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.     

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.     

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.     

The potential function for HCNCNH isomerization

The semirigid bender model (P. R. Bunker and D. J. Howe, J. Mol. Spectrosc.83, 288–303 (1980)) has been developed to fit the observed vibrational energy levels of the ground electronic state of $\text{HCN}\text{CNH}$ and $\text{DCN}\text{CND}$ allowing for the complete bending (internal rotation) of HCN into CNH and of DCN into CND. From the fit we have been able to determine the bending potential function and the contribution to the bending potential that arises from the effect of averaging over the two stretching vibrations. The results are compared with ab initio calculations.     

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.     

Derivation of the nonrigid rotation-large-amplitude internal motion Hamiltonian of the general molecule

The nonrigid (effective) rotation-large-amplitude internal motion Hamiltonian (NRLH) of the general molecule with one or more large-amplitude vibrations has been derived to the order of magnitude κ²T_VIB. The derivation takes advantage of the idea of a nonrigid reference configuration and uses the contact transformation method as a mathematical tool. The NRLH has a form fairly similar to that of the effective rotation Hamiltonian of semirigid (i.e., normal) molecules. From a careful examination of the Eckart-Sayvetz conditions and of the Taylor expansions of the potential energy surface in terms of curvilinear displacement coordinates, three types of large-amplitude internal coordinates of different physical meaning (effective large-amplitude internal coordinates, real large-amplitude internal coordinates, and reaction path coordinates) are described. To test the ideas and the formulas the effective bending potential function of the C₃ molecule in its ground electronic and ground stretching vibrational state is calculated from the ab initio potential energy surface given by W. P. Kraemer, P. R. Bunker, and M. Yoshimine (J. Mol. Spectrosc. 107, 191–207 (1984)). The calculations were carried out by using either the effective or the real large-amplitude bending coordinate of C₃. The NRLH theory is compared to the nonrigid bender theory at a theoretical level as well as through the results of the test calculations.     

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.     

An ab Initio Study of the ÃΠ State and the ÃΠ←X?Σ Electronic Transition of MgNC

We use the RENNER program system (see, for example, P. Jensen, G. Osmann, and P. R. Bunker, in “Computational Molecular Spectroscopy” (P. Jensen and P. R. Bunker, Eds.), Wiley, Chichester, 2000, and references therein) to make a detailed calculation of the rovibronic energies in the first excited electronic state, òΠ, of the MgNC radical. This calculation is based on ab initio data (supplemented here with points for larger bending displacements from linearity) calculated at the level of MR-SDCI(+Q)/[TZ3P+f(Mg), aug-cc-pVQZ (N and C)] by T. E. Odaka, T. Taketsugu, T. Hirano, and U. Nagashima (J. Chem. Phys.115, 1349-1354 (2001)). These authors employed ab initio derived spectroscopic constants to calculate vibronic energies using perturbation expressions (J. T. Hougen and J. P. Jesson, J. Chem. Phys.38, 1524-1525 (1963)), and their results suggested that an observed vibronic band belonging to the òΠ←X?²Σ⁺ electronic transition (R. R. Wright and T. A. Miller, J. Mol. Spectrosc.194, 219-228 (1999)) should be reassigned. The present work confirms this conclusion, which is further substantiated by the rotational structures calculated in the vibronic states and by Franck-Condon theory predicting relative intensities.     

HCNO as a semirigid bender: The degenerate ν4 state

The semirigid bender Hamiltonian for fulminic acid HCNO (Bunker, Landsberg, and Winnewisser, J. Mol. Spectrosc.74, 9–25 (1979)) is extended. The extended Hamiltonian describes the manifold of large amplitude vibrational states (due to the ν₅ HCN bending mode) superimposed on a high frequency vibrational state involving excited quanta of the ν₄ CNO bending mode. Such high frequency vibrational states may be degenerate when the large amplitude coordinate is zero, and the semirigid bender Hamiltonian is modified to account for the ν₄ vibrational angular momentum around the molecular axis in the linear limit, and for l-doubling effects. The extended Hamiltonian is used to fit HCN bending and rotation energy level separations for HCNO superimposed in the ν₄ fundamental level. It is found that the effective HCN bending potential in the ν₄ state is very similar to that in the high frequency vibrational ground state. The results obtained confirm the conclusion reached by Bunker, Landsberg, and Winnewisser: HCNO is linear at equilibrium.     

A new Morse-oscillator based Hamiltonian for H3+: Calculation of line strengths

In two recent publications [V. Špirko, P. Jensen, P. R. Bunker, and A. Čejchan, J. Mol. Spectrosc. 112, 183–202 (1985); P. Jensen, V. Špirko, and P. R. Bunker, J. Mol. Spectrosc. 115, 269–293 (1986)], we have described the development of Morse oscillator adapted rotation-vibration Hamiltonians for equilateral triangular X₃ and Y₂X molecules, and we have used these Hamiltonians to calculate the rotation-vibration energies for H₃⁺ and its X₃⁺ and Y₂X⁺ isotopes from ab initio potential energy functions. The present paper presents a method for calculating rotation-vibration line strengths of H₃⁺ and its isotopes using an ab initio dipole moment function [G. D. Carney and R. N. Porter, J. Chem. Phys. 60, 4251–4264 (1974)] together with the energies and wave-functions obtained by diagonalization of the Morse oscillator adapted Hamiltonians. We use this method for calculating the vibrational transition moments involving the lowest vibrational states of H₃⁺, D₃⁺, H₂D⁺, and D₂H⁺. Further, we calculate the line strengths of the low-J transitions in the rotational spectra of H₃⁺ in the vibrational ground state and in the ν₁ and ν₂ states. We hope that the calculations presented will facilitate the search for further rotation-vibration transitions of H₃⁺ and its isotopes.     

