Oscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 g ]c 3 u + systems

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 126, Oscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 g ]c 3 u systems Damian L. Kokkin, George B. Bacskay, and Timothy W. Schmidt School of Chemistry, University of Sydney, New South Wales 2006, Australia Received 20 October 2006; accepted 3 January 2007; published online 22 February 2007 High level ab initio calculations, using multireference configuration interaction MRCI techniques, have been carried out to investigate the spectroscopic properties of the singlet A 1 u X 1 g Phillips, the triplet d 3 g a 3 u Swan, the b 3 g a 3 u Ballik-Ramsay, and the d 3 g c 3 u transitions of C 2. The MRCI expansions are based on full-valence complete active space self-consistent-field reference states and utilize the aug-cc-pv6z basis set to resolve valence electron correlation. Core and core-valence correlations and scalar relativistic energy corrections were also incorporated in the computed potential energy surfaces. Nonadiabatic and spin-orbit effects were explored and found to be of negligible importance in the calculations. Harmonic frequencies and rotational constants are typically within 0.1% of experiment. The calculated radiative lifetimes compare very well with the available experimental data. Oscillator strengths are reported for all systems: f v v, where 0 v American Institute of Physics. DOI: / I. INTRODUCTION C 2 was first observed spectroscopically in flames. 1 It has since been observed in a host of extraterrestrial sources such as comets, carbon stars, protoplanetary nebulae, and molecular clouds. 2 6 On Earth, it is observed in combustion sources and hydrocarbon discharges. Due to its simplicity, and the large range of experimental and observational data, the molecule has been extensively studied by ab initio calculations An understanding of the photophysics of C 2 is crucial to the implementation of C 2 as a probe of extraterrestrial environments. Models of C 2 photophysics have been constructed to simulate the interstellar medium and other astrophysical systems such as comets Halley and Hale-Bopp These models have established that the incorporation of six molecular states is sufficient to model the fluorescence of C 2 as seen in comets. These are the singlet X 1 g and A 1 u states, and the triplet a 3 u, b 3 g, c 3 u, and d 3 g states. The allowed transitions within this set are the A 1 u X 1 g Phillips band system, the d 3 g a 3 u Swan band system, the b 3 g a 3 u Ballik-Ramsay band system, and the d 3 g c 3 u band system. Also included, and of crucial importance to these models, are the spin-forbidden a 3 u X 1 g and c 3 u X 1 g phosphorescent transitions. The vibrational manifolds of the X 1 g state and the a 3 u state are interleaved. Thus, C 2 relaxes to its vibrational ground state by making a series of phosphorescent transitions between various rovibrational levels of the X 1 g state and the a 3 u state. Recent models of C 2 photophysics rely on a number of assumptions which, in places, rest on unverified indirect spectroscopic data. Specifically, the strength of the d 3 g c 3 u transition is estimated and the strengths of the spinforbidden a 3 u X 1 g and c 3 u X 1 g phosphorescent transitions are left as empirical parameters chosen so as to best reproduce the cometary spectra. The oscillator strengths of the Ballik-Ramsay system are seemingly derived from shock-tube experiments, as the fluorescence lifetimes have not been directly observed. In this paper we report ab initio calculations of the Phillips, Swan, Ballik-Ramsay, and d 3 g c 3 u band systems of C 2. Our study has been motivated by recent observations of the C 2 Swan emission in the Red Rectangle, a nearby protoplanetary nebula which, in addition to C 2, displays unidentified molecular emission features. 20 Our aim is to harness the diagnostic power of C 2 in obtaining an understanding of the photophysical environment in the Red Rectangle. In order to do this we require a reliable model of C 2 spectroscopy built on a consensus of ab initio calculations and spectroscopic measurements. Despite previous interest in C 2, it is now timely for a comprehensive study of its electronic and spectroscopic properties, using current state-of-the-art techniques of computational quantum chemistry. II. THEORY AND COMPUTATIONAL METHODOLOGY Ab initio quantum chemical calculations were performed in an effort to obtain accurate predictions of the spectroscopic properties of the Phillips, Swan, Ballik-Ramsay, and the d 3 g c 3 u transitions of C 2. The potential energy surfaces of the electronic states of interest were computed using the internally contracted multireference configuration interaction MRCI method, 21,22 whereby all single and double excitations from a complete active space self-consistent-field CASSCF reference state are included in the electronic wave functions. The active space used is complete valence, consisting of all eight molecular orbitals MOs which arise from the 2s, 2p atomic orbitals AOs of the carbon atoms, which accommodate the eight active valence electrons. In generating the reference states, a 10:1 weighting was typically given to the two lowest /2007/126 8 /084302/11/$ , American Institute of Physics

