How to Deal with Large Perturbations

Intermediate states do not have to be physically meaningful, i.e., they do not have to correspond to systems that actually exist. As an example, assume that we want to calculate the difference in hydration free energies of a Lennard-Jones particle and an ion with a positive charge q of le. For simplicity, we further assume that the Lennard-Jones parameters remain unchanged upon charging the particle. Since a direct calculation of the free energy difference is not likely to succeed in this case, we construct intermediate states in which the particle carries fractional charges / such that qi q3 for i j and 0 q3 q. 

More generally, we can consider the Hamiltonian as a function of some parameter, A. Without loss of generality, we can choose 0 A 1, such that A = 0 and A = 1 for the reference and target states, respectively. A simple choice for the dependence of the Hamiltonian on A is a linear function 

If we recall the discussion about integrating out the kinetic term in the Hamiltonian in Sects. 1.2.1 and 2.2, then we can rewrite AA as 

In the example of charging a neutral particle, A, = qt/q is the linear parameter. Choosing intermediate states separated by a constant AX is, however, not a good choice for this problem because, as has been seen in Sect. 2.5, AA is a quadratic function of q. A better choice would be to decrease AX quadratically. Alternatively, one could define M A) as a quadratic function of A. Then, using a constant AX would be appropriate. 

Stratification is not specific to FEP - it is a universal strategy that improves the efficiency of many other methods for calculating free energies. Not surprisingly, we will return to this strategy several times, in particular in Chaps. 3 and 6. 

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