Umbrella sampling

It is usefiil to write down here the basic fomuilae for sampling with an additional weight fimction applied, sometimes called non-Boltzmaim or umbrella sampling, and for sampling when the selection of trial moves is done in a biased way, i.e., the a matrix is not syimnetrical. 

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]. 

The average of the step function, using the action for a Boltzmann weight can be pursued by standard statistical mechanics. It may require more elaborate sampling techniques such as the Umbrella sampling [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 ... 

Umbrella sampling can give free-energy differences, but not absolute free energies Usually done in NPT—isothermal/isobaric ensembles, including a water box... 

If additional, auxiliary constraints are present that are not part of the reaction coordinate (e.g., constraints on covalent bond lengths), the formulas are much more complicated, and the algebra becomes rapidly prohibitive. The same is true when qisa. multidimensional coordinate (e.g., a set of dihedrals). Umbrella sampling approaches (discussed in previous sections) are vastly simpler in such cases and appear to be the method of choice for all but the simplest reaction coordinates. 

The umbrella sampling and eonstrained dynamies teehniques are suffi-eiently general to investigate the free energy profile not only as a funetion of... 

This is achieved by coupling the system to a suitably defined order parameter that is sensitive to the crystal order (the stacking sequence of 111 planes in this case), and doing umbrella sampling with this quantity. The result of the simulation is the free energy difference between both candidate structures—and the winner is fed... 

Umbrella sampling method, see Free energy perturbation method... 

As explained didactically by King and Warshel (1990), a histogram for n/x) can be constructed in which the number of times the condition x = Ae occurs is collected for all x, and the functions P/x) and Pp(x) can be obtained. A technique called umbrella sampling is used for this purpose. Figure 35.3 shows curves for Ag ... 

FIGURE 35.3 Free-energy functions for reactant (AE) and product Ag (AE) of an electron transfer reaction as calculated using umbrella sampling within a simple dipolar diatomic solvent. AG° is the reaction free energy. Solid lines are polynomial fittings to the simulated points. Dashed lines are parabolic extrapolations from the minimum of the curves. (From King and Warshel, 1990, with permission from the American Institute of Physics.)... 

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... 

Figure 1-5. Free energy profile for the reaction from chorismate (RC 1.75) to prephenate (RC — 1.75), obtained using MSMD and Jarzynski s equality and pulling speeds of 2.0 A/ps (red) and 1.0 A/ps (green), and using umbrella sampling (blue)...

The coordinates involved are depicted in Scheme 1-2. Two-dimensional Potential Energy Surface (PES) scans of the r /r2 and r /rn surfaces were performed where the r values were restrained to values in a grid, and the geometry of the remaining atoms optimized, were followed by 10 ns windows of umbrella sampling... 

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