Internuclear coordinates

This striking result can be qualitatively understood as related to CB DOS-influenced changes in the 02 anion lifetime [118]. For a diatomic molecule with R as the internuclear coordinate, a transient anion state is described in the fixed nuclei limit [123,124] by an energy and i -dependent complex potential Vo i R,E ) = Fd(2 ) + A( i)—l/2 T( i), where Va R) = a R) + is the potential energy curve of the discrete state, Vg(R) is the... 

The solution to Eq. (1.6) for the electronic energy in, e.g., a diatomic molecule is well known. In this case there is only one internuclear coordinate and the electronic energy, Ei(R), is consequently represented by a curve as a function of the internuclear distance. For small displacements around the equilibrium bond length R = Ro, the curve can be represented by a quadratic function. Thus, when we expand to second order around a minimum at R = Ro,... 

First, let us consider what is meant by internuclear coordinates and, in particular, how many of these coordinates are needed in order to specify the electronic energy. We consider a collection of N atomic nuclei, which in this context are considered as point particles. In the following, we will for convenience refer to any collection of nuclei and electrons as a molecule . The atomic nuclei and the electrons may form one or more stable molecules but this is of no relevance to the following argument. The internuclear coordinates are defined as coordinates that are invariant to overall translation and rotation. These coordinates can, for example, be chosen as internuclear distances and bond angles. 

N coordinates are needed in order to completely specify the position of the nuclei. Three coordinates are needed in order to specify the position of the center of mass.1 Thus, 3N — 3 coordinates account for the internal degrees of freedom, that is, overall orientation and internuclear coordinates. The overall orientation can be specified by two coordinates for a linear molecule, say by the two polar angles (6, ). For a nonlinear molecule three coordinates are needed in order to specify the orientation. These coordinates are often chosen as the so-called Euler angles. Thus, for a molecule with N atomic nuclei,... 

Note that in a non-linear molecule, one of the vibrational modes of the linear molecule has been replaced by a rotational coordinate. As an illustration, let us consider two examples. For the stable linear triatomic molecule CO2, there are 3 x 3 — 5 = 4 internuclear coordinates, which corresponds to the vibrational degrees of freedom, namely the symmetric and antisymmetric stretch and two (degenerate) bending modes (see Appendix E). For the three atoms in the reaction D + H — H—> D — H + H, there are 3 x 3 — 6 = 3 internuclear coordinates. These coordinates can, for example, be chosen as a D H distance, the H H distance, and the I) II H angle. 

Fig. 3.1.5 A potential energy surface for a direct unimolecular reaction without a saddle point. The surface corresponds to a reaction like H2O — H + OH for dissociation along a fixed bond angle, where only two internuclear coordinates are required in order to specify the configuration. (Note that in this figure all energies above a fixed cut-off value Amax have been replaced by Fmax.)...

For applications to atomic systems, this approach is apparently working very well, whereas the applications to molecular systems still have a long way to go, depending on the fact that we are not yet familiar with the handling of the complex scaling of internuclear coordinates. 

This is because the electric dipole operators, ftp, act only on the electronic coordinates, the contribution to the transition moment from the nuclear coordinates being negligible. The resulting matrix elements, 9ge(Q), are still parametrically dependent on Q through the electronic states. This dependence is supposed to be a weak function of the internuclear coordinates and is therefore described by a rapidly converging Taylor series expanded about the equilibrium configuration, Q = 0, of the ground electronic state ... 

Potential energy surface Schematic two- or three-dimensional representation of the total potential energy of a chemical system as a function of internuclear coordinates. 

So far, we have been concerned with properties associated with the electronic structure of a molecule at a fixed geometrical structure. Within the Born-Oppenheimer approximation these properties depend on internuclear coordinates in a parametric fashion. That is, there exist property surfaces just as there are potential energy surfaces, and a recent review by Amos has focused on this issue. Because of this variation, vibrational and rotational motion play a role in determining the overall properties of a molecule. 

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