## Abstract

We review the theoretical work carried out on the tri-hydrogen ion in the electronic triplet state 1^{3}*E*′, which is split into *a* and 2^{3}*A*′ by vibronic interaction. We begin with an overview on analytical potential energy surfaces and calculations of rovibrational states by focusing on our own results, which are based on the most accurate potential energy surfaces available so far. This is followed by an examination of the selection rules and predictions of infrared transition frequencies. Finally, we discuss the Slonczewski resonance states supported by the upper sheet of the potential energy surface. Theoretical work reported here may be of interest for future experiments on the title ion.

## 1. Introduction

One of the simplest and most fundamental molecular ions is . Being a two-electron system, it offers an appealing prototype for accurate electronic structure and dynamics calculations. Furthermore, since the two electrons are spin-aligned in the triplet state of , it is probable that high accuracy can be achieved without employing explicitly correlated electronic wave functions (Preiskorn *et al*. 1991). Despite being such an attractive system, experiments have not been carried out on this ion, which to some extent may be attributed to the lack of accurate theoretical data on its rovibrational structure. In this work, we provide a short review of the existing theoretical work, and point out possible avenues for challenging issues still pending.

## 2. Potential energy surfaces

The lowest electronic states of can be derived easily from the electronic configurations. Assuming equilateral arrangement of the three nuclei, the three hydrogenic 1s atomic orbitals can be combined to yield a molecular orbital of symmetry and one of *e*′ symmetry. In the electronic ground state, , the two electrons occupy the lowest, i.e. the , molecular orbital. A single electronic excitation leads to the configuration from which a singlet state and a triplet state of *E*′ symmetry can be derived. These states are Jahn–Teller unstable. In the case of the triplet state, denoted as 1^{3}*E*′, vibronic interaction lifts the degeneracy and leads to a stabilization of the molecule at a symmetric linear structure. Hence, this lower adiabatic sheet of the 1^{3}*E*′ potential energy surface is denoted . The upper adiabatic sheet of the 1^{3}*E*′ surface, 2^{3}*A*′, forms a cone, which breaks into H^{+}+2H at high energies. The two sheets become degenerate at equilateral triangular configurations. Such configurations form a line of conical intersections.

Calculations have been reported for the two sheets of the lowest triplet potential energy surface. They will be discussed below.

The degenerate ^{3}*E*′ state has been investigated in the pioneering *ab initio* calculations of Hirschfelder (1938) and more recently by Kawaoka & Borkman (1971). In addition, Pfeiffer *et al*. (1967) used the semi-empirical diatomics-in-molecules (DIM) approach. No stable structures could be found in this early work.

The first calculations on the lower sheet of the ^{3}*E*′ state of have been reported in 1974 by Schaad & Hicks. They have found that the ion has an equidistant collinear structure but is very weakly bound with respect to dissociation into H atom and ion. Their results were confirmed by Ahlrichs *et al*. (1977). Table 1 compares these calculations with other more recent work that we describe below.

The first potential energy surface was reported in 1989 by Wormer & de Groot. These authors calculated 400 *ab initio* points with the full-configuration interaction (FCI) method and a 5s3p basis. Of them, 240 have been fitted to an analytical expression in hyperspherical coordinates (*ρ*, *θ*, *ϕ*). From the data points, the nuclear repulsion energy was first subtracted to remove singularities, with the remaining electronic energy being then expanded in terms of Wigner *D* functions. Their analytical expression assumed the form(2.1)with the surface being interpolated between the *ρ* grid points. Owing to the modest size of the basis in the electronic energy calculations (it comprises 42 basis functions), the surface does not have spectroscopic accuracy, as admitted by those authors. Perhaps, for this reason, this surface has never been used for bound state calculations. Nevertheless, it gave the first detailed information on the general topology of the surface. It was found that the surface is very flat and consequently the molecule is very floppy.

