The nonrigid bender Hamiltonian using an alternative perturbation technique

The derivation of the nonrigid bender Hamiltonian for the calculation of the rotation-vibration energies of a triatomic molecule was completed by P. Jensen and P. R. Bunker [J. Mol. Spectrosc. 99, 348–356 (1983)] using Van Vleck perturbation theory. This perturbation technique assumes that the bending vibration frequency is much less than the stretching vibration frequencies (such as in the ground electronic state of C₃). For molecules such as H₂O, for which this is not the case, an alternative formulation of the theory is possible in which allowance is made for the dependence of the perturbation theory energy denominators on the bending vibration quantum number v₂ and on the rotational quantum number K. This was pointed out by A. R. Hoy and P. R. Bunker [J. Mol. Spectrosc. 74, 1–8 (1979)], and some of the corrections were made by them. We now develop the perturbation theory expressions allowing for the dependence of all the energy denominators on v₂ and K.     

Rotation-vibration energy levels of H2O and C3 calculated using the nonrigid bender Hamiltonian

We have written a new computer program for diagonalizing the nonrigid bender Hamiltonian, and have based the program entirely on the theory as reviewed by P. Jensen [Comp. Phys. Rep. 1, 1–56 (1983)] and P. Jensen and P. R. Bunker [J. Mol. Spectrosc. 118, 18–39 (1986)]. Using this program we can calculate the rotation-vibration energy levels of a triatomic molecule from the potential energy function. The program is an improvement over an earlier version, particularly in the systematic treatment of all singular terms, and in the allowance made for the dependence of all perturbation energy denominators on the bending quantum number v₂ and rotation quantum number K. The new program can be used for symmetric and unsymmetric triatomic molecules. In the present paper we test the program by applying it to the calculation of the rotation-vibration energy levels of C₃ from an ab initio potential surface, and of H₂O from ab initio and experimental potential surfaces.     

A refined potential energy function for the electronic ground state of H2Se

The potential energy surface for the electronic ground state of the hydrogen selenide molecule has been determined previously by Jensen and Kozin [J. Mol. Spectrosc. 160 (1993) 39] in a fitting to experimental data by means of the MORBID computer program. We report here a further refinement of this surface, also made with the MORBID program. With the refined potential surface, we can make predictions of rotation-vibration transition wavenumbers for H₂Se, D₂Se, and HDSe, and with these predictions we can assign weak spectra of these molecules. We assign here two very weak bands of HD⁸⁰Se, ν₁+ν₂+ν₃ and 2ν₁+ν₃. The refinement of the potential energy surface was made possible because (1) the number of vibrational states characterized experimentally for various isotopomers of H₂Se has approximately doubled since 1993, and (2) we now have access to larger computers with which we can fit energy spacings of states with J?8, whereas Jensen and Kozin could only use J?5. In the present work, we fitted rotation-vibration energy spacings associated with 24 vibrational states of H₂⁸⁰Se with v₁?6, v₂?3, and v₃?2; 11 vibrational states of D₂⁸⁰Se with v₁?2, v₂?3, and v₃?2, and 17 vibrational states of HD⁸⁰Se with v₁?3, v₂?3, and v₃?3. The input data set comprised 3611 energy spacings. In the fitting, we could usefully vary 29 potential energy parameters. The standard deviation of the fitting was 0.12 cm⁻¹ and the root-mean-square deviation for 49 vibrational term values was 0.59 cm⁻¹.     

A precise solution of the rotation bending Schrödinger equation for a triatomic molecule with application to the water molecule

In this paper we report the results of improving the non-rigid bender formulation of the rotation-vibration Hamiltonian of a triatomic molecule [see A. R. Hoy and P. R. Bunker, J. Mol. Spectrosc., 52, 439 (1974)]. This improved Hamiltonian can be diagonalized as before by a combination of numerical integration and matrix diagonalization and it yields rotation-bending energies to high values of the rotational quantum numbers. We have calculated all the rotational energy levels up to J = 10 for the (v₁, v₂, v₃) states (0, 0, 0) and (0, 1, 0) for both H₂O and D₂O. By least squares fitting to the observations varying seven parameters we have refined the equilibrium structure and force field of the water molecule and have obtained a fit to the 375 experimental energies used with a root mean square deviation of 0.05 cm^?1. The equilibrium bond angle and bond length are determined to be 104.48° and 0.9578 Å respectively. We have also calculated these energy levels using the ab initio equilibrium geometry and force constants of Rosenberg, Ermler and Shavitt [J. Chem. Phys., 65, 4072 (1976)] and this is then the first complete ab initio calculation of rotation-vibration energy levels of high J in a polyatomic molecule to this precision. the rms fit of these ab initio energies to the experimental energies for the H₂O molecule is 2.65 cm^?1.     

