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Computing the Free Energy of Molecular Solids by the Einstein Molecule Approach: Ices XIII and XIV, Hard-Dumbbells and a Patchy Model of Proteins ͒ ͒ E

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Computing the Free Energy of Molecular Solids by the Einstein Molecule Approach: Ices XIII and XIV, Hard-Dumbbells and a Patchy Model of Proteins ͒ ͒ E THE JOURNAL OF CHEMICAL PHYSICS 129, 104704 ͑2008͒ Computing the free energy of molecular solids by the Einstein molecule approach: Ices XIII and XIV, hard-dumbbells and a patchy model of proteins ͒ ͒ E. G. Noya,a M. M. Conde, and C. Vegab Departamento de Química-Física, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain ͑Received 22 April 2008; accepted 24 July 2008; published online 11 September 2008͒ The recently proposed Einstein molecule approach is extended to compute the free energy of molecular solids. This method is a variant of the Einstein crystal method of Frenkel and Ladd ͓J. Chem. Phys. 81, 3188 ͑1984͔͒. In order to show its applicability, we have computed the free energy of a hard-dumbbell solid, of two recently discovered solid phases of water, namely, ice XIII and ice XIV, where the interactions between water molecules are described by the rigid nonpolarizable TIP4P/2005 model potential, and of several solid phases that are thermodynamically stable for an anisotropic patchy model with octahedral symmetry which mimics proteins. Our calculations show that both the Einstein crystal method and the Einstein molecule approach yield the same results within statistical uncertainty. In addition, we have studied in detail some subtle issues concerning the calculation of the free energy of molecular solids. First, for solids with noncubic symmetry, we have studied the effect of the shape of the simulation box on the free energy. Our results show that the equilibrium shape of the simulation box must be used to compute the free energy in order to avoid the appearance of artificial stress in the system that will result in an increase in the free energy. In complex solids, such as the solid phases of water, another difficulty is related to the choice of the reference structure. As in some cases there is no obvious orientation of the molecules; it is not clear how to generate the reference structure. Our results will show that, as long as the structure is not too far from the equilibrium structure, the calculated free energy is invariant to the reference structure used in the free energy calculations. Finally, the strong size dependence of the free energy of solids is also studied. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2971188͔ I. INTRODUCTION Einstein crystal with the constraint that the center of mass of the system is fixed in order to avoid a quasidivergence in the 1 Since the pioneering work of Hoover and Ree, deter- integral of the free energy change from the real solid to the mining the free energy of molecular solids has been an im- reference system. This constraint introduces some complex- 2–8 portant area of research. One of the most popular methods ity in the method. In particular, the derivation of some terms to compute the free energy of solids is the Einstein crystal that contribute to the free energy is somewhat involved.2,13 method, proposed by Frenkel and Ladd2 more than two de- The main idea behind the Einstein molecule approach is that cades ago. In this method, the free energy of a given solid is the derivation of the analytical expressions can be consider- computed by designing an integration path that links the ably simplified by fixing the position of one molecule instead solid to an ideal Einstein crystal with the same structure as of fixing the center of mass of the system. The Einstein mol- the real solid, for which the free energy can be analytically ecule approach has been successfully applied to compute the computed. This method was soon extended to molecular free energy of the hard spheres ͑HSs͒ and Lennard-Jones solids.3 In this case, in addition to the springs that bound ͑LJ͒ face centered cubic ͑fcc͒ solids. Here it will be shown each molecule to its lattice position, springs that keep the how it can be applied to molecular solids. particles in the right orientation must also be added.3,9 Using Moreover, even though the Einstein crystal method has this technique, the free energy of several atomic and molecu- been extended to molecular solids more than 20 years ago, lar solids has been computed.8–39 there are several subtle issues concerning the calculation of Quite recently a new method to compute the free ener- the free energy that are not clear yet. These difficulties are gies of solids which was denoted as “the Einstein molecule” common to the Einstein crystal and Einstein molecule ap- approach has been proposed.11,12 This method consists of a proaches. One of these issues concerns the shape of the slight modification of the Einstein crystal method. In the simulation box. For solids with noncubic symmetry, prior to Einstein crystal method, the reference system is an ideal the computation of the free energy for a given thermody- namic state, the solid structure must be relaxed to obtain the ͒ a Electronic mail: [email protected]. equilibrium unit cell corresponding to that thermodynamic ͒ b Electronic mail: [email protected]. state. This is not usually a problem in structures with cubic 0021-9606/2008/129͑10͒/104704/16/$23.00129, 104704-1 © 2008 American Institute of Physics Downloaded 22 Oct 2008 to 147.96.6.138. