Sampling saddle points using Gentlest Ascent Dynamics (GAD)

The method GAD MD-GAD Reference

We now extend GAD to perform finite temperature simulations using GAD. The underlying assumption is that the system evolves at sufficiently low temperatures such that the PES remain smooth. This is important for proper evaluation of the Hessian. The dynamical equations are :
\( \bf \dot{x} = v \)
\( \bf \dot{v} = F - 2\left(F, n\right)n \)
\( \bf \dot{n} = -Hn + \left(n, Hn\right)n \)
The stable fixed points of the above dynamical systems are index-1 saddle points of the original energy surface. The idea of MD-GAD is to explore the energy surface for saddle points and locally stable points. In contrast to ususal molecular dynamics simulations in which the system spends significant amount of time trapped in energy wells, in MD-GAD the system can easily move out of energy wells.

Also, as in GAD, one can even add a small parameter (\(\gamma\le 1\)) to control the timescales of the evolution of the position, velocity and direction vectors :
\( \bf \dot{x} = v \)
\( \bf \dot{v} = F - 2\left(F, n\right)n \)
\( \bf \gamma\dot{n} = -Hn + \left(n, Hn\right)n \)


The information obtained from such simulatons can be used to tune existing mean-field models which are generally based on information of locally stable states.
  • A. Illustrative examples: 2D potential (matlab code)

    Potential : \(V\left(x, y\right) = \sin\left(\pi x\right)\sin\left(\pi y\right)\) Simulations are performed with a randomly initialized direction vector.
    This simple example illustrates the evolution of a system starting from an initial local minimum. The simulations are performed with zero initial velocity. As the system moves out of the local energy well and moves towards a saddle point it gains kinetic energy. Once at the saddle point, it is this high kinetic energy that pushes the system towards the next local minimum. Once the system reaches the next local minimum the kinetic energy again goes to zero. Again, as the system moves towards the saddle point the same process is repeated. Thus in this way the system can easily explore the PES.

    This behavior of MD-GAD is quite different from standard molecular dynamics simulations where the system tends to get trapped at locally stable states and only rarely jumps over a saddle state to another local minimum.