Optimized Free-Energy Evaluation Using a Single Reversible-Scaling Simulation

Abstract

We present a method, for highly efficient free-energy calculations by means of molecular dynamics and Monte Carlo simulations, which is an optimized combination of coupling parameter and adiabatic switching formalisms. This approach involves dynamical reversible scaling of the potential energy function of a system of interest, and allows accurate determination of its free energy over a wide temperature interval from a single simulation. The method is demonstrated in two applications: crystalline Si at zero pressure and a fcc nearest-neighbor antiferromagnetic Ising model. PACS numbers: 02.70.Lq, 02.70.Ns, 65.50. + m In the study of thermodynamic properties of materials [1,2], free-energy calculation is a unique application of computer simulation techniques. For this purpose, the coupling parameter formalism [2,3] provides a powerful and robust framework which underlies state-of-the-art techniques such as thermodynamic integration (TI) [2] and adiabatic switching (AS) [4][5][6][7][8][9][10]. Standard application of this approach involves the evaluation of reversible work along a path connecting a physical system of interest to a reference. Usually, the path is constructed using a composite Hamiltonian H͑l͒ coupling the two systems through a parameter l. Upon varying l the coupled system evolves along a reversible trajectory, changing continuously from the system of interest to the reference. The reversible work done by the generalized force ≠H͞≠l along this path is then equal to the free-energy difference between the systems. While this approach is very powerful, it is not optimal in that only the initial and final points on the trajectory correspond to physically relevant systems. The information gathered at the intermediate states of the path has no physical meaning, serving only to connect the end points of the path. As a consequence, one obtains only one value of the desired free energy per simulation.In this Letter we describe a formulation which fully utilizes all the information available along a reversible path and thereby allows the evaluation of free energies over a wide temperature interval from a single simulation. This approach, which effectively exploits both the coupling parameter formalism and the adiabatic switching technique, is based on the use of a specific path which is defined by the introduction of a scaling factor l in the potential energy function of the physical system of interest. The fundamental difference from the usual coupling approach is that for this particular path all intermediate states provide physically relevant information. In fact, an exact relation between the partition functions of the original and the scaled systems shows that all the states along the scaling path correspond to the original physical system at different temperatures. The combination of this reversiblescaling concept with the dynamical variation of l within the adiabatic switching method results in a highly optimized technique with a significant efficiency gain without loss of accuracy.The idea of determ...

Efficient wide-range calculation of free energies in solids and liquids using reversible-scaling molecular dynamics

“…Consistent with the requirements of RSMD simulation [3], Eqs. (6)- (10) also implicitly assume that the forces that control ion motion in the system are independent of temperature.…”

“…The efficient calculation of accurate free energies in solids and liquids from molecular dynamics (MD) simulations is a challenging problem of long-standing interest and importance in condensed matter physics. Numerous specific computational schemes have been developed that depend to varying degrees on the principles of thermodynamic integration (TI) [1,2], adiabatic switching (AS) [2,3], and/or thermodynamic perturbation theory (TPT) [2,4,5]. Particularly challenging is the problem of efficiently calculating free energies in real materials over wide ranges of volume or pressure and temperature in multiple phases.…”

“…This capability is required to obtain multiphase equations of state and equilibrium phase boundaries to high (1-2%) accuracy from a given set of either quantum-mechanical or empirical interatomic forces. The simplifying and highly efficient method of reversible scaling molecular dynamics (RSMD) proposed by de Koning et al [3], which marries the principles of TI and AS, has proven to be useful in this regard, allowing one to obtain free-energy differences as a function of temperature at constant volume in a given phase from a single MD simulation. In their original paper, de Koning et al [3] showed only that their RSMD technique could efficiently reproduce the classical free energy of a simple solid as obtained from TI.…”

“…The simplifying and highly efficient method of reversible scaling molecular dynamics (RSMD) proposed by de Koning et al [3], which marries the principles of TI and AS, has proven to be useful in this regard, allowing one to obtain free-energy differences as a function of temperature at constant volume in a given phase from a single MD simulation. In their original paper, de Koning et al [3] showed only that their RSMD technique could efficiently reproduce the classical free energy of a simple solid as obtained from TI. In a later paper, de weak-coupling thermodynamic framework applicable to materials described by temperature-independent forces with additive zero-temperature, ion-thermal, and electron-thermal free-energy contributions.…”

Polymorphism and melt in high-pressure tantalum

“…[33]. This latter scheme, which utilizes the statistical mechanics principles of thermodynamic integration and reversible-scaling MD (RSMD) simulation [40], involves a combination of standard QHLD and MGPT-RSMD This contribution has been retained nonetheless and used in reported MGPT free-energy melt-curve calculations, both in the previously published Ta3 [2] and Ta4 [21,37] In all three cases, the MGPT-MD simulations used to determine A ion for the bcc and liquid phases were performed with 250-atom, constant-volume ensembles with periodic boundary conditions. The Ta4 melt curve is lowered significantly below the Ta3 result above 100 GPa, with the melt temperature at 400 GPa lower by about 1000 K. On the other hand, and as expected, the Ta4 and Ta6.8x melt results are quite close to one another, with the Ta6.8x melt curve calculated slightly lower, except near ambient pressure.…”