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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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°.     

Vibrational anharmonicity and the inversion potential function of NH3

The nonrigid bender formulation of the vibration-inversion-rotation Hamiltonian for an XY₃ pyramidal molecule (V. ?pirko, J. M. R. Stone, and D. Papou?ek, J. Mol. Spectrosc.60, 159 (1976)) is improved by allowing for anharmonicity in all the vibrations. To model the anharmonic potential function of an XY₃ molecule with a low barrier to inversion a Plíva-type empirical potential (J. Plíva, Collect. Czech. Chem. Commun.23, 777 (1958)) is used. A fitting procedure that involves the numerical integration of the effective inversion Schrödinger equation (the nonrigid bender equation) and diagonalization of some resonance matrices is used to determine the equilibrium structure and the anharmonic potential function of the ammonia molecule.     

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.     

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.     

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.     

The inversion potential and rotation-inversion energy levels of H3O+ and CH3−

The rotation-inversion-vibration energy levels of H₃O⁺ and CH₃^? using the ab initio potential surfaces and the nonrigid bender Hamiltonian are calculated. It is hoped that these results will be of use in the search for the spectra of these ions.     

Laser Spectroscopy of Cerium Monofluoride: Ligand Field Assignments of Some 4f5d6p← 4f5d6sTransitions

The techniques of selectively detected fluorescence excitation, dispersed fluorescence, and magnetic rotation spectroscopy have been applied to the CeF molecule and used to characterize one [17.6]4.5 electronic state in the ∼2 eV region and three electronic states withT₀< 2000 cm⁻¹:X(1)3.5 and (2)3.5 with Ω = 3.5 and (1)4.5 with Ω = 4.5. Two of these states, the [17.6]4.5 andX(1)3.5 states, had been observed previously [R. M. Clements and R. F. Barrow,J. Mol. Spectrosc.107,119 (1984); Y. Azuma, W. J. Childs, and K. L. Menningen,J. Mol. Spectrosc.145,413 (1991)]. Based on the pattern of Ω-values and the value of ΔG_1/2≈ 550 cm⁻¹, the three low-lying states are assigned to the Ce⁺(4f5d(³H)6s)F⁻superconfiguration. The upper state is assigned to the Ce⁺(4f5d(³H)6p)F⁻superconfiguration on the basis of computed ligand field monopoleB⁰₀(nl,nl) orbital destabilization energies and one-electron transition propensity rules.     

Analysis of the H2O molecule second-hexade interacting vibrational states

The fitting of the experimental vibration-rotation energy levels (J. Mol. Spectrosc., 75, 339–362 (1979)) of the H₂¹⁶O interacting states (031)-(111)-(130)-(210)-(012) was made. The sixth vibrational state (050) was also taken into account. The rotational and resonance constants obtained reproduce the experimental data with an average accuracy of about 0.04–0.05 cm⁻¹.     

τ Dependence of the vibrational zeropoint energy in the partially deuterated methyl alcohols,revisited

The contributions of the zeropoint energy from the 3N ? 7 other vibrations to the effective potential energy for internal rotation have been calculated for eight isotopic species of methyl alcohol. The basis of the calculation is the set of force constants determined by A. Serrallach, R. Meyer, and Hs. H. Günthard [J. Mol. Spectrosc.52, 94–129 (1974)] from infrared analyses. The results calculated for CH₂DOH are V₁ = 11.93 cm^?1 and V₂ = 0.17 cm^?1 with ΔV₃ < 0.05 cm^?1. These values agree favorably with the experimental results as previously determined from analysis of the microwave torsional-rotational spectra of CH₂DOH.