Orthogonality representations

Polyatomic Molecules in Spherical Polar Parameterization. I. Orthogonal Representations. 

Another method that may be used to generate the projection operator involves the use a matrix representation of the operator. In particular, we will use the orthogonal representation. First we must assign a Yamanouchi symbol to each tableau we have created. This is done by going through the numbers from 1 to n in each tableau and writing down in which row the number occurs. Thus if we assign names to the above tableaux ... 

The Hamiltonian of valence electrons (39), in the so-called orthogonal representation (or in the most localized representation, neglecting orbital overlap) can be mapped on a tight-binding form Hamiltonian... 

Figure 4. Orthogonal representation of ternary solvent systems ABC and BC D ...

Figure 7 Two orthogonal representations for each of the p-peptide 14-, 12-, and 12/10-helices (a, b, and c, respectively) formed by a AA or/and a AA, where R /R are methyl goups. The H-atoms, except the -CONH- H-atoms, are omitted for clarity.

M. Mladenovic, Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parameterization I. Orthogonal representations. J. Chem. Phys. 112, 1070-1081 (2000). 

We have shown that R is represented by an orthogonal matrix if we choose a unitary coordinate system. The matrices (2.7), (2.8) and (2.9) are examples of orthogonal matrices. Alternatively, we can choose the axes of our coordinate system to be a lattice base a, b, c, i.e. three primitive non-coplanar translations. The coordinates of the lattice points u, u, w are then integers and all the terms in the corresponding representation of R are thus also integers. Let us indicate the orthogonal representation by the matrix U, and the representation with integers by the matrix N. The matrices U and N are related by a similarity transformation because they represent the same operation. Hence, there must exist a matrix X such that N = X" UX. The matrix X transforms the coordinate system of the lattice to a unitary system. Moreover, U is similar to one of the matrices (2.7), (2.8) or (2.9). We know that similar matrices have the same trace. It thus follows that ... 

The wavenumber-dependent terms, A/4 (v) and AA" v), are known respectively as the in-phase spectrum and quadrature spectrum of the dynamic dichroism of the system. They represent the real (storage) and imaginary (loss) components of the time-dependent fluctuations of dichroism. Figure 1-6 shows an example of the in-phase and quadrature spectral pair extracted from the continuous time-resolved spectrum shown in Figure 1-5. These two ways of representing a DIRLD spectrum contain equivalent information about the reorientation dynamics of transition dipoles. However, the orthogonal representation of the time-resolved spectrum using the in-phase and quadrature spectra is obviously more compact and easier to interpret than the stacked-trace plot of the time-resolved spectrum. 

We have to note here that the properties of matrix P are different from those in an orthogonal representation. While in the latter case its trace gives the number of electrons, in a non-orthogonal basis we find ... 

In the late fifties, Eringen and his co-workers [1-3] have analyzed the responses of beams and plates to random loads. Since these pioneering works, response analysis of structures subjected to random excitations has attracted considerable attention in the past thirty years. An extensive review of the recent developments have been provided by Crandall and Zhu [4]. Most of the earlier studies on nonstationary random vibrations were concerned with the analysis of mean-square response statistics [5,6]. Recently, evaluation of the time-dependent power spectra of structural response has attracted considerable interest. Priestley [7] introduced the orthogonal representation of a random function. Hammond [5], Corotis and Vanmarcke [8] and To [9] have studied the time-dependent spectral content of responses of single- and multi-degree-of-freedom structures. 

The presented results show that the method of normal mode and orthogonal representations of random functions provides a useful technique for analyzing random responses of shear beam structures. The method provides, in addition to the mean-square response statistics, information concerning the time-dependent spectral content of the response. The presented examples also... 

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