Eigenvalues bound motion

If we are only interested in the frequency of the modulations in the vicinity of the zero field limit we may employ a different approach, used by Freeman et al,6,7 and Rau10. They used the fact that the motion in the direction is bound and found the energy separation between successive eigenvalues. Specifically, they used Eq. (8.8), the WKB quantization condition for the bound motion in the direction, and differentiated it to find the energy spacing between states of adjacent n1 or, equivalently, between the oscillations observed in the cross sections. Differentiating Eq. (8.8) with respect to energy yields... 

The eigenvectors of the Hessian matrix, H, correspond to the normal modes. Eigenvectors with positive eigenvalues indicate bound motions of the nuclei eigenvalues with negative eigenvalues indicate unbound dissociative motions. 

The GA and GT combinations are somewhat less strongly bound. It was noted by the authors that these structures are not true minima. The single negative eigenvalue probably refers to an out-of-plane motion of a side group (the authors enforced full planarity), and so is probably not a serious concern to this discussion. These two dimers have in common the presence of a pair of H-bonds nitrogen serves as the proton donor in all of these. 

The first situation we wish to consider concerns the error bounds in the spectral density of a pure dissipation operator T, whose eigenvalues y = ( in 0) lie on the negative imaginary axis. Consider the equation of motion of the type... 

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