Multiple-step thermodynamic perturbation

In a multiple-step thermodynamic perturbation calculation, the free energy difference, found by using the ensemble for Hamiltonian can be evaluated for the system described by the next intermediate Hamiltonian as well as for the system described by the preceding Hamiltonian This... 

The ability to evaluate forward and reverse perturbations simultaneously can be used to decrease the perturbation from one intermediate step to the next in multiple-step thermodynamic perturbations using the method of double wide sampling. In this method, free energy differences are evaluated from each ensemble for half a step in X forward and a half a step in X in reverse direction and taking the sum of all contributions as the final free energy difference,... 

In thermodynamic integrations and multiple-step thermodynamic perturbations it is necessary to define a path that connects the initial and final states. This is done by formulating a A-dependent Hamiltonian such that 3. = 0 and k = 1 define the initial and final states, respectively. If only the free energy difference is needed, any path can be taken since the free energy is a thermodynamic state function. In practice, however, calculated systematic and statistical errors will depend on the particular path chosen. 

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