On the vibronic level structure in the NO₃ radical: II. Adiabatic calculation of the infrared spectrum

Abstract

The infrared spectrum of the nitrate radical (NO₃) is studied on the basis of ab initio calculations that use the equation-of-motion coupled-cluster method known as EOMIP-CCSD. While the spectrum associated with vibrational levels is simple, with only the ν₂ fundamental having appreciable intensity, that for the e′ vibrational states is extremely rich. The first 20 levels of e′ symmetry (not including those that feature two quanta of excitation in the out-of-plane ν₂ mode) are calculated. The major features are: the ν₄ fundamental has a large (positive) anharmonicity, several combination bands involving ν₄ and its overtones are prominent in the spectrum, and the ν₃ fundamental is calculated to lie at 1067 cm⁻¹ and have a small intensity. In fact, of the 20 levels studied, the predicted dipole strength associated with the transition to ν₃ ranks nineteenth. The results are broadly consistent with an earlier study that was based on a diabatic representation that also indicated that ν₃ is well below the 1492 cm⁻¹ band to which it has been assigned. The present study suggests that the 1492 cm⁻¹ experimental feature is due to a strongly absorbing level that has mostly ν₃ + ν₄ character, with some mixing of ν₁ + ν₄.

Keywords:

• adiabatic
• vibronic
• coupling
• nitrate
• radical

Acknowledgements

Discussions with K. Kawaguchi (Okayama), T.A. Miller (Ohio State), H. Müller (Köln) and E. Hirota (Graduate University for Advanced Studies, Hayama) at the 63rd Ohio State University International Symposium on Molecular Spectroscopy (Columbus, OH, 2008) served as motivation for this work. Gratefully acknowledged are several discussions over the past year with M. Jacox (NIST) as well as a preprint of 10. Additional discussions with M. Okumura (Caltech), A. Krylov (USC), T. Ichino (Austin) and J. Vazquez (Austin) helped to shape this work. The research was supported by the US National Science Foundatation (Grant CHE-0710146) and the US Department of Energy, Basic Energy Sciences Division (Contract DE-FG01-05ER05-01).

It has been a pleasure to know, visit, and publish a few papers with Fritz Schaefer over the past twenty years. His contributions to quantum chemistry have been influential to me, and were a source of inspiration during my graduate days, and he has become a valued friend since that time. Thank you, Fritz.

Notes

Notes

1. This statement uses the language of the diabatic treatment. In the limit of zero coupling, the molecule is a rigid D _3h system. As the coupling strength increases (between the diabatic electronic states), the force constant for e′ distortion decreases. The ‘threshold’ referred to in the text is the point at which the lowest e′ force constant on the adiabatic potential surface becomes negative.

2. The totally symmetric fundamental is known to be near 1050 cm⁻¹, but this is clearly not seen in the infrared.

3. These include all determinants derived from the closed shell reference via removal of one electron, and those that can be generated from these ‘1h’ (one-hole) determinants by excitation of an additional electron from an orbital occupied in the Hartree–Fock self-consistent field solution for the reference state to one that is not occupied (2h1p determinants).

4. Using the diabatic potential parameters from 5 (which were used in 7), the normal coordinates associated with the lowest adiabatic surface (q ₃ and q ₄) are related to those of the anion ( and ) by the relationswhich implies a rotation of ca. 41 degrees away from the nearly pure stretch and bend coordinates of the anion. Thus, it is really not appropriate to refer to ν₃ as a stretching mode, although it is true that it contains slightly more stretch than bend.

5. All of the ‘skeleton’ force constants except the essential can be fit from energy calculations only for geometries that have C _2v and D _3h symmetries. The same is true for the dipole moment components, except the two elements. These require additional points for a numerical fit.

6. The basis sets in Table 3 and mentioned throughout the text are as follows. Basis sets designated as 10, 15 and 20 contain the corresponding number of basis functions for all six normal modes. That designated as A has 20 functions on ν₄ and ten on all other modes, while A′ is the same as A but omits the functions on ν₂. Basis sets B′ and C′ are similar to A′, but have thirty and forty functions on ν₄, respectively.

