Conical intersections theoretical principles

In Chapter VIII, Haas and Zilberg propose to follow the phase of the total electronic wave function as a function of the nuclear coordinates with the aim of locating conical intersections. For this purpose, they present the theoretical basis for this approach and apply it for conical intersections connecting the two lowest singlet states (Si and So). The analysis starts with the Pauli principle and is assisted by the permutational symmetry of the electronic wave function. In particular, this approach allows the selection of two coordinates along which the conical intersections are to be found. 

In principle, the theory of nonlinear spectroscopy with femtosecond laser pulses is well developed. A comprehensive and up-to-date exposition of nonlinear optical spectroscopy in the femtosecond time domain is provided by the monograph of Mukamel. ° For additional reviews, see Refs. 7 and 11-14. While many theoretical papers have dealt with the analysis or prediction of femtosecond time-resolved spectra, very few of these studies have explicitly addressed the dynamics associated with conical intersections. In the majority of theoretical studies, the description of the chemical dynamics is based on rather simple models of the system that couples to the laser fields, usually a few-level system or a set of harmonic oscillators. In the case of condensed-phase spectroscopy, dissipation is additionally introduced by coupling the system to a thermal bath, either at a phenomenological level or in a more microscopic maimer via reduced density-matrix theory. 

To illustrate the practical application of the theoretical formalism, we have discussed several representative examples that demonstrate the potential of femtosecond time-resolved spectroscopy. Since typically only a few degrees of freedom (say, 2-4) are involved in the ultrafast photodynamics via a conical intersection, explicit simulations of spectroscopic experiments for polyatomic molecules are nowadays possible on a routine basis. As an example of a state-of-the-art simulation of a femtosecond experiment on a polyatomic system, we have presented first-principles calculations of the time-resolved photoelectron spectra of pyrazine. Although in this case the interpretation was found to be quite complicated, the investigations nevertheless demonstrated the considerable potential of a joint experimental/theoretical study of the dynamics at conical intersections. Employing... 

In principle, one can carry out the analysis of conical intersections for any problem of n electrons and n orbitals in the manner discussed in the previous section. However, it does get increasingly rather complicated. Nevertheless, the 6 orbital with 6 electrons problem merits discussion because the VB structures of benzene in ground and excited states is such a fundamental part of basic chemistry. Accordingly, in other work, our strategy was to determine the complete space of Sq/Si conical intersections for benzene" and to analyze the computed conical intersection stmctures obtained at the ab initio level using the MMVB method. One could then deduce the VB analysis a posteriori by comparison with various theoretical hypotheses about the nature of the intersecting states. 

In Eq. (7.1), states A and B are the two electronic states ground and excited states associated with the conical intersection, is the yth mass-weighted Cartesian coordinate of the t th atom, the index i labels the N atoms and y the Cartesians components, x, y, and z. These quantities are in principle obtainable only from a theoretical calculation. Nevertheless, as we shall discuss subsequently, they have a simple interpretation and one can often make a reasonable guess as to the nature of these two vectors using qualitative valence bond theory. 

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