Hessian matrix eigenvalues

Point Stationary point ( i, 2) m Hessian matrix eigenvalues Classification... 

The local curvature properties of the surface G(m) in each point r of the surface are given by the eigenv ues of the local Hessian matrix. Moreover, for a defined reference curvature b, the number p,(r, b) is defined as the number of local canonical curvatures (Hessian matrix eigenvalues) that are less than b. Usually b is chosen equal to zero and therefore the number p(r, 0) can take values 0,1, or 2 indicating that at the point r the molecular surface is locally concave, saddle-type, or convex, respectively. The three disjoint subsets Ao, Ai, and A2 are the collections of the surface points at which the molecular surface is locally concave, saddle-type, or convex, respectively the maximum connected components of these subsets Ao, Aj, and A2 are the surface domains denoted by Do,, Diand D2, where the index k refers to an ordering of these domains, usually according to decreasing surface size. 

Tlie eigenvalues are A = 4 and A = 8. Thus both eigenvalues are positive and the point is minimum. At the point (0,0) the Hessian matrix is... 

The eigenvalues (coa of the mass weighted Hessian matrix (see below) are used to compute, for each of the 3N-7 vibrations with real and positive cOa values, a vibrational partition function that is combined to produce a transition-state vibrational partition function ... 

It is possible (see, for example, J. Nichols, H. E. Taylor, P. Schmidt, and J. Simons, J. Chem. Phys. 92, 340 (1990) and references therein) to remove from H the zero eigenvalues that correspond to rotation and translation and to thereby produce a Hessian matrix whose eigenvalues correspond only to internal motions of the system. After doing so, the number of negative eigenvalues of H can be used to characterize the nature of the... 

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... 

This equation determines a rank-1 matrix, and the eigenvector of its only one nonzero eigenvalue gives the direction dictated by the nonadiabatic couphng vector. In the general case, the Hamiltonian differs from Eq.(l), and the Hessian matrix has the form... 

The ratio of the largest to the smallest eigenvalue of the Hessian matrix at the minimum is defined as the condition number. For most algorithms the larger the condition number, the larger the limit in Equation 5.5 and the more difficult it is for the minimization to converge (Scales, 1985). 

According to Scales (1985) the best way to solve Equation 5.12b is by performing a Cholesky factorization of the Hessian matrix. One may also perform a Gauss-Jordan elimination method (Press et al., 1992). An excellent user-oriented presentation of solution methods is provided by Lawson and Hanson (1974). We prefer to perform an eigenvalue decomposition as discussed in Chapter 8. 

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. 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

Because the Hessian matrix is calculated in practice using finite-difference approximations, the eigenvalues corresponding to the translational and rotational modes we have just described are not exactly zero when calculated with DFT. The normal modes for a CO molecule with a finite difference displacements of 8b 0.04 A are listed in Table 5.2. This table lists the calculated... 

No first derivative terms appear here because the transition state is a critical point on the energy surface at the transition state all first derivatives are zero. This harmonic approximation to the energy surface can be analyzed as we did in Chapter 5 in terms of normal modes. This involves calculating the mass-weighted Hessian matrix defined by the second derivatives and finding the N eigenvalues of this matrix. 

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