Metropolis Monte Carlo algorithms are also well-suited for evaluation of probability distributions. For example, a slightly modified Monte Carlo program calculates a histogram of a distance distribution for a particle in harmonic potential. The potential energy parameters are chosen to approximate bond stretching potential in carbon monoxide. Such histograms illustrate, for example, that the carbon - oxygen distance adopts a range of values at any given temperature. We can also see that the range of values is getting broader as the temperature increases, reflecting increased amplitude of motion of atoms at higher temperature. A word of caution is due here, however. Monte Carlo sampling of harmonic potential gives classical probability distributions while bond vibrations in the real CO molecule are quantized. As a consequence, thermodynamic properties such as vibrational entropies or heat capacities of isolated molecules are not accurately reproduced by classical Monte Carlo simulations.
For systems with multiple potential minima such histograms reflect the probability of the system being in one or another minimum. For example, probability distributions for flexible molecules depict relative populations of different conformations. However, the relative probabilities from Monte Carlo simulation reflect true populations only when transitions from one minimum to another can occur frequently. This is true only when the energy barriers separating the minima are not much larger than RT, which at room temperature is 0.59 kcal/mol. A fairly long Monte Carlo simulation was required to obtain the well-converged probability distribution for ethanol, where the gauche and anti minima are separated by barriers of about 3.3 kcal/mol. Because of the tendency of Metropolis Monte Carlo methods to sample low energy regions of conformational space, Monte Carlo-based algorithms are powerful tools for finding important conformations of flexible biomolecules. For systems where potential enery minima are separated by barriers much higher than 2-3 kcal/mol the sampling efficiency can be increased by running the calculation at elevated temperatures or using simulated annealing techniques.
Monte Carlo simulations have been very valuable in understanding the structure and properties of liquids. For example, Monte Carlo simulations with accurate energy potentials can estimate liquid densities and heats of vaporization with few percent accuracy. Monte Carlo simulations can provide information about the structure of hydration shells around solutes and allow to estimate how different solvents alter the energy profiles in chemical reactions.
For macromolecules, thermodynamic parameters such as enthalpies, entropies, and free energies depend on many conformational degrees of freedom that these flexible molecules can take. We typically cannot estimate free energies of macromolecules in solution using Monte Carlo simulations, partially because transitions from one conformer to another occur infrequently. Furthermore, for macromolecules Molecular Dynamics simulations frequently offer more efficient sampling of conformational space. What we can do with Monte Carlo or Molecular Dynamics simulations, however, is to estimate free energy