Hessians potential energy surfaces

In such a case the last choice is to take the direction of the eigenvector of the only one nonzero eigenvalue of the rank one Hessian matrix of the difference between the two adiabatic potential energies [51]. In the vicinity of conical intersection, the topology of the potential energy surface can be described by the diadiabatic Hamiltonian in the form... 

Figure 6. Initial rovibrational state specified reaction probabilities. Solid line exact quantum mechanical numerical solution. Solid line with solid square generalized TSH with use of the nonadiabatic coupling vector. Solid line with open circle generalized TSH with use of Hessian. Sur= 1(2) means the ground (excited) potential energy surface. Taken from Ref. [51].

Second, every compound with a Group 14-Group 16 element double bond corresponds to a minimum on the potential energy surface, as confirmed from all positive eigenvalues of the Hessian matrix. This suggests that all the double bond compounds in Table I are synthetically accessible, if one can find an appropriate synthetic methodology. 

Local Minimum. Any Stationary Point on a Potential Energy Surface for which all elements in the diagonal representation of the Hessian are positive. 

The Hessian and gradient can also be used to trace out streambeds connecting local minima to transition states. In doing so, one utilizes a local harmonic description of the potential energy surface... 

The strategies for saddle point optimizations are different for electronic wave functions and for potential energy surfaces. First, in electronic structure calculations we are interested in saddle points of any order (although the first-order saddle points are the most important) whereas in surface studies we are interested in first-order saddle points only since these represent transition states. Second, the number of variables in electronic structure calculations is usually very large so that it is impossible to diagonalize the Hessian explicitly. In contrast, in surface studies the number of variables is usually quite small and we may easily trans-... 

Density functional theory (DFT) calculations have been carried out to elucidate the structure and energetics of the various isomeric (Si-N) rings as well as the corresponding anions and dianions. The local minima on the potential energy surfaces were verified by computation of the eigenvalues of the respective Hessian matrices. From these, harmonic vibrational frequencies and the zero-point vibration corrected energetics were calculated. The calculations were carried out for isolated molecules in the gas phase. The theoretical results are expected to be reliable for molecules in non-polar or weakly aprotic polar solvents. 

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.)... 

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

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