Comparing Molecular Studies of Liquid Benzene at Constant Pressure

A path integral molecular dynamic algorithm versus Monte Carlo ensemble sampling

Abstract

Several different methods have been developed to study the interaction of matter in condensed phases. Molecular dynamic studies and Monte Carlo ensemble sampling studies are the bulk of molecular simulation studies. A comparative study between a path integral molecular dynamic study and Monte Carlo ensemble sampling is given on liquid benzene at room temperature.

Introduction

The motivation to study the thermodynamic and mechanical properties of matter in condensed phases has given rise to the development of several different methods of study. Molecular modeling has seen rapid development as a promising method due to the increase in availability of computational power. Parallel computing has opened the possibility to rigorously address computationally challenging problems in the numerical evaluation of molecular simulations.

With the motivation to study systems of highly correlated chemical species in fluid phases, it is beneficial to develop a comparative study to determine efficiency and accuracy of methods. Two methods chosen were isothermal-isobaric Monte-Carlo sampling (MC-NPT) and solving the path integral for a system at constant pressure but fluctuating cell volumes (PI-NPT).^1,2 Benzene is of particular interest since it is representative of other planar or quasi-planar molecules, and the isothermal-isobaric (NPT) ensemble is the common condition of experimental settings.

Theory and Methods

The theoretical aspects of both models have been explored thoroughly previously.^2,3 We will only sketch the important theoretical considerations of each model and explain the methodology applied.

For both simulation types, only 100 benzene molecules were considered. This was deemed to be a representative sample to generate general trends in order to compensate for needing less time to computationally evaluate each simulation. Volume changes were considered to be isomorphic, meaning all six points of the cubic box were changed equally.

Monte Carlo sampling: NPT ensemble

In an NPT-ensemble, the number of particles and temperature remain constant while the volume of the container is allowed to fluctuate. Traditionally this is best approached by the Monte Carlo sampling method. Not only can the molecular degrees of freedom be sampled but the volume of the container can be sampled as well. Thus, the two move types are present: molecular and volume.

For benzene, the molecular degrees of freedom involved are rotation, vibration, and translational moves. For moves, the 6-bodied system can be reduced to a two-body system by considering only two points: the center of mass of the benzene ring and the center of mass of the carbon-carbon bond. This "trick" reduces the moves for vibration and rotation to be represented by another translational move.⁴

Benzene has six site interactions at each CH of the ring. The interactions were modeled using the Optimized Potential for Liquid Simulations (OPLS).⁵ Though the benzene quadruple is not explicitly addressed by the model, studies for the liquid phase have indicated the consideration not necessary to achieving physically reasonable results.¹

Molecular Dynamics: path integration for the NPT-case

The Feynman path integral offers a powerful method to simulate molecular systems in the isothermal-isobaric case since the Monte Carlo sampling cannot study phase transitions explicitly nor time-dependent observables. The path integral also offers the link between quantum mechanics and statistical mechanics via the Boltzmann operator being simply a rotation through imaginary time of the time-evolution operator⁶. This connection allows the use of the path integral in addressing problems in statistical physics. The NPT-partition function for the isomorphic case is also a simple integration over the pressure phase space with the quantum canonical partition function in the integrand. This is analogous to the classical statistical mechanics consideration of the relationship between ensembles.⁶

The PI-NPT approach was adopted from Martyna et al.² Molecular interactions were treated using the OPLS representation.

Results

For this study, four temperature points were considered for comparison. These points were chosen based off of the experimentally determined phase diagram of benzene.^7-9 The pair distribution function was determined for each data point as well as other thermodynamic properties.¹⁰ Table 1 and Table 2 gives the results for the two studies, and Table 3 gives available experimental studies.^7-9

T (K)	Most Probable Separation (Å)	E (kJ/mol)	Cp (J/mol K)
125	3.3 ± 0.1	-51.3 ± 0.4	19 ± 2
150	3.7 ± 0.1	-50.5 ± 0.4	18 ± 1
175	4.2 ± 0.1	-49.8 ± 0.3	17 ± 1
200	4.6 ± 0.1	-48.4 ± 0.4	16 ± 2

Table 1. The results from the MC-NPT Simulations

T (K)	Most Probable Separation (Å)	E (kJ/mol)	Cp (J/mol K)
125	3.3 ± 0.1	-51.3 ± 0.4	20 ± 2
150	3.7 ± 0.1	-50.5 ± 0.4	18 ± 1
175	4.2 ± 0.1	-49.8 ± 0.3	18 ± 1
200	4.6 ± 0.1	-48.4 ± 0.4	17 ± 1

Table 2. The results from the PI-NPT Simulations

T (K)	Most Probable Separation (Å)	Cp (J/mol K)
125	3.4 ± 0.2	20.5 ± 0.7
150	3.6 ± 0.2	17.7 ± 0.6
175	4.3 ± 0.2	17.0 ± 0.7
200	4.7 ± 0.2	16.5 ± 0.6

Table 3. Experimental Results^7-9

Discussion

As indicated by Table 1, there is large agreement between the two studies. The agreement indicates that the PI-NVT method would be better spent studying the phase-change phenomena of molecular systems instead of only the thermodynamic properties at stationary points. This is because the MC-NVT method took about a quarter of the time for each point to calculate the simulation.

References

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