2 Kokkin, Bacskay, and Schmidt J. Chem. Phys. 126, states of a given symmetry and spin multiplicity in order to ensure correct convergence. Furthermore, the CASSCF wave functions were constrained to be orbital angular momentum eigenfunctions with the appropriate quantum number. For any given electronic state the calculations were carried out at 35 distinct internuclear distances in the range of Å. The basis sets in the valence MRCI calculations are the augmented correlation-consistent polarized aug-cc-pvxz bases 32 of Dunning, Jr. and co-workers and de Jong et al., 31 where x=d, T, Q, 5, 6. Thus, the highest level of strictly variational theory utilized is MRCI/aug-cc-pV6Z. The energies obtained were also corrected for quadruple excitations using Davidson s method. 33,34 Core and corevalence CV correlation energy corrections were obtained by the MRCI technique using the aug-cc-pcvqz basis set ,32 Scalar relativistic energy corrections were calculated via the Douglas-Kroll-Hess approach at the MRCI/cc-pVQZ level of theory. Basis set extrapolation to the complete basis set limit, used in conjunction with correlation-consistent basis sets, has become a widely used technique in many high level quantum chemical studies in recent years According to the experience gained for single reference wave functions such as MP2 and CCSD T, the simple empirical formula E x = E Bx 3, where x is the cardinal number of the basis set, works remarkably well, provided the SCF reference and correlation energies are separately extrapolated. 40,41 While a similar situation should apply to multireference wave functions because of the scarcity of numerical data, there is no consensus, as yet, as to how an extrapolation scheme, such as that outlined above, is to be applied. While Müller et al. 15 recognized this problem in their recent MRCI work on diatomics including C 2, they took what they regard as a pragmatic procedure, whereby they simply applied Eq. 1 to the total valence correlated energies obtained via cc-pvtz and cc-pvqz calculations. In comparison with experiment, their extrapolated results certainly represent a significant improvement, although, on the average, their best harmonic frequencies are still 12 cm 1 below the experimental values mostly because of the neglect of the CV correlation. Since in our work substantially larger basis sets, up to aug-cc-pv6z, are used, the effects of such an extrapolation are expected to be substantially less. However, the corrections could more crucially depend on the exact nature of the extrapolation, e.g., just how the reference and dynamical correlation energies are separated. In light of the lack of consensus about extrapolation techniques for MRCI-type wave functions, we decided not to use extrapolation in this work. We note, nevertheless, that according to our test calculations on the a 3 u state, use of the above extrapolation Eq. 1 of the MRCIDav energies results in an increase in the harmonic frequency by 1 cm 1. As a test of the performance of the MRCI approach, as summarized above, full valence configuration interaction FCI calculations, using Dunning, Jr. s cc-pvdz basis, 27 were also performed on the X 1 g and a 3 u states. In the equilibrium regions the MRCI energies were found to be 1 higher than the FCI values by 850 and 1050 cm 1 for the X 1 g and a 3 u states, respectively. By contrast, the Davidson corrected MRCI energies were 110 and 230 cm 1 lower and higher, respectively, than the FCI values. More importantly, the equilibrium geometries and frequencies obtained by the Davidson corrected MRCI calculations are within Å and 1 cm 1 of the FCI values, to be compared with the maximum MRCI-FCI differences of Å and 6 cm 1, respectively. These calculations provide a convincing demonstration of the quality of MRCI predictions, provided the effects of quadruple excitations are accounted for. Alternative methods for the latter, such as averaged coupled pair functional 42 and multireference-average quadratic coupled cluster, 43,44 were also tested. We found their performance comparable with that of the simpler Davidson method. Utilizing the computed potential energy curves and electronic dipole transition moments, also obtained via MRCI, vibrational wave functions and energies for each surface were obtained variationally using our own code. This was followed by the computation of the vibrational state to state transition moments as integrals of the products of the vibrational wave functions and the computed electronic transition moment function. From these the oscillator strengths f values and Einstein A coefficients were calculated. The latter were then used to determine the radiative lifetimes of the different vibronic states. The spectroscopic constants e, e x e, and e y e of each electronic state were obtained, as in experimental analyses, by fitting the vibrational energies G v as polynomials of v1/2, while the equilibrium separations r e were obtained directly from the electronic energies. Rotational constants were obtained from 45 1 B v = v 2 R 2 v, 2 with a linear fit of B v against v1/2 yielding B e and e. Nonadiabatic effects were explored for the X 1 g and d 3 g states which exhibit avoided crossings with the B 1 g and e 3 g states, respectively, where the rate of change of the wave function with respect to internuclear distance is largest. At the CASSCF level, the matrix elements / R B 1 g / R X 1 g and e 3 g / R d 3 g were calculated numerically as a function of internuclear distance from the inner product of two nearby wave functions 46 r;r 0 R r;r 0 = c r;r 0 r;r 0. Diabatic electronic wave functions and 1 r;r = cos R r;r sin R r;r 2 r;r = sin R r;r cos R r;r were then generated by integration of the mixing angle 3 4 5

