Monte-Carlo simulations of the structure of complex liquids with various interaction potentials

1. Monte-Carlo simulations of the structure of complex liquids with various interaction potentials Aljaž Godec Advisers: prof. dr. Janko Jamnik and doc. dr. Franci Merzel National Institute of Chemistry

2. Contents 1. Introduction 2. Statistical mechanics of complex liquids 3. Spherical multipole expansion of the electrostatic interaction energy 4. Monte-Carlo simulations of ensembles of anisotropic particles 5. How to present the results of MC simulations? 6. Conclusions and considerations for future work

3. Introduction complex liquid simple liquid What are complex liquids? hard spheres, Lennard-Jones particles- SIMPLE ISOTROPIC POTENTIALS anisotropic particles, COMPLEX POTENTIALS Importance of complex liquids? ΔU = 0 Hydrophobic interactions ΔU > 0 ≈ρvapour ΔF= ΔU-TΔS ρbulk

4. Introduction hard sphere in LJ fluid ≈ρvapour S S Angular correlations completely ignored!! ρbulk

5. Statistical mechanics of complex liquids Assumption: separable Hamiltonian (intermolecular interactions have no effect on the quantum states) H=Hclass+Hquant classical: centre of mass and the external rotational degrees of freedom quantum mechanicalvibrational and internal rotational degrees of freedom two sets of independent quantum states (i.e. eigenstates can be taken as a product) with energy En=Encl+Enqu The partition function factorises Q = Q cl Q qu and individual molecular energy Consequence of the above assumption: the contributions of quantum coordinates to physical properties are independent of density

6. Statistical mechanics of complex liquids Probability density for the classical states The classical Hamiltonian can be split into kinetic and potential energy H=Kt+Kr+U(rNωN) Iα In Monte-Carlo calculations we need only the configurational probability density, but we introduce a new distributionP'(rNpNωNJN)

7. Statistical mechanics of complex liquids new probability density it is convinient to introduce a new distributionP(rNpNωNJN) We can write

8. Statistical mechanics of complex liquids We can now directly factorize theps and Js of different molecules are uncorrelated furthermore Λt=h/(2πmkT)1/2 thus we can directly integrate Λr=(h/(8π2IxkT)1/2)× (h/(8π2IykT)1/2)(h/(8π2IzkT)1/2) Ω=8π2 (4π)

9. Spherical multipole expansion of the electrostatic interaction energy a molceule= a distribution of charges (placed in the atomic centres); Atoms have finite sizes and also interact with polarization interactions Pair potential energy: dispersion polarisation electrostatic interaction exchange repulsion (finite size of atom) q2 electrostatic interaction between two molceules= interaction between two charge distributions · z r2 q1 · spherical harmonic expansion of r12-1=|r+r2-r1|-1 r r1 y potential of a charge distribution: x

10. Spherical multipole expansion of the electrostatic interaction energy mth component of the spherical multipole moment of order l: interaction between two charge distributions= ∑(interactions of components of multipole moments of charge distributions) Introduction of body-fixed coordinate frame: z z z’ z’ y’ z’ y’ Ω r y x’ x’ y’ x’ y x x

11. Spherical multipole expansion of the electrostatic interaction energy z z’’ q2 Relation between multipole moments in the space-fixed and body-fixed coordinate frames: · r2 x’’ q1 z’ y’ · y’’ r r1 y x’ xyz: space-fixed x’y’z’ and x’’y’’z’’: body-fixed x TIP5P water model calculated only once What is gained? example: molecule consisting of four charges Spherical multipole expansion 5 (10) terms /pair 17 terms /pair

12. Monte-Carlo simulations of ensembles of anisotropic particles What do we do in a MC calculation? P(x)... probability density function Monte-Carlo: perform a number of trials τ: in each trial choose a random number ζ from P(x) in the interval (x1,x2) How to choose P in a way, which allows the function evaluation to be concentrated in the region of spacethat makes importatnt contributions to the integral? Construction of P(x) by Metropolis algorithm generates a Markov chain of states 1. outcome of each trial depends only upon the preceding trial 2. each trial belongs to a finite set of possible outcomes

13. Monte-Carlo simulations of ensembles of anisotropic particles a state of the system m is characterized by positions and orientations of all molecules probability of moving from m to n= πmn πmn constitute a N×N matrix, π N possible states each row ofπ sums to 1 probability that the system is in a particular state is given by the probability vector ρ=(ρ1, ρ2, ρ3,...,ρm, ρn,...,ρN) probability of the initial state = ρ(1) equilibrium distribution Microscopic reversibility (detailed balance): Metropolis:

14. Monte-Carlo simulations of ensembles of anisotropic particles exp(-βΔU) How to accept trial moves? Metropolis: - allways accept if Unew≤Uold - if Unew>Uold choose a random number ζfrom the interval [0,1] 1 reject ζ1 × allways accept accept × ζ2 0 ΔUnm Unew-Uold How to generate trial moves? How many particles should be moved? translation sampling efficiency: ~kT reasonable acceptance rotation 1. N particles, one at a time: CPU time ~ nN 2. N particles in one move: CPU time ~ nN sampling efficiency down by a factor 1/N

15. Monte-Carlo simulations of ensembles of anisotropic particles How to represent results (especially angular correlations)? we introduce a generic distribution function: we further introduce a reduced generic distribution function: ideal gas: homogenous isotropic fluid:

16. How to present the results of MC simulations? angular correlation function, g(rhωh) : generally: pair correlation function: δ(ω)=δ(φ) δ(cosθ) δ(χ) spherical harmonic expansion of the pair correlation function in a space fixed frame: linear molecules: intermolecular frame ω=0φ :

17. How to present the results of MC simulations? removing the m dependence: θ2 φ EXAMPLE: dipoles in LJ spheres r reconstruction

18. Conclusions and considerations for the future - we have briefly reviewed the statistical mechanics of complex liquids - in order to reduce the number of interaction terms that have to be evaluated in each simulation step a spherical multipole expansion of the electrostatic interaction energy was made - the basics of the Monte-Carlo method for simulation of ensembles of anisotropic particles were provided along with useful methods for representing the results of such simulations. - finally results of MC simulations of dipoles embedded in Lennard-Jones spheres were briefly presented. - employ such simulations to study biophysical processes, such as the hydrophobic effect - possibility of including polarization effects  basis for developing a polarizable water model for biomolecular simulations