Chaotic evolution of the energy of the electron orbital and the hopping integral in diatomic molecule cations subjected to harmonic excitation

Highlights

• Chaotic dynamics of the electronic parameters.

• Dynamics of diatomic cations with symmetric and asymmetric charge distribution.

• Harmonic excitation of strong correlated systems with nonlinear effects.

Abstract

We analysed the dynamics of the positively charged ions of diatomic molecules (X²⁺ and XY⁺), in which the bond is realized by the single electron. We assumed that the atomic cores separated by the distance $R$ were subjected to the external excitation of the harmonic type with the amplitude $A$ and frequency $\Omega$. We found the ground states of ions using the variational approach within the formalism of second quantization (the Wannier function was reproduced by means of Gaussian orbitals). It occurred that, on the account of the highly non-linear dependence of the total energy on $R$, the chaotic dynamics of cores induced the chaotic evolution of the electronic Hamiltonian parameters (i.e. the energy of the electron orbital $\epsilon$ and the hopping integral $t$). Changes in cation masses or in the charge arrangement does not affect qualitatively the values of Lyapunov exponents in the $A$-$\Omega$ parameter space.

Introduction

While analysing the results of the conventional deterministic chaos theory, one can notice that the complicated dynamics of the physical system does not necessarily result from its intricate structure. Much more significance should be attached to the presence of the nonlinear interactions in the examined system, because they can lead to the exponential divergence of the initially close trajectories in the phase space. Let us recall the example of the three-body system with gravity interactions, which was examined in detail e.g. by Poincare [1] (see also [2]). Similar simple system (Lorenz equations) was successfully applied also to the phenomenon of thermal convection [3], or to the dynamics of the Belousov–Zhabotinsky chemical reaction [4]. And the extremely simple system of this kind is the periodically accelerated pendulum for which the gravity force component implying the motion is proportional to the sine of the swing angle [5].

The conventional chaos theory is fairly well established by now and its results are presented in many scientific treatises [6], [7], [8]. On the other hand, for the case of quantum systems, we cannot speak of full understanding of their dynamics yet [9], [10], [11], [12], [13]. This is due to the fact that these systems are described by much more complex formalism than Newton formalism. Specifically, fermions are described by the Schrödinger equation [14], [15], [16], [17], its relativistic version, i.e. the Dirac equation [18], [19], and the quantum electrodynamics [20], [21]. Boson systems are also subject to the quantization procedure (see e.g. [22], [23]). However, in the case under consideration there is a classical limit, which facilitates the interpretation of the obtained results. The researches carried out so far seem to suggest that there is no quantum system which would behave in the chaotic way (i.e. the one exhibiting the continuous power spectrum or deterministic diffusion) [6]. This fact can be checked by studying the example of the Arnold’s quantum transformation [24] or stricken quantum rotator [25]. However, these quantum systems, which on approach to the boundary of validity of the conventional theory exhibit chaotic behaviour, have their wavefunctions distinctly different from the systems with regular behaviour at the same boundary. The wavefunctions of the free particle in the stadium and in the circle can be compared as the example [26]. The reason for suppression of chaos in quantum system is said to be the finite value of the Planck constant ($h$), which along with the Heisenberg uncertainty principle introduces the indistinguishability of points in the $2N$-dimensional phase space contained within the ${(h∕2\pi )}^N$ volume.

Currently developed research directions related to quantum chaos are based on the methods for solving quantum problems, where the perturbation cannot be considered small [13], [27], [28], [29]. In particular, the statistical descriptions of energy levels are used [30], [31], [32], [33], [34], [35]. The starting point for considerations is the distribution of level spacing between eigenlevels: $P\left(s\right)=〈\delta \left(p-E_j+E_{j+1}\right)〉$. For regular systems, $P\left(s\right)$ has the universal form of the Poisson distribution [36]: $P\left(s\right)=e^{-s}$, i.e. the successive energy levels are not correlated. The universal nature of the distribution means that it is valid for systems belonging to the same symmetry class and does not depend on their individual properties. Based on the Random Matrix Theory, it was shown that in the case of quantum chaotic systems three basic universal distributions can be distinguished: the Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble [37], [38], [39], [40]. Other chaotic criteria were also obtained, e.g. the spectral stiffness [41], the autocorrelation function of energy levels velocity [42], [43], and the noise of $1∕f$ type [44]. Please note that the $1∕f$ feature is universal, i.e. this behaviour is the same for all kinds of chaotic systems, independently of their symmetries. Another approach is based on the semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with the quantum features [45]. In addition, studies that directly refer to the correspondence principle are worth emphasizing [13], [29]. Recently, the out-of-time-order correlators (OTOC) have been intensively discussed as a measure for quantum chaos [46], [47], [48], [49], [50], [51]. The OTOC are useful to quantify quantum chaos by defining the quantum analogue of the Lyapunov exponent [52], [53]. The correlators were first studied in the context of theory of superconducting state [54]. Note that the discussed method of analysis is very universal, as it was also used in context of the quantum gravity, the anti-de Sitter/conformal field theory correspondence, the field theories, and the many-body physics (including the many-body localization [47], [55], [56], [57]).