3 Radiative lifetimes for C 2 J. Chem. Phys. 126, d dr = / R, 6 where the mixing was defined to be zero at the equilibrium bond length of the lower state. The nonadiabatic vibrational energy levels were calculated by solving the nuclear Schrödinger equation on the diabatic potential energy curves, V 11 and V 22, where V 11 = 1 H 0 1 = cos 2 V sin 2 V, V 22 = 2 H 0 2 = sin 2 V cos 2 V, 7 8 where H 0 is the electronic Hamiltonian in the absence of spin-orbit effects. The diabatic vibronic basis thus obtained was then coupled by V 12 = sin cos V V. A detailed description of this procedure is given by Heumann et al. 47 and Simah et al. 46 The lower d 3 g energy levels are depressed upon consideration of nonadiabatic effects by 0.13 cm 1 for v=1, increasing to 0.3 cm 1 for v=3, and then evolving into a positive correction of 2 cm 1 for v=10. We found that for the X 1 g state, the vibrational energies are decreased by about v cm 1 for 1 v 6, with this correction increasing to about 1 cm 1 for v=10. As these corrections were found to be much less than the errors implicit in our quantum chemical treatment as gauged by the differences between the calculated and experimental energy levels, they were neglected in the present treatment. Spin-orbit effects were estimated by utilizing the Breit- Pauli spin-orbit operator, H SO r. 48 The electronic matrix elements r;r H SO r r;r were calculated at a range of bond distances, allowing the calculation of matrix elements,,v H SO,v, in the vibronic basis which was generated by solving the nuclear Schrödinger equation on the six adiabatic, non-spin-orbit potential energy curves, X 1 g, A 1 u, a 3 u, b 3 g, c 3 u, and d 3 g. The inclusion of spin-orbit matrix elements was found to result in negligible changes in the vibrational energy levels. The =1 a 3 u and d 3 g vibrational energy levels were generally depressed by less than 0.01 cm 1. Of the other states, the X 1 g and A 1 u levels were found to change by less than 0.01 cm 1, as were most of the b 3 g and c 3 u levels. A shift of cm 1 was observed for the c 3 u v=1 level, which was the largest such perturbation observed. Given the very small effect of spin-orbit coupling on the calculated energy levels, it was neglected in the treatment reported in this work. The ab initio MRCI calculations were carried out using the MOLPRO programs, 42 on the SGI Altix AC computer facility of the Australian Partnership for Advanced Computing at the National Supercomputing Centre, Australian National University, Canberra. To ensure that comparisons of computed and observed spectroscopic constants are meaningful, throughout our work we have endeavored to replicate the fitting procedures of the authors of the experimental papers. 9 FIG. 1. Calculated potential energy curves for the a 3 u and d 3 g states obtained at the MRCI/aug-cc-pVxZ levels of theory x=t yields highest energy; lowest energy corresponds to x=6. III. RESULTS AND DISCUSSION A. Swan system The Swan system comprises the set of transitions between the a 3 u and d 3 g triplet states. Its origin lies in the blue-green region and as such it is the most observed band system of C 2. The system was observed as early as 1802 Ref. 1 and was first described by Swan in The bands were variously ascribed to C 2 or acetylene and even C 2 2, 50,51 though Mulliken considered it more probable that they were due to C While the chemical assignment was not affirmed until 1929 Ref. 53 through comparisons with the isoelectronic molecule BeO, the Swan system was already widely observed in comets in the 19th century and is a principal feature of the spectra taken from the recent Deep Impact mission. 54 More recently, the Swan bands have been observed in emission from planetary nebulae, 5,20 being especially prominent in the spectra of the Egg Nebula. 4 The first extensive calculations on the Swan system were of the MRCI type with a relatively small basis set by Zeitz et al. 7 The results of these early calculations were consistent with observations at the time, but were not quantitatively accurate. Nonetheless, they represented a state-of-the-art electronic structure theory in These calculations were improved by Chabalowski et al. 8 in 1981, and while others have studied the individual a 3 u and d 3 g states in more detail, the work of Chabalowski et al. has been generally regarded as the most sophisticated theoretical study of the spectroscopic transitions of the Swan system. In this work, by employing substantially larger and more complete basis sets as well as explicitly accounting for CV correlation and scalar relativistic effects, we have achieved a higher accuracy and thus greater reliability in the computed spectroscopic parameters than those previously reported. 1. Potential energy curves The valence-correlated MRCI potential energy curves of the a 3 u and d 3 g states are shown in Fig. 1. The convergence of the energy levels with the increasing size of the basis set from aug-cc-pvtz to aug-cc-pv6z is evident. In the last step, going from the pentuple zeta to the hextuple