Ab initio SCF-Cl studies on the large amplitude bending motion of the water molecule: Rigid,semirigid, and nonrigid bender models for the Hamiltonian

Self-Consistent Field (SCF) and Configuration Interaction (CI) studies are performed on the bending mode of the water molecule using a double zeta plus polarization basis set. The ab initio points are fitted to a three-parameter double minimum potential consisting of a quadratic plus Lorentzian terms. The vibration-rotation energies are then evaluated using the large amplitude Hamiltonian developed by P. R. Bunker and co-workers at various levels of approximations. It is found that the calculated frequencies improve significantly as one proceeds from approximate H_b⁰⁰(ρ) to rigid bender H_b⁰(ρ) [P. R. Bunker and J. M. R. Stone, J. Mol. Spectrosc.41, 310–332 (1972)] to semirigid bender H_b⁰(r, ρ) [P. R. Bunker and P. M. Landsberg, J. Mol. Spectrosc.67, 374–385 (1977)] Hamiltonian. With H_b⁰(r, ρ), the ab initio calculated bending frequency ν₂ differs from the observed value (1595 cm^?1) by 30 cm^?1 and the barrier height is 12 229 cm^?1. It is also shown that ν₂ and its first four overtones are better calculated by 45–98 cm^?1 when the ab initio potential is used directly instead of the three-parameter analytic potential fitted to ab initio data. Finally, rotation bending energy levels are calculated for v₂ ≤ 3 and J ≤ 10 on the basis of a nonrigid bender Hamiltonian of A. R. Hoy and P. R. Bunker [J. Mol. Spectrosc.74, 1–8 (1979)], using the ab initio quadratic force field of P. Hennig, W. P. Kraemer, G. H. F. Diercksen, and G. Strey, [Theor. Chim. Acta47, 233–248 (1978)]. These results show that the accuracy of calculated force constants and frequencies is critically dependent not only on the size of the basis set but also on the number and spacing of the ab initio points used to derive the force field.     

HCNO as a semirigid bender

The semirigid bender Hamiltonian [Bunker and Landsberg, J. Mol. Spectrosc., 67, 374–385 (1977)] is used to fit the rotation-vibration energy level separations in the fulminic acid (HCNO) molecule. The allowance made in the model for the variation of the CH and CN bond lengths with the HCN bending angle proves to be very important, and as well as achieving a good fit we are able to make a detailed investigation of the shape of the HCN bending potential function.From the results we conclude that the equilibrium structure of HCNO is linear but that excitation of the ν₁ or ν₂ stretching vibrations gives rise to an effective HCN bending potential function having its minimum at a nonlinear configuration. Even in the ground state the zeropoint vibrational contributions from ν₁ and ν₂ to the effective HCN bending potential give a small barrier (11.5 cm^?1) to linearity, and we determine that the zero-point HCN bending vibrational amplitude is ±34°.     

The effect of the breakdown of the Born-Oppenheimer approximation on the rotation-vibration Hamiltonian of a triatomic molecule

The rotation-vibration-electronic Hamiltonian of a triatomic molecule has been derived in a manner similar to that used by J. T. Hougen, P. R. Bunker, and J. W. C. Johns [J. Mol. Spectrosc.34, 136 (1970)] in deriving the rotation-vibration Hamiltonian. An effective rotation-vibration Hamiltonian for the ground electronic state has been obtained from this, by using the perturbation technique of P. R. Bunker and R. E. Moss [Mol. Phys.33, 417 (1977)], in order to account for the effect of the breakdown of the Born-Oppenheimer approximation to second order. The same form of effective rotation-vibration Hamiltonian, in which the breakdown of the Born-Oppenheimer approximation is allowed for, will be obtained for any molecule. This Hamiltonian contains effective moments of inertia (these involve rotation g-factor corrections) and effective nuclear masses (likely to be close to the atomic masses). Following the procedure of A. R. Hoy and P. R. Bunker [J. Mol. Spectrosc.74, 1 (1979)] the effective rotation-bending Hamiltonian is derived from the effective rotation-vibration Hamiltonian, and this could be used to fit the rotation-bending energy levels.     

