Eigenvalue, vibrational

Solving the vibrational problem, as presented in this chapter, provides the eigenvalues (vibrational frequencies) and the eigenvectors (atomic displacements) that are needed to calculate 5 (g,o)). Recently we have developed and released ACLIMAX [39] as fi-eeware . This takes ab initio outputs to generate a spectrum fi-om a low-bandpass spectrometer ( 3.4.2.3). The program can be downloaded from ... 

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. 

The simplest way to write down the 2 x 2 Hamiltonian for two states such that its eigenvalues coincide at trigonally symmetric points in (x,y) or (q, ( )), plane is to consider the matrices of vibrational-electronic coupling of the e Jahn-Teller problem in a diabatic electronic state representation. These have been constructed by Haiperin, and listed in Appendix TV of [157], up to the third... 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

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. 

Motion along each normal coordinate is described by each atom vibraiingin phase with one another with th e sam e frequency. Th e vibration frequency, v, is related to the eigenvalues, X, by... 

The hi, b 1 and a2 bloeks are formed in a similar manner. The eigenvalues of eaeh of these bloeks provide the squares of the harmonie vibrational frequeneies, the eigenveetors provide the normal mode displaeements as linear eombinations of the symmetry adapted... 

So, of the 9 eartesian displaeements, 3 are of ai symmetry, 3 of b2,2 of bi, and 1 of a2- Of these, there are three translations (ai, b2, and b i) and three rotations (b2, b i, and a2). This leaves two vibrations of ai and one of b2 symmetry. For the H2O example treated here, the three non zero eigenvalues of the mass-weighted Hessian are therefore of ai b2, and ai symmetry. They deseribe the symmetrie and asymmetrie streteh vibrations and the bending mode, respeetively as illustrated below. 

The method of vibrational analysis presented here ean work for any polyatomie moleeule. One knows the mass-weighted Hessian and then eomputes the non-zero eigenvalues whieh then provide the squares of the normal mode vibrational frequeneies. Point group symmetry ean be used to bloek diagonalize this Hessian and to label the vibrational modes aeeording to symmetry. 

The eigenvalues (coa of the mass weighted Hessian matrix (see below) are used to compute, for each of the 3N-7 vibrations with real and positive cOa values, a vibrational partition function that is combined to produce a transition-state vibrational partition function ... 

The squares of the desired harmonie vibrational frequeneies co are thus given as eigenvalues of the mass-weighted Hessian H ... 

There were some problems with the eigenvalue following transition-structure routine jumping from one vibrational mode to another. The semiempirical geometry optimization routines work well. 

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. 

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. 

Figure 1. Eigenvalues of the scaled champagne bottle Hamiltonian (Eq. (2)) for p = 0.00625, in the energy, , and angular momentum, k map. The eigenvalues, represented by points, are joined (a) by lines of constant bent vibrational quantum number, vt, and (b) by lines of constant linear quantum number, v = 2vt + k. ...

Figure 6. The Bohr-Sommerfeld phase corrections t)(8, ) for k = 0, 1, and 2. The ratio r z,k)lK estimates of the error of primitive Bohr-Sommerfeld eigenvalues as a fraction of their local vibrational spacing.

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