Eigen energies

For example, after the drum head has been hit, its shape will evolve spatially and in time in a manner that depends on how it was prepared by the initial blow. However, if one carries out an experiment to detect and frequency-analyze the sound that emanates from this drum, thereby measuring differences in the eigen-energies of the system, one finds a... 

Diagonalization of the matrix formed by the multipliers fiy yields the Kohn-Sham orbitals and their eigen-energies ... 

Negative of the Highest Occupied Orbital Eigen Energies (Atomic Units) of Some Atoms Calculated with the Potential Wx Equation 7.34... 

Bound states are readily included in the line shape formalism either as initial or final state, or both. In Eq. 6.61 the plane wave expression(s) are then replaced by the dimer bound state wavefunction(s) and the integration(s) over ky and/or kjj2 are replaced by a summation over the n bound state levels with total angular momentum J n or J . The kinetic energy is then also replaced by the appropriate eigen energy. In this way the bound-free spectral component is expressed as [358]... 

The result is transparent - the function Aap(e) is nonzero only at particle eigen-energies, so that... 

This expression can be reduced to the diagonal form then, the resulting Hamiltonian H n will be the sum of the oscillator Hamiltonians with normal coordinates q. The eigen energy of the system is the sum of the energies of the independent oscillators,... 

Here, Wv( p ) is the eigen function of Hamiltonian H( p ) with the eigen energy ev and the summation occurs on all states including the integration on the continuous spectra, 8 is the infinitesimal defining the rule of the pole bypass in expression (51). 

V c r)- In the second case, HE as well as DFT-LDA eigen-states and eigen-energies may be used as starting points for the implementation of the GW approximation. 

The above problem is the so-called resonance problem when the positive eigen-energies lie under the potential barrier. We solve this problem, using the technique fully described in refs. 12, 26, 30-32. 

Figure 8. Surfaces of eigen-energies (in units of 8) as functions of Qp/8 and 0 /8 when the dressed states (denoted A-i, X-2, and A-3, respectively connected to E, Ei, and 3 for fields off) are such that X2 < < X3, with...

Figure 9. Surfaces of eigen-energies (in units of 8) as functions of Hp/8 and fi /8 for the case 3 with Ap — 38/2 and As — 8/2. Paths (a) and (b) (with pulses of the same length and peak amplitude) correspond respectively to the intuitive and counterintuitive pulse sequences in A or ladder systems.

The potential energy surfaces (LEPS A from Refs. (8.10)). the effective potentials U (n=0 gerade and ungerade symmetry) and the bound states of the collinear IHI and lOI are shown in Fig. 4. As argued for the Cl+HCl system the reaction is very adiabatic, and thus the effective potentials and the adiabatic eigen-energies e (r) coincide within 10 eV. It should be noticed that isotopic suBstitution of the hydrogen atom by its heavier isotope deuterium tends to destabilize the vibrational bonded molecule. 

It is interesting to recognize that the familiar perturbation method for the analysis of the rotational spectra and the exact evaluations of the eigen-energies and eigenfunctions just reviewed in this section use the same tool. 

Thus, exploiting the built-in v-dependence of the Morse oscillator eigen-energies (Eq. (9.4.107)),... 

Although in classical mechanics E may be varied continuously, the trajectories are displayed only for the values of E that are quantum eigen-energies. The dotted /z(V0 line is a separatrix, which is a dividing surface that no trajectory can cross and which divides the accessible phase space into qualitatively distinct regions (exhibiting normal vs. local mode behavior) filled with qualitatively different trajectories. 

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