Non-Born–Oppenheimer nuclear and electronic densities for a three-particle Hooke–Coulomb model

Highlights

• A three-particle quantum system is investigated within the non-Born–Oppenheimer regime.

• The method for the calculation of NBO one-particle densities is described.

• The described method is applied to the H⁻, ${\text{H}}_2^{+}$, and Ps⁻ systems.

• It is not possible to derive a unique molecular structure form NBP electronic densities.

Abstract

Non-Born–Oppenheimer, nBO, one-particle nuclear and electron densities for a Hooke–Coulomb model of a three-body system are presented. These densities are obtained using exact closed-form analytic solutions to this problem as well as variational solutions. Moreover, the densities are calculated using different reference points, such as the global center of mass [cm], the geometric centers between both identical [gc12] and non-identical particles [cm13], and the location of the non-identical particle [p3]. It is shown that the topology of these nBO densities depends upon the choice of the reference points. This result is in turn used to argue that in a nBO regime the topological properties of the one-particle density cannot be univocally correlated with molecular structure, in the way it is done for the Born–Oppenheimer approximation.

Introduction

The Born–Oppenheimer, BO, approximation has been, without any doubt, a fundamental concept underlying the development of quantum-mechanical approaches to the electronic structure of molecular and crystalline systems. However, the BO approximation breaks down for many systems and processes of chemical and physical interest (see, for example, Refs. [1], [2], [3], [4], [5], [6], [7], [8]).

Thus, in sight of this breakdown of the BO approximation and bearing in mind that some important chemical and physical phenomena occurring at the atomic level strongly depend on nuclear quantum effects, recent efforts in the field of quantum theory have been directed toward the development of methods based on a non-Born–Oppenheimer, nBO, regime [2], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. In the particular case of chemistry, the attention placed on nBO treatments has reopened the debate of whether molecular structure, a basic concept in chemistry [25] can be derived from wave functions that include the behavior of non-clamped nuclei [22], [26], [27], [28], [29], [30], [31].

The problem is that the average distances and angles obtained from nBO wave functions generally are not sufficient to extract from them molecular structure [32], [33]. However, let us mention that as shown in recent work of Mátyus et al. [31] the triangular structure of the ${\text{H}}_2{\text{D}}^{+}$ ion can be retrieved from the angular density corresponding to a highly accurate nBO variational wave function.

In the context of Bader’s Quantum Theory of Atoms in Molecules, QTAIM [34] a very basic relation has been shown to exist between the concept of molecular structure and the topology of the BO one-electron density, obtained either through theoretical or experimental means (in the latter case for solids) [34], [35], [36], [37]. In QTAIM, the one-electron density comes from a clamped-nuclei wave function. Recently, however, this restriction has been removed in the TC-QTAIM where topological atoms are constructed from a joint density made up from electronic and nuclear ones calculated in a nBO context [38], [39], [40], [41], [42].

Since nBO treatments are becoming available, the open question is whether from the topological features of nBO one-particle densities (i.e., electronic and nuclear) we may obtain molecular structure. This question, however, does not have an easy answer since the inclusion of the nuclear motion in the Schrödinger equation makes its solution much more difficult to obtain. As a result, analytical and variational wave functions, and associated one-particle densities, have been computed only for a small group of simple systems consisting of few particles: ${\text{H}}_2^{+}$ [22], [30], [43], [44] ${\text{HD}}^{+}$ [45] ${\text{HT}}^{+}$ [46] ${\text{H}}_2$ [22] ${\text{H}}_2{\text{D}}^{+}$ [47] ${\text{LiH}}^{+}$ [48] ${\text{H}}_3$ [49] and $\text{HD}$ [50]. Moreover, in these calculations there has been an implicit use of a given reference point from which the density is defined. As different reference points have been employed in the above calculations, it is difficult to compare the topological features of the ensuing densities [51], [52].

An alternative way for examining this question is provided by the use of exactly solvable models for few-particle systems as they yield analytic expressions for the one-particle densities. In particular, the Hooke–Calogero three-particle model has been previously employed in order to examine whether molecular structure arises in a nBO regime [53], [51]. However, as pointed out by Ludeña et al. [51] the topology of the nBO one-particle density depends upon the selection of the reference point. This non-uniqueness of the one-particle densities, observed for the Hooke–Calogero model showed that there does not exist a one-to-one relationship between density and molecular structure.

In the present work, we go beyond the Hooke–Calogero model [51] and introduce a more realistic Hooke–Coulomb model for a three-particle system where the harmonic interaction between identical particles are replaced by the true Coulomb potential. As in the present case, we can also obtain analytic solutions for the nBO Schrödinger equation, we calculate and analyze close-form expressions for electronic and nuclear one-particle densities for this three-particle system. However, since the analytic solutions are restricted to selected values of the coupling constant, we also obtain general variational solutions to this problem. We examine for both types of wave functions the effect that the selection of the reference points has on the topologies of the nuclear and electronic one-particle densities. In Section 2, we describe the Hooke–Coulomb model employed in the present work, In Section 3, we discuss the calculation of one-particle densities for different reference points, and consider the specific cases of the ${\text{H}}^{-}$, ${\text{H}}_2^{+}$, and ${\text{Ps}}^{-}$ systems. Finally in Section 4, we present some conclusions.