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Why Simulation Molecular Simulation Background 1 Predicting properties of new materials 2 Understanding phenomena on a molecular scale 3 Simulating known phenomena Example computing the melting point of ice Why Simulation 1 Predicting properties of new materials 2 Understanding phenomena on a molecular scale 3 Simulating known phenomena Why bother This works better 1 The Monte Carlo Method Classical limit replace the SUM over quantum states by an INTEGRAL of phase space Aim to compute thermal averages of equilibrium systems Where H is the Hamiltonian of the system and 1 kT Where i labels all eigenstates of the system and Problem We cannot compute the sum over all quantum states because there are so many And we cannot compute the classical integral either except the integration over momenta In replacing the sum by an integral we have attributed a volume h3N to every quantum state Similar problem but much less serious Measure the depth of the Nile by quadrature Consider normal numerical integration 100 particles 3 dimensions 10 points in every direction Requires 10300 points for a very poor estimate of the integral Microsoft owerPoint Presentatio 2 We wish to perform a RANDOM WALK in configuration space such that BETTER STRATEGY The number of times that each point is visited is proportional to its Boltzmann weight IMPORTANCE SAMPLING Then Whatever our rule is for moving from one point to another it should not destroy the equilibrium distribution That is in equilibrium we must have How do we achieve that 3 Now we construct the transition probabilities Stronger condition Then detailed balance implies that For every pair n o Detailed Balance Often we choose Then it follows that Metropolis Rosenbluth Rosenbluth Teller and Teller choice 4 Practical issues 1 Boundary conditions In small systems boundary effects are always large 2 Reduced units 1000 atoms in a simple cubic crystal 488 boundary atoms 3 Time saving devices 1000000 atoms in a simple cubic crystal still 6