Text_Template

Module 2 : Molecular Structure

Lecture 11 : Non-covalent interactions 2 : Structures of Liquids

11.4

The Monte Carlo (MC) Method
One of the methods for obtaining the bulk equilibrium properties from intermolecular forces is known as the Monte Carlo method. In this method, the initial configuration (set of coordinates of all the N molecules in the box of volume L³ ) is chosen and the total energy of the system is calculated using eq (11.5). Then each molecule is moved by a small amount ( x_i, y_i and z_i). If is the size (radius or the extent in one direction) of a molecule, then the maximum x_i is about 0.1 . Now, all the molecules have obtained a new configuration. Let the energy of the old configuration be denoted U_old and let U_new be the energy of the new configuration. If U_new is less than U_old, then the new configuration is accepted and the coordinates are written into the hard disc. If the new energy is greater than U_old then the ratio f

f = e ^{- U new} ^{/ k_BT}/ e^{- U old / k_BT}	(11.6)
is compared with a random number r which lies between 0 and 1. You can generate a few random numbers on your calculator and verify that they lie between 0 and 1. These are called uniform random numbers (because, they span the range between 0 and 1 uniformly) and there are several standard algorithms to generate these. If the value of f is less than r, the the new configuration is rejected and the old one is written to the disc once again. After this, all the molecules in the current configuration are again moved by ( x_i, y_i, z_i) and the new f is compared with another random number r' to see whether the move is acceptable. Using this algorithm about 10⁵ to 10⁶ configuration are generated and written to the disc. This method is called the Metropolis Monte Carlo method. The values of ( x_i, y_i, z_i) are chosen such that about half the moves are accepted. If x_i are very large, very few moves get accepted and if x_i are too small, a very large number of moves get accepted.

The configurations so generated are referred to as the members of a canonical ensemble. For each member or the configuration that is saved, N, V (volume) and T (the temperature) is the same, but of course the coordinates are different. The distribution of energies in the members has been found to obey the standard Boltzmann distribution (wherein the probability of finding a member with energy U is proportional to exp (-U/k_BT).

You may have noticed that in the above method, velocities are not used nor have we considered time. The Monte Carlo method is useful for calculating the equilibrium properties such as E, p, and so on. E.g., the average energy of the system under consideration can be obtained as

<E> = ( 1 / N_E )	(11.7)
Here, E_is are the energies of the members of the ensemble ( N_E) generated by the MC algorithm described above.