9.4. Umbrella Sampling

.....Umbrella sampling improves the sampling probabilities of the low probable states. In this sampling, an additional potential, called the umbrella potential , E^u is added to the existing potential of the system. The addition of E^u improves the acceptance probability of the low probable states. E^u is often expressed by a harmonic potential function of the form: are constants and  Φ is the instantaneous value. Φ₀ can be considered as the equilibrium (or most probable value, which can be obtained from the distribution function of Φ).  The value of K may be set by trial – for each value of K, one has to run the simulation and construct the histogram of E.  If the successive histograms have significant overlap, then the chosen parameters can be accepted, otherwise, trial should continue till the significant overlap of the histograms is achieved. Similar strategy for Φ₀ could be followed.

9.4.1. Parameter Φ

Parameter Φ is usually the reaction coordinate (RC). RC is such a parameter, which is specific to a specific system and capable of characterizing a system completely. An order parameter can be a very good reaction coordinate. The value of the order parameter is close to 0 in a disorder state and close to 1 in order state. In some cases, especially in polymer, the angle of rotation (dihedral angle) is considered as a good reaction coordinate.

To illustrate this concept, let us look at the energy profile diagram of n -butane with respect to the rotation across C₂ - C₃ bond in the following figure including its stereo-chemical representation. The total potential energy depends on the relative position of two CH₃ group. It possesses low energy states as the distance between two CH₃ group increases. Here Φ is the RC.

For every system, we have to find out the correct parameter for RC to carry out simulation.

Reweighting: Since, in this method, we are applying a biased potential in the form of E^u, all the thermodynamic averaged quantities need to be reweighted to get the correct value.