Energy levels of a coupled-rotors model
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Energy levels of a coupled‑rotors model Paolo Amore1 · Francisco M. Fernández2 Received: 24 November 2019 / Accepted: 1 October 2020 © Springer Nature Switzerland AG 2020
Abstract We calculate the energy levels of a coupled-rotors model that has proved useful for the analysis of the multiple tunnelling peaks in the inelastic neutron scattering from 4-methylpyridine crystals. We resort to the Rayleigh–Ritz variational method with a basis set for each irreducible representation. The energy levels in terms of the rotor coupling strength exhibit a rich pattern of crossings between states of different symmetry and avoided crossings between states of the same symmetry. Keywords Coupled rotors · 4-methylpyridine · Avoided crossings · Neutron scattering
1 Introduction Rotational symmetry is commonly studied by means of simple models based on effective Hamiltonians for properly chosen restricted rigid rotors [1–6]. In some cases a single rotor model provides an acceptable description of the experimental data but in others one has to resort to a set of coupled rotors. The Schrödinger equation for such models has been solved in more than one way [4, 6]. In a recent paper [7] we applied a well known algorithm for the solution of band matrices [8–11] that is a convenient alternative approach to the iterative matrix inversion proposed several years ago [4]. We showed that the C3 symmetry of the interaction potential enables one to avoid the problem posed by a small rotational barrier (or large quantum numbers) that may lead to numerical errors due to almost degenerate rotational states [7]. * Francisco M. Fernández [email protected] Paolo Amore [email protected] 1
Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, Mexico
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División Química Teórica, INIFTA, Blvd. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina
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Vol.:(0123456789)
Journal of Mathematical Chemistry
In a somewhat earlier paper Khazaei and Sebastiani [6] (KS from now on) studied the influence of rotational coupling between a pair of methyl rotors on the tunneling spectrum in condensed phase. They considered the simplest model in which two hindered C3 rotors interact by means of a periodic potential. They calculated some of the eigenvalues of the model by means of the Rayleigh–Ritz variational method based on a Fourier basis set. The authors analyzed the lowest energy levels in different regions of coupling strength and found two avoided crossings between two pairs of them. According to KS the avoided crossing closest to the origin is due to a pair of energy levels of symmetry AE and EE. However, states of different symmetry are expected to undergo actual crossings and not avoided ones. Avoided crossings take place between states of the same symmetry [12–14] (and references therein). This confusion probably arose from the fact that KS did not calculate the energy levels for each irreducible representation (irrep) separately but used a Fourier basis set that contains al
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