Monte Carlo calculations represent an entirely different type of simulation from molecular dynamics. The name ``Monte Carlo'' comes from the random-chance nature of the simulations, akin to the games of chance at Monaco's gambling resort. Rather than being a deterministic method like molecular dynamics, where the physical properties of a system (e.g., coordinates, interatomic forces) determine its time evolution, Monte Carlo simulations are stochastic and use random numbers to generate a sample population of the system from which properties can be determined. Monte Carlo simulations are by no means limited to molecular systems, but are used in such diverse areas as integrated-circuit design and solving differential equations. But Monte Carlo calculations are very widespread in chemical simulations, primarily in studies of gases and fluids, where the random nature of the technique is readily employed.
An extremely important Monte Carlo algorithm for molecular systems was developed by Metropolis et al.. One can calculate a molecular property from a canonical ensemble, using the equation
where is the Boltzmann constant, is the system temperature, and is a volume element in phase space. The integral is generally too complex to solve analytically, but it can be estimated by a computer simulation using a sufficiently large sample. A simulation with sample configurations has properties calculated from:
The straightforward approach to calculating Equation () by generating numerous configurations and weighting them by , has problems in the (common) case where most generated configurations have high energies. The great majority of the conformations will be those which are least important, i.e., they have the lowest weighting factors. The Metropolis algorithm avoids this problem by generating conformations according to the probability and weighting them all equally. This ideal distribution is established by giving each conformation a conditional probability of being accepted into the average. Each conformation is perturbed in some way to produce conformation . If the energy of the new conformation, is smaller than that of , the new conformation is accepted. If its energy is higher, the probability of it being accepted is where . The standard method for enforcing this probability is to generate a random number and to accept the new conformation if . Otherwise, the new conformation is rejected and the previous one is restored and it is included again in the summation. Although many enhancements have been made to Monte Carlo theory since the Metropolis algorithm was derived, it still has very wide popularity. In some fields, the Metropolis algorithm has practically become the definition of Monte Carlo simulations.
The random nature of Monte Carlo simulations makes them useful for sampling conformational space. Although they are generally not as efficient as molecular dynamics simulations for sampling conformational space, Monte Carlo simulations can incorporate large conformational changes which cannot be simulated by molecular dynamics. For instance, the Dihedral Probability Grid Monte Carlo method of Chapter 3 can rotate a dihedral angle in a single step without regard to an energy barrier which might prevent the same rotation in molecular dynamics. In general, we have found that Monte Carlo simulations are excellent for coarse-grained sampling of conformational space while molecular dynamics and minimization techniques are excellent for performing the complementary role of local conformational optimization.