Eleven years later, a new potential energy surface was obtained by Friedrich (2000) and used for the first rovibrational calculations of the system (Friedrich *et al*. 2001). On a grid defined in hyperspherical coordinates, 2155 *ab initio* points were calculated using the FCI method and the cc-pV5Z basis set of 165 functions for the three atoms. As the hyperspherical method makes full use of the symmetry of the potential, the points need to cover only one-sixth of the configuration space. To obtain an analytical representation of the data points, the sum of nuclear Coulomb interaction energies was first subtracted, following work by Wormer & de Groot (1989), as it was the energy of the conical intersection line. The remaining flat surface, i.e. a surface free of Coulomb singularities, was then written as a fraction of symmetry-adapted polynomials such that(2.2)

In this approach, the functions sin^{j}3*ϕ* reflect the permutational symmetry of the potential. The coefficients are determined at given values of the hyperradius, *ρ*, while intermediate values are interpolated by splines. Note that the form of the *θ* expansion functions guarantees the correct behaviour of the surface near the conical intersection. Consequently, if one extends the range of the hyperspherical angle *θ* to include negative values, i.e. *π*/2≥*θ*≥−*π*/2, such that those negative values of *θ* correspond to the upper sheet of the triplet potential, the two sheets can be represented simultaneously by equation (2.2). Indeed, Friedrich (2000) in his thesis reported one fit only for the lower sheet and one for the two sheets. Unfortunately, accurate inclusion of the atom–diatom dissociation channels proved difficult.

Another potential energy surface was presented in 2001 by Sanz *et al*. Those authors calculated, on a grid defined in internal coordinates, a set of 7689 *ab initio* points with the FCI method and a (11s6p2d)/[8s6p2d] basis comprising 108 basis functions for the three atoms. Of these, 405 points, located close to the minimum region, were further calculated with the large cc-pV6Z basis set of 271 functions. The potential energy surface was then constructed by the DIM method. In this case, a 3×3 DIM matrix was set up in terms of three diatomic molecule potential curves, , and . The lowest eigenvalue of this matrix yields a DIM-approximation, *V*_{DIM}, of the adiabatic potential energy surface of , which was then corrected by addition of a three-body term (Last & Baer 1981) to give(2.3)

The explicit form of the three-body term must be chosen such as to retain permutational symmetry. Two years later, Cernei *et al*. (2003) reported a global potential energy surface for the lowest adiabatic sheet of the triplet state which also correctly describes the dissociation channels. They employed a double many-body expansion (DMBE)(2.4)whose coefficients were calibrated from a fit to Friedrich *et al*.'s (2000, 2001) *ab initio* points. Within this expansion, the two-body terms, *V*^{(2)}(*R*_{i}), and the three-body term, *V*^{(3)}(** R**), are expressed as sums of extended Hartree–Fock (EHF) and dynamical correlation energies. To describe correctly the behaviour of the potential energy surface near the conical intersection, the EHF part of the three-body term is written as a superposition of two polynomials,(2.5)where

*Γ*

_{2}, in front of the second polynomial, denotes the Jahn–Teller active coordinate. As in equation (2.3), the three-body term is subject to permutational constraints.

We now turn our attention to the second adiabatic sheet of the electronic triplet state, 2^{3}*A*′. The first potential energy surface was obtained by Friedrich (2000) as described above. Later, Viegas *et al*. (2004) refitted the *ab initio* points to an analytical DMBE expression, see equations (2.4) and (2.5), with the sign in front of *Γ*_{2} inverted. However, individual fits of the two adiabatic sheets of the potential energy surface do not necessarily assure the degeneracy at the *D*_{3h} intersection line. As a remedy, Varandas *et al*. (2005) worked out a novel DMBE representation. The problem in the individual DMBE fits is that the two-body contributions for the two sheets are different at the intersection line since the underlying diatomic curves, for the lower sheet and for the upper, are different. In the new DMBE approach, the two-body contributions are invalidated at the intersection line and folded into the three-body term. For each sheet, the line of degeneracy is thus described solely by the first polynomial, *P*_{1}, in equation (2.5), while the splitting due to the Jahn–Teller coordinate *Γ*_{2} is controlled by the polynomials *P*_{2}. For their representation of the double-valued potential energy surface, those authors used a total of 3621 *ab initio* points of cc-pV5Z quality for each adiabatic surface. Table 1 gathers some relevant characteristics of the *ab initio* calculations reported so far on the title system. Note that the dissociation energy in the lowest branch, , is −1.1026342*E*_{h}. In turn, the dissociation channel 2H(^{2}*S*)+H^{+} in the upper branch lies at −1.0000*E*_{h}, with the energy minimum being −1.034590*E*_{h} at an equilateral triangular configuration with side lengths of 3.61011*a*_{0}.

## 3. Rovibronic states on the lower sheet

The lower sheet of the double-valued 1^{3}*E*′ potential energy surface is characterized by three equivalent minima at symmetric linear configurations produced by Jahn–Teller distortion. The rovibronic wave functions are therefore superpositions of wave functions localized in the respective minima. The superpositions and their spin statistical weights determine the rovibronic structure of the states.

