Potential-energy surfaces zero order

Another use of frequency calculations is to determine the nature of a stationary point found by a geometry optimization. As we ve noted, geometry optimizations converge to a structure on the potential energy surface where the forces on the system are essentially zero. The final structure may correspond to a minimum on the potential energy surface, or it may represent a saddle point, which is a minimum with respect to some directions on the surface and a maximum in one or more others. First order saddle points—which are a maximum in exactly one direction and a minimum in all other orthogonal directions—correspond to transition state structures linking two minima. 

Maxima, minima and saddle points are stationary points on a potential energy surface characterized by a zero gradient. A (first-order) saddle point is a maximum along just one direction and in general this direction is not known in advance. It must therefore be determined during the course of the optimization. Numerous algorithms have been proposed, and I will finish this chapter by describing a few of the more popular ones. 

As briefly stated in the introduction, we may consider one-dimensional cross sections through the zero-order potential energy surfaces for the two spin states, cf. Fig. 9, in order to illustrate the spin interconversion process and the accompanying modification of molecular structure. The potential energy of the complex in the particular spin state is thus plotted as a function of the vibrational coordinate that is most active in the process, i.e., the metal-ligand bond distance, R. These potential curves may be taken to represent a suitable cross section of the metal 3N-6 dimensional potential energy hypersurface of the molecule. Each potential curve has a minimum corresponding to the stable... 

Let us consider the possible relations of LS and HS potential energy surfaces as shown schematically in Fig. 9. As long as the zero-order or diabatic surfaces are considered, the eleetrons remain localized on the particular spin state, no eleetron transfer being possible. In order that a conversion between the LS and HS state takes place, electronic coupling of the states is required. This coupling effectively removes the degeneracy at the interseetion of the zero-order surfaces... 

Fig. 15. Section of the zero-order (...) and first-order (—) potential energy surfaces along the reaction coordinate in cases where stretching of the cleaving bond is the dominant factor of nuclei reorganization.

Fig. 16 (a) R (D + RX) and P (D,+ + R + X ) zero-order potential energy surfaces. Rc and Pc are the caged systems, (b) Projection of the steepest descent paths on the X-Y plane J, transition state of the photoinduced reaction j, transition state of the ground state reaction W, point where the photoinduced reaction path crosses the intersection between the R and P zero-order surfaces R ., caged reactant system, (c) Oscillatory descent from W to J on the upper first-order potential energy surface obtained from the R and P zero-order surfaces. 

The minimum on the intersection parabola is the saddle point corresponding to the transition state of the dark reaction, denoted J in Figs 16b and 16c. The first-order potential energy surfaces involve an upper surface associating the portions of the R and P zero-order potential energy surfaces situated above the intersection parabola and a lower surface associating the portions of the R and P zero-order potential energy surfaces situated below the intersection parabola. 

It may not at first be obvious that the Jahn-Teller theorem applies to transition states (40). The proof rests on the fact that the matrix element of the distortion gives a first-order change in energy and hence is linear in Q. In other words there must be a non-zero slope in some direction and this is incompatible with the definition of a transition point as a saddle point on the potential energy surface. 

We have carried out DFT (B3LYP/6-31G(d)) calculations (the basis set comprises 312 cGTOs) in order to establish the energetic order of the different possible isomers of (Me2Si-NH)4, OMCTS (Fig. 15). At the local minima on the potential energy surfaces, the Hessian matrices were computed. Harmonic vibrational frequencies were used to calculate the zero-point vibration-corrected energetics. (Results are collected in Table I and Fig. 16.)... 

Fig. 9. Interaction between the zero-order potential energy surfaces in the intersection region. E+ and E are the first-order surfaces. HRp is the interaction energy...

Si and Su are the slopes of the zero-order potential energy surfaces at the intersection (Si = —Sn for an exchange reaction), and v is the velocity with which the point representing the system moves through the intersection region. For typical conditions it is found that p 1 for interactions EIfII of more than 0.5 kcal mol"1 (50). Under these conditions the reactions will be adiabatic, and the square root relation is expected to hold provided EitU is not too large. However, for small EltJ1 ... 

In chemical dynamics, one can distinguish two qualitatively different types of processes electron transfer and reactions involving bond rearrangement the latter involve heavy-particle (proton or heavier) motion in the formal reaction coordinate. The zero-order model for the electron transfer case is pre-organization of the nuclear coordinates (often predominantly the solvent nuclear coordinates) followed by pure electronic motion corresponding to a transition between diabatic electronic states. The zero-order model for the second type of process is transition state theory (or, preferably, variational transition state theory ) in the lowest adiabatic electronic state (i.e., on the lowest-energy Bom-Oppenheimer potential energy surface). 

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