Hessian matrix, 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... 

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. 

Once the geometries of reactants and products are defined, the transition state can be located. These are points on the potential energy surface that are characterized by one, and only one, negative eigenvalue of the second derivative (Hessian) matrix. Finding such points that determine the barriers to chemical reactions remains a complicated process, but there are now several powerful techniques available. Most of the more successful methods require... 

Hessian matrix A matrix of second-order partial derivatives that determine curvature used in calculations to test for minima on a potential energy surface. 

Scuseria and Schaefer [202a] followed up some earlier work by attempting to determine whether the open trimer of HF represents a true minimum in the potential energy surface. Within the context of their double-C basis set, it was found that the cis open trimer is indeed a minimum (Hessian matrix has all positive eigenvalues), whereas the trans structure is a transition state. However, the situation is decidedly different when polarization functions are added to the basis set wherein the trans geometry becomes a transition state and the cis type does not correspond to a stationary point of any order. Nor does relaxing the fully planar restraint lead to a cis stationary point. Vibrational frequencies and intensities were reported for the stationary points. 

Potential Energy Surfaces of Chemical Reactions matrix H referred to as the Hessian ... 

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