Bound primitive semiclassical diatomic electronic states

Abstract

We report the first examples of bound molecular electronic states computed using the primitive semiclassical approximation. This method is based directly on classical actions derived from classical trajectories, and as such is closely related to the Bohr method for atoms. The examples chosen are neutral one-electron diatomic molecules, with fractional nuclear charges; both polar and non-polar cases are considered. It is the reduction of the nuclear repulsion from that experienced by one-electron cations which allow this semiclassical method to find bound states. The errors in the bond length and energies decrease to less than a percent as the quantum numbers are increased; Rydberg states are computed with much higher accuracy. The implications for interpretation of chemical bonding are briefly discussed.

Introduction

Some time ago Strand and Reinhardt [1] presented the canonical semiclassical description of the electronic states of the hydrogen molecule cation H₂⁺; we refer to this work as S&R. For several states, they reported total energies less than the energy of the asymptotic atomic state [2], an indication that the state is stable (ground state) or metastable (excited states) from the classical point of view. However, the classical action J_η suffers a discontinuity at some value of the internuclear separation R for every state of H₂⁺; in the primitive semiclassical approximation (PSC), the energy eigenvalue, which depends on this action, is then also discontinuous. Usually the discontinuity leads to a gap, a range of values of the internuclear distance R for which no PSC approximation to the electronic Born–Oppenheimer energy is available. The discontinuity occurs at an internuclear distance corresponding to the onset of molecule formation from the separated atoms. As a result, no state in which the primitive semiclassical total energy has a minimum (dE_tot/dR=0) was located. Since the discontinuity occurs for all electronic states of H₂⁺, there is little chance that any state possessing a minimum will ever be located (except for Rydberg type states, as discussed below).

We extended the primitive semiclassical analysis to polar molecules [3]. The structure of the classical phase space for the polar one-electron molecules differs from that of the non-polar cation H₂⁺. In polar molecular ions such as HeH²⁺ and LiH³⁺, some semiclassical states avoid the discontinuity in the action and consequently have electronic energy curves continuous at all R. However, none of these states have been found to be bonding (again, except for Rydberg states). Some other states of the polar cations were found to possess total energies less than the corresponding asymptotic atomic energy and so might be classified as “bonding”. However, all display the same type of energy gap as do the “bonding” states of H₂⁺. Once again, no minimum in the total energy curve occurs. For example, the 2pσ state of HeH²⁺ state is a metastable state which has been observed [4], [5] experimentally. For some internuclear distances, PSC molecular electronic energies are less than the asymptotic atomic energy of the state to which it correlates (in this case, H(1s)). But the semiclassical solution fails to exist in the gap region, which is where the energy minimum would have been attained.

One primary reason for the lack of stable states is that the large nuclear charges of the highly polar one-electron ions tend to blow the diatomic molecule apart. Therefore we report here on two artificial cases with fractional nuclear charges chosen to give neutral polar and non-polar one-electron diatomic molecules. The earliest reference to such molecules we have found is by Bates et al. [6]. As described below, the smaller nuclear charges and the movement of a classical boundary to larger internuclear distances contribute to improved PSC results. We are now able to report the first examples in which the primitive semiclassical method yields electronic states with true minima, dE_tot/dR=0. We show that the primitive semiclassical method provides reasonable approximate values for equilibrium bond lengths and energies for all of the molecular states, as well as providing accurate estimates of molecular parameters for Rydberg states. Furthermore, the purely classical analysis which determines the molecular or atomic character correctly identifies all the bonding states considered here. In particular, a PSC energy minimum is found for the ground state of the non-polar compound. Its electron trajectory provides a molecular analog to the circular Bohr orbits of the old quantum theory. Because the semiclassical trajectories scale with total charge, the bonding trajectory was first calculated by S&R [1].

The PSC method applied to the neutral molecules still suffers from the discontinuity mentioned in connection with the cations. The structure of the classical phase space and in particular the discontinuity is due to the failure of the PSC method to account for the tunnelling which occurs between the two atomic wells. As shown below, in the neutral molecules the discontinuity is moved to larger values of R, away from the immediate vicinity of the minimum in the total energy. Some of the states of the polar example with energy minima avoid the discontinuity and thus have no gap in the energy as a function of the internuclear distance; the results for the cases examined show that all these to be best classified as Rydberg type molecular states.

Improved semiclassical methods are able to treat these one electron states. Specifically, the uniform semiclassical method [1], [7] provides good approximations to the quantum results for H₂⁺, leading to good approximations to bond energy and bond length. The uniform semiclassical method [8], [9], [10] treats tunnelling between the wells and eliminates the discontinuity in the action and consequently in the energy.

So why even consider the PSC approach? The PSC method retains the direct connection to the structure of the classical phase space, a connection which allows a classical interpretation of chemical bonding to be brought forward. There are many other discussions of the origin of chemical bonding in the literature, far too many to review in the current context, so we mention only a few recent articles. A recent set of papers [11], [12], [13], [14] shows the continued interest in the question of the proper interpretation of the source of chemical bonding. Many other papers deal with the full quantum descriptions [15], but here we focus on qualitative interpretations of chemical bonding. Many of the presentations trace their roots back to the work of Rudenberg [16] and others, reviewed by Kutzelnigg [17]. But this “quasiclassical” approach treats the atoms as quantum mechanical objects, obtaining the proper density function for the electrons before considering the classical interaction of the charge distributions. Alternate paradigms, such as that discussed recently by Maslen [18], approach the classical interaction of quantum distributions. Many analyses are based on the work of Bader [19] on critical points of electronic density functions, and their role in distinguishing various regions of space; this work is fully quantum in its approach. We consider the critical point analysis in a classical context elsewhere [20]. There are many studies of the role of electron correlation in chemical bond formation (e.g., see [21]), a topic beyond the scope of this paper. Here we consider the motion of the electrons classically, in the atom and in the molecule; only the determination of the eigenvalues by the WKB method is pasted on top of the classical description. The advantage of the classical method is that the interpretation of the results is not subject to the ambiguities of interpretation that infect quantum results.

While the motivation of the current work is to delve into the connection between the global structures of the classical and quantum descriptions, potential application include approximate treatments of Rydberg molecules, both polar and non-polar. The polar parameter employed here is arbitrarily taken to have the value $\text{1}\text{3}$, which is probably too large to represent polar rather than ionic molecules. Extension to smaller values of the parameter is straightforward.

We review the PSC method for electronic states in the following section, emphasizing the structure of the classical phase space in chemical terms. We even more briefly review the well-known quantum treatments for one-electron diatomic molecules, mainly to establish notation. Application to and results for the neutral molecules are then presented.