9.5. WHAM
WHAM was originally developed by Ferrenberg and Swendsen [5,6] and is based on the use of multiple histograms generated from NVT simulation. A histogram is essentially a probability distribution of the total energy of the system.  In the multiple histogram technique, histograms at different parameters (viz., one histogram at a particular value of the specific parameter) are combined to form a single histogram. It has been employed in various investigations to determine the free energy across a range of parameters. For example, Kumar and co-workers [7] have applied WHAM to estimate the potential of mean force of the pseudo-rotation phase angle of the sugar ring in deoxyadenosine. It has also been implemented in many other investigations such as, estimating the energy landscape for ligand-protein binding energy [8] , to generate free energy profiles for the conformational transition of protein molecules [9] , to estimate surface tension by calculating surface free energy [10] , to calculate the chain elasticity and free energy profile for polyethylene chain [11] , etc. WHAM has also been used in combination with other simulation tools to extend the applicability of the tools to a wider parameter space. For example, Gallicchio and co-workers [12] combined replica exchange MD simulation with WHAM to estimate the potential of mean force for a peptide molecule. Chodera and Dill [13] have combined parallel tempering and simulated annealing with WHAM, to estimate the potential of mean force for alanine dipeptide in both, implicit and explicit solvent condition. Chen et al. [14] combined WHAM with aggregation-volume-biased Monte Carlo and adaptive umbrella sampling, to study the vapor-liquid water nucleation over the temperature range, 200K to 300K.

We now describe the principle of WHAM and explore the possibility of using WHAM to study crystal nucleation in a polymer melt. As already mentioned in the introduction, the sampling probability for energy states at the top of the nucleation barrier is very low and thus, reweighting by combining multiple histograms, improves the sampling probability of those conformations at the top of the barrier. From the combined histogram, we estimate the density of states (DOS), as described later in this section. Once we have the DOS, we can calculate any thermodynamic property of the system for a wide range of.parameters.