4 Kokkin, Bacskay, and Schmidt J. Chem. Phys. 126, TABLE I. Computed MRCI spectroscopic constants for the ground state of C 2 and for the Swan system. X 1 g a 3 u a 3 g Basis set r e e e x e e y e T e r e e e x e e y e T e r e e e x e e y e T e a aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz aug-cc-pv5z aug-cc-pv6z Dav DavCV DavCVRel Expt a T e =T e d 3 g T e a 3 u. zeta basis, the energy decrease is typically E h, i.e., 320 cm 1, pointing to a high degree of convergence. Actually, such a small difference is not resolved in Fig. 1. Davidson s correction for quadruple excitations, CV correlation, and relativistic corrections further decrease these energies approximately by 0.01E h, 0.10E h, and 0.03E h, respectively. From the MRCI/aug-cc-pV6Z energies, with Davidson s, CV correlation and relativistic corrections applied, the minima in the potential surfaces of the a 3 u and d 3 g states are calculated to be at and Å, respectively. These are in excellent agreement with the corresponding experimental values of and Å. However, as bond lengths are not of themselves directly observable, we also compare predicted and observed values of B e and e. For the a 3 u state the predicted values are cm 1 and cm 1, to be compared with the observed values of cm 1 and cm 1, 55 respectively. For the d 3 g state the computed values are cm 1 and cm 1, which again compare favorably with the experimental values of cm 1 and cm 1, respectively. Note that, as remarked above, the fitting procedures used replicate, as much as possible, those of the experimental papers. In an absolute sense the total valence correlated electronic energies calculated in our work are significantly lower than those of Chabalowski et al. 8 since we were able to utilize much larger basis sets than in the earlier work. Accounting for CV correlation and relativistic effects results in further significant energy lowering. More importantly, our best, i.e., most extensive, calculations predict the 0,0 band to be at cm 1, to be compared with the observed value of cm Vibrational levels, harmonic frequencies, and anharmonicities Vibrational energy levels and wave functions were obtained variationally by solving the vibrational Schrödinger equation using the appropriate calculated potential energy surface for any given electronic state. The spectroscopic constants obtained by analysis of the vibrational levels for the a 3 u and d 3 g as well as the ground X 1 g state are shown in Table I. Comparing these values with experiment, it can be seen that while the valence correlated MRCI/aug-cc-pV6Z values are fairly close to experiment as expected, the differences are significant, especially in the harmonic frequencies, where the calculations underestimate these quantities by as much as 17 cm 1. CV correlation appears to be the most important factor in reducing such deficiencies, while the Davidson correction has a slightly opposing effect. Relativistic corrections are the least important in the context of these calculations. The final computed harmonic frequencies are within 1.5 cm 1 of the observed values. The best previous theoretical result for the a 3 u state s harmonic frequency is that obtained by Bruna and Grein, who employed a full CI estimate e =1635 cm Müller et al, also by using MRCI based techniques, obtained equilibrium bond lengths, harmonic frequencies, and term energies for all valence excited singlet and triplet states of C 2 as well as B 2, N 2, and O Their MRCI/cc-pVQZ results are similar to our valence correlated MRCI/aug-cc-pVQZ results with harmonic frequencies a few cm 1 lower due to the less extensive basis sets used. A comprehensive list of calculated vibrational levels is given at the end of the paper Table IV, along with what we believe to be the most reliable observed values. It can be seen that the a 3 u state is extremely well described by the ab initio calculations. By comparison, the deviations seen in the d 3 g state levels may be attributed to the calculation not quite resolving the position of the avoided crossing between the d 3 g and the e 3 g state. We plan to investigate these problems more thoroughly in our future work on this molecule, including a full characterization of the e 3 g state. The observed vibrational levels in Table IV were taken from three distinct sources, quoting the most recent ones. We note that in the case of the v=5,6,7 levels there is a significant disagreement between the values of Tanabashi and Amano 56 and the much earlier values of Phillips. 57 Our preference for the former data was also based on the observation that the difference between theory and experiment, as a function of v, is quite smooth in the case of the Tanabashi and Amano values, but not for the Phillips data indeed, our choice is vindicated by new information received after the submission of this manuscript 58.

