Localized eigenvector

Let us assume that we are at a point on the manifold and that there is some perturbation H. The perturbation can be decomposed into its components in the local eigenvector basis, i.e. into two parts - one part, describing the rate of change in the slow subspace and one describing the rate of change in the fast subspace. 

C and eJ are local eigenvectors and eigenvalues of the subsystem. And /x is the chemical potential for the molecule. For each subsystem, the C and ej satisfy the eigenvalue equation for the subsystem. 

To find maximum-occupancy orbitals for atom A, one first finds the local eigenvectors of the one-center atomic block... 

The classification of critical points in one dimension is based on the curvature or second derivative of the function evaluated at the critical point. The concept of local curvature can be extended to more than one dimension by considering partial second derivatives. d2f/dqidqj, where qt and qj are x or y in two dimensions, or x, y, or z in three dimensions. These partial curvatures are dependent on the choice of the local axis system. There is a mathematical procedure called matrix diagonalization that enables us to extract local intrinsic curvatures independent of the axis system (Popelier 1999). These local intrinsic curvatures are called eigenvalues. In three dimensions we have three eigenvalues, conventionally ranked as A < A2 < A3. Each eigenvalue corresponds to an eigenvector, which yields the direction in which the curvature is measured. 

Fig. 2. The quantum mechanics of the two-state prpblem provide a paradigm for the much more extensive electronic state space of a real molecular or macromolecular system. The eigenvectors c, of the Hamiltonian are symmetric and antisymmetric linear combinations of the localized basis vectors with an eigenvalue splitting of 2A, where s is the overlap integral and A is the direct coupling (the only kind possible in this case)...

The small parameter of the expansion is the mode displacement SQi, because it is always evaluated for the mode ground state or a low-energy state. Both are very localized. Let us consider the eigenvector Qi of the mode i. Then the Hamiltonian can be expanded in a Taylor series on the displacements 5Qi. 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

The symmetry eigenvectors of any local NRG may be advantageously used as basis functions in a pre-diagonalization of the complete Hamiltonian matrix. For this purpose, the local NRG has only to be a larger group than the complete NRG. In the present case we have for example ... 

In the present paper, symmetry eigenvectors which factorize the Hamiltonian matrix into boxes are given for the single rotation in phenol (24) for double rotations in benzaldehyde (29), pyrocatechin (34) and acetone (44-46), for double rotation and inversion in non-planar pyrocatechin (40) and pyramidal acetone (49-51). In the same way, symmetry eigenvectors deduced in the local approach are deduced for some of these non-rigid systems (79), (83), and (89). Symmetry eigenvectors for the double internal Czv rotation in molecules with frame of any symmetry are given in reference [36]. 

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