Vibronic definition

The absoiption spectra of these three materials are shown in the bottom panel of Figure 9-16. From these spectra it becomes clear that the m-LPPP shows the longest effective conjugation length 23 the best resolution of vibronic progression, and the steepest onset of absorption [231. Therefore, one would assume the m-LPPP to be a material of the highest chemical definition. This is indeed con-... 

Now, in the statistical limit for which we can treat our final manifold of closely spaced twisted triplets as an effective continuum, the sum in eq. (12-24) can be replaced by a definite integral whose weight function is taken to be the density of vibronic states, p(E). Thus we write... 

The near IR spectra of the tetrakis(cumylphenoxy)phthalocyanines have not been reported before. The absorption in the Cu complex and one of the absorptions in the Co complex lie close to bands which have been tentatively assigned to trip-multiplet transitions in other phthalocyanines.(14) However, the other absorption bands shown in Table 1 have not been previously reported for phthalocyanines with no peripheral substitution. The small absorption cross sections of these bands in the cumylphenoxy phthalocyanines suggest that they are forbidden transitions. Possible assignments for these bands include a symmetry forbidden electronic transition (like the MLCT transitions in NiPc discussed above) becoming vibronically allowed, d-d transitions on the metal ion, or trip-multiplet transitions. Spectroscopic studies are in progress to provide a more definitive assignment of these absorptions. 

Equations (6) and (7) refer to vibronic couplings with totally symmetric vibrations. For other couplings see Ref. [19], Different formulas in literature refer to alternate definitions of Af,... 

Vibronic spectra, definition, 23 Visible spectra—See Absorption spectra, Emission spectra... 

A convenient definition of second-order vibronic RFs useful in the context of the FC approximation can be found in Ref. [4] based on earlier work [5]. The RFs will be expressed in the form K j rm 0 / ) for electronic perturbations V of symmetries 1 and rm where the symmetry label Af G T 0 /] . The electronic perturbation Hamiltonian within an orbital triplet 7] can be written as ... 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

The aim of this work is to elucidate these problems. To this end, we calculate the effective spin Hamiltonian of the 5f2—5f2 superexchange interaction between the neighboring U4+ ions in the cubic crystal lattice of UO2 and we calculate T5 <%> eg, rs f2g(l) ancl r5 f2g(2) linear vibronic coupling constants. These data are then used to draw a more definite conclusion about the driving force of the phase transition and especially about the actual mechanism of the spin and orbital ordering in U02. 

In the crude Born-Oppenheimer approximations, the oscillator strength of the 0-n vibronic transition is proportional to (FJ)2. Furthermore, the Franck-Condon factor is analytically calculated in the harmonic approximation. From the hamiltonian (2.15), it is clear that the exciton coupling to the field of vibrations finds its origin in the fact that we use the same vibration operators in the ground and the excited electronic states. By a new definition of the operators, it becomes possible to eliminate the terms B B(b + b ), BfB(b + hf)2. For that, we apply to the operators the following canonical transformation ... 

In the definition of line broadening it is necessary to exercise some discrimination. On the one hand spectral linewidths of less them 0.17 cm-1 are observed for some of the vibronic bands of the lowest singlet system of benzene 1f 2 - -1diff in the vapor phase W, while on the other hand many electronic spectra have been encountered, in particular in higher excited singlet and triplet systems, for which few or no vibrational features are apparent. In crystal spectra at 4 K, linewidths as sharp as 0.5 cm-1 are often obtained for the lowest excited state of any multiplicity, despite coupling with the lattice modes, which may be expected to lead to considerable broadening. Nevertheless, these crystal linewidths are considerably more than the linewidths observed in the vapor phase and certainly more than the natural radiative widths. 

The competition between intramolecular vibrational relaxation and chemical reaction has been discussed in terms of the applicability of transition state theory to the kinetic analysis [6], If the environment functions mainly as a heat bath to ensure thermalization among the vibrational modes in the excited complex, then transition state theory is a good approximation. On the other hand, when the reaction is too fast for thermalization to occur the rate can depend upon the initial vibronic state. Prompt reaction and prompt intersystem crossing are, by definition, examples of the latter limit. 

The Jahn-Teller effect [7-9,25] originates from vibronic coupling [19], In this chapter, we discuss the definition of vibronic coupling with emphasis on its difference from non-adiabatic coupling. 

The vibronic coupling constant is expressed in other forms depending upon the definition of the variables or operators of vibrations. 

The following equations for the Eg (g> Sg and T2g Sg vibronic interaction of a d cation and some definitions supply the necessary background from theory ... 

It is worthy to give a direct definition of CITE. CITE is virtual phonon exchange at electron orbital degeneracy, leading to the correlation of local distortions and selfconsistent correlation of electrons. The virtual phonon exchange is the result of electron-phonon (vibronic) interaction and of phonon dispersion. 

The OOA was not designed for and does not apply to temperature dependencies of any kind in JT crystals. In particular, one cannot expect a reasonable estimate of the temperature of phase transitions in crystal lattice (structural), electron orbital, and/or spin system. This follows from the partitioning procedure that includes averaging over vibrational degrees of freedom. One can see the same reason from another perspective. The pseudo spin of a JT site, as the basic concept used in the OOA, operates in the basis of degenerate ground state wave functions. Excited vibronic states are beyond the pseudo spin setup. Therefore, in the OOA, by its very definition, temperature population of excited states does not make sense. 

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