Free energy calculations equilibration

UU is the Hamiltonian difference (the perturbation) the angle brackets represent a canonical ensemble average performed on an equilibrated system designated by the subscript. Usually the kinetic component of the Hamiltonian is not included in the free energy calculation, and A- //, n can be replaced by the potential energy difference AU = Ui - U0. 

The establishment of chemical potential equilibrium (with respect to either a setpoint or phase coexistence) is the central component of most Monte Carlo schemes for simulation of the phase behavior and stability of molecular systems. Simulation of the chemical potential (or chemical potential equilibration) in a polymeric system requires more effort than the corresponding calculation for a simple fluid. The reason is that efficient conformational sampling of the polymer is implicitly required for a free-energy calculation and, in fact, the ergodicity problems described in earlier sections are often exacerbated. 

H. J. C. Berendsen, in Proceedings of the Whistler Conference on Proteins, V. Re-nugopalakrishan, Ed., ESCOM, Leiden, 1990, pp. 385-392. Incomplete Equilibration A Source of Error in Free Energy Calculations. 

Dynamic simulations need to be applied to longer trajectories and continuing difficulties with force fields, equilibration and free energy calculations need to be addressed. 

In practice, the free energy is calculated by performing a series of free energy calculations for successive values of X(i). Each time the value of X (and hence V(X)) is changed, the system must be allowed to re-equilibrate to the new potential function before we commence accumulation of ensemble statistics. Thus, the procedure is select X, equilibrate, collect ensemble data, update X(i) -> X(i + 1), repeat until AG has been determined for each window in the chain. The ensemble average is calculated from the MD or MC ensemble by simple averaging over the number of moves or integration steps ... 

Insufficient equilibration is a source of systematic errors. This is especially a problem in SCTI in which the Hamiltonian is changed at every step without equilibration. But this time lag problem also exists in other implementations of free energy calculations. " ... 

FEP calculations for paths A, B and C were performed with a 40 ps equilibration run prior to the sampling for all points along the path. The free energy contributions were sampled for 20 ps for each point on the MEP. In all cases a time step of 2.0 fs was employed, maintaining a constant temperature of 300 K. The SHAKE [47] algorithm was used to constrain all bonds involving hydrogen atoms. 

Rao and Singh32 calculated relative solvation free energies for normal alkanes, tetra-alkylmethanes, amines and aromatic compounds using AMBER 3.1. Each system was solvated with 216 TIP3P water molecules. The atomic charges were uniformly scaled down by a factor of 0.87 to correct the overestimation of dipole moment by 6-31G basis set. During the perturbation runs, the periodic boundary conditions were applied only for solute-solvent and solvent-solvent interactions with a non-bonded interaction cutoff of 8.5 A. All solute-solute non-bonded interactions were included. Electrostatic decoupling was applied where electrostatic run was completed in 21 windows. Each window included 1 ps of equilibration and 1 ps of data... 

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