Efficient approach to the free energy of crystals via Monte Carlo simulations: Application to continuous and orientation-dependent potentials

We extend a recently proposed approach to determine the absolute free energy of crystalline phases [Phys. Rev. E 92, 022103 (2015)] to systems with continuous and orientation-dependent potentials. The method is based on the Monte Carlo thermodynamic integration method, and it provides a general procedure to obtain a simple, analytical, and exact reference free energy, together with an easy and accurate thermodynamic coupling integral. The approach is free from the difficulties of the standard method based on the harmonic coupling potential. A comparative discussion of both approaches is provided.

Figure 1

$\ln \langle n\rangle$ vs $\beta \epsilon$ for the hard spherocylinder model. (a) Aspect ratio $L/D=5$ and reduced density $\rho D^3=0.86$. (b) $L/D=5$ and $\rho D^3=0.76$. (c) $L/D=3$ and $\rho D^3=0.82$. The number of particles was $N=576$, and $\beta {\epsilon }^{*}=10$. $R_{\mathrm{SW}}$ was chosen to have the optimal value (see the text). Straight lines are fits at high values of $\beta \epsilon$. The arrow indicates the value of $\beta {\epsilon }^{*}=10$. Solid lines are a guide to the eye.

Figure 2

The Aziz potential $\beta \phi \left(r\right)$ as a function of distance $r/r_0$ for ${}^4\mathrm{He}$. Arrows indicate the range of reference potential $d_{\mathrm{ref}}$ and the nearest-neighbor distance $d$. The inset is a zoom of the potential in the region about the minimum.

Figure 3

$\ln \langle n\rangle$ vs $\beta \epsilon$ for the Aziz potential at density $\rho r_0^3=2.6511$ and temperature $T=327.04$ K (circles). The range of reference potential ${\phi }_{\mathrm{ref}}$ is equal to $0.711r_0$, and $R_{\mathrm{SW}}=0.05r_0$. The number of particles is $N=256$, and the maximum value of $\beta {\epsilon }^{*}$, indicated with an arrow, is $\beta {\epsilon }^{*}=8$. Data corresponding to the reference potential ${\phi }_{\mathrm{ref}}$ only are given by squares. The straight line is a fit for high values of $\beta \epsilon$. Solid lines are a guide to the eye.

Figure 4

Integrand $〈n-(U-U_{\mathrm{ref}})/N{\epsilon }^{*}〉$ of $I_2$, given by Eq. (16) (triangles), $〈n〉$ for the whole potential (squares) and $〈n〉$ for only the reference potential ${\phi }_{\mathrm{ref}}$ (circles), all plotted as a function of $\beta \epsilon$. The density is $\rho r_0^3=2.6511$ and the temperature $T=327.04$ K. The range of the reference potential is $0.711r_0$, and $R_{\mathrm{SW}}=0.050r_0$. The number of particles is $N=256$, and the maximum value of $\beta {\epsilon }^{*}$, indicated with an arrow, is $\beta {\epsilon }^{*}=8$. The dashed curve is an exponential fit of $ln〈n〉$ for high values of $\beta \epsilon$. Solid lines are a guide to the eye.

Figure 5

Standard deviation ${\sigma }_{\gamma }$ vs force constant $\beta k_{\mathrm{EC}}/2$ at density $\rho r_0^3=3$ and temperature $T=327.04$ K for the Aziz potential. Several cases with different values of $\beta k_{\mathrm{EC}}^{*}/2$ (indicated with labels) are shown; $\beta k_{\mathrm{EC}}^{*}/2=160$ (circles), 320 (squares), and 640 (triangles). Solid lines are a guide to the eye.

Figure 6

Fraction of particles inside the potential well. Full circles: square well. Open circles: quadratic well. Stars: linear well. Open squares: square-root well.

Figure 7

Potential energy per particle in units of $\epsilon$. Symbols as in Fig. 6.