^{1}, Á. Vibók

^{2}, R. Baer

^{3,a)}and M. Baer

^{4,b)}

### Abstract

Some time ago we published our first article on the Renner-Teller (RT) model to treat the electronic interaction for a triatomic molecule [J. Chem. Phys.124, 081106 (2006)]. The main purpose of that Communication was to suggest considering the RT phenomenon as a topological effect, just like the Jahn-Teller phenomenon. However, whereas in the first publication we just summarized a few basic features to support that idea, here in the present article, we extend the topological approach and show that all the expected features that characterize a three (multi) state RT-type'3 system of a triatomic molecule can be studied and analyzed within the framework of that approach. This, among other things, enables us to employ the topological matrix [Phys. Rev. A62, 032506 (2000)] to determine, *a priori*, under what conditions a three-state system can be *diabatized*. The theoretical presentation is accompanied by a detailed numerical study as carried out for the HNH system. The -matrix analysis shows that the two original electronic states and (evolving from the collinear degenerate doublet), frequently used to study this Renner-Teller-type system, are insufficient for diabatization. This is true, in particular, for the stable ground-state configurations of the HNH molecule. However, by including just one additional electronic state—a state (originating from a collinear state)—it is found that a rigorous, meaningful three-state diabatization can be carried out for large regions of configuration space, particularly for those, near the stable configuration of . This opens the way for an accurate study of this important molecule even where the electronic angular momentum deviates significantly from an integer value.

Three of the authors (M.B., Á.V., and G.J.H) acknowledge the US-Israel Bi-national Science Foundation for partly supporting this study. One of the authors (Á.V.) acknowledges the OTKA Grant Nos. T037994 and M041537 and the computational resources provided by the John-von-Neumann Institute, Research Centre Juelich (Project ID ehu01).

I. INTRODUCTION

II. THEORY

A. Background comments

B. The NACM and the diabatic potential matrix

C. The Renner-Teller topological matrix

1. The two-state Hilbert subspace

2. The three-state Hilbert subspace

III. NUMERICAL RESULTS

A. Potential energy curves

B. Nonadiabatic coupling terms

C. The topological -matrix and the single-valued diabatic potentials

IV. ANALYSIS AND CONCLUSIONS

### Key Topics

- Subspaces
- 13.0
- Jahn Teller effect
- 6.0
- Potential energy surfaces
- 4.0
- Angular momentum
- 3.0
- Group theory
- 3.0

## Figures

*Ab initio* RT nonadiabatic coupling terms, and , as calculated for the collinear and planar HNH molecule. The results are presented as a function of q—the vertical distance of the rotating atom from the fixed (collinear) axis—for different configurations: (a) the rotating atom is nitrogen and the calculations are done for (the symmetry); (b) the rotating atom is nitrogen and the calculations are done for ; (c) the rotating atom is hydrogen and the calculations are done for (—) ; (⋯) .

*Ab initio* RT nonadiabatic coupling terms, and , as calculated for the collinear and planar HNH molecule. The results are presented as a function of q—the vertical distance of the rotating atom from the fixed (collinear) axis—for different configurations: (a) the rotating atom is nitrogen and the calculations are done for (the symmetry); (b) the rotating atom is nitrogen and the calculations are done for ; (c) the rotating atom is hydrogen and the calculations are done for (—) ; (⋯) .

Energy curves related to three electronic states: the state the state (both evolving from the two degenerate states) and the state which evolves from the (collinear) state to become the . These three states are the lower ones for the collinear arrangement and at regions close to it: (⋯) ; (—) ; (---) .

Energy curves related to three electronic states: the state the state (both evolving from the two degenerate states) and the state which evolves from the (collinear) state to become the . These three states are the lower ones for the collinear arrangement and at regions close to it: (⋯) ; (—) ; (---) .

The two-state and the three-state -matrix elements. Two curves are shown. Oone represents the (1,1) element of the matrix and the second the average value, , of the three diagonal elements of the matrix [see Eq. (20)]: (⋯) ; (—) .

The two-state and the three-state -matrix elements. Two curves are shown. Oone represents the (1,1) element of the matrix and the second the average value, , of the three diagonal elements of the matrix [see Eq. (20)]: (⋯) ; (—) .

The three-state -matrix elements: (⋯) ; (—) ; (---) .

The three-state -matrix elements: (⋯) ; (—) ; (---) .

The position of the four JT cis in configuration space as found for : (◆) ; (×) . The (1,2) is located at a distance from the collinear axis. The (2,3) is located at a distance from the collinear axis. The two (2,3) twins are located at a distance from the collinear axis and at a distance of from the symmetry line. The numerical value near each stands for the topological phase as calculated for [see Eq. (20)].

The position of the four JT cis in configuration space as found for : (◆) ; (×) . The (1,2) is located at a distance from the collinear axis. The (2,3) is located at a distance from the collinear axis. The two (2,3) twins are located at a distance from the collinear axis and at a distance of from the symmetry line. The numerical value near each stands for the topological phase as calculated for [see Eq. (20)].

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