Self-orthogonality

In Eqs. (3-5) above we note that a occurs in the bra-position in order to produce matrix elements that are analytic in the parameter a. Eq. (4), however, displays the hope or wish that the matrix can be diagonalised and self-orthogonality avoided. 

In these extended applications, the possible occurrence of degeneracies, with Segre characteristics larger than one, could result in an unwelcome calculational breakdown - usually considered as a numerical accident brought forward by the self-orthogonality of the transformed vectors. The objective of a successful computation was to stay away from them. 

We will now give a more detailed description of the electron trapping phenomena and show that it is associated with the self-orthogonality phenomenon. It is known that the resonance of H2 which is responsible for the vibrational excitation in the low energy region, is an extremely short-lived resonance with a lifetime comparable with the duration of the non resonant scattering [17]. 

Laibinis P E, Flickman J J, Wrighton M S and Whitesides G M 1989 Orthogonal self-assembled monolayers— alkanethiols on gold and alkane carboxylic-acids on alumina Science 245 845-7... 

Martin B R efa/1999 Orthogonal self-assembly on colloidal gold-platinum nanorods Adv. Mater. 11 1021... 

Let the mid-surface of the Kirchhoff-Love plate occupy a domain flc = fl Tc, where C is a bounded domain with the smooth boundary T, and Tc is the smooth curve without self-intersections recumbent in fl (see Fig.3.4). The mid-surface of the plate is in the plane z = 0. Coordinate system (xi,X2,z) is assumed to be Descartes and orthogonal, x = xi,X2)-... 

In a general case parameters re, XdP and y must be determined by self-consistent two-parameter fitting. Owing to the property of orthogonality of Laguerre polynomials, one has for the spectral band shapes... 

A self-adjoint operator A in the space Rn possesses n mutually orthogonal eigenvectors, ... We assume that all the j. s are normalized, that is, Mi II = 1 for k = I,..., n. Then ( j, i,) = The corresponding eigenvalues are ordered with respect to absolute values ... 

If a linear operator A given on Rn possesses n mutually orthogonal eigenvalues, then A is a self-adjoint operator A = A. ... 

Some micro- and mesoporous materials exhibit anisotropic pore structures, which may yield different values for the diffusivities in the three orthogonal spatial directions. In such systems, the self-diffusion should be described by a diffusion tensor rather than by a single scalar self-diffusion coefficient. By measuring over a... 

For the conduction electrons, it is reasonable to consider that the inner-shell electrons are all localized on individual nuclei, in wave functions very much like those they occupy in the free atoms. The potential V should then include the potential due to the positively charged ions, each consisting of a nucleus plus filled inner shells of electrons, and the self-consistent potential (coulomb plus exchange) of the conduction electrons. However, the potential of an ion core must include the effect of exchange or antisymmetry with the inner-shell or core electrons, which means that the conduction-band wave functions must be orthogonal to the core-electron wave functions. This is the basis of the orthogonalized-plane-wave method, which has been successfully used to calculate band structures for many metals.41... 

Compared to US and its subsequent variants, the ABF method obviates the a priori knowledge of the free energy surface. As a result, exploration of is only driven by the self-diffusion properties of the system. It should be clearly understood, however, that while the ABF helps progression along the order parameter, the method s efficiency depends on the (possibly slow) relaxation of the collective degrees of freedom orthogonal to . This explains the considerable simulation time required to model the dimerization of the transmembrane domain of glycophorin A in a simplified membrane [54],... 

In order to construct localized orbitals for molecules, it is necessary to define a measure for the degree of localization of an arbitrary set of molecular orbitals. The localized orbitals are then defined as that set of orthogonal molecular orbitals obtained by a transformation of the type given in Eq. (5), for which the measure of localization has the maximum value. It is clear that the resulting localized orbitals will depend, at least to some degree, upon the choice of the localization measure. In the present work the localized molecular orbitals are defined as those self-consistent-field orbitals which maximize the localization sum 14)... 

For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

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