C3O2 as a semirigid bender: The degenerate ν5 state

The semirigid bender Hamiltonian for carbon su?ide C₃O₂ [P. R. Bunker, J. Mol. Spectrosc.80, 422–437 (1980)] is extended in a manner similar to the extension previously described for HCNO [P. Jensen, J. Mol. Spectrosc.101, 422–439 (1983)]. The extended Hamiltonian describes the manifold of large-amplitude vibrational states (due to the ν₇ CCC bending mode) superimposed on a high-frequency vibrational state involving excited quanta of the CCO bending modes ν₅ and ν₆. The extended model is used to fit CCC bending and rotation energy level separations for¹²C₃¹⁶O₂ superimposed on the ν₅ fundamental level. Due to the severely limited experimental data it is not possible to unambiguously determine the effective CCC bending potential energy function in the ν₅ state, but estimates of the potential energy parameters are obtained by determining them in two limiting cases.     

Laser spectroscopy of perturbed states of BaO in the region above 32 000 cm−1

The excitation spectrum of BaO in the region above 32 000 cm⁻¹ was investigated with a frequency-doubled pulsed dye laser. We have observed fully developed rotational structures of the C¹Σ⁺-X¹Σ⁺ transition. The analysis of the vibrational states v′ = 0 through 7 leads to a large number of perturbations. This spectroscopic information in combination with the observation and rotational analysis of transitions to several new electronic states allows a systematic summary, which gives more than eight electronic states in the investigated region. Besides the known states B, C, D and c, we find four new bound states, designated by E, F, G, and H. For all states molecular constants are given. The discussion of possible molecular electron configurations leads to classifications of the molecular electronic states. Our results on the vibrational levels v′ = 0 to 3 are in reasonable agreement to the optical-optical double resonance work of R. A. Gottscho, P. S. Weiss, and R. W. Field [J. Mol. Spectrosc. 82, 283–309 (1980)], but show several new details.     

A Theoretical Study of MgNC and MgCN in the X?Σ Electronic State

The MgNC radical was the first Mg-containing species to be observed in interstellar space. This fact has stimulated considerable spectroscopic interest in this molecule, and in its isomer MgCN, but nevertheless the only rotationally resolved spectroscopic data presently available for X?²Σ⁺ MgNC comprise the rotational spectrum (K. Kawaguchi et al., 1993, Astrophys. J.406, L39-L42; K. Ishii et al., 1993, Astrophys. J.410, L43-L44; M. A. Anderson and L. M. Ziurys, 1994, Chem. Phys. Lett.231, 164-170; E. Kagi et al., 1996, J. Chem. Phys.104, 1263-1267; E. Kagi and K. Kawaguchi, J. Mol. Spectrosc. 2000, 199, 309-310) together with a few vibronic bands, all originating in the vibronic ground state and belonging to the òΠ←X?²Σ⁺ electronic transition (R. R. Wright and T. A. Miller, 1999, J. Mol. Spectrosc.194, 219-228). For MgCN, only the rotational spectrum in the vibrational ground state is known (M. A. Anderson, T. C. Steimle, and L. M. Ziurys, 1994, Astrophys. J.429, L41-L44). We report here potential energy surfaces calculated by the Averaged Coupled-Pair Functional (ACPF) method with TZ3P+f (Mg), TZ2P+f(N,C) basis sets including core-valence correlation due to the Mg 2s and 2p electrons. The ab initio results are used for determining the standard spectroscopic constants of X?²Σ⁺ MgNC and MgCN. Also, we report variational calculations of the rotation-vibration energies, and variational simulations of the lowest rotation-vibration bands, carried out with the MORBID program system (P. Jensen, 1988, J. Mol. Spectrosc.128, 478-501). We hope that our theoretical results will encourage and facilitate further characterization of X?²Σ⁺ MgNC and MgCN by high-resolution spectroscopy.     

