Molecular Dynamics in the Canonical Ensemble

Notice that if s(t) = 1, then this extended Lagrangian reduces exactly to Eq. (9.11). Formally, we can determine the equations of motion associated with this extended Lagrangian using Eq. (9.12). These equations were written by Hoover in a convenient form using slightly different variables than the extended Lagrangian above  

These equations are not as hard to understand as they might appear at first glance. The first and second equations are the same as we saw in Eq. (9.5) apart from the addition of a friction term in the second equation that either increases or decreases the velocity, depending on the sign of The third equation controls the sign and magnitude of The meaning of this equation is clearer if we rewrite Eq. (9.7) as 

This equation acts as a feedback control to hold the instantaneous temperature of the atoms in the simulation, 7md close to the desired temperature, T. If the instantaneous temperature is too high, is smoothly adjusted by Eq. (9.15), leading to an adjustment in the velocities of all atoms that reduces their average kinetic energy. The parameter Q determines how rapidly the feedback between the temperature difference rMD — T is applied to 

By applying the Taylor expansion as we did in Eq. (9.8), it is possible to derive an extension of the Verlet algorithm that allows these equations to be integrated numerically. This approach to controlling the temperature is known as the Nose-Hoover thermostat 

There are two aspects of classical MD calculations that are important to understand in order to grasp what kinds of problems can be treated with these 

---

Because Newton s equations of motion are conservative, the natural ensemble is NVE (micro-canonical), Aat is one in which the internal energy rather than the temperature is held constant. This is inconvenient if one wishes to compare with experiment where it is the temperature that is generally controlled. In order to perform molecular dynamics in the canonical ensemble, a thermostat must be applied to the system. This is accomplished by constructing a pseudo-Lagrangian. Many forms for temperature-conserving Lagrangians have been proposed, most of which can be written in a form that adds a frictional (velocity-dependent) term to the equations of motion (Allen and Tildesley 1989). Physically, the thermostat can be thought of as a heat bath to which the system is coupled. In the NPT ensemble, in which the pressure is held constant, the cell size and shape fluctuates. The choice of dynamical variables is critical. If the lattice parameters are chosen as in the method of Parrinello and Rahman (1981), the time evolution may depend on the chosen size or shape of the supercell. This difficulty is... 

---