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Eigenvector

The exact ground-state eigenvalue and corresponding eigenvector... [Pg.47]

Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... [Pg.48]

In practice, the matrix (iL+R+K) is diagonalized first, with a matrix of eigenvectors, U, as in equation (B2.4.15)), to give a diagonal matrix. A, with the eigenvalues, X, of L down the diagonal. [Pg.2096]

It is convenient, for simple systems, to have explicit expressions for equation (B2.4.17). Since the original matrix is non-Hemiitian, the matrix fomied by the eigenvectors will not be unitary, and will have four independent complex elements. Let them be a, b, c and d, so that U is given by equation (B2.4.20). [Pg.2097]

For slow exchange, a convenient matrix of eigenvectors is given by equation (B2.4.23). [Pg.2098]

After coalescence, a possible set of eigenvectors is given in equation (B2.4.24). If these are substituted into (B2.4.22), the results are pure real, reflecting the fact that iP - 8 is now positive. [Pg.2098]

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. [Pg.2100]

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... [Pg.2100]

The quantities (U and (UF ) in B2.4.34 are projections of the eigenvector J along 1. From the above equations, this can be interpreted as follows. The temi (UF ) is the amount that the transition J received from... [Pg.2101]

E.( ), possesses a complete set of eigenfiinctions, the matrix F whose dimension Mis equal to the number of atomic basis orbitals, has M eigenvalues e. and M eigenvectors whose elements are the. . Thus, there are... [Pg.2170]

CISD Yes/No A/ transformed integrals n N to solve for one Cl energy and eigenvector... [Pg.2190]

Aq becomes asymptotically a g/ g, i.e., the steepest descent fomuila with a step length 1/a. The augmented Hessian method is closely related to eigenvector (mode) following, discussed in section B3.5.5.2. The main difference between rational fiinction and tmst radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the fonuer it is imposed smoothly and is automatically reduced to zero as convergence is approached. [Pg.2339]

There is no simple genei al form for the adiabatic eigenvectors, except in the limits, k = 0 and i = 0, wlien, for example. [Pg.22]

At this stage, we are ready to prove that Kramers theorem holds also for the total angular momentum F. We will do it by reductio ad absurdum. Then, let Itte) be the eigenvector of H with eigenvalue E,... [Pg.564]

As noticed from Figure 15, the two surfaces u and 2 ate conelike potential energy surfaces with a common apex. The corresponding eigenvectors are... [Pg.715]


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A — Eigenvalues and Eigenvectors of the Rouse Matrix

Column eigenvectors

Column-eigenvector

Coulomb eigenvector

Design Using Eigenvectors

Determining eigenvalues and eigenvectors

Differentiability eigenvector

Dynamic equations, eigenvectors

Dynamical Matrix and Eigenvectors

Effective eigenvectors

Effective operators eigenvectors, norms

Eigenvalue analysis eigenvector

Eigenvalue-eigenvector method

Eigenvalue/eigenvector problem

Eigenvalue/eigenvector problem generalized matrix

Eigenvalues and Eigenvectors of the Matrix

Eigenvalues and Eigenvectors of the Rouse-Mooney Matrix

Eigenvalues and Eigenvectors, Diagonalizable Matrices

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors defined

Eigenvalues and eigenvectors of a symmetric matrix

Eigenvalues, eigenvectors and band

Eigenvector Following optimization

Eigenvector Following optimization method

Eigenvector analysis

Eigenvector associated with Hessian matrix

Eigenvector degeneracy

Eigenvector expansion

Eigenvector expansion Hamiltonian diagonalization

Eigenvector following

Eigenvector following Transition structure

Eigenvector following method

Eigenvector generalized

Eigenvector map

Eigenvector matrix decomposition and basis sets

Eigenvector normalized

Eigenvector orthonormality

Eigenvector phases

Eigenvector plot

Eigenvector positive

Eigenvector problem

Eigenvector problem, solutions

Eigenvector projection

Eigenvector quantitation methods

Eigenvector reconstruction

Eigenvector rotation

Eigenvector, definition

Eigenvector, vibrational

Eigenvector-following, potential energy

Eigenvectors

Eigenvectors Floquet Hamiltonian

Eigenvectors auxiliary system

Eigenvectors degenerate

Eigenvectors description

Eigenvectors diagonalization transformation

Eigenvectors equations

Eigenvectors hydrogen bonds

Eigenvectors linear independence

Eigenvectors noise

Eigenvectors significant

Eigenvectors structure

Eigenvectors, hybrid orbitals

Eigenvectors, theorems

Electronic stress tensor eigenvectors

Error eigenvector

Generalised eigenvectors

Generalized column-eigenvectors

Geometric phase theory eigenvector evolution

Geometric phase theory, eigenvector

Graph drawing eigenvectors

Hessian eigenvector

Jacobi eigenvectors

Left-hand eigenvectors

Linear algebra eigenvector

Liouville eigenvectors

Localized eigenvector

Matrices eigenvalues/eigenvectors

Matrices, Eigenvalues, and Eigenvectors

Matrix eigenvector

Matrix eigenvectors

Matrix of eigenvectors

Metric null eigenvectors

Model Updating Using Eigenvalue-Eigenvector Measurements

Momentum space eigenvectors of the Dirac operator

Newton eigenvector method

Normalized eigenvectors

Numerical calculation of eigenvalues and eigenvectors in MATLAB

Orthogonal eigenvectors

Orthonormal eigenvectors

Perturbed eigenvector

Primary eigenvectors

Properties of Eigenvalues and Eigenvectors

Properties of Eigenvectors

Reduced Eigenvector Space

Reduced eigenvectors

Representation by the Floquet Eigenvectors

Right-hand eigenvectors

Row eigenvectors

Row-eigenvector

Spin eigenvector

Structural eigenvector

Symmetry eigenvectors

Symmetry eigenvectors pyrocatechin

Tensor eigenvector

The Eigenvector

Unperturbed eigenvector

Updating eigenvectors

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