Umbrella Sampling

Umbrella sampling attempts to overcome the sampling problem by modifying the potential function so that the unfavorable states are sampled adequately. The modification of the potential function $u(r)$ can be written as

$$u'(r)=u(r)+W(r)$$ (24)

where $W(r)$ is a weighting function, which often takes a quadratic form;

$$W(r)=k_{W}(r-r_{0})^{2}.$$ (25)

For configurations far from the equilibrium state $r_{0}$ the weighting function will be large and so a simulation using the method energy function $u'(r)$ will be biased along some relevant reaction coordinate away from the configuration $r_{0}$. The resulting distribution will, of course, be non-Boltzmann. The corresponding Boltzmann averages can be extracted from the non-Boltzmann distribution by [28]

$$\langle A\rangle=\frac{\langle A(\{\vec{R}_{i}\})\exp[+W(r)/k_{B}T]\rangle_{W}}{\langle\exp[W(r)/k_{B}T]\rangle_{W}}.$$ (26)

The subscript $W$ indicates that the average is based on the probability $P_{W}(\{\vec{R}_{i}\})$, which in turn is determined by the modified energy function $u'(r)$. As shown in Fig. 12, we can use the distribution function with the weighting function which would be determined and then corrected to give the true radial distribution function. The free energy can be calculated as a function of the separation. It is usual to perform an umbrella sampling calculation in a series of stages, each of which is characterized by a particular value of the coordinate and an appropriate value of t. However, if the weighting function is too large, the denominator in Eq. (26) is dominated by contributions from only a few configurations with especially large values of $\exp[W(r)]$ so that the averages may take too long to converge.

Figure 12: We can calculate the real distributions by several umbrella sampling windows.

In our case, we add a weighting function to the distance between two methane molecules, to simulate and get many piece of PMF curve. Then, we reduce the weighting function and combine these pieces of curve to get the PMF.