Diabatic State Model

Explanations for the existence of hypervalent species started with Pauling s proposal of d orbital hybridization [3] where, for example, a set of sp d hybrid orbitals on sulfur was put forward to account for the hexavalence of SFg. Chemical computations [4] ruled out the participation of d atomic orbitals. A later model, the Rundle-Pimentel three-center, four-electron (3c-4e) bonding model [5], does not require d orbital participation and, so, is consistent with chemical computations. Other hypervalency models include the diabatic state model of Dixon and coworkers [6] and the democracy principle of Cooper and coworkers [7]. These various models provide useful insights into some aspects of the electronic structure of hypervalent molecules, but they don t provide an overarching description of these molecules that enables connections to be drawn between hypervalency and related molecular phenomena. 

Tbe model is that tbe ground-state PES is first altered by the electronic excitations (on-diagonal coupling leads to a change in equilibrium geomeby and frequency), and these smooth diabatic states are then further altered by vibronic (off-diagonal) coupling. 

The motivation comes from the early work of Landau [208], Zener [209], and Stueckelberg [210]. The Landau-Zener model is for a classical particle moving on two coupled ID PES. If the diabatic states cross so that the energy gap is linear with time, and the velocity of the particle is constant through the non-adiabatic region, then the probability of changing adiabatic states is... 

Appendix C On the Single/Multivaluedness of the Adiahatic-to-Diahatic Transformation Matrix Appendix D The Diabatic Representation Appendix E A Numerical Study of a Three-State Model Appendix F The Treatment of a Conical Intersection Removed from the Origin of Coordinates Acknowledgments References... 

Let us next turn to Model II, representing the C —> B —> X internal-conversion process in the benzene cation. Figure 2 demonstrates that this (compared to the electronic two-state model, Model I) more complicated process is difficult to describe with a MFT ansatz. Although the method is seen to catch the initial fast C —> B decay quite accurately and can also qualitatively reproduce the oscillations of the diabatic populations of the C- and B-state, it essentially fails to reproduce the subsequent internal conversion to the electronic X-state. Jn particular, the MFT method predicts a too-slow population transfer from the C- and B-state to the electronic ground state. 

Figure 9. Population of the initially prepared diabatic state for Model IVb. Shown are results of a generalized MET method where all modes (dashed line), the two lower frequency modes (dashed-dotted line), and the mode with the lowest frequency (dotted line) have been treated classically, respectively. The full line shows the quantum-mechanical result.

Figure 13. Comparison of quantum (thick hues), QCL (thin lines), and SH (dashed lines) results as obtained for the one-mode two-state model IVa [205], Shown are (a) the adiabatic excited-state population P i), (b) the corresponding diabatic population probability and (c) the...

The quantum results (thick lines) of the sequential model (a) are seen to completely decay in the diabatic electronic ground state of the system. For short times, furthermore, the model predicts transient electronic population in the intermediate state. In the superexchange model (b), on the other hand, there is a nonvanishing long-time population of the initial diabatic state, whereas the intermediate state is hardly ever populated. We have performed classical mapping simulations with ZPE correction (a) y = 0.6 and (b) y = 0.34. The ZPE corrections have been calculated via Eq. (98) and are only employed to the... 

The GMH method of Cave and Newton [39, 40] is based on the assumption that the transition dipole moment between the diabatic donor and acceptor states vanishes, i.e., the off-diagonal element of the corresponding dipole moment matrix is zero. Thus, in the localization transformation one diagonalizes the dipole moment matrix of the adiabatic states ij/i and ij/z. For a two-state model, the rotation angle ft) can be expressed with the help of the transition dipole moment and the difference of the dipole mo-... 

Let a crossing of diabatic surfaces of potential energy occur in a certain point R0. Taking into account only the linear expansion term of the difference between the energies of the diabatic states near the crossing point (the Landau-Zener model)... 

Fig. 10 For the 3-state model of Sec. 5.2., projections of the coupled diabatic XT, CT and IS potential surfaces (E configuration) onto the XT-CT branching plane are shown. The white and black circles indicate the conical intersection and Franck-Condon geometry, respectively. Reproduced from Ref. [52]. Copyright 2008 by the American Physical Society.

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