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First described for atomistic simulations in 1953 by Metropolis et al. [7], the MC method cannot account for the time evolution of the system investigated. By calculating instead its physical properties from repeated samplings made on the basis of a Boltzmann energy distribution of equilibrium states, it differs from a search algorithm with which the energy of the system would occasionally increase even though a steady decrease would be sought after at every step. As already stated, the method is thus inappropriate to tackle any nonequilibrium or history‐dependent phenomenon. Over MD simulations, its main advantage is to shorten the calculation time needed to arrive at the equilibrium structure if a suitable sampling method is employed.

Within the framework of the canonical ensemble, the partition function Q(N,V,T) is, for example,

(9)

where is the thermal de Broglie wavelength, β the reciprocal temperature (1/k_B T), and N, r, U_p, m, k_B, and T are the number of atoms, atomic coordinates, potential energy of a system, atomic mass, Boltzmann's constant, and temperature, respectively.

From the partition function it follows that the probability (P) of finding a configuration r ^N is

(10)

To fulfill Eq. (9), the standard Metropolis procedure in MC simulations consists of

1 setting up an initial configuration in a periodic boundary cell,

2 calculating the energy of this configuration,

3 selecting an atom at random and moving it randomly along all coordinate directions,

4 accepting the new configuration resulting from the move if it lowers the energy of the system, because the new state is more probable than the former, but keeping the former configuration otherwise only in case its Boltzmann factor is higher than a real number drawn randomly between 0 and 1.

In other words, the MC method does not weight configurations selected randomly according to their Boltzmann factor to calculate the properties of the system, as was done earlier, but weight evenly instead configurations selected with the probability exp[−U_p(r ^N )]. Its trick thus is to concentrate on the sampling in the regions of the phase space that contribute the most to the partition function. Although they will not be described here, there are several other sampling methods in the standard MC calculations to accelerate convergence to the equilibrium state [3].