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 104704-2 Noya, Conde, and Vega J. Chem. Phys. 129, 104704 ͑2008͒ symmetry, as the equilibrium structure is determined system sizes, so that the value of the free energy at the ther- uniquely by the lattice parameter a. However, in structures modynamic limit can be obtained by extrapolation to N go- with lower symmetry, it is convenient to first perform a ing to infinity. However, this procedure requires performing simulation in which both the edges and the angles that define many simulations to compute the free energy of a solid at the simulation box are allowed to relax to the equilibrium just one thermodynamic state. Therefore, it would be useful structure. This can be achieved by, previously to the free to introduce finite size corrections ͑FSCs͒, i.e., a simple energy calculation, performing a Parrinello–Rahman NpT recipe that would allow one to estimate the value of the free simulation. Other alternative would be to perform a simula- energy in the thermodynamic limit from simulations of a tion at constant volume but where the shape of the simula- system of finite size. In a previous paper, we have proposed tion box is allowed to change, i.e., a variable-shape constant several FSCs whose performances were assessed for simple volume ͑VSNVT͒ simulation.33,34,40,41 We would like to atomic models, namely, the HS and LJ model potentials.11 stress the importance of using the equilibrium structure to The best performance was obtained by the so-called compute the free energy, otherwise the solid could be under asymptotic FSC, in which the free energy in the thermody- some stress that will lead to an increase in the free energy. namic limit is estimated from the free energy at a finite size This has already been noted previously,3,9 but, due to its N by taking the limit when N tends to infinity in the expres- importance, we believe that it is worthy to review this point. sion used to compute the free energy. Depending on how this Another difficulty that one might encounter when com- limit was taken, three different variants were proposed, and puting the free energy of molecular solids concerns the ref- all of them give quite reasonable estimates of the free energy erence structure that is used either in the Einstein crystal or in the thermodynamic limit. However, these results might not in the Einstein molecule approaches. In simple solids, in be general and, therefore, it would be of interest to check which all the particles exhibit the same orientation, this does whether the FSC works well also for molecular solids. not pose a problem, as the reference structure is chosen sim- In this paper we will address all these issues concerning ply as a solid where all the particles lie on their lattice posi- the computation of the free energy of solids. It is our hope tions and are perfectly oriented. However, for more complex that this will contribute to encourage other authors to com- solids, where not all the molecules exhibit the same orienta- pute free energies. This paper will be structured in the fol- tion, the choice of the reference structure might be a subtle lowing way. First, the recently proposed Einstein molecule issue. This is the case, for example, of some solid phases of approach will be extended to the case of molecular solids, water that exhibit complex unit cells. In this situation several and it will be shown that the results obtained for all the choices are possible. One might choose to build the reference solids studied ͓i.e., a hard-dumbbell ͑HD͒ solid, a solid made structure by using experimental data to obtain the position of anisotropic particles with octahedral symmetry, and the and orientation of the molecules or, alternatively, one might two recently discovered solid phases of water, ices XIII and ͔ choose to perform an energy minimization, so that each mol- XIV are in agreement, within statistical uncertainty, with the ecule will be located as to minimize the potential energy.42 results obtained with the Einstein crystal method. Second, Another reasonable choice would be to calculate the average the free energy of ices XIII and XIV using the rigid nonpo- positions and orientations at the particular thermodynamic larizable TIP4P/2005 model of water will also be calculated.
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