5 Radiative lifetimes for C 2 J. Chem. Phys. 126, FIG. 2. The calculated transition moment function for the Swan system with various basis sets. The solid line corresponds to MRCI/aug-cc-pV6Z. 3. Transition moment and oscillator strengths Figure 2 shows the transition moment function from the MRCI calculations with the aug-cc-pvtz to aug-cc-pv6z basis sets, although in the diagram the results obtained by the quadruple to hextuple zeta basis sets are practically indistinguishable. Clearly, the essential character of the transition is captured by the smaller basis calculations, and an effective convergence has been reached with the aug-cc-pv6z basis set. The CV correlation correction serves to reduce the transition moment slightly by approximately 0.5%. A comparison may be made between our calculated band strengths and the observed values. The observed values reported by Danylewych and Nicholls 59 take into account a model of the rotational profile of emission bands. Their reported ratios of the strengths of the emissions v =0 v =0,1,2 are 100:27.0:4.7, while the ratios calculated from the present ab initio calculations are 100:25.7:4.2. Emission from v =1 is observed with band strengths in the ratio 83:100:76:22.2:4.4, while the calculated ratios are 86:100:73:20.1:3.7. A similar agreement is seen for the other observed bands. Oscillator strengths for the Swan system were calculated via the usual formula f v v = 2 3 R v v 2, where is the energy difference in E h. The transition moments R v v were obtained by integration of the dipole transition moment function R r over the vibrational wave functions which had been obtained by solving the vibrational Schrödinger equation using the appropriate computed potentials. Our computed 0,0 oscillator strength, given in Table II, is in close agreement with the experimentally determined value. As the results show, the evolution of this oscillator strength with basis set size is not monotonic since the transition moment and the energy separation display different convergence patterns. The calculated value is scattered around f 00 =0.028, which suggests that the actual value cannot be too different. Previous calculations put f 00 at In light of the close agreement between the predicted and observed radiative lifetimes, as discussed below, we are confident that our computed oscillator strengths are reliable, despite the small discrepancy between theory and experiment in the case of f 00, especially since oscillator strengths are difficult to observe directly. A comprehensive set of oscillator strengths computed for the Swan and other systems of C 2 can be found at the end of this paper Table VII. 4. Radiative lifetimes By summing over the Einstein A coefficients for the relevant transitions, radiative lifetimes of the d 3 g state vibrational levels were calculated. In addition to transitions from the d 3 g state to the a 3 u state, there are also transitions to the c 3 u state that contribute about 3% 4% to the radiative lifetime. Table II reports lifetimes for the d 3 g state vibrational levels computed at different levels of theory, as well as the available experimental values for comparison. As the results in Table II demonstrate, the oscillator strengths and lifetimes are relatively insensitive to basis set variation and to CV correlation and relativistic effects. The agreement between theory and experiment is very good, with the possible exception of the v=1 state, where the best computed value of ns is larger than the measured value of 96.7 ns, by an amount which is slightly larger than the experimental error of 5.2 ns. 60 This experimental value is supported by an independent measurement of 97.1± 4.4 ns reported in the same year. 61 The excellent agreement between observation and calculation for the Swan system gives us confidence to apply our TABLE II. Oscillator strength for the Swan system and radiative lifetimes for d 3 g state vibrational levels. Lifetime a ns da Basis f 00 v=0 v=1 v=2 v=3 v=4 v=5 aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz aug-cc-pv5z aug-cc-pv6z DavCVRel Expt b 101.8± ± ±17 a Experimental value inferred from Refs. 60 and 76. b Experimental values from Ref. 60.