### (a) Homonuclear isotopologues

For the case of a homonuclear isotopologue, like or , the superpositions transform as irreducible representations of the three-particle permutation inversion group or *S*_{3}×*I*, a group isomorphic to the molecular symmetry group *D*_{3h}(*M*) (Bunker & Jensen 1998). Of the three localized states, a one-dimensional representation(3.1)and a two-dimensional representation(3.2)(3.3)are obtained where *ω*=e^{(2πi)/3}. A classification of states in *D*_{3h}(*M*), which is the symmetry group of the tunnelling molecule, is exact. In contrast, a classification according to the irreducible representations of the linear molecule symmetry group, *D*_{∞h}(*M*), is only approximate as tunnelling is not accounted for.

Linear or have the following four normal modes: the symmetric stretching vibration, *ν*_{1}; the twofold degenerate bending vibration, *ν*_{2}; and the antisymmetric stretching vibration, *ν*_{3}. Associated with the bending vibration is the vibrational angular momentum , which takes the values . is equal to the internal projection of the angular momentum *N*, where . Neglecting the electronic spin *S*, the eigenkets of the localized states can now be written as(3.4)for and(3.5)for . In the linear combinations of localized functions, equations (3.1)–(3.3), the superimposed functions have identical linear molecule quantum numbers. Therefore, as long as these superpositions are good zero-order functions for a diagonalization of the rovibronic Hamiltonian, a classification of each component, of *A* or *E* symmetry, in terms of linear molecule quantum numbers and *N* is sensible. Note that for each set of linear molecule quantum numbers, the one-dimensional (*A*) and two-dimensional (*E*) components are split in energy. If the splitting is small, then the linear molecule quantum numbers hold.

The theoretical investigation of the rovibrational states of such systems must allow for the possibility of isomerization of the individual linear molecules and hence a choice of coordinates must be made such as to treat the three equivalent structures on equal footing. A natural choice being hyperspherical coordinates. Indeed, in all theoretical results reported so far, hyperspherical coordinates were employed.

The first calculations on vibrational (Sanz *et al*. 2001) and rovibrational (Friedrich *et al*. 2001) states were published independently by the two groups in 2001. The former authors used their new potential energy surface reported in the same paper, while the latter ones based their calculations on an unpublished surface (Friedrich 2000). The results are in good agreement. Sanz *et al*. (2001) reported a total of 17 vibrational states, each having one component of *A* symmetry and one of *E* symmetry, up to the energy of dissociation into . On the other hand, Friedrich *et al*. (2001) studied the rotational structure of the vibrational ground state and the singly excited states. They also examined the effect of the geometrical phase and found that no such topological effect is present, which they explained by the localization of states in their respective minima. The most exhaustive investigation of the rovibrational states of triplet is based on the potential energy surface by Cernei *et al*. (2003) and was reported by Alijah *et al*. (2003). Those authors calculated, with the method of hyperspherical harmonics, the 19 lowest bands for *J*≤10 and provided spectroscopic assignments.

The physically allowed states of must be antisymmetric with respect to an exchange of two protons and thus transform as or in *D*_{3h}(*M*). The symmetry of a rovibronic state is given by the product(3.6)where *Γ*_{el}, *Γ*_{rovib} and *Γ*_{nspin} denote the electronic, rovibrational and nuclear spin symmetry, respectively. In , the three protons can be coupled to a quartet state (*I*=3/2) of symmetry and a doublet (*I*=1/2) of *E*′ symmetry. Since the electronic symmetry is , the spin statistical weights for rovibronic states of symmetry *A*_{1}, *A*_{2} and *E*, ‘prime’ or ‘double prime’, are 0, 4 and 2, giving rise to missing states.

A summary of the vibrational states is shown in figure 1. Each of those states has two symmetry components. The missing states of symmetry are drawn with a dashed line. The figure shows that, of the vibrational states, only the two lowest are located below the barrier of isomerization, and only the three lowest are below the electronic energy of the dissociation channel. Thus, the majority of states is located within the zero-point energy interval of , i.e. within 1145 cm^{−1}.

Of the bound states of the isotopologue , only vibrational states have been reported by Cuervo-Reyes *et al*. (2002). Owing to the high nuclear masses, seven vibrational states are located below the barrier of isomerization. The deuteron has a nuclear spin of *i*=1, so that the total wave function is required to be symmetric with respect to an exchange of two nuclei. The nuclear spin states and their respective symmetry designations are , , *I*=2(*E*′) and . Hence, all rovibronic states can be realized in contrast to what has been found for . States of symmetry *A*_{1}, *A*_{2} and *E* have statistical weights of 10, 1 and 8. In their article, Cuervo-Reyes *et al*. (2002) compared the energy splitting between the *A* and the *E* components of the vibrational states for and . Noticeable splitting begins above the isomerization barrier once the zero-point energy of the transition state is exceeded, which in the case of occurs at a lower energy than in the case of .

