MTD Algorithm

Once the CV have bee chosen, the MTD method can be applied in two different fashions.
Direct MTD: The simplest approach is to define the time dependent potential as function of $\mathbf{S}$ , $V(t,\mathbf{S})$ , and apply it directly onto the involved degrees of freedom. In this case, the equations of motion of the dynamic variables of the system, $\mathbf{R},\mathbf{\phi},\mathbf{h}$ , will include an additional term in the total forces, due to the contribution of $V(t,\mathbf{S})$ . The disadvantage of this simplified version is that there is scarce control on the dynamics in the space defined by the CV (CV-space), which is a projection of the space of all the possible configurations. In general, we would like to span thoroughly the CV-space, and to acquire information about the underlying potential. Often, this means that we need a slow dynamics in this space, where, for each set of values of the CV, we allow the system to equilibrate and to choose the configuration with the highest probability. Only in this way we will be able to construct a reasonable probability distribution in the configurational space that has been explored and consequently we will be able to reproduce the Free Energy surface.
Lagrangian MTD: This formulation is based on the method of the extended Lagrangian. In addition to the dynamic variables that characterize your system, a new set of variables $\mathbf{s}=\{s_{\alpha}\}$ is introduced. Each $s_{\alpha}$ is associated to one of the selected $S_{\alpha}$ , it has a fictitious mass $M_{\alpha}$ and velocity $\dot{s}_{\alpha}$ . The equations of motion for the $s_{\alpha}$ variables are derived by a properly extended Lagrangian, where we add the fictitious kinetic energy and the potential energy as a function of $\mathbf{s}$ . Therefore the total potential energy includes two new terms, a sum of harmonic potentials, which couple the $s_\alpha$ to the respective $S_{\alpha}(\mathbf{R},\mathbf{\phi},\mathbf{h})$ , $\sum_{\alpha}k_{\alpha}(S_{\alpha}(\cdots)-s_{\alpha})^{2}$ , and the time dependent potential, which now is a function of $\mathbf{s}$ , $V(t,\mathbf{s})$ . The coupling constants $\{k_{\alpha}\}$ and the fictitious masses $\{M_{\alpha}\}$ are the parameters that determine the dynamics of the $\{s_{\alpha}\}$ in the CV-space. Please notice that the units of $k$ are Hartree divided by the square power of u.s., the characteristic units of the specific CV (if CV is a distance it will be $a.u.^2$ , if an angle $radiants^2$ , etc.). In analogy, the units of the fictitious mass are $Hartree ((t)/(u.s.))^2$ , where $t$ indicates the unit of time. Some guide lines on the choice of these parameters will be given in the following paragraphs. By choosing the temperature $T_\mathbf{s}$ , the velocities of the components of $\bf s$ can be initialized giving via a Boltzmann distribution. Moreover, the velocities can be kept in a desired range by the activation of a temperature control algorithm (at the moment only the rescaling of velocity is implemented).