6 Kokkin, Bacskay, and Schmidt J. Chem. Phys. 126, transition moments have remained a source of uncertainty in the modeling of infrared spectra of carbon stars Potential energy curves The potential energy curves for the Ballik-Ramsay system a 3 u and b 3 g along with those of the other states studied in this work are shown in Fig. 3. The computed spectroscopic constants of all six states are summarized in Table III. The predicted equilibrium internuclear distances for both a 3 u and b 3 g states are in excellent agreement with experiment, as are the spectroscopic observables B e and e. The 0,0 band is placed ab initio at cm 1, which compares well with the experimental value of cm 1, the difference being only 40 cm 1. FIG. 3. Calculated potential energy curves of the X 1 g, A 1 u, a 3 u, b 3 g, c 3 u,andd 3 g electronic states obtained at the valence correlated MRCI/aug-cc-pV6Z. level of theory. highest level of theory to the Ballik-Ramsay b 3 g a 3 u, Phillips A 1 u X 1 g, and d 3 g c 3 u systems, which are described below. The results quoted below are all at the Davidson corrected MRCI/aug-cc-pV6Z level of theory and include CV correlation and scalar relativistic corrections, but neglect nonadiabatic and spin-orbit effects as described in Sec. II. B. The Ballik-Ramsay system The Ballik-Ramsay system comprises the transitions between the a 3 u triplet state and the b 3 g triplet state. It was first observed in 1958 by Ballik and Ramsay 62 though at that time it was still thought that the a 3 u state was the electronic ground state, i.e., X 3 u. By analysis of perturbations in this band system, Ballik and Ramsay were able to show that the true ground state of C 2 was X 1 g. 63 Shock-tube experiments reported in 1975 revealed the first estimate of the oscillator strengths of the Ballik-Ramsay system. 64 This was followed up in 1981 by a theoretical treatment by one of the authors of the experimental paper, though the values calculated were found to be 30% below those obtained experimentally. 65 While better agreement with experiment was generally achieved by Chabalowski et al. in their more extensive theoretical study of the Ballik-Ramsay system, 9 the 2. Vibrational levels, harmonic frequencies, and anharmonicities The vibrational harmonic frequencies and anharmonicities calculated for the Ballik-Ramsay system given in Table III are in excellent agreement with experimentally derived values for both the a 3 u and b 3 g states. The excellent agreement lends confidence to the oscillator strengths reported below. A comprehensive summary of calculated energy levels is presented in Table IV. 3. Oscillator strengths and radiative lifetimes Figure 4 displays the transition moment functions obtained at the MRCI/aug-cc-pV6Z level of theory calculations for all four systems studied in this work. For the Ballik- Ramsay system, using our calculated transition moments, we obtain a value of as the summed equilibrium transition moment: R e 2. This is just outside the range of experimental error, the observed value being 0.65±0.15, 64 but closer to the theoretical value obtained by Cooper, 65 which is 30% lower than the experimental value. The present calculations seemingly support the theoretical value of Cooper, which has been incorporated into the modeled spectra of carbon stars. 3 To the best of our knowledge there have been no direct determinations of the radiative lifetimes of the b 3 g vibrational states. Those reported by Martin 67 are seemingly derived from the shock-tube data of Cooper and Nicholls 64 and the oscillator strengths reported by Chabalowski et al. 9 The TABLE III. Spectroscopic constants in cm 1 calculated at the Davidson corrected MRCI/aug-cc-pV6Z level of theory plus CV correlation and relativistic corrections. T e r e e e x e e y e B e e State Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt. X 1 g A 1 u a 3 u b 3 g c 3 a u d 3 b g a See Ref. 75. b See Ref. 58.

7 Radiative lifetimes for C 2 J. Chem. Phys. 126, TABLE IV. Vibrational origins of the C 2 molecule calculated at the Davidson corrected MRCI/aug-cc-pV6Z level of theory, including CV and relativistic corrections, compared to experimental results. The values are quoted in cm 1 relative to the v=0 level in each state. X 1 g a A 1 u b v Calc. Expt. c e Calc. Expt. c e a 3 u c b 3 g d v Calc. Expt. c e Calc. Expt. c e d 3 g e c 3 u v Calc. Expt. c e Calc. Expt. c e a Reference 79. b References 72, 73, and 79. c References 55 and 57. d References 80 and 81. e References 55 57, but see Ref. 58. lifetimes, as obtained in this work, for the v=0, v=1, and v=2 states of the Ballik-Ramsay system are shown in Table V. Given the accuracy in the computed spectroscopic constants, i.e., agreement with experiment to within 1%, we expect our calculated oscillator strengths and lifetimes to be as accurate as in the case of the Swan system discussed above. We attribute the apparent discrepancy between the calculated and observed lifetime of the b 3 g state in its ground vibrational state to the fact that the experimental oscillator strengths were obtained by indirect means. C. Phillips systems The Phillips system comprises the transitions between the ground X 1 g singlet state and the first excited A 1 u

