Potential hypersurface

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The potential hypersurface (outdated potential hypersurface ) is the hypersurface that describes the potential energy of a quantum mechanical system of atoms depending on the geometry ( bond lengths , possibly also bond and torsion angles ). The potential energy surface of an n -atomic system (with n ≥3) is 3 n −6-dimensional and is plotted in a 3 n −5-dimensional space. In the limiting case of a diatomic system, the potential energy surface is called a potential curve and is a one-dimensional curve in a two-dimensional coordinate system.

The term potential surface is only mathematically correct in the borderline case of a linear three-atom system (e.g. collinear approximation of an atom to a diatomic molecule), but is often used synonymously based on the English term potential energy surface .

Dimensions

In three-dimensional space, each atom has 3 degrees of freedom of movement ( translation in x, y and z directions). To describe a system of n atoms, 3 n coordinates are therefore required . These can be the coordinates of the respective atoms. More favorable, however, is the generalized representation in the form of degrees of freedom of movement of the system center of gravity , degrees of freedom of rotation of the overall system around the system center of gravity and degrees of freedom of oscillation. Since the potential energy of the system does not change when the center of gravity is moved or rotated around the center of gravity, only the degrees of freedom of the oscillation have to be taken into account in the context of the potential hypersurface. There are 3 degrees of freedom for the translation of the entire system, 2 degrees of freedom for the rotation around the center of mass for linear systems and 3 for non-linear systems. Since the potential hypersurface is supposed to represent the general geometry, only diatomic ones are necessarily linear, for n ≥3 they are not applicable always 3 degrees of freedom on the rotation. For a system of n coordinates, there remain 3 n −6 degrees of freedom for the oscillation (3 · 2−5 in the case of a 2-atom system), depending on which the potential energy must be represented. The potential energy surface of an n -atomic system is thus 3 n −6-dimensional. The potential energy is applied in an additional dimension so that the representation takes place in a 3 n −5 dimensional space (better: hyperspace ).

application

The potential energy surface of a system allows predictions about the course of elementary chemical reactions . The following reaction is considered as an example:

AB + C → A + BC

The potential energy surface of the system of atoms A, B and C describes (among other things) the bond geometry of the ABC molecule in the course of the reaction. The initial state ( reactants ) corresponds to the borderline case of an AB molecule with a carbon atom that is infinitely distant. The final state ( products ) corresponds to the borderline case of a BC molecule with an A atom infinitely distant. During the reaction, various states are passed through, with C approaching B and in the sequence of which the binding interaction between B and C increases. In return, the bond between A and B becomes longer and therefore weaker. The reaction is over when A has moved very far from B (the AB bond has been broken) and the distance between B and C corresponds to the bond length in the BC molecule.

Trajectory and reaction coordinate

Of particular interest is the ratio in which the bond lengths (and angles) change during the reaction. For statistical reasons that reaction path (the trajectory) on the potential hypersurface is preferred which is given by the energy gradient between the initial and final state. With increasing energy of the system ( kinetic energies of the atoms relative to the system center of gravity ), however, more deviating reaction paths are also possible. The description of the reaction using only this one reaction path is therefore an approximation, which is necessary in many cases, since there are an infinite number of reaction paths for every reaction, which only deviate infinitesimally from the most probable and are also only infinitesimally less probable .

The point on the trajectory with the highest energy corresponds to the transition state of the reaction. A reaction can only take place if the kinetic energy of the system in the initial state is greater than the activation energy (the difference in potential energies between the transition state and the initial state).

If the trajectory of a chemical reaction is known, it is sufficient to describe a point on the potential energy surface in the course of this reaction by specifying the position along the trajectory, which is referred to as the reaction coordinate. The reaction coordinate, since it is only one-dimensional, can be plotted against another variable without any problems. Energy diagrams are therefore widespread, in which the energy of a chemical system is plotted depending on the reaction coordinate and which are much easier to represent and interpret than the representations of the potential energy surface itself.

calculation

To calculate the potential energy surface, the Schrödinger equation is solved for different bond geometries. For a given bond geometry there are several solutions, of which the energetically lowest corresponds to the ground state . For many reactions only this is relevant. For photochemical reactions and reactions that are carried out at high temperatures, on the other hand, the energetically higher-lying excited states can also be important.

Since the potential energy surface does not contain a time scale, the energy of a molecule in the electronic state of equilibrium is inevitably described. This corresponds to the approximation that the electrons react very quickly to the changing arrangement of the atomic nuclei ( Born-Oppenheimer approximation ). In addition, the oscillation and rotation states of the system are often neglected.

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