LSSC2011 Optimization of intermolecular interaction potential energy parameters for Monte-Carlo and Molecular dynamics simulations using Genetic Algorithms (GA)

Optimization of intermolecular interaction potential energy parameters for Monte-Carlo and Molecular dynamics simulations using Genetic Algorithms (GA) Dragan Sahpaski [email_address] Institute of Informatics, Faculty of Natural Sciences University “Ss. Cyril and Methodius” Skopje, Macedonia Ljupco Pejov [email_address] Institute of Chemistry, Faculty of Natural Sciences University “Ss. Cyril and Methodius” Skopje, Macedonia Anasas Misev [email_address] Institute of Informatics, Faculty of Natural Sciences University “Ss. Cyril and Methodius” Skopje, Macedonia *This work is supported by the FP7 project HP-SEE

Introduction Condensed-phase systems are of substantial importance in both fundamental natural sciences and technology. Theoretical modeling of such systems has been shown to be of crucial importance for a thorough understanding of their properties. It has been often demonstrated that theoretical model can be complementary to the experimental studies of condensed phases.

Introduction Theory can sometimes predict certain properties or systems’ behavior which hasn’t been observed yet (or, in certain cases, is not even observable with the current experimental techniques). The most widely used theoretical methods for modeling of condensed phases are Monte Carlo and molecular dynamics techniques.

Introduction We aim to propose a general methodology (approach) for optimization of the interaction potentials, using genetic algorithms We analyze the performances and drawbacks of non-optimized potentials and emphasize the need for a very careful construction of general-purpose potentials. As a particular example, we focus our attention on liquid carbon tetrachloride (CCl 4 ).

Computational Details and Algorithms Introduction Monte Carlo simulations . The Optimization Problem Representation of the Solution The Optimization Procedure Results Conclusion

Monte Carlo simulations Chosen system: liquid CCl 4 (of broad interest for chemistry and technology as one of the most frequently used organic solvents). To generate the structure of liquid, first a series of Monte-Carlo (MC) simulations were performed, using the statistical mechanics code DICE. MC simulations of 500 carbon tetrachloride molecules placed in a cubic box with side length of 43.36 Å, imposing periodic boundary conditions.

Monte Carlo simulations We have also carried out a Monte Carlo simulation of pyrrole solution in CCl 4 : is important for modeling the solvent effects on the vibrational N-H stretching band of pyrrole in liquid carbon tetrachloride 1 pyrrole molecule solvated by 412 carbon tetrachloride molecules relatively accurate experimental data are available

Monte Carlo simulations DICE FORTRAN NOT Parallel K. Coutinho and S. Canuto, DICE: A Monte Carlo program for molecular liquid simulation, University of São Paulo, Brazil, version 2.9 (2003).

Monte Carlo simulations Intermolecular interactions were described by a sum of Lennard-Jones 12-6 site-site interaction energies plus Coulomb terms: where i and j are sites in interacting molecular systems a and b , r ij is the interatomic distance between sites i and j , while e is the elementary charge.

Monte Carlo simulations The following combination rules were used to generate two-site Lennard-Jones parameters  ij and  ij from the single-site ones:

Monte Carlo simulations Table 1 . Lennard-Jones interaction potential parameters used initially in Monte-Carlo simulations Atom q / e ε /(kcal mol -1 ) σ /Å C Cl Cl Cl Cl 0.248 -0.062 -0.062 -0.062 -0.062 0.050 0.266 0.266 0.266 0.266 3.800 3.470 3.470 3.470 3.470

Results with the standard LJ parameters We have chosen the following physical quantities as representative to test the quality of the used LJ potential energy parameters: the average density of the liquid ( ρ ), the thermal expansion coefficient ( α P ), isothermal compressibility ( β T ) molar heat capacity at constant pressure ( C P ,m ) of the liquid.

