Eigenvalue equation modes

The eigenvalue equation of the representation of the effective Hamiltonian operators (28) in the base of the number occupation operator of the slow mode is characterized by the equation... 

The most essential properly of acoustic vibrations in a nanoparticle is the existence of minimum size-quantized frequencies corresponding to acoustic resonances of the particle. In dielectric nanociystals, the Debye model is not valid for evaluation of the PDOS if the radius of the nanociystal is less than 10 nm. The vibrational modes of a finite sphere were analyzed previously by Lamb (1882) and Tamura (1995). A stress-free boundary condition at the surface and a finiteness condition on both elastic displacements and stresses at the center are assumed. These boundary conditions yield the spheroidal modes and torsional modes, determined by the following eigenvalue equations ... 

Besides, for situations involving indirect damping, the eigenvalue equation for the non-Hermitean effective Hamiltonian corresponding to the ground state of the high frequency mode is... 

The eigenvalue equation of the non-Hermitean Hamiltonian (297) may be written in order to make explicit the angular frequency oo° of the fast mode and the corresponding direct damping parameter y° ... 

The eigenvalue Eqs. 34 and 35 are transcendental equations for imknown modal propagation constants. After solving the eigenvalue equations, the field profiles can be determined by substituting the values of modal propagation constants fi into the boundary conditions and calculating the amplitudes and a i for TE modes and fcf and hj for TM modes (i = 1,2,3). 

The modes of a dielectric-metal-dielectric waveguide can be found by solving the eigenvalue (Eq. 35). Numerical solutions of this eigenvalue equation for a symmetric waveguide structure ( di = d2) are shown in Fig. 8. For any thickness of the metal film, there are two coupled surface plasmons, which are referred as to the symmetric and antisymmetric surface plasmons. 

A normal mode analysis can also be performed in curvilinear internal coordinates (Wilson et al., 1955). The approach is the same as that described above for Cartesian coordinates with one major modification. In an internal coordinate normal mode analysis, the internal coordinate cannot be simply scaled by the masses as is done for the Cartesian coordinates, Eq. (2.27), so that the masses become an explicit part of the eigenvalue problem. Thus, in an internal coordinate normal mode analysis, one does not solve Eq. (2.45), but instead solves the eigenvalue equation... 

Solving the eigenvalue equation, Eq. (2.54), for this operator gives the normal mode energy levels... 

For a molecule of N atoms with its structure at a local energy minimum, the normal modes can be calculated from a 3A x 3A1 mass-weighted second derivative matrix H, the Hessian matrix, defined in a molecular force field such as CHARMM [32-34] or AMBER [35-38]. For each mode, the eigenvalue X and the 3A x 1 eigenvector r satisfy the eigenvalue equation. Hr = Ar. 

The prompt mode, is the solution of the time-absorption eigenvalue equation... 

The random phase approximation (RPA) was first introduced into many-body theory by Pines and Bohm.This approximation was shown to be equivalent to the TDHF for the linear opticcd response of many-electron systems by Lindhard. ° (See, for example, Chapter 8.5 in ref 83. The electronic modes are identical to the transition densities of the RPA eigenvalue equation.) The textbook of D. J. Thouless contains a good overview of Hailree—Fock and TDHF theory. 

When all redundancies between internal coordinates are removed, the size of the eigenvalue equation to be solved is equal to the number of normal modes (3N - 6) expected for the molecule under study. If the molecule has some symmetry it belongs to a given symmetry point group g group theory provides the structure of the irreducible representation T of g, i.e., the number of normal modes in each symmetry species T,. By a suitable hnear and orthogonal transformation... 

The separability of the positive and negative Hessian eigenvalues led Simons and coworkers to suggest that the problem be reformulated into [4 principal modes to be maximized and (n - fi) modes to be minimized. There would be two shift parameters, Ap and A , one for modes along which the energy is to be maximized and the other for which it is minimized. The RFO eigenvalue equation (O Eq. 10.11) could thus be partitioned into two smaller P-RFO equations, and each one solved separately. 

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