Common Eigenvalues

Hiickel molecular orbital calculations, of course, today are mostly of historical and educational significance, but they should not be completely dismissed as not being any more relevant in chemistry. HMO continues to be of some interest in a few specific problems, one of them being the packing interaction of folded model protein, outlined in the previous section. The characteristic polynomial and the eigenvalues of the characteristic polynomial thus continue to be of some interest as a tool for comparative study of structurally related systems. 

In comparison to the golden ration cp, the nnmber jt, if written so that aU digits appear at least once, is 

MRA is not the only area of chemical science in which numerology may be detected. Theoretical chemistry, and in particular quantum chemical calculations, have also been vulnerable in this respect. As Hoffinann et al. pointed out in an appeal, Predicting Molecules— More Realism, Please [29], people often report results of calculations on an unnecessarily larger number of digits than are warranted. This apparently applies to Gaussian ab initio MO calculations [30] and even more so to density functional theory (DFT) [31]. 

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Such transformations superficially change the individual elements of the matrix, but they leave invariant certain deep features of the matrix (the eigenvalues ) that are its most important qualities. Matrices A, A that are related by (S9.1-14) (and thus share common eigenvalues) are indeed fundamentally similar, in a sense to be described below. 

The fact that certain numbers frequently occur among the eigenvalues of benzenoid systems was observed a long time ago. Hall [28] proposed an explaination of such regularities based on symmetry-relations. It should be noted, however, that such common eigenvalues are found in both symmetric and non-symmetric... 

The article [36] illustrates how to use the basis of spherical harmonics to evaluate the common eigenvalues and eigenfunctions of the operators in Equations (49) and (50). On the other hand, the article [37] uses the complementary choices of... 

For additional illustrations of subspectral conjugated molecular systems, see a paper by Dias [40] on a series of molecular graphs having a preponderance of common eigenvalues in which, for a collection of subspectral structures, their eigenvalues are tabulated. 

Here W is an m x m matrix whose columns are the eigenvectors of the Hermitian matrix MM, while U is an n x n matrix constructed similarly using M the n x n matrix D is diagonal, its elements being the square roots of the common eigenvalues of MM and M M. The weaker form of... 

Another step that is common to most, if not all, approaches that compute orbitals of one fomi or anotiier is tlie solution of matrix eigenvalue problems of the fomi... 

The BCUT descriptors (Buiden-CAS-University of Texas eigenvalues) [52], are commonly used, are eigenvalue-based, and include 3D information also. 

As far as the operators A[ and A 2 are commuting, other operators Ai, A2, A and T possess common systems of eigenfunctions. By utilizing this fact we denote by A. (T ) the eigenvalues of the operator T and take into account that... 

Fig. 31.13. Schematic example of three common algorithms for singular value and eigenvalue decomposition.

A little reflection shows that the commutation relationships, recognized as one of the fundamental differences between classical and quantum systems, are common to all forms of angular momentum, including orbital, polarization and spin. It is of interest to note that the eigenvalues for all forms of angular momentum can be obtained directly from the commutation rules, without using special differential operators. To emphasize the commonality, angular momentum M of all forms will be represented here by three linear operators Mx, My and Mz, that obey the commutation rules ... 

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