The Calculation of Free Energy Differences

Consider two well-defined states X and Y. For example, X could be a system comprising a molecule of ethanol in a periodic box of water and Y could be ethane thiol in water. X contains N particles interacting according to the Hamiltonian Jf x X contains N particles interacting according to The free energy difference (AA) between the two states is as follows  

Equation (11.4) can be written in terms of an ensemble average, as foUows  

The subscript 0 indicates averaging over the ensemble of configurations representative of the initial state X. If the averaging is over the ensemble corresponding to the final state Y (indicated by the subscript 1) then we are effectively simulating the reverse process, from which AA can be determined by  

This approach to the calculation of free energy differences. Equation (11.6), is generally attributed to Zwanzig [Zwanzig 1954]. To perform a thermodynamic perturbation calculation we must first define and then run a simulation at the state X, forming the 

If X and Y do not overlap in phase space then the value of the free energy difference calculated using Equation (11.6) will not be very accurate, because we will not adequately sample the phase space of Y when simulating X. This problem arises when the energy difference between the two states is much larger than k T - 3tx bT- How then 

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W. K. den Otter and W. J. Briels, The calculation of free-energy differences by... 

Prom a computational point of view the Jarzynski equality is interesting, because it permits the calculation of free energy differences from simulations in which a control parameter is switched at arbitrary speed. To be more specific, consider a system with Hamiltonian H x, A) depending on the phase space point X and the control parameter A. By changing A continuously from its initial value Aq to its final value Ai the Hamiltonian H x,Xo) of the initial state is transformed into that of the final state H x,Xi). The free energy difference... 

A third approach for the calculation of free energy differences from computer simulation is the slow growth method. Here, the Hamiltonian changes by a very small, constant amount at... 

In certain contexts, the calculation of free energy differences is difficult to access computationally. Examples would include problems in which a large amount of solvent would need to be displaced in a chemical association, one in which a large conformational change occurs, or one in which a complex chemical intermediate is present. In these cases, one of which is described below in the context of acid-base chemistry, the method can often be readily combined with a hypothetical free energy cycle[46], with individual legs that are each readily evaluated computationally. 

Theodorou DN (2006) A reversible minimum-to-minimum mapping method for the calculation of free-energy differences. J Chem Phys 124 034109... 

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