# Stationary Points of U(x,y) | Locate Minima, Maxima & TS

In summary, the conversation is about a potential function U(x,y) with variables A, B, Lx, and Ly that has minima, maxima, and transition states. The periodicity of U(x,y) along the two directions is discussed and the TS barriers for an atom diffusing on the surface are derived as a function of A and B. The shape of U(x,y) is also considered in proposing possible analytical curves that separate the basins of attraction of adjacent minima. The conversation also mentions the need to show workings as per forum rules.
U(x,y)=Asin(pi*x/Lx) + Bcos(pi*y/Ly) where A,B,Lx,Ly are positive

1 Locate all the stationary points (i.e. minima, maxima and transition states
(TS)) for this potential.

a. What is the periodicity of U(x,y) along the two directions?

b. Derive the TS barriers that an atom diffusing on the surface would
have to surmount during a “jump” between two minima as a function
of A and B.

c. Using information about the shape of U(x,y), propose possible
analytical curves separating the basin of attraction of adjacent minima.

Per the rules of the forum, you have to show some workings...

## 1. What is a stationary point?

A stationary point is a point on a graph where the slope is zero. This means that the function is neither increasing nor decreasing at that point.

## 2. What is the significance of stationary points in scientific research?

Stationary points are important in finding the minimum or maximum values of a function. In scientific research, these points can indicate the most stable or optimal conditions for a certain system or process.

## 3. How do you locate the minima and maxima of a function?

To locate the minima and maxima of a function, you can use the first and second derivative tests. The first derivative test involves finding the critical points of the function, where the derivative is equal to zero. The second derivative test involves evaluating the second derivative at these critical points to determine if they are minima or maxima.

## 4. What is a transition state and how is it related to stationary points?

A transition state is a point on a graph where the potential energy of a system is at a local maximum. It is related to stationary points because it is a type of critical point where the derivative is zero, but it is not a minima or maxima. Transition states are important in studying chemical reactions and other dynamic processes.

## 5. Can a function have multiple stationary points?

Yes, a function can have multiple stationary points. These could be minima, maxima, or transition states. The number and type of stationary points a function has can provide valuable information about the behavior of the function.

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