Monte Carlo umbrella sampling

Torrie G M and Valleau J P 1977 Nonphysical sampling distributions In Monte Carlo free energy estimation umbrella sampling J. Comput. Phys. 23 187-99... 

What has been developed within the last 20 years is the computation of thermodynamic properties including free energy and entropy [12, 13, 14]. But the ground work for free energy perturbation was done by Valleau and Torrie in 1977 [15], for particle insertion by Widom in 1963 and 1982 [16, 17] and for umbrella sampling by Torrie and Valleau in 1974 and 1977 [18, 19]. These methods were primarily developed for use with Monte Carlo simulations continuous thermodynamic integration in MD was first described in 1986 [20]. 

Umbrella sampling attempts to overcome the sampling problem by modifying the potenti function so that the unfavourable states are sampled sufficiently. The method can be use with both Monte Carlo and molecular dynamics simulations. The modification of tf potential function can be written as a perturbation ... 

Monte Carlo simulations were performed in 6/V-dimensional phase space, where N = 120-500 atoms [5]. The Metropolis algorithm was used with umbrella sampling. The weight density was... 

One weakness of this treatment, however, is that it neglects entropic contributions. Entropic contributions were considered in the free energy profiles (FEP) calculated earlier using umbrella sampling [58] and Monte Carlo Free energy Perturbation [59], both using a QM/MM scheme and the AMI Hamiltonian for the QM part. Our group used the same SIESTA DFT-based QM/MM method described above... 

Torrie, G. M. Valleau, J. R, Nonphysical sampling distributions in Monte Carlo free energy estimation Umbrella sampling, J. Comput. Phys. 1977, 23, 187-199... 

Lee, C. Y. Scott, H. L., The surface tension of water a Monte Carlo calculation using an umbrella sampling algorithm, J. Chem. Phys. 1980, 73, 4591 1596... 

To circumvent this problem avoiding such inefficient exploration of the configurational space, several methods have emerged. Two particularly useful approaches are the Umbrella Sampling and the Statistical Perturbation Theory. Both methods can be used either with Monte Carlo or with Molecular Dynamics simulations. 

In both solvents, the variational transition state (associated with the free energy maximum) corresponds, within the numerical errors, to the dividing surface located at rc = 0. It has to be underlined that this fact is not a previous hypothesis (which would rather correspond to the Conventional Transition State Theory), but it arises, in this particular case, from the Umbrella Sampling calculations. However, there is no information about which is the location of the actual transition state structure in solution. Anyway, the definition of this saddle point has no relevance at all, because the Monte Carlo simulation provides directly the free energy barrier, the determination of the transition state structure requiring additional work and being unnecessary and unuseful. 

In considering fluids, a variety of approaches have been used to compute entropy and free energy a free-volume method,111-112 thermodynamic perturbation theory,113-116 thermodynamic integration,117-121 umbrella sampling,122-124 and a Monte Carlo recursion method.125-126 The entropy of association of two protein molecules in water has also been computed.127... 

G. M. Torrie and J. P. Valleau, /. Comput. Phys., 23, 187 (1977). Nonphysical Sampling Distribution in Monte Carlo Free-Energy Estimation Umbrella Sampling. 

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