Non-Born-Oppenheimer electronic and nuclear densities for a Hooke-Calogero three-particle model: Non-uniqueness of density-derived molecular structure

Abstract

We consider the calculation of non-Born-Oppenheimer, nBO, one-particle densities for both electrons and nuclei. We show that the nBO one-particle densities evaluated in terms of translationally invariant coordinates are independent of the wavefunction describing the motion of center of mass of the whole system. We show that they depend, however, on an arbitrary reference point from which the positions of the vectors labeling the particles are determined. We examine the effect that this arbitrary choice has on the topology of the one-particle density by selecting the Hooke-Calogero model of a three-body system for which expressions for the one-particle densities can be readily obtained in analytic form. We extend this analysis to the one-particle densities obtained from full Coulomb interaction wavefunctions for three-body systems. We conclude, in view of the fact that there is a close link between the choice of the reference point and the topology of one-particle densities that the molecular structure inferred from the topology of these densities is not unique. We analyze the behavior of one-particle densities for the Hooke-Calogero Born-Oppenheimer, BO, wavefunction and show that topological transitions are also present in this case for a particular mass value of the light particles even though in the BO regime the nuclear masses are infinite. In this vein, we argue that the change in topology caused by variation of the mass ratio between light and heavy particles does not constitute a true indication in the nBO regime of the emergence of molecular structure.

Molecular structure calculations: A unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation

“…Note also that (27) and the zero overall momentum of the basis function in the LF Cartesian representation is guaranteed if c u = 0. The condition c u = 0 is equivalent to the requirement It is convenient to consider various coordinates for the calculations, for at least two reasons.…”

Some Remarks on the Mass Density Distribution

“…However, we can define some auxiliary tools which may be helpful in understanding the process of emerging the molecular shape from a shapeless, spherically-symmetric structure given by the square of the many-body wavefunction. One of these tools is the operator of the oneparticle density of mass 4,12 exemplified in Eqs. 10 and 58.…”

“…3. Very recently, in two very interesting works 4,5 Ludẽna et al have demonstrated that the one-particle density of mass distribution, for a state described by a given wavefunction, strongly depends on the choice of the reference point. In a way, this issue may be traced back to the very beginning of the modern science: the difference between the Ptolemean and the Copernican representations of the Solar System may also be reduced to a difference in the choice of the reference point.…”

“…Last but not least, the form of the function which describes the mass distribution depends on the choice of the reference point in the laboratory frame. [3][4][5] This technical problem, appearing also in the celestial mechanics, adds another difficulty to the way the transition between the free and frozen nuclei models may be performed.…”

“…[8][9][10][11][12][13][14][15][16] By an easy tracing the mass-dependence of the analytical solutions they lead to a better understanding of some specific features of the BO approximation and allow to trace the process of the emergence of the molecular structure from the spherically-symmetric non-BO objects. 4,5,[12][13][14][15][16] In this report we investigate the distribution of the density of mass in three exactly solvable model systems: two particles bound by the Coulomb force, and two and three particles connected by the Hooke force. In particular, the relation between the degree of the localization of a particle and its mass is described in some detail.…”