## Potential energy surface As we are able to express the potential as a function of $r$, i.e., [[AMBER force field|force field]], we may generate a potential energy surface describes the energy of the system as a function of the coordinates. This is in principle known as soon as the interaction model is defined. >[!Notice] > When we see "geometry optimization", we typically mean "energy optimization". ![[Drawing 2023-09-25 20.03.08.excalidraw.svg]] Zero gradient points correspond to **product**, **reactant**, or **transition states**, (corresponds to max, min, saddle point), And in many occasions, there are of our interest. Exploring the landscape of such a system could be very difficult. Before entering specific techniques to determine the (local) minima and saddle points, let's first make some general discussion. ^d5abce ### Disconnectivity graphs One way to represent the transit state and minima for such landscape, the graph looks like below. ![[Drawing 2023-09-25 21.14.15.excalidraw.svg]] ### Reaching global minimum It could be difficult to reach global minimum of energy. Difference between local min are large. One way is so-called “Basin hopping method”. This method modified the landscape so that each basin or minimum gets constant energy. >[!Note] > Other ways could be using different Monte Carlo probability or like simulated temporary. >[!Notice] > This is different from the umbrella method. We get the following curve, so for later search one jump far to next minimum instead of in current one. > ![[Drawing 2023-09-25 21.23.28.excalidraw.svg]] ### Number of transit states and a number of local minima It can be shown that both number of transit states and number of minima follows an exponential relation with total number of particles. But their ratio is linear with N, i.e., $\frac{\text{\# of TS}}{ \text{\# of minima}} \propto N$ >[!Note] >$\text{\# of TS}$ is large, $\propto Ne^{\xi N}$, while $\text{\# of minima} \propto e^{\xi N}$. This gives the linear ratio. Here N is the particle number.