7. This is simply the strategy long used by the German group 5 for calculating spectral intensities. In their work, the trial vector usually has a simpler form, as–for example–photoelectron spectra using the neutral (or anion in the case of negative ion PES) as the diabatic reference can be calculated with Lanczos driven by a trial vector with a single nonzero element. For the adiabatic calculation of infrared spectra, the author has not seen this method used, but it is entirely possible that it has.

8. Isotope shifts calculated for the ¹⁵NO₃ and ¹⁵N¹⁸O₃ species, relative to the normal isotopomer are (respectively, in cm⁻¹): −8 and −23 (A); −15 and −50 (B); −17 and −50 (C); −22 and −72 (D); −13 and −83 (E); −20 and −80 (F); −27 and −100 (G); −22 and −101 (H); −35 and −101 (I); −16 and −113 (J); −29 and −117 (K); −29 and −108 (L); −33 and −117 (M); −39 and −151 (N); and −31 and −119 (O).

9. There are, of course, e′ levels in the region investigated in this paper that involve two quanta of excitation in 2ν₂. Using the basis with twenty functions per mode, these are found at 1999 and 2442 cm⁻¹ and can be unambiguously denoted as 2ν₂ + ν₄ and 2ν₂ + 2ν₄, respectively. However, both of them have intensities of ca. 10⁻⁴ or less than that corresponding to level F, and are unlikely to be seen experimentally for some time. These levels are excluded from discussion in this paper.

10. Using the diabatic Hamiltonian of 5, the vibronic level with the character of ground state ν₃ appears at 994 cm⁻¹. While it was stressed in 7 that this is only an approximate position with an error bar of about 100 cm⁻¹, this qualification may have been overlooked by others that have referred to this as an ab initio estimate. Indeed, it was not based on an ab initio calculation but rather a model potential. The predictions from this work can properly be termed ab initio, however.

11. In 7, it was suggested that the feature at 753 cm⁻¹ and labelled as 3 in the most recent dispersed fluorescence study 11 and assigned to ν₂ is instead 2ν₄. The analogous band was also seen at 760 cm⁻¹ in the pioneering dispersed fluorescence study of NO₃ by Ishiwata et al. 23 and assigned there to 2ν₄.

12. Reported positions (none of these are from high-resolution experiments nor do they discriminate between band origins and band centers) for ν₁ are: 1051 cm⁻¹ 22; 1053 cm⁻¹ 11; 1057 cm⁻¹ 23; and 1060 cm⁻¹ 22. Those for ν₄ include: 360 cm⁻¹ 22 (this was, however, assigned to the lowest fundamental mode of a C _2v structure of NO₃); 361 cm⁻¹ 4; 368 cm⁻¹ 11 and 380 cm⁻¹ 23. The values used in the text are roughly averages of these values.

13. This is the reason why the convergence of the energy of level L (see Table 3) is–by far–the fastest of any level above 1600 cm⁻¹. The still small, but greater, relative contribution of ν₃ + 2ν₄ to its partner in the effective dyad (level O) causes its energy to drop by more than 100 cm⁻¹ when going from 10 to 15 basis functions on ν₄. The corresponding drop in level L is just 23 cm⁻¹.

14. Group theoretical considerations are important here. The positions of the calculated combination levels involving ν₂ are in fact quite different from the sum of ν₂ and the e′ levels documented in Table 3. However, such combination levels have e″ symmetry, and are dipole forbidden. It is the a ₁′ parts of the degenerate levels that combine with ν₂ to give states that can be accessed by dipole-allowed transitions.

15. The kinetic energy operator.

16. The relevant harmonic frequencies are: 1108 [CCSD(T)/ANO0], 1123 [EOMIP-CCSD/ANO0], 1051 [EOMIP-CCSDT/ANO0], 1141 [CCSD(T)/ANO1] and 1170 [EOMIP-CCSD/ANO1]. All of these are within 120 cm⁻¹ and in keeping with the author's general observation that this mode has a harmonic frequency that is substantially less sensitive to the level of theory than is ν₄. This, in turn, reflects the fact that ν₄ is affected to a much greater degree by vibronic coupling with the state. The UHF-based calculations are based on a reference Hartree-Fock wavefunction that transforms properly as the irreducible representation of the D _3h point group (i.e. not a symmetry-broken reference).