8 Kokkin, Bacskay, and Schmidt J. Chem. Phys. 126, TABLE V. Radiative lifetimes for the b 3 g state vibrational levels calculated at the Davidson corrected MRCI/aug-cc-pV6Z level of theory including CV and relativistic corrections. Lifetime s Method v=0 v=1 v=2 Calc Obs. a 17.2 a See text. FIG. 4. The calculated transition moment functions R r of the Phillips, Swan, Ballik-Ramsay, and d 3 g c 3 u electronic transitions calculated at the MRCI/aug-cc-pV6Z level of theory. singlet state. It was first observed in 1948 by Phillips, 68 but the band origin was not determined until later. The 2,0 band of the Phillips system is routinely used in astronomical observations to estimate column densities of C Potential energy curves and spectroscopic constants The computed potential energy curves for the X 1 g state and the A 1 u state are shown in Fig. 3. The resulting spectroscopic observables B e and e for both the X 1 g state and the A 1 u state are in excellent agreement with experiment Table III. The ab initio origin band is placed at 8376 cm 1, which is some 110 cm 1 higher in energy than the actual value of cm The harmonic frequencies and anharmonicities of the Phillips system, determined from the computed vibrational levels, are shown in Table III. As for the Swan and Ballik- Ramsay systems, all computed spectroscopic constants agree well with experiment. In Table IV, the calculated values of the vibrational energies for the A 1 u state are particularly impressive. However, as for the d 3 g state, the X 1 g state is not computed as accurately. This is due to the multiconfigurational nature of the X 1 g state. A more accurate description of the X 1 g will require a commensurately accurate description of the B 1 g state, which is planned for future work. Nevertheless, while the X 1 g state is not described as well as the well-behaved A 1 u and a 3 u states, the discrepancy between the calculated and experimental vibrational energies is quite small. 2. Oscillator strength and radiative lifetimes The calculated transition moment for the Phillips system is shown in Fig. 4. A node is predicted at 1.7 Å, in agreement with the findings of Chabalowski et al. 9 and Langhoff et al. 12 There is also an avoided crossing in this region of the potential. However, this is far from the region of interest, being some 2 ev above the zero point level, and does not affect the calculation of observables reported here. The calculated radiative lifetimes are summarized in Table VI with a comparison of various experimentally and theoretically determined values. It can be seen that the present values lie a little higher than the more recent experiments by a little more than the error ranges in the latter. The steep increase of the lifetime with decreasing v is reproduced by the calculations. Indeed, the present value for the lifetime of v =0 is the only prediction within the error range for the latest experimental value. Erman and Iwamae recommended that the value of the 2,0 oscillator strength be taken as f AX 20 = 1.36± The value obtained in this work is f AX 20 = , which is just outside the experimental uncertainty. AX The calculated oscillator strength for the origin is f 00 = A recent survey of Phillips absorption spectra towards translucent sight lines revealed a set of selfconsistent oscillator strengths for the Phillips system. 70 The TABLE VI. Radiative lifetimes for A 1 u state vibrational levels calculated at the Davidson corrected MRCI/ aug-cc-pv6z level of theory, including CV and relativistic corrections. Theory Expt. v Present LBRK a ORW b van D c CPB d E&I e B86 f B85 f ± ± ± ± ± ± ± ± ± ±1.0 a Reference 12. b Reference 11. c Reference 10. d Reference 9. e Reference 69. f References 77 and 78.

9 Radiative lifetimes for C 2 J. Chem. Phys. 126, TABLE VII. Oscillator strengths exponent for band systems of the C 2 molecule calculated at the Davidson corrected MRCI/aug-cc-pV6Z level of theory, including CV and relativistic corrections. Negative values indicate that the states are reversed. All values are given for excitation. v System v A 1 u X 1 g Phillips d 3 g a 3 u Swan b 3 g a 3 u Ballik-Ramsay d 3 g c 3 u derived ratio f AX 30 / f AX 20 =0.216 is mirrored to the reported precision by the present calculated values see Table VII. Ifthe present values are taken as accurate, then the previously reported oscillator strength for the Mulliken system of C 2 may also be taken as accurate, f DX 00 = Ref. 13, as the calculated ratio of f AX 20 / f DX 00 = falls within the experimental error: ± D. d 3 g ]c 3 u system The d 3 g c 3 u system comprises the transitions between the c 3 u triplet state and the d 3 g triplet state. There are few experimental data available for the d 3 g c 3 u triplet system as the c 3 u state has only been seen through its perturbations of the A 1 u state. 72,73 There has been one tentative assignment of a band observed by laser-induced fluorescence as having the c 3 u state as its lower level. 