Matrix vibronic

The approach developed by Jungen and Merer (JM) [24] is of a similar level of sophistication. The main difference is that IM prefer to remove the coupling between the electronic states by a transformation of the Hamiltonian matrix (i.e., vibronic energy matrix), rather that of the Hamiltonian itself. They first calculate the large amplitude bending functions for one of the adiabatic potentials, as if it belonged to a E electronic state. These functions are used as... 

It follows that the only possible values for la + Ip are S A and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number I = la + Ip. The off-diagonal elements of [second and third of Eqs. (60)] connect the basis functions with I — la + Ip and I — l + l — l 2A. 

In the lowest optieally excited state of the molecule, we have one eleetron (ti ) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaetion and can either form a singlet 5 state (5 = 0), or a triplet T state (S = 1). Since the electric dipole matrix element for optical transitions — ep A)/(me) does not depend on spin, there is a strong spin seleetion rule (AS = 0) for optical electric dipole transitions. This strong spin seleetion rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the w eak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

Techniques other than UV-visible spectroscopy have been used in matrix-isolation studies of Ag see, for example, some early ESR studies by Kasai and McLeod 56). The fluorescence spectra of Ag atoms isolated in noble-gas matrices have been recorded (76,147), and found to show large Stokes shifts when optically excited via a Si j — atomic transition which is threefold split in the matrix by spin-orbit and vibronic interactions. The large Stokes shifts may be explained in terms of an excited state silver atom-matrix cage complex in this... 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

The interpretation of the results was hindered to some extent by the inability to observe more than the gz absorption, except in the Kr matrix, but it was found that although the Fermi contact term, k0, showed some lattice dependence, the main feature of the results was that whilst A ranged from 286 cm-1 in Ne to 668 cm-1 in Kr, the vibronic overlap term varied only between 0.30 and 0.25, thus bearing out the authors predictions concerning the origin of the static distortion, A. 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

A final study that must be mentioned is a study by Hartmann et al. [249] on the ultrafast spectroscopy of the Na3F2 cluster. They derived an expression for the calculation of a pump-probe signal using a Wigner-type density matrix approach, which requires a time-dependent ensemble to be calculated after the initial excitation. This ensemble was obtained using fewest switches surface hopping, with trajectories initially sampled from the thermalized vibronic Wigner function vertically excited onto the upper surface. 

Terms involving Majorana operators are nondiagonal, but their matrix elements can be simply constructed using the formulas discussed in the preceding sections. The total number of parameters to this order is 15 in addition to the vibron numbers, N and N2- This has to be compared with 4 for the first-order Hamiltonian (4.91). For XY2 molecules, some of the parameters are equal, Xi,i = X2,2 XU2 = X2,12, Y112 = Y2 U, A] = A2, reducing the total number to 11 plus the vibron number N = Aj = N2. Calculation of vibrational spectra of linear triatomic molecules with second-order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.8. 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

It is seen from Equation 19 that the electronic transitions take place without changing the equilibrium positions of the nuclei, and the electronic component of the dipole transition moment is non-zero only if there is no change of the vibronic state during this transition. Dg is non-zero only if the transitions occur between the vibronic states within one electronic state, and the selection rules of Equation 16 are derived from the conditions for a non-vanishing matrix element in Dg. ... 

Alexandrite, the common name for Cr-doped chrysoberyl, is a laser material capable of continuously tunable laser output in the 700-800 nm region. It was established that alexandrite is an intermediate crystal field matrix, thus the non-phonon emitting state is coupled to the 72 relaxed state and behaves as a storage level for the latter. The laser-emitted light is strongly polarized due to its biaxial structure and is characterized by a decay time of 260 ps (Fabeni et al. 1991 Schepler 1984 Suchoki et al. 2002). Two pairs of sharp i -lines are detected connected with Cr " in two different structural positions the first near 680 nm with a decay time of approximately 330 ps is connected with mirror site fluorescence and the second at 690 nm with a much longer decay of approximately 44 ms is connected with inversion symmetry sites (Powell et al. 1985). The group of narrow lines between 640 and 660 nm was connected with an anti-Stokes vibronic sideband of the mirror site fluorescence. 

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