Eigenvector normalized

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

For a nonlinear molecule composed of N atoms, 3N—6 eigenvalues provide the normal or fundamental vibrational frequencies of the vibration and and the associated eigenvectors, called normal modes give the directions and relative amplitudes of the atomic displacements in each mode. 

The eigenvalues and eigenvectors of the mass-weighted force matrix can be obtained by diagonalizing equation (21.5). Then each eigenvalue corresponds to its normal coordinates, Qj,... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

A particular advantage of the low-mode search is that it can be applied to botli cyclic ajic acyclic molecules without any need for special ring closure treatments. As the low-mod> search proceeds a series of conformations is generated which themselves can act as starting points for normal mode analysis and deformation. In a sense, the approach is a system ati( one, bounded by the number of low-frequency modes that are selected. An extension of th( technique involves searching random mixtures of the low-frequency eigenvectors using Monte Carlo procedure. 

Compute normal modes. These represent primarily harmonic motions internal to the molecule. There are 3N—6 displacement eigenvectors, where N is the number of degrees of freedom of the system. The associated eigenvalues are the frequencies. 

The determination of some of the eigenvalues and eigenvectors of a large real symmetric matrix has a long history in numerical science. Of particular interest in the normal mode... 

The normal mode refinement method is based on the idea of the normal mode important subspace. That is, there exists a subspace of considerably lower dimension than 3N, within which most of the fluctuation of the molecule undergoing the experiment occurs, and a number of the low frequency normal mode eigenvectors span this same subspace. In its application to X-ray diffraction data, it was developed by Kidera et al. [33] and Kidera and Go [47,48] and independently by Diamond [49]. Brueschweiler and Case [50] applied it to NMR data. 

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

The normal modes for solid Ceo can be clearly subdivided into two main categories intramolecular and intermolecular modes, because of the weak coupling between molecules. The former vibrations are often simply called molecular modes, since their frequencies and eigenvectors closely resemble those of an isolated molecule. The latter are also called lattice modes or phonons, and can be further subdivided into librational, acoustic and optic modes. The frequencies for the intermolecular modes are low, reflecting, the... 

To avoid numerical differentiation (which is inherently unstable) one uses the fact that an eigenvalue can be expressed as Ai = v Tvf where are the corresponding normalized left and right eigenvectors. Differentiation of the eigenvalue with respect to any parameter is then equivalent to the differentiation of the transfer matrix, and one finds... 

This becomes even clearer when we examine the alternate version gf this normal mode included later in the output, labeled as the eigenvector of the Hessian ... 

In order to find the normal modes of vibration, I am going to write the above equations in matrix form, and then find the eigenvalues and eigenvectors of a certain matrix. In matrix form, we write... 

This is clearly a matrix eigenvalue problem the eigenvalues determine tJie vibrational frequencies and the eigenvectors are the normal modes of vibration. Typical output is shown in Figure 14.10, with the mass-weighted normal coordinates expressed as Unear combinations of mass-weighted Cartesian displacements making up the bottom six Unes. 

In the new coordinate system A is a diagonal matrix, and the (normalized) e vector is a new coordinate axis. The diagonal elements in A are therefore directly the eigenvalues, and since e = U e, the columns in the U matrix are the eigenvectors. 

Thus, if we wish to compare the eigenvectors to one another, we can divide each one by equation [57] to normalize them. Malinowski named these normalized eigenvectors reduced eigenvectors, or REV". Figure 52 also contains a plot of the REV" for this isotropic data. We can see that they are all roughly equal to one another. If there had been actual information present along with the noise, the information content could not, itself, be isotropically distributed. (If the information were isotropically distributed, it would be, by definition, noise.) Thus, the information would be preferentially captured by the earliest... 

D.—There exists a set of three operators, Qk, h — 1,2,3 corresponding to the measurement of position q = (x,y,z). There exists a continuum of eigenvectors of these operators, q>, with the following normalization properties (cf. Eq. (8-32)) ... 

Here the symbol q, indicates the 3-dimensional vector position of the f particle. To apply Hilbert space concepts to this theory we now postulate a one-to-one correspondence between the rays of and the points of the Schrodinger 3N hyperspace. Thus there exists a continuum of normalized eigenvectors in represented by the symbol 4i.4a.". > ... 

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