Rotational and ro-vibrational transitions of He2C2+

Ab initio calculations of transition frequencies and line strengths have been calculated for the “forbidden” rotational spectrum and ν2 fundamental ro-vibrational of the 1A1 state of He2C2+.

For the “pure” rotational spectrum, the vibration ground state transitions are very weak, with the intensity of the strongest line, 331–440, being only 2.15 × 10−22 cm molecule−1.

A group of significant line strengths is attributed to ro-vibrational transitions between (〈0,2,0〉 ←〈0,1,0〉) and (〈0,1,0〉 ← 〈0,0,0〉) bands.

These band centers are calculated at 309.5 and 330.4 cm−1 respectively.

The strongest band is identified with the (〈0,2,0〉 ← 〈0,0,0〉) transition with a band center of 625.1 cm−1 and line strength of 2.58 × 10−19 cm molecule−1 for the (440 ← 441) transition.


Over the past several decades there has been considerable interest in using ab initio techniques in order to calculate the infrared spectra of molecular ions, which are of interest in ion-molecule reaction schemes pertinent to the production of interstellar molecules.1–6

The simplest molecular ion, H3+, has played a critical role in the synergy between theory2,4,7 and experiment8–9 in unravelling the existence of this ion in interstellar space.10–11

It is well known that helium is the second most abundant element in the interstellar medium.12

Helium is involved in several important gas-phase reactions.

The charge transfer recombination of multiply charged ionic species in collision with atomic helium is important in interstellar processes.

For example, more than eighty charge exchange reactions involving the He+ have been identified in astrophysical plasmas.12

Moreover, recent ion measurements from the Jovian magnetosphere have indicated the presence of helium, oxygen and sulfur ions.13

It is therefore not surprising that the electronic structure of multiply charged seeded helide cations of form, HenXm+ (where n = 1 2, X = C,N,O and m = 1,2), have been extensively investigated using ab initio techniques by Hughes and von Nagy-Felsobuki.5–6

It is hope that these calculations will assist the interstellar as well as the laboratory detection of these ions.

Of the triatomic seeded helides ions, He2C2+ is the most likely candidate for detection in the interstellar medium.12

The χ 1A1 state of He2C2+ has been extensively investigated by Koch and Frenking14 and Hughes and von Nagy-Felsobuki.4–5

This state is predicted to be thermodynamically stable with a dissociation energy of ca. 57 kJ mol−1.

It is lower in energy than the 3B1 state.

Frenking and Cremer15 have also postulated that in this state the binding force of this ion is electrostatic in nature.

At the CCSD(T)_AE/cc-pCVTZ level of theory, Hughes and von Nagy-Felsobuki4–5 have determined that its optimised structural parameters (rC–He; θHe–C–He) are (1.570 Å; 84.1°), respectively.

These workers have also calculated the low-lying ro-vibrational states of this ion,4 although so far no dipole moment surface or line strengths for the rotational and ro-vibrational transitions have been determined (such information would further assist the identification of this ion in a laboratory or interstellar setting).

As our extension of our earlier work on the electronic structure of He2C2+,4–5 we wish to report the rotational and ro-vibrational spectrum of this ion.

In doing so, we shall detail the most accurate ro-vibration transition frequencies, dipole moment surface and line strengths reported to-date, thereby assisting its laboratory and interstellar detection.

Computational procedure

The potential energy surface used in the ro-vibrational calculations has been detailed elsewhere by Hughes and von Nagy-Felsobuki.4

The potential energy function is a Padé (4,5) power series expansion, which has been embedded in the Eckart–Watson Hamiltonian, with the resultant eigenvalue problem solved with a variational methodology using an algorithm developed by Searles and von Nagy-Felsobuki.16

The basic approach to the variational solution is the construction of a “full” three-dimensional vibration configuration basis set from 20 × 20 × 20 one-dimensional (1D) eigenfunctions (which are finite-element solutions of a 1D Hamiltonian expressed in terms of a single rectilinear t co-ordinate).

The “full” ro-vibrational wave function is given by a linear combination of the products of the three 1D configurational bases functions (yielding 8000 vibration basis functions in total) together with a complete set of symmetric-top rotor functions.

The ro-vibrational matrix spanned by this trial basis set is then diagonalised yielding ro-vibrational eigenenergies and eigenfunctions.

The ab initio dipole moment surface (DMS) of a molecule is usually generated in conjunction with the calculation of the potential energy surface.

The dipole moment calculations were performed in the centre-of-mass system using the QCSID_AE/aug-cc-pCVTZ level of theory.

A total of 75 ab initio points have been calculated on the 1A1 state of He2C2+ hypersurface using the GAUSSIAN suite of programs.17

A table containing these points is available as electronic supplementary information (ESI).

This discrete surface has been fitted to an analytical function generated using a fifth order power series, which was formulated by Gabriel et al18. in the following manner,The functional form of the DMS is in terms of two bond lengths and included bond angle (ρ1 = R1R1e, ρ2 = R2R2e, ρ3 = θθe).

The (χ2)1/2 for fifth order expansion is ca. 10−3 au for all the μx and μy components.

The dipole moment function has been graphically inspected in order to ensure that the regions of electrical anharmonicity are within the expectation values of the ab initio dipole moment calculations.

Table 1 gives a full description of the dipole moment function.

The general strategy in the evaluation of radiative properties has been detailed elsewhere.19

Nevertheless it is pertinent to point out some of the following features of the procedure.

