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The Journal of Chemical Physics

Semiclassical behavior at a quantum avoided crossing

Marc Joyeux

Published Feb 15, 1995• 7 citations
Publisher

Abstract

For a polynomial potential with resonant fundamental frequencies (1:2 and 1:3 resonances), quantum avoided crossings can occur when quantum eigenvalues are plotted versus a parameter in the Hamiltonian. In the present paper, primitive (EBK) semiclassical behavior at the quantum avoided crossing is reinvestigated, using the exact analytical calculation of the action integrals, which was devised recently [Chem. Phys. 185, 263 (1994)] for an approximate resonance Hamiltonian that can be deduced from the exact polynomial Hamiltonian by low order perturbation theory. The previously reported behavior, that is semiclassical levels passing through the intersection instead of avoiding each other, is shown to happen if there exist two superimposed branches in the plot of the second action integral ℐ2 as a function of the energy. These results are interpreted in terms of semiclassical diabatic basis and of quantum dynamical tunneling. In contrast, if the semiclassical system enters the (anti)crossing region with semiclassical quantum numbers ℐ2 which do not lie on superimposed branches of the plot, it is shown that at least one, and possibly two, level(s) must cross the separatrix, that is pass from the inside to the outside of the resonance region (or conversely) in order to adapt to the quantum avoided crossing. This causes (i) corresponding semiclassical quantum number ℐ2 to change (ii) the close correspondence between quantum and semiclassical mechanics to break down.

DOI
10.1063/1.468659

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nlin.CDCross-listedarXiv:2602.04793•Feb 5, 2026
For a polynomial potential with resonant fundamental frequencies (1:2 and 1:3 resonances), quantum avoided crossings can occur when quantum eigenvalues are plotted versus a parameter in the Hamiltonian. In the present paper, primitive (EBK) semiclassical behavior at the quantum avoided crossing is reinvestigated, using the exact analytical calculation of the action integrals, which was devised recently [Chem. Phys. 185, 263 (1994)] for an approximate resonance Hamiltonian that can be deduced from the exact polynomial Hamiltonian by low order perturbation theory. The previously reported behavior, that is semiclassical levels passing through the intersection instead of avoiding each other, is shown to happen if there exist two superimposed branches in the plot of the second action integral ℐ2 as a function of the energy. These results are interpreted in terms of semiclassical diabatic basis and of quantum dynamical tunneling. In contrast, if the semiclassical system enters the (anti)crossing region with semiclassical quantum numbers ℐ2 which do not lie on superimposed branches of the plot, it is shown that at least one, and possibly two, level(s) must cross the separatrix, that is pass from the inside to the outside of the resonance region (or conversely) in order to adapt to the quantum avoided crossing. This causes (i) corresponding semiclassical quantum number ℐ2 to change (ii) the close correspondence between quantum and semiclassical mechanics to break down.
The quantal and classical/semiclassical behavior at an isloated avoided crossing are compared. While the quantum mechanical eigenvalue perturbation parameter plots exhibit the avoided crossing, the corresponding primitive semiclassical eigenvalue plots pass through the intersection. Otherwise, the eigenvalues agree well with the quantum mechanical values. The semiclassical splitting at the intersection is calculated from an appropriate Fourier transform. In the quasiperiodic regime, a quantum state near an avoided crossing is seen to exhibit typically more delocalization than the classical state. However, trajectories near the ‘‘separatrix’’ display a quasiperiodic ‘‘transition’’ between two zeroth order classical states.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.
Abstract We study the Bohmian trajectories of a perturbed two-dimensional quantum harmonic oscillator and examine how avoided crossings in the quantum energy spectrum influence their dynamical behaviour both in non-resonant and resonant cases. Using superpositions of a finite number of eigenstates of the perturbed Hamiltonian, we show that avoided crossings strongly enhance the prevalence of chaotic trajectories within the Born-rule distribution. The use of eigenstates of the full Hamiltonian allows ordered Bohmian trajectories to persist, in contrast to constructions based on unperturbed states. We further compare the Bohmian behaviour with the corresponding classical dynamics at comparable energies, identifying parameter regimes in which the two descriptions exhibit markedly different sensitivity to perturbations.
A serious difficulty in the semiclassical Ehrenfest theory for nonadiabatic transitions is that a path passing across the avoided crossing is forced to run on a potential averaged over comprising adiabatic potential surfaces that commit the avoided crossing. Therefore once a path passes through the crossing region, it immediately becomes incompatible with the standard view of "classical trajectory" running on an adiabatic surface. This casts a fundamental question to the theoretical structure of chemical dynamics. In this paper, we propose a non-Born-Oppenheimer path that is generated by an anti-Hermitian Hamiltonian, whose complex-valued eigenenergies can cross in their real parts and avoid crossing in the imaginary parts in the vicinity of the nonadiabatic transition region. We discuss the properties of this non-Born-Oppenheimer path and thereby show its compatibility with the Born-Oppenheimer classical trajectories. This theory not only allows the geometrical branching of the paths but gives the nonadiabatic transition amplitudes and quantum phases along the generated paths.
In this paper, we examine the behavior of the quantum energy levels of a coupled oscillator system as the zero-order frequencies are varied to carry the corresponding classical system through a resonance. We find that the levels exhibit a pattern that is characteristic of the resonance. This pattern consists of clusters of levels, each containing a number of curves that run roughly parallel to one another and a number of curves that undergo pairwise narrowly avoided crossings. Adiabatic switching calculations show that the ‘‘parallel’’ curves are associated with states within the classical resonance region, while the narrowly avoiding curves are associated with states that are outside this region. It is further shown that the curves describing resonance states are formed from zero-order nonresonant curves by the overlap of many avoided crossings. This reorganization of multiply intersecting lines into parallel curves reflects the classical reorganization of phase space at a resonance.
Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, this leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li⇆\leftrightarrows⇆Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to ℏ=0\hbar=0ℏ=0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit ℏ→0\hbar\to0ℏ→0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.