Eigenvalue equation structure

In order to solve the electronic structure problem for a single geometry, the energy should be minimized with respect to the coefficients (see Eq. (5)) subject to the orthogonality constraints. This leads to the eigenvalue equation ... 

The second complication is that the equation, as traditionally interpreted, only handles point particles, but produces eigenfunction solutions of more complex geometrical structure. By analogy with electromagnetic theory the square of the amplitude function could be interpreted as matter intensity, but this is at variance with the point-particle assumption. The standard way out is to assume that ip2 represents a probability density rather than intensity. Historical records show that this interpretation of particle density was introduced to serve as a compromise between the rival matrix and differential operator theories of quantum observables, although eigenvalue equations, formulated in either matrix or differential formalism are known to be mathematically equivalent. 

Specific for ISC and other predissociative curve-crossings is that the response function approach can be afflicted by instabilities which has to be treated with some care. The instabilities encountered for the MCLR eigenvalue equation near curve crossings is a structural problem of the method itself. Partitioning the MCSCF Hessian to orbital and configuration parts on one hand, and excitation and deexcitations on the other, gives the structure... 

The modes of a dielectric-metal-dielectric waveguide can be found by solving the eigenvalue (Eq. 35). Numerical solutions of this eigenvalue equation for a symmetric waveguide structure ( di = d2) are shown in Fig. 8. For any thickness of the metal film, there are two coupled surface plasmons, which are referred as to the symmetric and antisymmetric surface plasmons. 

Fig. 15 Sensitivity of the real propagation constant (—) and effective index (---) of a surface plasmon on a metal-dielectric interface to a bulk refractive index change as a function of wavelength calculated rigorously from eigenvalue equation and using the perturbation theory. Waveguiding structure gold-dielectric (nj = 1.32)...

Written in matrix-form, the new eigenvalue equation has a simple structure, with all r-dependent parts in the off-diagonal of the matrix ... 

Other kinds of approximants can also be used. For example, Olsen et al. [20] analyzed MP series convergence using a 2 X 2 matrix eigenvalue equation [38,39], which implicitly incorporates a square-root branch point. It is of course possible simply to explicitly construct an approximant as an arbitrary function with the singularity structure that E z) is expected to have. We suggest, for example, approximants of the form ... 

For a molecule of N atoms with its structure at a local energy minimum, the normal modes can be calculated from a 3A x 3A1 mass-weighted second derivative matrix H, the Hessian matrix, defined in a molecular force field such as CHARMM [32-34] or AMBER [35-38]. For each mode, the eigenvalue X and the 3A x 1 eigenvector r satisfy the eigenvalue equation. Hr = Ar. 

When all redundancies between internal coordinates are removed, the size of the eigenvalue equation to be solved is equal to the number of normal modes (3N - 6) expected for the molecule under study. If the molecule has some symmetry it belongs to a given symmetry point group g group theory provides the structure of the irreducible representation T of g, i.e., the number of normal modes in each symmetry species T,. By a suitable hnear and orthogonal transformation... 

Tlie method is particularly useful for the automatic removal of all redundancies in structurally complex systems. Indeed if m > 3N — 6 is the size of the eigenvalue equation... 

Since Uq is a unitary matrix, it preserves the orthonormality condition. Ho is then diagonalized by a standard hermitean eigenvalue solver. The eigenvalue equation has the same structure as Eq. (14.35) with the primed labels (L) and (S) replaced by doubly primed ones, ( )" and (S)", to indicate the change of basis by the free-particle Foldy-Wouthuysen transformation. The X matrix in this basis representation is obtained by Eq. (14.36) with the same label replacement. Analogously, the renormalization matrix reads R" = I + The final decoupling transformation... 

The four-component DHF LCAO equations for ID-, 2D- and 3D-periodic systems were at first presented by Ladik [547]. The resulting somewhat compUcated generaUzed matrix eigenvalue equation for solids is described (for details we refer the reader to [547]). It was also shown for ID and 2D systems how MP2 methods could be applied iu their relativistic form. With the help of these, on the one hand, the total energy per unit cell (including correlation effects) can be computed. On the other hand, the relativistic band structure can also be corrected for correlation. Note that the symmetry of crystaUine orbitals changes, compared with the nonrelativistic case, as the symmetry of the DHF Hamiltonian is described by double space groups. Finally,... 

The analysis of the step-profile fiber is readily extended to multilayered fibers with a uniform refractive index in each layer. An example of this is the W fiber, which has three layers with a smaller index in the centre layer than in the two adjacent layers [6], The modal fields and eigenvalue equations for such structures are constructed from the appropriate solution of Eq. (12-11) in each layer, ensuring that the tangential fields are continuous at each interface. 

Snyder, A. W. and de la Rue, R. (1970) Asymptotic solution of eigenvalue equations for surface waveguide structures. I.E.E.E. Trans. Microwave Theory Tech., 9,650-1. 

The above-mentioned statistical characteristics of the chemical structure of heteropolymers are easy to calculate, provided they are Markovian. Performing these calculations, one may neglect finiteness of macromolecules equating to zero elements va0 of transition matrix Q. Under such an approach vector X of a copolymer composition whose components are X = P(M,) and X2 = P(M2) coincides with stationary vector n of matrix Q. The latter is, by definition, the left eigenvector of this matrix corresponding to its largest eigenvalue A,i, which equals unity. Components of the stationary vector... 

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