Sub-bands

Looking at (110), it appears that the spectral density /Sf(oo, aG = 0) ex is composed of two sub-bands, as is also shown by the four sample spectra of Fig. 10 which were computed for various parameters A and A. The frequency and intensity of these two sub-bands do not depend on the temperature, and are similar to those which may be obtained within the simpler undamped treatment. Solving the eigenvalue problem (111), we obtain... 

We may observe that the two sub-bands of the damped spectral density (110), as well as the two peaks involved by the undamped case, must be of the same intensity in the resonant case (A = 0), which may be verified by looking at the... 

Fig. 10(a). On the contrary, their intensities differ in the nonresonant case (A / 0), but they must become closer as the Fermi coupling parameter A increases. Note that the frequency ft x of the Evans hole which separates the two sub-bands is independent of A, because it is given by... 

Some sample calculations are displayed in Fig. 10. As may be seen, the spectral density (124) involves two sub-bands in the vs (X-H) frequency region, like it is observed within the exchange approximation. Note that other submaxima appear at overtone frequencies (near 2oo0, 3co0,. ..) with a much lower intensity (less than 0.1% of the doublet intensity) and will not be studied here. 

We may finally conclude that, with the purpose of comparison with experiments, one has to be careful and must remember the following (i) Two sub-bands of the same intensity may not be the consequence of a resonant situation 0, (ii) The frequency of each submaxima is governed by the three parameters co0, o>0. and A, and (iii) The frequency of the Evans hole, which appears between the two sub-bands, is given within the exchange approximation by the average frequency j ( 0 + 2o>0), but it is dependent on A within the full treatment of Fermi resonances. 

Figure 12. Hydrogen bond involving a Fermi resonance damping parameters switching the intensities. The lineshapes were computed within the adiabatic and exchange approximations. Intensities balancing between two sub-bands are observed when modifying the damping parameters (a) with y0 =0.1, and y5 = 0.8 (b) with ya = ys = 0.8 (c) with yB — 0.8 and ys — 0.1. Common parameters oto = 1, A = 150cm, 2g)5 = 2850cm-1, and T = 30 K.

If the experimental lineshapes do not exhibit sub-bands and are asymmetric, or if they involve sub-bands, but with intensity anomalies with respect to the Franck-Condon progression law, then, together with the dephasing mechanism, damping of the slow mode ought to be also considered as occurring in a sensitive competitive way. 

The energy difference between the highest occupied tt sub-band and the lowest unoccupied tt sub-band defines the tt-tt energy gap Eg. 

The densities of states of Ni5, Nig, and Ni7, decomposed into d and sp contributions, are compared in Figure 5. The occupied states of the majority-spin sub-band (t) have mainly d character, except for a small peak with the sp character at the Fermi level on the other hand, d holes are present in the minority-spin sub-band (f). Integration of the density of states gives average d... 

Table 2. Positions and FWHM of gaussian sub-bands in the OH stretching region of H2O...

In Fig. 7.3 the d bands are shown split into sub-bands. The lower d bands are derived from the d orbitals that overlap least with the chalcogen p orbitals. (This is because the d bands are formed from antibonding combinations of d and p orbitals, and those with more overlap are pushed higher in energy.) The details of the splitting depend on how the... 

In physical terms the site inhomogeneous component may be conceived of either as slightly different protein binding sites for chromophores or as a distribution of protein conformational substates [127] at any one site. In operational terms the spectral forms are represented by the sub-bands of a gaussian or lorentzian decomposition analysis of absorption or fluorescence spectra. 

A. The splitting of the band energy E° into two sub-bands, for spin-up and spin-down... 

Fig. 14 a, b. Hubbard sub-bands for the localized ("f or j.) and polar states in a narrow band solid... 

We consider first (Fig. 14 a) what happens at very large distances. The Hamiltonian (11) (without the correction (35)) would then give rise to a very narrow band (a level). With the correction (35), the band splits into two separate sub-bands (two energy levels) Eo and Eq + Uh (see Fig. 14 a). These two sub-bands containing each M (and not 2M) states, represent, respectively, a state in which each core holds one spin and a state in which half of the cores hold two antiparallel spins, and the others empty (polar states). The two sub-bands are separated by a gap which is exactly Uh- Without excitation to the highest sub-band, in this conditions the lower sub-band is fully occupied. It represents the insulator s state in which all electrons are sitting in the cores, i.e. all electrons are fully localized. 

When the cores are approached, the sub-bands split, acquiring a bandwidth, and decreasing the gap between them (Fig. 14 a). At a definite inter-core distance, the subbands cross and merge into the non-polarized narrow band. At this critical distance a, the narrow band has a metallic behaviour. At the system transits from insulator to metallic (Mott-Hubbard transition). Since some electrons may acquire the energies of the higher sub-band, in the solid there will be excessively filled cores containing two antiparallel spins and excessively depleted cores without any spins (polar states). 

We may therefore assume that the 5 f non-spin-polarized band splits into two subbands because of spin-polarization. Approximation of the two sub-bands, according to Friedel s model, by two rectangular ones, having densities of state N+(E) = N (E) = 7/ Wf, and occupation numbers n+ and n, leads to the following expression for the total Pspd pressure ... 

The integration within the sub-bands is performed to a common Fermi level for the two spin-populations, which is the usual Stoner thermodynamic condition (an neglecting the Nspd density of state). The occupation numbers n- are therefore ... 

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