Hypersurfaces eigenvalues

Comparing the PE spectroscopic ionization pattern with the results of MNDO hypersurface calculations, at a first glance, satisfactory agreement is observed. The MNDO eigenvalue sequence obtained for the total energy minimum predicted at to = 86° corre-... 

From this hypersurface survey, incomplete with respect to approxir mate barriers for kinetic aspects e.g. concerning the ready polymerization of thiocarbonyl derivatives (19g), we tentatively concluded that the PE spectroscopic detection of thioacroleine (18) should be feasible, that it would be less stable than its oxygen analogue, acroleine, and that it could be identified by its PE spectroscopic ionization pattern because of the considerable differences in MNDO eigenvalue sequences to all nearby C4H3S isomers (Figure 2). 

A point K of M where the gradient of E(K) vanishes [where the tangent hyperplane to E(K) is "horizontal"], is a point where the force of deformation is zero, i.e., point K represents an equilibrium configuration. Such a point is called a critical point, and is denoted by K(A,i). Here, the first derivatives being zero, the second partial derivatives of the energy hypersurface are used to characterize the critical points. The first quantity in the parentheses, X, is the critical point index (and not the "order of critical point" as it is sometimes incorrectly called). The index A, of a critical point is defined as the number of negative eigenvalues of the Hessian matrix H(K(A,i)), defined by the elements... 

According to MP2/6-31G(d,p) calculations, the He-B distance in HeBBHe is rather short at 1.270 A, and may be compared with the standard value for a B-H bond (1.21 A). Inspection of the diagonalized force-constant matrix showed one degenerate negative eigenvalue for the linear HeBBHe molecule at all levels of theory. Geometry optimization without linear constraints resulted in dissociation. This means that HeBBHe is not a minimum on the respective potential energy hypersurface [11]. 

It is remarked that the use of internal coordinate models is also of interest for normal mode calculations since a reduced number of variables simplifies the eigenvalue problem and also eliminates the high frequency movements associated with bond stretching, which are only weakly coupled to the low frequency collective modes. It has also been demonstrated that the harmonicity of the energy hypersurface can be assumed over a wider range in dihedral angle space than in Cartesian space. This approach, however, requires a special treatment to exclude any motion of the center of mass of the system. ... 

In a closed system, if the simulation is started from an arbitrary point in concentration space, it will finally end up at the equilibrium point, whilst the values of conserved variables remain constant. The equilibrium point is determined by the conserved properties, which are defined by the initial state of the system. If in an isothermal system there are Ng species and Nc conserved properties, then the trajectory of the system will move on a hypersurface with dimension Ns Nc- As time elapses, active modes will collapse, with the fastest mode relating to the largest negative eigenvalue relaxing first. The trajectory then approaches a hypersurface with dimension 1. The relaxation will be approximately according to an... 

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