HDO absorption spectrum above 11 500 cm: Assignment and dynamics

Assignment of an HDO line list extracted from a recently measured H₂O/HDO/D₂O Fourier transform absorption spectrum recorded in the 11 600-23 000 cm⁻¹ region by Bach et al. (M. Bach, S. Fally, P.-F. Coheur, M. Carleer, A. Jenouvrier, A.C. Vandaele, J. Mol. Spectrosc. 232 (2005) 341-350.) is presented. More than 94% of the 3256 lines are given quantum number assignments and ascribed to line absorption by HDO; most of the remaining lines are actually due to D₂O. High accuracy variational predictions of line positions and intensities are used for the spectral assignment process. Assignments to the ν₁ + 5ν₃, 2ν₂ + 5ν₃, ν₁ + ν₂ + 3ν₃ and ν₁ + 6ν₃ bands are presented for the first time. Comparisons are made with published ICLAS spectra covering the same spectral region and suggestions made for its recalibration. The results are used to illustrate the dynamical behaviour of highly excited vibrational states of HDO and to discuss previous vibrational assignments to high lying rotation-vibration states of this system.     

The ν2, 2ν2, 3ν2, ν4, and ν2 + ν4 bands of 15NH3

The ir absorption of gaseous ¹⁵NH₃ between 510 and 3040 cm^?1 was recorded with a resolution of 0.06 cm^?1. The ν₂, 2ν₂, 3ν₂, ν₄, and ν₂ + ν₄ bands were measured and analyzed on the basis of the vibration-rotation Hamiltonian developed by V. ?pirko, J. M. R. Stone, and D. Papou?ek (J. Mol. Spectrosc.60, 159–178 (1976)). A set of effective molecular parameters for the ν₂ = 1, 2, 3 states was derived, which reproduced the transition frequencies within the accuracy of the experimental measurements. For ν₄ and ν₂ + ν₄ bands the standard deviation of the calculated spectrum is about four times larger than the measurements accuracy: a similar result was found for ν₄ in ¹⁴NH₃ by ?. Urban et al. (J. Mol. Spectrosc.79, 455–495 (1980)). This result suggests that the present treatment takes into account only the most significant part of the rovibration interaction in the doubly degenerate vibrational states of ammonia.     

The development of a new Morse-oscillator based rotation-vibration Hamiltonian for H3+

The rotation-vibration Hamiltonian for an equilateral triangular X₃ molecule is derived in terms of the three curvilinear stretching coordinates Δr_i, and expanded in the form of a power series in the variables y_i = 1 ? exp(-aΔr_i), where a is a molecular parameter obtained from the potential function. The reason for the use of the variable y_i is twofold: Stretching potentials exhibit a much stronger convergence in the y_i than in the Δr_i, and a Hamiltonian expressed in the y_i can be diagonalized in a straightforward fashion using a Morse-oscillator basis set as we do here. Using a published ab initio potential surface we have expanded it as a polynomial in the y_i, and have calculated variationally the rotation-vibration energies of H₃⁺ and D₃⁺ using a symmetry-adapted Morse-oscillator-rigid symmetric top basis set. The results indicate that an expansion of the potential function to quartic terms in the y_i might be adequate, and that satisfactorily converged energies can be obtained with a relatively small basis set. The molecule H₃⁺ is the simplest polyatomic molecule. Inspection of the Appendix of this paper shows that the rotation-vibration Hamiltonian used here is one of the more complicated ones.     

Formaldehyde as a semirigid bender: The rotation-inversion spectrum in its Ã1A2 excited state

The semirigid bender Hamiltonian [Bunker and Landsberg, J. Mol. Spectrosc.67, 374–385 (1977)] was used to fit the rotation-inversion energy level separations in the $\text{A}\text{?}{}^1\text{A}{}_2$ excited state of formaldehyde. We fix the r⁰(CH) bond length and allow the R(CO) bond length and (H?H) bond angle to vary with the inversion angle ρ. The fit to 64 rotation-inversion energies (with v₄ and J < 4) is significantly better with a standard deviation of 0.199 cm^?1 than when the rigid bender [Bunker and Stone, J. Mol. Spectrosc.41, 310–332 (1972)] is used. The barrier height to planarity is 358 cm^?1 and the equilibrium ρ_e = 34.7°. The CO bond length is found to decrease by 0.034 from 1.3670 Å and the H?H angle by about 6 from 122.4° as the molecular configuration changes from planar to pyramidal. The rigid bender model developed earlier by Moule and Rao for formaldehyde [J. Mol. Spectrosc.45, 120–141 (1973)] is then used to fit the 32 rotation-(out-of-plane) bending energy levels (with v₄ = 0 and 1) of the $\text{X}\text{?}{}^1\text{A}{}_1$ ground electronic state of H₂CO. For this, a simple potential consisting of quadratic and quartic terms is used and the standard deviation of the fit is 0.148 cm^?1.     