The vt=1 Rovibrational Hamiltonian for C3v Symmetric Top Molecules Containing One Quadrupolar Nucleus: Reductions and Application to AsF3

The reduction of the effective Hamiltonian has been derived for C_3v symmetric top molecules in the v_tequals;1 state with extension to nuclear quadrupole coupling. The theory has been applied to the hyperfine structures of rotational transitions observed for AsF₃ (v₄equals;1) [H. Bürger, J. Demaison, P. Dréan, C. Gerke, L. Halonen, H. Harder, H. Mäder, J. Radtke, and H. Ruland, J. Mol. Spectrosc.180, 85-99 (1996)]. The data have been analyzed using two reduced forms of the effective Hamiltonian which follow the reductions Q and D [E. I. Lobodenko, O. N. Sulakshina, V. I. Perevalov, and Vl. G. Tyuterev, J. Mol. Spectrosc.126, 159-170 (1987)] in the pure rovibrational case. The unitary equivalence of the derived parameter sets has been demonstrated.     

Renner-Teller and Spin-Orbit Interactions between the ÃA1 and the X?B1 States of NH2: The Stretch-Bender Approach

The stretch-bender model, developed originally to describe the effects of stretch-bend interactions and Renner-Teller coupling in symmetrical triatomic molecules, has been extended to incorporate the effects of spin-orbit interaction and of overall rotation. A comparison is made between the treatment of spin-orbit interaction and of overall rotation in our model and in the MORBID approach of P. Jensen, M. Brumm, W. P. Kraemer, and P. R. Bunker (J. Mol. Spectrosc.171, 31-57 (1995)).     

The microwave spectrum of propyne in the 17- to 52-GHz region for the v10 = 4 vibrational state

The rotational spectra in the v₁₀ = 4 vibrational state of propyne have been determined in the frequency range 17–52 GHz. Molecular constants for this vibrationally excited state have been determined from more than 16 observed rotational transitions. Agreement between experimentally measured frequencies are presented and compared with those calculated using the results of basic perturbation theory. Matrix elements for the diagonal and nondiagonal terms were obtained from a previous paper by A. Bauer, G. Tarrago, and A. Remy. [J. Mol. Spectrosc. 58, 111–124 (1975)].     

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°.     

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.     

Simultaneous Analysis of Rovibrational and Rotational Spectra of thev5= 1 andv8= 1 Vibrational Levels of Propyne

The microwave and submillimeter wave spectra of propyne between 17 and 358 GHz were measured and the rotational transitions in thev₈= 1 excited vibrational state of the CH₃rocking vibration were assigned. About 1050 wavenumbers of the ν₈vibration–rotation fundamental band and about 600 wavenumbers of the ν₅fundamental band of the[formula]stretching vibration were assigned from the infrared spectrum between 910 and 1130 cm⁻¹which was used previously (G. Graneret al., J. Mol. Spectrosc.161,80–101 (1993)) in the analysis of the combinationv₉=v₁₀= 1 and thev₁₀= 3 overtone levels (ν₉being the[formula]bending and ν₁₀the[formula]bending vibrations). The rovibrational and rotational data corresponding to the two fundamental levels were analyzed simultaneously in least-squares fits using a model which treats together all the vibrational levels in the region around 1000 cm⁻¹with their strong anharmonic and vibration–rotation resonances. The refined parameters reproduce the infrared and submillimeter wave data of thev₅= 1 level with standard deviations of 0.32 × 10⁻³cm⁻¹and 59 kHz, respectively, while for thev₈= 1 level the standard deviations were 0.41 × 10⁻³cm and 290 kHz. The refined parameters of the combination and overtone levels provide reliable predictions for future submillimeter wave studies.     

Spectrum of H218O in the 2900 to 3400 cm−1 region

Measurements of line center positions of H₂¹⁸O in the 2900 to 3400 cm^?1 region have been made at high resolution. This region contains absorptions of the (020) band and P-branch absorptions of the (100) and (001) bands of H₂¹⁸O. Values of the energy levels of the (020) state were determined in which ground state energy levels derived by Fraley, Rao, and Jones [J. Mol. Spectrosc.29, 312 (1969)] and Williamson, Rao, and Jones [J. Mol. Spectrosc.40, 372 (1971)] were used in the analysis. A new set of ground state levels was obtained by an iterative procedure.     

A simultaneous analysis of ν5, ν6, and the ground state of hydrazoic acid

The pure rotational spectrum in the vibrational ground state [J. Bendtsen and F. M. Nicolaisen, J. Mol. Spectrosc. 119, 456–466 (1986)] and the infrared spectrum of the fundamental bands ν₅ and ν₆ [J. Bendtsen, F. Hegelund, and F. M. Nicolaisen, J. Mol. Spectrosc. 118, 121–131 (1986)] of HN₃ have been simultaneously analyzed using a three-level model taking into account resonance between the ground state and ν₅ due to centrifugal distortion [K. Yamada, J. Mol. Spectrosc. 81, 139–151 (1980)] and a- and b-Coriolis interactions between ν₅ and ν₆. A set of ground- and upper-state constants have been obtained and values for the centrifugal distortion parameter C₅^ab and the Coriolis coupling constant ζ_5,6^b are derived. A complete set of ground-state energies for J ≦ 50 and K_a ≦ 10 is tabulated.