Continuum resonances

The original derivation of the Fano profile in Ref. [29] is more general. It applies to any transition matrix for a final state with a QBS interacting with a background continuum. Resonance photoionization cross sections, for example, is describable by Eq. (18) with some background augmented. Fano [29] also generalizes his derivation for either more than one QBS or more than one continuum, but not for both simultaneously, which is the most difficult case and will be discussed in Section 2.3 of this article. 

Atabek, O., Lefebvre, R., and Jacon, M. (1980a). Continuum resonance Raman scattering of light by diatomic molecules. I. The role of radiative crossings between the potentials of the dressed molecule, J. Chem. Phys. 72, 2670-2682. 

Hartke, B. (1991). Continuum resonance Raman scattering in bromine comparison of time-dependent calculations with time-independent and experimental results, J. Raman Spec. 22, 131-140. 

Zhang, J.Z.H. and Miller, W.H. (1990). Photodissociation and continuum resonance Raman cross sections and general Franck-Condon intensities from 5-matrix Kohn scattering calculations with application to the photoelectron spectrum of H2F- + hv —> H2 + F, HF + H+e, J. Chem. Phys. 92, 1811-1818. 

Figure 6.1-3 Schematic representation of continuum resonance Raman scattering for the Br2 molecule. The incident laser frequency (o o) is in resonance with the continuous states of the repulsive 77 excited state and the repulsive part of the bound B(- 77o-i- ) state, which is above the dissociation limit at around 20 000 cm (Baierl and Kiefer, 1981).

Figure 6.1-4 Illustration of Eq. 6.1-19, the time-dependent approach to continuum resonance Raman scattering. Shown is a 2 > 1> vibrational Raman transition in Bra for Aq = 457.9 nm excitation. As examples, (A), (B) and (C) show the potential curves of the relevant ground (X = continuous line) and excited (B = 7o+m, dashed line, and 77 = 7T , dotted line) electronic states, together with the absolute values of the coordinate representations of the initial state It >= 1 >, final state ]f >= 2 >, and the time-dependent state i(r) > at times / = 0, 20 and 40 fs, respectively. The excitation and de-excitation processes and the related unimolecular dissociations are indicated schematically by vertical and horizontal arrows. For clarity of presentation, the energy gap between state (> and f> is expanded (Ganz et al., 1990).

Figure 6.1-5 Stokes transition for continuum resonance Raman scattering in from the initial vibrational state />= 0> to the final state f >= 6> via electronic state B( 77o+). (A) Absolute value of the time overlap < 6 0(t) > as a function of time, (B) excitation profile of this transition [square of the half Fourier transform of < f i t) > as a function of energy] (Ganz and Kiefer, 1993 a).

Based on the two approaches to continuum resonance Raman theory, many spectra of diatomic molecules could be successfully simulated (Hartke, 1989, 1991 Ganz and Kiefer, 1993a, 1994). Besides the good simulations of continuum resonance Raman band... 

Finally, we like to mention that equivalent to the conventional energy frame KHD formulation, the time-dependent theory of Raman scattering is free from any approximations except the usual second order perturbation method used to derive the KHD expression. When applied to resonance and near resonance Raman scattering, the time-dependent formulation has shown advantages over the static KHD formulation. Apparently, the time-dependent formulation lends itselfs to an interpretation where localized wave packets follow classical-like paths. As an example of the numerical calculation of continuum resonance Raman spectra we show in Fig. 6.1-7 the simulation of the A, = 4 transitions (third overtone) of D excited with Aq = 488.0 nm. Both, the KHD (Eqs. 6.1-2 and 6.1-18) as well as the time-dependent approach (Eqs. 6.1-2 and 6.1-19) very nicely simulate the experimental spectrum which consists mainly of Q- and S-branch transitions (Ganz and Kiefer, 1993b). 

Figure 6.1-7 Experimentally observed continuum resonance Raman spectrum for the An = 4...

As discussed above, continuum resonance occurs when the excitation laser energy is higher than the dissociation limit of an excited, bound electronic state or directly with purely repulsive states. Continuum resonance Raman spectra of gaseous molecules are very sensitive to the position and shape of the potential functions involved in this type of light scattering as well as to the electronic transition moments between ground and excited states. Since it is possible to calculate the relevant spectra using both the KHD... 

Figure 6.1-9 Continuum resonance Raman spectra of the fourth overtone (An = 5) of excited with Ap = 363.8 nm. A experiment B, C, D calculated (KHD-approach) with potentials 1, 2, and 3 as shown in Figure 6.1-8, respectively (Strempel and Kiefer, 1992).

The methods of level dynamics can be extended to resonance dynamics in case the energy levels En acquire a width in the presence of a continuum. Resonance dynamics of the one-dimensional helium atom is discussed in Section 10.5.2. 

The very different spectra of iodine obtained under continuum and discrete resonance-Raman conditions are illustrated in Fig. 11 for resonance with the B state, whose dissociation limit is 20,162 cm . In the case illustrated of discrete resonance-Raman scattering, Xl =514.5 nm, and specific re-emission results from an initial transition from the v" = 1 vibrational, J" = 99 rotational level of the X state to the v = 58, J = 100 level of the B state, i.e. the transition is 58 - l" R(99). Owing to the rotational selection rule for dipole radiation, AJ = 1, a pattern of doublets appears in the emission. Clearly, the continuum resonance-Raman spectrum of iodine (Xl = 488.0 nm) is very different from the discrete case spectrum. The structure, which arises from the 0,Q, and S branches of the multitude of vibration-rotation transitions occurring, can be analysed in terms of a Fortrat diagram, as done for gaseous bromine (67). 

Continuum resonance-Raman scattering can be observed under discrete resonance-Raman scattering conditions only if the resonance fluorescence is quenched, either with an inert gas, or (in the case of condensed phase studies) by the solvent or matrix. Thus, on excitation of a liquid, solution, or solid within the contour of an absorption band, the Raman spectrum observed has the characteristics of the continuum rather than the discrete case or, in other terminology, of resonance Raman, rather than resonance fluorescence spectra. Such spectra provide unique information on the spectroscopic properties of radical cations and ions, some of which species are unstable in air. Particularly noteworthy have been the studies by Andrews et al. (65) which have... 

B. Continuum resonance-Raman scattering fromj2 gas with excitation at 488.0 nm which is above the dissociation limit of the B( ITou) <- X( Sg) transition. The fine structure of each vibrational overtone is attributable to the Q, O and S branches of the multitude of rotational transitions occurring... 

Fig. 2. Photon molecule interaction processes. (A) Normal Raman scattering, (B) discrete resonance Raman scattering, (C) continuum resonance Raman scattering. All these processes are amenable to direct scattering experiments generally, only (B) can be easily studied by time-resolved observation.

The large amplitude of the continuum resonance states is a direct result of the non Hermitian properties of Hamiltonian (i.e. the resonance eigenfunctions which are associated with complex eigenvalues are not in the Hermitian domain of the molecular Hamiltonian). Let us explain this point in some more detail. As was mentioned above Moiseyev and Priedland [7] have proved that if two N x N real symmetric matrices H and H2 do not commute, there exists at least one value of parameter A = such that matrix H + XH2 possesses incomplete spectrum. That is at A A there are at least two specific eigenstates i and j for which ]imx j ei — ej) =0 and also lim ( i j) — 0- Since and ipj are orthogonal (within the general inner product definition i.e., i/ il i/ j) = = 0 not in the... 

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