Simulated annealing trajectory

Figure 7. Nuclear dynamics of the simplified model defined in Equation (5) in which the wave function parameters are determined by solution of the damped equation of motion (14). The parameters are chosen so that, for artistic reasons, the simulated annealing and exact Born-Oppenheimer trajectories are distinguishable to the reader. Further diminution of the parameter mass m and damping coefficient 7 would bring the simulated annealing trajectory arbitrarily close to the exact result.

Quenched dynamics is a combination of high temperature molecular dynamics and energy minimization. This process determines the energy distribution of con formational families produced during molecular dynamics trajectories. To provide a better estimate of conformations, you should combine quenched dynamics with simulated annealing. 

Restrained MD (rMD) is followed by the use of MD in explicit solvent, i.e. the conformation as determined above is taken into a box containing many solvent molecules around the molecule. Subsequently, simulated annealing (SA) and energy minimizahon steps are performed to draw the molecule into the global energy minimum. An MD run (the so-called trajectory) over at least 150ps to Ins is followed and a mean structure is calculated from such a trajectory. The con-formahon must be stable under this condihon even when the experimental constraints are removed. 

Figure 4. Nuclear trajectories generated by the simulated annealing molecular dynamics defined by the Lagrangian in Equation(4) using masses for all parameters equal to either 0.1 or 0.01 times the proton mass. For comparison, the exact nuclear trajectory on the Born-Oppenheimer surface is shown.

Figure 5. Simulated annealing nuclear dynamics under the holonomic constraint (6). The parameter masses are set to half the proton mass, chosen, for artistic reasons, so that some small deviation from exact Born-Oppenheimer dynamics would be visible to the reader. Further reduction of the parameter mass would lead to nuclear dynamics indistinguishable from the exact Born-Oppenheimer trajectory, also shown in the figure.

Figure 6. Trajectories of the wave function parameters corresponding to the simulated annealing nuclear trajectory shown in Figure 5.

Our trajectories are sampled with the help of a simulated annealing protocol. But how can we test that the sampling is appropriate One measure that can help us to assess the quality of the simulation is the distribution of the orientation of the initial momentum vector. If we sample effectively the space of initial conditions (by sampling complete trajectories), then the momentum vectors should cover all of the orientation space. Alternatively, the vectors of the initial direction of the momentum behave as random vectors (with norm of one). In... 

Figure 32 Trajectory of a simulated annealing experiment, which can end up in two different minima on the energy surface.

A simulated annealing protocol can be used to optimize the target function subjected to the overall translation and rotation constraints (Eq. [35]). We can denote the variable components of the initial guess for the trajectory Y° i and optimize the trajectory for K st s solving the second-order differential equation for the trajectory Z =... 

Figure 7.15 Partial trajectory of a simulated annealing run of a cost function with multiple minima.

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