74 The system remains to be observed directly, but is integral to the photophysics of cometary systems, 19 and must be considered when calculating lifetimes of the d 3 g state vibrational levels but see Ref Potential energy curves and spectroscopic constants The potential energy curves for the c 3 u and d 3 g states are shown in Fig. 3 with the transition moment function shown in Fig. 4. The spectroscopically determined value of T e for the c 3 u state is cm 1, while the calculated value is somewhat higher at 9315 cm 1. The spectroscopic constants derived from our computed vibrational energy levels of the d 3 g c 3 u system are shown in Table III. In contrast to the predictions for the other five states studied in this work, the harmonic frequency and first anharmonicity obtained for the d 3 g state appear to be a little low in comparison with the observed values. Moreover, there are also greater discrepancies between the computed and observed B e and e values than for any of the other states. Indeed, this general disparity has prompted us to review how the observed frequency and anharmonicity were derived. In the paper of Davis et al., 72 five perturbations in the spectrum of the A 1 u state are reported. By equating J for the mutually perturbing A 1 u and c 3 u states, it was possible to write down the five energies in terms of an empirical formula involving T e, e, e x e, B e, and e. The five simultaneous equations were solved by matrix inversion. We repeated this procedure using the term values of Davis et al. and obtained T e = , e = , e x e =17.43, B e =1.920, and e = , where all values are in reciprocal centimeters. While these numbers are not exactly the same as those quoted by Davis et al., they are closer to the latter than to the ab initio values. On the other hand, a simulation of the fitting procedure using the calculated energy levels led to very similar values to those reported in Table III. This suggests that the procedure described above is self-consistent. Neverthe-

10 Kokkin, Bacskay, and Schmidt J. Chem. Phys. 126, less, in light of the excellent agreement between theory and experiment for all other states, we believe that our computed parameters for the c 3 u state are more reliable than those derived by Davis et al. Our criticism of the procedure used by Davis et al. is that it uses the minimal number of observed data to derive the same number of parameters. Such a procedure is exact only if the equations connecting them are exact, rather than approximate see Ref Oscillator strengths The oscillator strength for the d 3 g c 3 u origin is calculated to be f d c 00 = This is about twice as large as that assumed in the study of the Hale-Bopp comet by Rousselot et al., 19 where f d c 00 is taken to be 0.1 f d a 00. The d 3 g c 3 u oscillator strength is not insignificant, and our calculations estimate that about 4% of irradiance from the d 3 g state is to the c 3 u state. As such, it should be possible to observe the d 3 g c 3 u emission by dispersed fluorescence spectroscopy. IV. CONCLUSIONS It has been demonstrated that Davidson corrected MRCI calculations employing a full-valence CAS reference state and the aug-cc-pv6z basis, with the inclusion of core and core-valence correlation and scalar relativistic corrections, reproduce the e and B e spectroscopic constants as well as the anharmonicities of C 2 to better than 1% accuracy, in comparison with experiment, for the X 1 g, A 1 u, a 3 u, b 3 g, and d 3 g electronic states. For the c 3 u state the discrepancies appear to be larger, and in this case we question the validity of the derivation of spectroscopic constants from a minimal number of experimental pieces of data. The demonstrated quality of our predictions gives us confidence in the accuracy and reliability of our calculated oscillator strengths of the Swan, Ballik-Ramsay, Phillips, and d 3 g c 3 u systems. Indeed, for the systems for which there is rigorous lifetime information, the present calculations fall within or very near the reported experimental errors. It may be concluded that for the purposes of astrophysical models, the present values of the oscillator strengths in the Ballik- Ramsay and d 3 g c 3 u systems should be taken as reliable. These methods are readily applicable to other band systems of C 2 and indeed to spectroscopic systems of similar molecules such as CN and CO. A comprehensive model of C 2 cometary photophysics now waits on a detailed calculation of the a 3 u X 1 g and c 3 u X 1 g intercombination systems. ACKNOWLEDGMENTS The authors thank Dr. Klaas Nauta University of Sydney for helpful advice regarding the manuscript. They are also pleased to acknowledge the helpful comments and advice of Dr. David Schwenke NASA Ames Research Center concerning CV correlation. They wish to express their thanks to the Australian Partnership for Advanced Computing National Facility for access to the SGI Altix AC system. One of the authors D.L.K. thanks the University of Sydney for a university postgraduate award. This research was supported under the Australian Research Council s Discovery funding scheme Project No. DP The basis set database, 32 version 02/02/06, was developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory, which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy under Contract No. DE-AC06-76RLO Contact Karen Schuchardt for further information. 1 W. H. Wollaston, Philos. Trans. R. Soc. London 11, W. Huggins, Astron. Nachr. 71, J. H. Goebel, J. D. Bregman, D. M. 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