The transition probabilities spanned by the variational wave functions have been determined using the Harris et al20. quadrature scheme.

The ro-vibrational transition probabilities have been transformed into the space-fixed co-ordinate system from the molecule-fixed framework, using the formulation as outlined by Zare21 and then appropriately evaluated.

The transition probability evaluation procedure and ensuing results have been carefully benchmarked for H2O and are in excellent agreement with methodologies using more traditional algorithms.19

All calculations have been performed using Compaq Alpha workstations under the UNIX operating system.

Results and discussions

The ro-vibrational wave function presented here differs significantly from that previously determined by Hughes and von Nagy-Felsobuki.4

The latter variational calculation used a nodal cut-off criterion in order to select only 560 vibration configuration basis functions from a possible configuration list of 8000.

The “pure” rotational and ro-vibrational transitional energies listed in Tables 2 and 3 have been calculated using the complete list of 8000 vibration basis functions.

The energy levels throughout these Tables are converged to ±0.02 cm−1 (based on a study of the variation of the energy levels as a function of configurational size).

Hence, these are the most accurate energy levels reported to date in the literature.

Table 2 gives the “pure” rotational transitions associated with the ground electronic and vibrational state of He2C2+.

The limiting case for the rotational levels is Mulliken’s prolate symmetric-top.

The rotational levels given in Tables 2 and 3 are assigned within this framework and are calculated up to the J = 4 level.

The assignments of the ro-vibrational states in these tables are based on the expansion densities determined from the coefficients of the variational wave function.

For the ro-vibrational states higher than the fourth excited rotational state, the resultant wave functions are heavily mixed due to basis functions, which belong to the same irreducible representation.

Hence, the assignment of a wave function in terms of a single diagonal representation becomes problematical for the more excited states.

For simplicity we have restricted the analysis to energy levels whose wave functions (according to the expansion density analysis) have unequivocal assignments; that is, for wave functions which are not severely delocalised.

Therefore, the assignments in Tables 2 and 3 are restricted to all transitions, up to and including the ν = 4 and J = 4 level.

A comprehensive assignment of a more extensive list of transitions between delocalised states is available from the authors.

The ro-vibrational spectral line intensities of a transition from an initial state |i〉 to a final state |j〉 can be evaluated via22where the usual definitions for k, h, c, R, T and NA apply.

Here CA is the isotopic abundance, gnsi is the nuclear statistical weight of the initial state, ωij is the frequency of transition (cm−1), Ei and Ej are the energy of the initial state |i〉 and final state |j〉 in cm−1, QV and QR are the vibration and rotation partition functions, respectively.

For He2C2+ the gnsi value for the symmetric and anti-symmetric levels is 1 and 0, respectively.

The R2 term is the transition probability summed in terms of the Wigner rotation matrices, with the transition moment integrals evaluated using the Harris et al20. quadrature scheme.

It should be noted that in this work the ro-vibrational transition probability (R2) is calculated exactly (i.e. by employing the “full” ro-vibrational wave function and the transformed DMS).

Therefore, there is no need to impose selection rules, such as ΔJ = 0,±1, since these are reflected within the calculations.

According to the ro-vibrational selection rules, the total number of allowable transitions up to and including the v = 4 and J = 4 level is 1250.

From ab initio calculations of He2C2+ spectrum, many of these transitions are still not significant in terms of their intensities (i.e. well below 10−30 cm molecule−1).

Furthermore, a number of these transitions are forbidden (i.e. the value of the line intensities are zero) due to the fact that the nuclear spin statistic of lower level is zero (e.g.

512 transition are spin forbidden from this consideration alone).

Hence, for He2C2+ the total number of transitions with intensities above 10−30 cm molecule−1 is 637.

Fig. 1 gives the ro-vibrational spectrum up to and including the v = 4 and J = 4 level using a line strength threshold of 1.0 × 10−30 cm molecule−1.

Table 2 details the line strengths for all the “pure” rotational transitions up to J = 4 level associated with the ground electronic and vibrational state of He2C2+.

From Fig. 1 and Table 2 it is evident that for the intensities of the “pure” rotational transition of the vibration ground state (with a transition energy less than 25.0 cm−1) are very weak (e.g. the intensity of the strongest line, 33 1–44 0, is only 2.15 × 10−22 cm molecule−1).

The next group of significant line strengths is attributed to ro-vibrational transitions between (〈0,2,0〉 ← 〈0,1,0〉) and (〈0,1,0〉 ← 〈0,0,0〉) bands.

The band centers are calculated to be 309.5 and 330.4 cm−1 respectively.

The strongest band is assigned to the (〈0,2,0〉 ← 〈0,0,0〉) transition, with the band center at 625.1 cm−1 and a line strength of 2.58 × 10−19 cm molecule−1 (which is assigned to the 44 0–44 1 transition).

Table 3 details the dipole-allowed ro-vibrational transitions for He2C2+ in the ν2 fundamental using a cut-off in intensity of greater than 1.0 × 10−20 cm molecule−1.

Using the Einstein Aul coefficient it also possible to calculate the fluorescence lifetime of a particular state in the absence of an external radiation field (where only spontaneous emission occurs).

The fluorescence lifetime, τfu, can be calculated via23where the Einstein A coefficient is defined as

Tables 2 and 3 list the Einstein A coefficients.

The selection rules discussed above lead to three or fewer possible decay routes for the levels presented in Tables 2 and 3 and so the radiative lifetimes using eqn. (2) can be easily determined.