Normal matrix eigenvalue properties

Fitted parameters are often strongly correlated. This can be a consequence of the structure of the data set or it can be an intrinsic property of the Heff model (for example, 7 and Ad in a 2II state, Brown, et al., 1979). The former effect can be minimized by supplementing the data set. A combined fit to optical and microwave data or to two electronic transitions sharing a common state (e.g., 2II — 2E+ and 2E+ — 2E+ systems) can be very effective. Another approach is to replace the two correlated parameters by two new parameters that are the sum and difference of the original parameters (e.g., B + B" and B — B"). The correlation matrix [Eq. (4.4.15)] and the magnitudes of the eigenvalues of the normal matrix [Eq. (4.4.11)] provide useful insights (Albritton, et al, 1976 Curl, 1970). Isotope relationships, computed D -values, and other semiempirical constraints are frequently used to minimize both data set and intrinsic correlation effects. 

EVA descriptors were recently proposed by Ferguson et al. [Ferguson et al, 1997 Turner et al., 1997] as a new approach to extract chemical structural information from mid- and near-infrared spectra. The approach is to use, as a multivariate descriptor, the vibrational frequencies of a molecule, a fundamental molecular property characterized reliably and easily from the potential energy function. The EigenVAIue (EVA) descriptor is a function of the eigenvalues obtained from the normal coordinate matrix they correspond to the fundamental vibrational frequencies of the molecule, which can be calculated using standard quantum or molecular mechanical methods from -> computational chemistry. 

A normal mode calculation is based upon the assumption that the energy surface is quadratic in the vicinity of the energy minimum (the harmonic approximation). Deviations from the harmonic model can require corrections to calculated thermodynamic properties. One way to estimate anharmonic corrections is to calculate a force constant matrix using the atomic motions obtained from a molecular d)namics simulation such simulations are not restricted to movements on a harmonic energy surface. The eigenvalues and eigenvectors are then calculated for this quasi-harmonic force-constant matrix in the normal way, giving a model which implicitly incorporates the anharmonic effects. 

Obtained in this way, a will share with p the necessary properties required of a physically meaningful density matrix, namely, hermiticity (a- = o- ), normalization [Tr(o-) = l], and positivity. Positivity is the property that the eigenvalues of a- are nonnegative this is necessary since the diagonal matrix elements of [Pg.82]

These reduced matrices represent the properties of the superelement as seen at its interface with adjacent structures. In case of mass normalised normal modes, the matrices [Kqq] and [Mqq simplify to respectively a diagonal matrix with the eigenvalues and a unity matrix ... 

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