Effect of the geometric phase on nuclear dynamics at a conical intersection: Extension of a recent topological approach from one to two coupled surfaces On the characterization of three state conical intersections: A quasianalytic theory using a group homomorphism approachThe properties of the five-dimensional branching space of conical intersections of three states of the same symmetry ͑denoted i,j,k͒ are considered. The results of a perturbative model are compared with multireference configuration interaction calculations for three spectroscopically observed states of the allyl radical. Of particular interest is the three-dimensional subspace of the branching space where two states remain degenerate. The energies, derivative couplings and geometric phase effect are studied in the neighborhood of this degeneracy subspace. The degeneracy subspace includes two kinds of conical intersections, i,j and j,k. The existence of a three-state intersection impacts the phase of the wave functions ͑and the derivative coupling͒ traversing a closed loop. For example, in the branching space, the number and kind of conical intersections in a surface bounding the closed loop is constrained if the closed loop contains the three-state intersection.
The location and consequences of linked seams of conical intersections, conical intersections of states (J,K) and (K,L), are considered. We show that this class of conical intersections gives rise to the induced geometric phase effect, as a result of which the derivative couplings may be double-valued. This double-valuedness has important consequences, some limiting others not. We show, using an analysis based on branch cuts, that if the derivative coupling is double-valued, its circulation, its line integral about a closed loop, is not a unique function of the path, being starting point dependent. On the other hand the change from single-valued to double-valued derivative couplings can be used to search for linked intersections.
The decay mechanisms of the metastable 2,3 3Πg states of Al2 are investigated. Both nonadiabatic radiationless decay to the dissociative 1 3Πg state and radiative decay to the ground X 3Πu state are considered. The 1,2,3 3Πg states are described using state averaged multiconfiguration self consistent field/configuration interaction wave functions [ψam(r,Q)]. The derivative couplings famn(Q)≡〈ψam(r, Q)‖(d/dQ)ψan(r,Q)〉r are determined and used to construct a rigorous diabatic basis for this strongly interacting three state problem. The 2 3Πg state and somewhat surprisingly the 3 3Πg state are rapidly predissociated by the dissociative 1 3Πg state. The lifetimes for nonradiative decay of the vibrational levels of the 2 3Πg state are on the order of picoseconds while those of the 3 3Πg state are on the order of nanoseconds being reduced from the direct coupling (3 3Πg∼1 3Πg) rate of milliseconds by indirect coupling through the 2 3Πg state, (3 3Πg∼2 3Πg∼1 3Πg). Radiative decay is found to be on the order of 102 and 30 ns for the 2 3Πg and 3 3Πg states, respectively, so that radiationless decay is principal decay mechanism. Significant variation in the lifetimes of the individual vibrational levels of the 2,3 3Πg states is expected. This is attributed to the mechanism of the predissociation which involves nonadiabatic interactions near the ‘‘inner walls’’ of the 1,2 3Πg states. Although avoided crossings strongly affect the properties of the 1,2,3 3Πg states the adiabatic basis is preferred over the diabatic basis both conceptually and computationally.
The description of resonances originating from several coupled electronic states in a diabatic or approximate diabatic basis can offer both conceptual insights and computational challenges. In a three-state problem, two bound electronic states strongly coupled to a single dissociative continuum, large resonance energy shifts (thousands of cm−1), and linewidths varying over 4 orders of magnitude can be encountered. In this work a nonperturbative computational approach is developed to treat this class of resonances. Expressions for both the radiative and radiationless decay rates are developed. Although the approach is nonperturbative, the linewidth is expressed in a Golden-Rule-type formula. The resonance energy is obtained from the iterative solution of an eigenvalue problem in the bound state space. These attributes enable efficient determination both narrow and broad linewidths and large resonance energy shifts. The approach is used to characterize both radiative and radiationless decay of the 2,3 3Πg states of Al2 using a rigorous three-state diabatic basis. Lifetimes ranging from tenths of picoseconds to nanoseconds are determined. The corresponding resonance energy shifts are on the order of 4000 cm−1.
The idea of dilation, or dilatation, analyticity with respect to complex scaling of the interparticle distances for nonrelativistic atomic or molecular electronic Hamiltonians is now over 20 years old and the first major reviews are just over 10. The method continues to be a fruitful source of new theoretical and computational results. Under the scale transformation r = re", the usual "spectrum" of bound states is exactly preserved, and scattering continua are "rotated" off the real axis by an angle of -2Re(e) about their respective thresholds. Useful features of this transformation are (1) that resonances are exposed, and, thus, (complex) resonance eigenvalues are easily calculated as the wave functions are L2, and standard results of Kato-type perturbation theory can thus be applied to them; (2) that utilization of this technique to study atoms in ac and dc fields was an early, and still evolving extension of the original theory; and (3) the fact that the "continua" are rotated off the real energy axis allows scattering information to be extracted from computations carried out entirely L2 in bases as the usual resolvents of scattering theory are no longer singular on the real axis. After a brief survey of the technique and its applications, these ideas are illustrated by discussion of positive energy-bound states and resonances, the extension of the theory to include the dc Stark effect, and a review of the resolution of the initially perplexing problem of atomic and molecular bound states in continua. These theoretical results are followed by a discussion of some very recent computational results, allowing computation of atomic partial photoionization cross sections with no specific coordinate space enforcement of boundary conditions, a highly advantageous situation for calculation of the partial cross section for three-body breakup, as in the process ho + He * He2++ e-+ e-. 0
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