Vibrational eigenvalue problem

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. 

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. 

The eigenfunction of the vibrational ground state is calculated on the ab initio 2D So potential energy surface by solving the eigenvalue problem. 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

In the course of conventional vibrational spectroscopic calculations in internal coordinates the following eigenvalue-problem has to be solved for the molecule under consideration (FG-method terminology of Wilson, Decius, and Cross (5)) ... 

With these approximations, the rotational and vibrational motion is completely separated and the eigenvalue problem... 

In general, a matrix equation in the form A x = 0 will have solutions other than x = 0 only if dot A = 0. In the case of vibrations, there will be non-trivial solutions only if det(7 — w M) = 0. This is an example of an eigenvalue problem. 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

The normal modes of vibration u are obtained as solutions of the Hessian eigenvalue problem,... 

As was already mentioned in Section 3.4, we can calculate the vibration—inversion-rotation energy levels of ammonia by solving the Schrodinger equation [Eq. (3.46)]. We are of course primarily interested in the determination of the potential function of ammonia from the experimental frequencies of transitions between these levels (Fig. 11), Le. we must solve the inverse eigenvalue problem [Eq. (3.46)]. 

In the non-rigid bender approximation, we solved the inverse eigenvalue problem described by Eq. (5.4), i.e. we determined the potential function parameters given in Table 3 for NX3 (X = H, D, T). We have used the experimental infrared frequencies of transitions from the ground state to the i>2,2 2 > 2. and 41 2 inversion states and the zero-order frequencies of vibrations (Table 4). The zero-order frequencies have been obtained from the observed fundamental frequencies of NH3 [Ref. >], ND3 [Ref. °>], NTg [Refs." and [Ref.- 3)] corrected for... 

In the traditional theory of the cooperative JT effect, its significant part is one-center JT problem in a low-symmetry mean field (see the last paragraph of Sect. 2.2). In particular, it includes the eigenvalue problem for the Hamiltonian, similar to (7), operating in an infinite manifold of vibrational one-center states. Compared to this relatively complex step, in the OOA, the mean-field approximation is much simpler. In the OOA, one has to solve just a finite-size matrix (2 x 2 in this case) or, for other JT cases, a somewhat larger matrix but finite anyway. In the theory of the cooperative JT effect, this important advantage of the OOA allows to proceed farther than... 

As pointed out by Gwinu and Gaylord19), the solutions of the eigenvalue problems associated with a vibration-rotation problem do not and must not depend on the choice of the rotating axis system as long as an adequate Hamiltonian is used. What do depend on the axis system used are the numerical values of elements of the inertial tensor or vibration-rotation interaction constants determined from analysing the data. 

Figure P14.8 also shows the error norm, ej, versus the number of Ritz vectors (from Problem 14.7). The error is smaller when Ritz vectors are used, because they are derived from the force distribution. Ritz vectors are useful for dynamic analysis of large systems with classical damping, since the vibration properties of the system can be obtained by solving, a smaller eigenvalue problem of order 7, instead of original eigenvalue problem of size N. It must be noted that the resulting frequencies and mode shapes are approximations to the...

A matrix eigenvalue problem is obtained by multiplying by and integrating over electronic and vibrational coordinates. Note that the electronic functions form an orthonormal set for a given Q. For the evaluation of the integrals, see the original paper by Klimkans and Larsson (2000). The vibrational functions are a set of harmonic functions centered at Q = -Qo, Q = 0, and Q = Qq. The eigenvalue problan is therefore of the form ... 

This equation, along with the boundary conditions that the midplane slope and deflection vanish at the cantilevered end of the beam and that the internal shear force and bending moment vanish at the free end, lead to an eigenvalue problem for free transverse vibration in which the eigenvalues are the squares of the natural frequencies of vibration. The fundamental natural frequency is... 

For the assignment of the different vibrational peaks to certain hydrogen sites a simple lattice dynamical model was used. The frequencies of localized hydrogen vibrations in metals are obtained by solving the eigenvalue problem... 

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