Monte Carlo methods (II) Simulating different ensembles

E1 E0 Accept with probability exp[-(E2-E1)/kBT] Accept E1

Configuration Xo, energy Eo Perturb Xo: X1 = Xo + DX Compute the new energy (E1) E1<Eo ? N Draw Y from U(0,1) Y Compute W=exp[-(E1-Eo)/kT] A:=A+A(Xo) W>Y? Y Xo=X1, Eo=E1 N

periodic bondary contidtions Infinite systems: periodic bondary contidtions Minimum image convention (a particle is not supposed to interact with its image). Spherical cut-off satisfying minimum-image convention

New positions in periodix box lx

Space-filling polyhedra that can serve as periodic boxes Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15, 1843-1850

Example: hexagonal prism/elongated dodecahedron Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15, 1843-1850

Example: solute molecules in non-cubic boxes Qian. Strahs, Schlick, J. Comput. Chem., 2001, 15, 1843-1850

Cut-off on short-range interactions Simple truncation Truncation and shift

Truncation correction (LJ potential)

Characteristic function Ensemble types Type Parameters Characteristic function Microcanonical N, V, E ln W Canonical N, V, T ln Q Isothermal-isobaric N, p, T ln D Grand canonical m, V, T ln X

Microcanonical ensemble N, V, E defined Canonical ensemble N, V, T defined

Isothermic-isobaric ensemble Grand canonical ensemble N ,T, p defined Grand canonical ensemble m , T, V defined

NVE Monte Carlo simulations V(x1) Ed1 accept V(x1) Ed0 V(x0) Ed0 reject V(x1) Ed1 accept M. Creutz, Phys. Rev. Lett., 1983, 50, 1411-1414

NPT Monte Carlo sampling Scaled variables

Acceptance criterion For coordinate change with keeping the box dimensions – as in canonical MC. For change of box dimensions keeping the scaled coordinates constant It should be noted that even though the scaled coordinates remain constant under this move, the actual coordinates don’t. Therefore U(sN,Vold)<>U(sN,Vnew)

Reference algorithms for MC/MD simulations (Fortran 77) M.P. Allen, D.J. Tildesley, „Computer Simulations of Liquids” , Oxford Science Publications, Clardenon Press, Oxford, 1987 http://www.ccp5.ac.uk/software/allen_tildersley.shtml F11: Monte Carlo simulations of Lennard-Jones fluid.

mVT Monte Carlo simulations Applicable, e.g., in studying adsorption phenomena when equilibration with the reservoir of gas/liquid would take years of computation.

Acceptance criterion For coordinate change with keeping the number of molecules constant – as in canonical MC. For insertion/deletion of a molecule The larger the molecule, the less is the probability of accepting insertion.

Ergodicity

Computing averages with Metropolis Monte Carlo It should be noted that all MC steps are considered, including those which resulted in the rejection of a new configuration. Therefore, if a configuration has a very low energy, it will be counted multiple times.

Importance of proper counting Analytical (solid lines) and simulated (symbols) equation of state of LJ fluid. Units are atomic units corresponding to scaled coordinates. Open squares: only new accepted configurations counted. Solid squares: all configurations (old and new) counted after a move.

Detailed balance (Einstein’s theorem) old new

Importance of detailed balance Analytical (solid lines) and simulated (symbols) equation of state of LJ fluid. Units are atomic units corresponding to scaled coordinates. Open squares: detailed balance not satisfied. Solid squares: detailed balance satisfied.

MC Simulations of chain molecules: moves endmove spike crankshaft

More moves

Configurational-bias Monte Carlo w=2/3*1/3 w=2/3 Rosenbluth and Rosenbluth, J. Chem. Phys., 1955, 23, 1955-1959