1. Monte-Carlo simulations of the structure of complex liquids with various interaction potentials Alja ž 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 What are complex liquids? simple liquid anisotropic particles, COMPLEX POTENTIALS hard spheres, Lennard-Jones particles- SIMPLE ISOTROPIC POTENTIALS Importance of complex liquids? ≈ ρ vapour ρ bulk complex liquid ΔF= ΔU-TΔS ΔU = 0 ΔU > 0 Hydrophobic interactions

4. Introduction ≈ ρ vapour ρ bulk S hard sphere in LJ fluid S Angular correlations completely ignored!!

5. Statistical mechanics of complex liquids Assumption: separable Hamiltonian (intermolecular interactions have no effect on the quantum states) H=H class +H quant two sets of independent quantum states (i.e. eigenstates can be taken as a product) with energy E n =E n cl +E n qu 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 classical: centre of mass and the external rotational degrees of freedom quantum mechanical vibrational and internal rotational degrees of freedom

6. Statistical mechanics of complex liquids Probability density for the classical states The classical Hamiltonian can be split into kinetic and potential energy H=K t +K r +U ( r N ω N ) IαIα In Monte-Carlo calculations we need only the configurational probability density, but we introduce a new distribution P ' ( r N p N ω N J N )

7. Statistical mechanics of complex liquids new probability density it is convinient to introduce a new distribution P ( r N p N ω N J N ) We can write

8. Statistical mechanics of complex liquids We can now directly factorize the p s and J s of different molecules are uncorrelated furthermore thus we can directly integrate Λ t =h/(2πmkT) 1/2 Λ r =(h/(8π 2 I x kT) 1/2 )× (h/(8π 2 I y kT) 1/2 )(h/(8π 2 I z kT) 1/2 ) Ω = 8 π 2 ( 4 π )

9. Spherical multipole expansion of the electrostatic interaction energy electrostatic interaction a molceule= a distribution of charges (placed in the atomic centres); Atoms have finite sizes and also interact with polarization interactions electrostatic interaction between two molceules= interaction between two charge distributions spherical harmonic expansion of r 12 -1 =|r+r 2 -r 1 | -1 Pair potential energy: dispersion polarisation exchange repulsion (finite size of atom) z y x r · · q1q1 q2q2 r1r1 r2r2 potential of a charge distribution:

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

11. z’’ y’’ x’’ x’ z’ Spherical multipole expansion of the electrostatic interaction energy z y x y’ r · · q1q1 q2q2 r1r1 r2r2 xyz: space-fixed x’y’z’ and x’’y’’z’’: body-fixed calculated only once What is gained? example: molecule consisting of four charges 17 terms /pair 5 (10) terms /pair Spherical multipole expansion TIP5P water model Relation between multipole moments in the space-fixed and body-fixed coordinate frames:

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 (x 1,x 2 ) How to choose P in a way, which allows the function evaluation to be concentrated in the region of space that 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 = π mn N possible states π mn constitute a N×N matrix, π each row of π sums to 1 probability that the system is in a particular state is given by the probability vector ρ =( ρ 1, ρ 2, ρ 3,..., ρ m, ρ n,..., ρ N ) probability of the initial state = ρ(1) equilibrium distribution Microscopic reversibility (detailed balance): Metropolis:

14. Monte-Carlo simulations of ensembles of anisotropic particles How to accept trial moves? Metropolis: - allways accept if U new ≤ U old - if U new > U old choose a random number ζ from the interval [0,1] 0 1 exp(-βΔU) U new -U old ΔU nm ζ 1ζ 1 ζ 2ζ 2 × × allways accept accept reject How to generate trial moves? translation rotation How many particles should be moved? sampling efficiency: ~kT reasonable acceptance 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? generally: pair correlation function: spherical harmonic expansion of the pair correlation function in a space fixed frame: δ ( ω )= δ ( φ ) δ (cos θ ) δ ( χ ) linear molecules: intermolecular frame ω = 0 φ : angular correlation function, g ( r h ω h ) :

17. removing the m dependence: reconstruction EXAMPLE: dipoles in LJ spheres r φ θ2θ2 How to present the results of MC simulations?

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