Dear Reza,

An example of molecules containing nitrogen is a reaction between alkyl halides and amines:

R-X + NH2-R' = R-NH-R' + HX ; where X is halide and R and R' are alkyl groups

Minima: Reactants (R-X + NH2-R', R-NH-R' and HX)

SN2 (transition state), Maximum: [R'-NH2-----R-----X]

Stationary points of PES

There are a number of minima (Reactant & Product) on this potential surface. A minimum is the bottom of a valley on the potential surface. From such a point, motion in any direction-a physical metaphor corresponding to changing the structure slightly-leads to a higher energy.

A minimum can be a local minimum, meaning that it is the lowest point in some limited region of the potential surface, or it can be the global minimum, the lowest energy point anywhere on the potential surface. Minima occur at equilibrium structures for the system, with different minima corresponding to different conformations or structural isomers in the case of single molecules, or reactant and product molecules in the case of multi-component systems. Peaks and ridges correspond to maxima on the potential energy surface. A peak is a maximum in

all directions (i.e., both along and across the ridge). A low point along a ridge - a mountain pass in our topographical metaphor - is a local minimum in one direction (along the ridge), and a maximum in the other. A point which is a maximum in one direction and a minimum in the other (1st order saddle point) (or in all others in the case of a larger dimensional potential surface) is called a saddle point (based on its shape). For example, the saddle point in the diagram is a minimum along its ridge and a maximum along the path connecting minima on either side of the ridge. A saddle point corresponds to a transition structure connecting the two equilibrium structures. The slice along the “lowest-energy” path connecting the global minimum, the local minimum and the transition state, that is, along the “reaction coordinate” gives a one-dimensional energy surface. Here the transition state (saddle point) looks like a maximum. Note that the horizontal axis (i.e. the reaction coordinate) is a combination of the two coordinates (bond length and angle) from the original plot.

Together, minima, maxima and saddle points are called stationary points. They have in common that at any stationary point (minimum, maximum) the surface is flat, i.e. parallel to the horizontal line corresponding to the one geometric parameter (or to the plane corresponding to two geometric parameters etc). A marble placed on a stationary point will remain balanced, i.e. stationary, while at any other point on a potential surface the marble will roll toward a region of lower potential energy.

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Hoping this will be helpful,

Rafik

Hi Reza!

If you mean a general molecule with N denoting the number of atoms in it, and you have an atomic potential (or a force field, if you like) that gives you the total cohesive energy E=E({xia}) of the molecule as a function of the atomic coordinates xia, where the subscript i is for the three coordinate directions and the subscript a indexes the N atoms, then a stationary point (i.e.,  a critical point) is one for which the first partial derivative of E with respect to xia is equal to zero for all i and a. The stationary point is a energy minimum if the corresponding (3N-by-3N) Hessian matrix is positive definite, and energy maximum if it is negative definite. If the Hessian is neither definite or semidefinite, the stationary point is a saddle point. If the Hessian is semidefinite (but not definite), further investigation is required to find the nature of the stationary point.

Thank you so much for your answering