### (b) Mixed isotopologues

Of the mixed isotopologues, the lowest rovibrational states of H_{2}D^{+} have been calculated by Alijah & Varandas (2006), based on the potential energy surface of Varandas *et al*. (2005) These states can be attributed to one of the two isomers, HDH^{+} and HHD^{+}. Of the latter, there are two equivalent structures of which superpositions may be formed which are symmetric or antisymmetric with respect to an exchange of the two protons. Thus, the rovibrational states of HHD have two, one-dimensional components that transform as the irreducible representations (*A*_{1}, *B*_{2}) or (*A*_{2}, *B*_{1}) of the two-particle permutation inversion group, *S*_{2}×*I*. This group is isomorphic with *C*_{2v}. All the rovibrational states of the two isomers, HDH^{+} and HHD^{+}, have non-zero spin statistical weights of one (A states) or three (B states). Alijah & Varandas (2006) considered vibrational states with up to twofold excitation for *J*≤3. Only the vibrational ground state and the vibrationally singly excited states are below the barrier of isomerization.

### (c) Dipole transitions

To derive selection rules for dipole transitions, the dipole operator has to be classified within the permutation inversion group of identical particles. Clearly, such an operator is antisymmetric with respect to inversion of the coordinate system and symmetric with respect to an exchange of identical particles. In *S*_{3}×*I*, it transforms as , while in *S*_{2}×*I* it transforms as *A*_{2} (Alijah *et al*. 1995; Alijah & Beuger 1996). Thus, we obtain the following selection rules:(3.7)for and and(3.8)for H_{2}D^{+} and D_{2}H^{+}. In addition, we may assume conservation of the nuclear spin state. Finally, the usual selection rule Δ*J*=0,±1 applies. As calculated spectra have not been reported so far, we list in table 2 the lowest frequencies of (far) IR transitions originating from the vibrational ground states of , HDH^{+} and HHD^{+}.

## 4. Rovibronic states on the upper sheet

The conical potential of the upper sheet, 2^{3}*A*′, has a depth of 7596.3 cm^{−1}, sufficient to support rovibrational states. However, these states suffer non-adiabatic coupling with the lower sheet and therefore are not stable. They are known as Slonczewski resonances. For their correct description, the effect of the geometrical phase cannot be neglected. As Slonczewski (1963) has shown, the geometrical phase produces a centrifugal-type barrier even for the non-rotating molecule, which hinders the vibrational wave function from reaching the conical intersection where the non-adiabatic coupling becomes infinite, thus stabilizing the resonance states. For an accurate calculation of their positions and lifetimes, in general, a vibronic calculation involving the two coupled sheets of the potential energy surface would be required. However, in the case of the present system, a single surface calculation should be sufficient for the determination of the resonance positions, as here the resonance states are degenerate with continuum states on the lower sheet. In a model study, Alijah & Nikitin (1999) found that non-adiabatic coupling to a continuum does not significantly alter the structure of the wave function of a resonance state but merely leads to its depopulation.

The rovibronic structure of resonance states in a conical potential has been recently investigated by Alijah & Varandas (2004). The resonance states can be represented in terms of spectroscopic quantum numbers, neglecting the electronic spin, as(4.1)(4.2)In equation (4.2), the geometrical phase factor e^{iαϕ} has been extracted from the basis kets to obtain an expression similar to that given by Mead (1980). The spectroscopic quantum numbers are that of a symmetric triatomic molecule of *D*_{3h} structure, like in the electronic ground state. *v*_{1} is the symmetric stretch and *v*_{2} the degenerate bending vibration. Associated with *v*_{2} is the vibrational angular momentum . Hougen (1962) and Watson (1984) have shown that *k* and are not conserved separately and introduced a new quantum number, *G*. Their quantum number has been generalized by the present authors (Alijah & Varandas 2004) to take into account the effect of the geometrical phase. The generalized *G* quantum number is defined as(4.3)where(4.4)*α* is zero if no geometrical effect is present (no geometrical phase (NGP) case), one half otherwise (geometrical phase (GP) case). A correlation between NGP and GP states shows that the geometrical phase has a severe, qualitative effect on their rotational structure (Alijah & Varandas 2004).

