Abstract.We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Front. Phys. Chem. Chem. Phys. 2, 1 (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -40 in total to be exact -analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [Am. J. Phys. 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index κ, as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters η and ζ. It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given κ, the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of κ, and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.

In this work, we show that van der Waals molecules X-RG (where RG is the rare gas atom) may be created through direct three-body recombination collisions, i.e., X + RG + RG → X-RG + RG. In particular, the three-body recombination rate at temperatures relevant for buffer gas cell experiments is calculated via a classical trajectory method in hyperspherical coordinates [Pérez-Ríos et al., J. Chem. Phys. 140, 044307 ( 2014)]. As a result, it is found that the formation of van der Waals molecules in buffer gas cells (1 K ≲ T ≲ 10 K) is dominated by the long-range tail (distances larger than the LeRoy radius) of the X-RG interaction. For higher temperatures, the short-range region of the potential becomes more significant. Moreover, we notice that the rate of formation of van der Walls molecules is of the same order of the magnitude independent of the chemical properties of X. As a consequence, almost any X-RG molecule may be created and observed in a buffer gas cell under proper conditions.

We study, analytically as well as numerically, the dynamics that arises from the interaction of a polar polarizable rigid rotor with single unipolar electromagnetic pulses of varying length, ∆τ , with respect to the rotational period of the rotor, τ r . In the sudden, non-adiabatic limit, ∆τ τ r , we derive analytic expressions for the rotor's wavefunctions, kinetic energies, and field-free evolution of orientation and alignment. We verify the analytic results by solving the corresponding timedependent Schrödinger equation numerically and extend the temporal range of the interactions considered all the way to the adiabatic limit, ∆τ τ r , where general analytic solutions beyond the field-free case are no longer available. The effects of the orienting and aligning interactions as well as of their combination on the post-pulse populations of the rotational states are visualized as functions of the orienting and aligning kick strengths in terms of populations quilts. Quantum carpets that encapsulate the evolution of the rotational wavepackets provide the space-time portraits of the resulting dynamics. The population quilts and quantum carpets reveal that purely orienting, purely aligning, or even-break combined interactions each exhibit a sui generis dynamics. In the intermediate temporal regime, we find that the wavepackets as functions of the orienting and aligning kick strengths show resonances that correspond to diminished kinetic energies at particular values of the pulse duration. * burkhard.schmidt@fu-berlin.de † bretislav.friedrich@fhi-berlin.mpg.de arXiv:1806.11329v3 [quant-ph] 9 Aug 2018 basis set for expanding the time-dependent wavefunction, with the resultwhere E J = J(J + 1). Note that the expansion coefficients, C J J 0 ,M 0 , are time-independent, in consequence of the fact that the time dependence in Eq. (10) only arises from the e −iJ 2 τ term.The final form of the wavefunction in the sudden limit is given by (for a detailed derivationare the Clebsch-Gordan coefficients and only c J is a function of the kick strengths, cf. Eq. (A5).Throughout the remainder of this paper, we restrict ourselves to the case when the free rotor is initially in its ground state, J 0 = M 0 = 0. As a result, the C J J 0 ,M 0 coefficients in Eq. (11) reduce toThe coefficients C J J 0 ,M 0 arising in Eq. (11) can be found by expanding the time-independent term in Eq. (10) in terms of spherical harmonics,with Γ the gamma function; we applied the Legendre duplication formula [49] to achieve the final form of Eq. (A4).By making use of Eq. (A2) and Eq. (A5) we can evaluateC J J 0 ,M 0 in Eq. (A1) as follows,where J 1 M 1 , J 2 M 2 |J 3 M 3 are the Clebsch-Gordan coefficients, which vanish unless |J −J 0 | ≤ J ≤ J + J 0 and J + J + J 0 is an even integer [49]. Finally, we obtainIn all the analytic results presented in this paper, Eqs. (A5) and (A7) were found to converge satisfactorily for κ and J up to 80 and 50, respectively (e.g. for P η = P ζ = 8 the 80th term of the sum over k in Eq. (A5) is close to 10 −30 for C 50 0,0 ≈ 1...

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We study the role of pairwise long-range interactions in the formation of van der Waals molecules through direct three-body recombination processes A + B + B → AB + B, based on a classical trajectory method in hyperspherical coordinates developed in our earlier works [J. Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014); M. Mirahmadi and J. Pérez-Ríos, J. Chem. Phys. 154, 034305 (2021)]. In particular, we find the effective long-range potential in hyperspherical coordinates with an exact expression in terms of dispersion coefficients of pairwise potentials. Exploiting this relation, we derive a classical threshold law for the total cross section and the three-body recombination rate yielding an analytical expression for the three-body recombination rate as a function of the pairwise long-range coefficients of the involved partners.

We present a study on ion-atom-atom reaction A+A+B$^+$ in a wide range of systems and collision energies ranging from 100~$\mu$K to 10$^5$~K, analyzing the two possible products: molecules and molecular ions. The dynamics is performed via a direct three-body formalism based on a classical trajectory method in hyperspherical coordinates developed in [J. Chem. Phys. \textbf{140}, 044307 (2014)]. Our chief finding is that the dissociation energy of the molecular ion product acts as a threshold energy separating the low and high energy regimes. In the low energy regime, the long-range tail of the three-body potential dictates the fate of the reaction and the main reaction product. On the contrary, in the high energy regime, the short-range of atom-atom and atom-ion interaction potential dominates the dynamics, enhancing molecular formation for the low energy regime.

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