Results with the standard LJ parameters

The Optimization Problem Find a set of values for S = { q Cl , ε Cl , σ Cl , q C , ε C , σ C }, such that the cost function is minimal. The function relerr gives the relative error of the parameter computed by the simulation procedure and the experimental value for the parameters ρ , α P , β T and C P, m . c 1 , c 2 , c 3 and c 4 are integer constants defining the weights in which each relative error affects the cost function.

The Optimization Problem C Cl C Cl q ε σ

Outline of a Genetic Algorithm (GA) Create a random starting population of chromosomes Calculate the fitness of each chromosome Select the next generation Crossover Mutation ? >= N generations? YES END NO

Representation of the Solution Chromosome S = { q Cl , ε Cl , σ Cl , q C , ε C , σ C }, q Cl ε Cl σ Cl ε C σ C 1 0 0 . . . 0 1 1 GENE CHROMOSOME

Mutation 1 0 . . 0 1 0 . . GENE 1 1 1 0 . . 0 0 0 . . 1 1 MUTATED GENE FLIP a BIT in a Random position with a certain probability

Crossover Pick a random position and swap all subsequent genes in the two parents q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C

Selection Select the best individuals with higher probability: *Always select the fittest q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C q Cl ε Cl σ Cl ε C σ C

The Optimization Procedure on GRID http://jgap.sourceforge.net

The Optimization Procedure on GRID

The standard LJ parameters . Table 1 . Lennard-Jones interaction potential parameters used initially in Monte-Carlo simulations Atom q / e ε /(kcal mol -1 ) σ /Å C Cl Cl Cl Cl 0.248 -0.062 -0.062 -0.062 -0.062 0.050 0.266 0.266 0.266 0.266 3.800 3.470 3.470 3.470 3.470

Results with the standard LJ parameters Table 2 . Comparison of the density, thermal expansion coefficient, molar heat capacity at constant pressure and isothermal compressibility of liquid carbon tetrachloride computed from the MC simulation with the standard (non-optimized) LJ potential parameters with the available experimental data. Parameter MC Experimental Rel. error % ρ / (g cm -3 ) 1.5697 1.5867 10.7 C P ,m / (J K -1 mol -1 ) 80.65 129.35 37.6 β T / Pa -1 1.126·10 -9 1.034·10 -9 8.9 α P / K -1 4.6199·10 -3 1.236·10 -3 273.8

The optimized LJ parameters . Table 3 . The optimized Lennard-Jones for CCl4 parameters by the genetic algorithm Atom q / e ε /(kcal mol -1 ) σ /Å C Cl Cl Cl Cl 0.412 -0.103 -0.103 -0.103 -0.103 0.025 0.374 0.374 0.374 0.374 3.328 0.149 0.149 0.149 0.149

Results with the optimized LJ parameters Table 4 . Comparison of the density, thermal expansion coefficient, molar heat capacity at constant pressure and isothermal compressibility of liquid carbon tetrachloride computed from the MC simulation with the standard (non-optimized) LJ potential parameters with the available experimental data. Parameter MC - GA Experimental Rel. error % ρ / (g cm -3 ) 1.5884 1.5867 0.10 C P ,m / (J K -1 mol -1 ) 122.13 129.35 5.6 β T / Pa -1 3.459·10 -12 1.034·10 -9 99.6 α P / K -1 3.3522·10 -3 1.236·10 -3 171.2

Conclusions and Directions for Future work We have efficiently implemented a genetic algorithm to optimize the interaction potential energy parameters of liquid CCl 4 to be used in statistical physics simulations of the pure liquid, as well as of various solutions thereof. We have demonstrated that it is possible to improve the values of certain parameters characterizing the static and dynamical properties of the liquid by the approach that we have adopted. It is also tempting to apply such novel approach to the problem of construction and optimization of intermolecular interaction energy parameters for various types of simulations of a number of molecular liquid systems.

LSSC2011 Optimization of intermolecular interaction potential energy parameters for Monte-Carlo and Molecular dynamics simulations using Genetic Algorithms (GA)

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