Perturbation theory conical intersections

D. Perturbation Theory, Time-Reversal Symmetry, and Conical Intersections... 

In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons are reviewed. Section II presents the mathematical basis for these developments, which exploits a degenerate perturbation theory previously used to describe conical intersections in nonrelativistic systems [11,12] and Mead s analysis of the noncrossing rule in molecules with an odd number of electrons [2], Section III presents numerical illustrations of the ideas developed in Section n. Section IV summarizes and discusses directions for future work. 

This analysis is heuristic in the sense that the Hilbert spaces in question are in general of large, if not infinite, dimension while we have focused on spaces of dimension four or two. A form of degenerate perturbation theory [3] can be used to demonstrate that the preceding analysis is essentially correct and, to provide the means for locating and characterizing conical intersections. 

We caution that this level of theory was chosen in part for its computational efficiency CASSCF does have some deficiencies, and in particular is unable to describe dynamic electron correlation. The PESs for ethylene from SA-3-CAS(2/2) are qualitatively similar to the PESs from higher level calculations, but dynamics calculations with higher levels of theory (e.g., including perturbation theory corrections with MS-CASPT2 [27] and/or including diffuse orbitals to represent the Rydberg states [28]) suggest that there is a tendency for SA-3-CAS(2/2) dynamics to overestimate the importance of the ethylidene conical intersection. 

All the analysis and discussion of the preceding subsection can now be carried over to the present situation. If perturbation theory is valid and real electronic wavefunctions are used, the lowest order contributions to the energy in growing powers of k listed in Sec. 5.1 apply also here. One, of course, has to identify the quasi-rigid motion and the soft motion in Sec. 5.1 with vibrational and rotational motion, respectively. Then, the discussion in Sec. 5.1 for cases in which perturbation theory breaks down, in particular in the presence of conical intersections, also remains valid. Where are the differences between the general analysis in Sec. 5.1 and the present one for quasi-rigid molecules First, mass polarization, see Eq. (48), contributes here in the order of. This contribution is obviously missing in Sec. 5.1, where the translational motion has not been separated off a priori. However, as discussed there, the translational motion starts to contribute... 

The singular behaviour of the adiabatic energies, wave functions and derivative couplings near a conical intersection makes a formal analysis of that region highly desirable. This analysis is accomplished using a generalization of the perturbation theory developed by Mead in his seminal treatment of X3 molecules. ... 

As will be outlined in more detail below, the much higher complexity of the d3mamics at conical intersections calls for a new strategy for the calculation of absorption and emission signals, which differs from the established formalism of nonlinear optics, based on higher-order (t3 ically third-order) perturbation theory in the laser-matter interaction. Independently of... 

In this subsection, we employ standard time-dependent perturbation theory with respect to the laser-matter interaction i/mt(i) in the wave-function formalism. Time-dependent perturbation theory may alternatively be formulated in the density-matrix formalism. The latter is more general in that it allows for the inclusion of finite temperature effects and a phenomenological description of relaxation phenomena. In this chapter, we are concerned with the description of the dynamics at conical intersections in a fully microscopic manner, and temperature effects play a minor role. The wave-function formalism is therefore appropriate for our purposes. Writing the time-dependent wave function as... 

Levine BG, Coe JD, Martinez TJ (2008) Optimizing conical intersections without derivative coupling vectors application to multistate multireference second-order perturbation theory (MS-CASPT2). J Phys Chem B 112 405... 

Quantum dynamics in the conical intersection To assist in the analysis of the quantum dynamics in internal conversion (IC) processes a number of papers have been presented which analyse the process in terms of perturbation theory, in terms of effective modes and quadratic coupling, careful analysis of the geometric phase effects near the CIX, or propose simple models to bring in a clearer picture of the fundamentals of the process. 

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