Electric anharmonicity in XY3 pyramidal molecules: The dipole moment function of 14NH3 and 15NH3

This paper is concerned with the determination of the shape of the electric dipole moment function of a pyramidal XY₃ molecule with a low barrier to inversion over a wide range of values for the inversion coordinate. The effective inversion-rotation Hamiltonian [V. ?pirko, J. M. R. Stone, and D. Papou?ek, J. Mol. Spectrosc.60, 159–178 (1976)] is used to explain the anomalous vibrational dependence of the electric dipole moment of ¹⁴NH₃ [F. Shimizu, J. Chem. Phys.52, 3572–3576 (1970)], and of ¹⁵NH₃ [B. J. Orr and Takeshi Oka, J. Mol. Spectrosc.66, 302–313 (1977)]. The experimental data of Orr and Oka are used to fit the μ_z component of the total dipole moment function and the fitted function is used to predict the transition moments in the nν₂ inversion sequence of ¹⁴NH₃ and ¹⁵NH₃. To illustrate the measure of the rotational dependence of the transition moments, two examples, involving ground and excited (v₂ = 1) states, are also presented.     

Reduction of the effective rotational Hamiltonian for pyramidal XY3 molecules with a large-amplitude inversion motion

The previously described reduction of the effective rotational Hamiltonian for semirigid molecules of C_3v or D₃ point group symmetry [M. R. Aliev and V. T. Aleksanyan, Opt. Spectrosc.24, 201–206 (1968)] has been extended to nonrigid molecules with a significant inversion splitting of the energy levels (e.g., NH₃, H₃O⁺, CH₃^?, or SiH₃ molecules). Although for semirigid molecules like PH₃ or AsH₃, the parameters α and η′₃ which appear in the terms α[J₊³ + J_?³, J_z]₊ and η′₃(J₊⁶ + J_?⁶) are almost completely correlated, the effects of the inversion splitting and the accidental resonance which can occur between the interacting rotational levels in nonrigid molecules make it possible to determine α and η′₃ separately. The results of fitting the experimental data for ¹⁴NH₃ and ¹⁵NH₃ [?. Urban, Romola D'Cunha, K. Narahari Rao, and D. Papou?ek, Canad. J. Phys.62, 1775–1791 (1984); Romola D'Cunha, ?. Urban, K. Narahari Rao, L. Henry, and A. Valentin, J. Mol. Spectrosc.111, 352–360 (1985)] are in agreement with this conclusion. The possibility of the determination of the sign of η₃ from a simultaneous analysis of the allowed and Δk = ±3 forbidden transitions in semirigid XY₃ molecules has been discussed.     

Transition dipole matrix elements for 14NH3 from the line intensities of the 2ν2 and ν4 bands

Line intensities as well as self- and nitrogen-broadening coefficients have been determined for 20 transitions in the 2ν₂ and ν₄ bands of ¹⁴NH₃ using a diode laser spectrometer. Vibrational-inversional transition moments have been determined for transitions from the ground state to the ν₂, 2ν₂ and ν₄ states by a least-squares fit to the line intensities, taking into account Coriolis and l-type interactions between the nν₂ (n = 1, 2, 3), ν₄ and ν₂ + ν₄ states [?. Urban, V. ?pirko, D. Papou?ek, R. S. McDowell, N. G. Nereson, S. P. Belov, L. I. Gershtein, A. V. Maslovskij, A. F. Krupnov, J. Curtis, and K. Narahari Rao, J. Mol. Spectrosc.79, 455–495 (1980)]. The values of these transition moments have been combined with the previously obtained transition moments for NH₃ and its isotopomers to obtain an improved fit to the μ_z component of the electric dipole moment function of ammonia [cf. V. ?pirko, J. Mol. Spectrosc.74, 456–464 (1979)].     

Spontaneous radiative dissociation in molecular hydrogen

The spontaneous radiative dissociations of the discrete vibrational levels of the B¹Σ⁺_u electronic states of H₂, HD and D₂ of the C¹Π_u electronic state of H₂ into the vibrational continuum of the ground X¹Σ⁺_g state are calculated as a function of the emission wavelength. The fluorescent spectra of HD in the Lyman system and of H₂ in the Werner system resulting from an excitation source uniform in wavelength are predicted. The vibrational radiative lifetimes are tabulated as are the fractions of radiative decays that lead to dissociation. The effects of centrifugal distortion are discussed briefly. An appendix describes a sum rule used to check the numerical accuracy of the calculations.