The positions of the vibrational resonance states of have been calculated by Viegas *et al*. (2004, 2005) for the two potential energy surfaces (Viegas *et al*. 2004; Varandas *et al*. 2005). As their calculations are based on a single sheet of the potential energy surface, no lifetimes could be determined but, as discussed before, the positions are expected to be rather accurate.

## 5. Conclusions

During the last five years, theoretical investigations have provided quite a lot of data on the rovibrational states of triplet and some of its isotopologues. It is expected that such data will stimulate experiments aiming at the detection of these molecules.

## Discussion

B. J. McCall (*University of Illinois at Urbana-Champaign, Urbana, USA*). As you mentioned, the transition from the ground state to the triplet state is likely to be very weak. Is it possible to calculate, even to within an order of magnitude, the oscillator strength for this transition?

A. Alijah. Such transitions should be very weak for two reasons. Firstly, because the two electronic states have very different equilibrium geometries, equilateral triangular configuration with bond lengths of *r*=1.65*a*_{0} in the electronic ground state and linear configuration with the two equal bond lengths *r*=2.454*a*_{0} in the lowest triplet state, the Franck–Condon factors should be very small. Secondly, spin–orbit coupling should be very small. At present, due to the lack of information on the spin–orbit coupling, it is not possible to give an estimate of the oscillator strength.

C. H. G. Greene (*University of Colorado, Boulder, USA*). You mentioned that you have treated the Berry phase issues associated with the conical intersection that arises in your triplet H_{3}^{+} calculations. However, I would have expected that the divergent non-Born–Oppenheimer couplings would pose an even greater problem in the adiabatic representation. Have you explored the use of diabatic representations, which tend to simplify the calculations that are affected by conical intersections?

A. Alijah. No, our results were obtained in the adiabatic representation. Actually, in our vibronic calculations of the Slonzwewski resonance states, we used a single potential energy surface only. In such a treatment, the phase change of the electronic wave function when transported adiabatically around the conical intersection is compensated by imposing proper cyclic boundary conditions on the nuclear wave function, such that the product of electronic and nuclear wave functions remains single-valued. In a two-surface adiabatic calculation, one would have to deal with the adiabatic coupling terms, of which the ∂/∂*ϕ* term, as you have said, becomes singular at the intersection. Nikitin and myself (Alijah & Nikitin 1999) have shown how to cope with this singularity within the framework of the adiabatic representation. Alternatively, one may solve the vibronic problem in a diabatic representation in which the ∂/∂*ϕ* term has been transformed away. We have not explored this possibility.

J. Hinze (*University of Bielefeld, Germany*). The generation of a diabatic surface, in more than one coordinate, is problematic.

A. Alijah. This is true as long as one is interested in *strictly* diabatic states (Mead & Truhlar 1982). However, if one is satisfied with approximate (but well converged) diabatic states, it is enough to transform away the tangential component ∂/∂*s* of the non-adiabatic coupling term, which in case of polar coordinates is the angular component ∂/∂*ϕ* (Pacher *et al*. 1989). As Baer (1975) has shown the corresponding transformation matrix is obtained as a solution of a first-order differential equation involving the non-adiabatic coupling matrix, *τ*, provided the electronic states form, at the region under consideration, a Hilbert subspace. Alijah & Baer (2000) recently studied numerically the diabatization process for a model potential. This study led to the formulation of a theoretical criterion for the validity of the diabatization based on the diagonality of the topological * D*-matrix (Baer & Alijah, 2000) defined as:

*=exp(∮*

**D**_{Γ}

*τ*

_{s}d

*s*) (here,

*s*is the tangential component of

*τ*along a closed contour,

*Γ*). The validity of this criterion is further discussed by Baer

*et al*. (2000), where it was shown that it forms a quantization condition for the non-adiabatic coupling matrix. Later, studies of three-state

*ab initio*molecular systems (e.g. C

_{2}H) revealed the existence of this criterion for realistic situations (Mebel

*et al*. 2002). Thus, it is quite possible to construct a meaningful diabatic potential energy matrix for the title system provided the derivative couplings in the adiabatic representation are known.

## Acknowledgments

This work has the support of the Fundação para a Ciência e a Tecnologia, Portugal. We are also grateful to the John von Neumann Institut für Computing, Jülich, for the provision of supercomputer time on the IBM Regatta p690+ (Project EPG00).

## Footnotes

One contribution of 26 to a Discussion Meeting Issue ‘Physics, chemistry and astronomy of ’.

- © 2006 The Royal Society