The presented paper analyses the dynamics of positively charged ions of diatomic molecules (${\mathrm{X}}_2^{+}$ or ${\mathrm{XY}}^{+}$), whose atomic cores are subjected to the harmonic excitation. The chaotic dynamics analysed in the study is that of the classical mechanical system of the nonlinear oscillator subjected to the harmonic force, whose potential is classically given by the ordinary nonlinear function (which is quantum-mechanically evaluated beforehand by the adiabatic approximation [58]). Let us note that at present the theoretical description of small molecular systems is so developed that it is possible to calculate their physical or chemical properties with high accuracy [58], [59], [60], [61], [62], [63]. From the point of view of the issues discussed in the paper, the particularly noteworthy are the results obtained for such molecules as: ${\mathrm{Li}}_2^{+}$, ${\mathrm{Na}}_2^{+}$, ${\mathrm{LiNa}}^{+}$, ${\mathrm{K}}_2^{+}$, and ${\mathrm{LiH}}^{+}$; the cation ${\mathrm{Cu}}_2^{+}$ or other examples [64], [65], [66], [67], [68], [69], [70], [71], [72], [73]. The high accuracy of obtained theoretical predictions for the small molecular systems results from the possibility of exact diagonalization of the electron Hamiltonian, since these systems contain at most a few electrons. As standard, calculations are made under the formalism of the second quantization [74], which enables the strict consideration of many-body interactions in issues related to chemistry and solid state physics [75]. In the case at hand, the electron Hamiltonian is conveniently written in the form proposed by Hubbard [76], [77] due to the fact that this approach makes it possible to easily distinguish one- or two-body contributions to the electron energy. The diatomic molecules ${\mathrm{X}}_2^{+}$ or ${\mathrm{XY}}^{+}$ are rather unique from the chaos theory point of view, because they consist of two correlated subsystems of very simple structure: the conventional one (the atomic cores or – in the extreme case of hydrogen – protons) and the purely quantum one (the electron). Let us notice that this description of the molecule is based on the Born–Oppenheimer approximation [58], which makes use of the fact that the atomic cores can have thousands of times greater mass than the single electron. Therefore their motion is slower by several orders of magnitude than the motion of electrons. Reversely, electrons adapt themselves ‘immediately’ to the changed position of cores. Because of this one can examine the influence of the chaotic dynamics of atomic cores, resulting from the existence of the highly nonlinear internuclear potential, on the time evolution of the parameters of the electronic Hamiltonian (energy of the electron orbital $\epsilon$ and the hopping integral $t$). This does not mean that the quantum subsystem (the electronic one) will evolve in the chaotic manner. Nevertheless it will respond to the behaviour of atomic cores, so that the model described here can serve as the basis for investigation of the changing dynamics of electrons in order to determine the influence of the chaotic evolution of the core subsystem on the quantum electronic system.

The structural simplicity of the considered systems is well worth attention, since it plays the considerable role. It enables to perform complicated quantum-mechanic calculations with the utmost accuracy (which is demanded in quantum chemistry) [58], [59], [60], [63]. We performed calculations for the presented work with an accuracy to six decimal places.

It is worth noticing that during the performed analysis we took into account cations with different core masses and the asymmetric charge distribution between the cores. It allowed us to show the universal character of the chaotic behaviour of cores in the whole family of diatomic cations with the molecular bond realized by the single electron.