system. “**This work**” refers to the curve already presented in Fig. 2(b) with 50 × 10^{9} MC steps. “**WL** ^{+}” refers to the result of a Metropolis calculation (100 × 10^{9} MC steps) following a Wang-Landau process (27 × 10^{9} MC steps).

Evolution of the number of MC steps with ln *f* for a 432 atom system. Three energy ranges are investigated: [81, 171 eV] (solid), [150, 210 eV] (transition zone), and [231, 1530 eV] (liquid).

Evolution of the number of MC steps with ln *f* for a 432 atom system. Three energy ranges are investigated: [81, 171 eV] (solid), [150, 210 eV] (transition zone), and [231, 1530 eV] (liquid).

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