P.R. Bunker

The vibration-rotation problem in triatomic molecules allowing for a large-amplitude bending vibration

Abstract In this paper we derive an expression for the vibration-rotation Hamiltonian of a triatomic molecule. In the derivation we use a curvilinear bending coordinate and two rectilinear stretching coordinates in such a way that the Hamiltonian obtained is applicable for any triatomic molecule, linear or bent, and allows for large displacements of the bending coordinate (sometimes said to result from the molecule being “quasi-linear” but, in fact, of general occurrence). We derive a zeroth-order Hamiltonian to describe the energy levels associated with the bending vibration, and are able to fit the experimental results on HCN and DCN better than if we had used the standard formalism of rectilinear (and small) displacements. We also use the formalism to describe the dependence of the rotational constant B on the bending vibrational quantum number and apply the results to the microwave data on CsOH and CsOD.

Far infrared laser magnetic resonance of singlet methylene: Singlet–triplet perturbations, singlet–triplet transitions, and the singlet–triplet splittinga)

We have observed and assigned a number of far infrared laser magnetic resonance spectra of CH2 arising from rotational transitions within the lowest vibrational state of the a 1A1 electronic excited state and from transitions between such singlet levels and vibrationally excited levels of the X 3B1 electronic ground state. The singlet–singlet transitions are magnetically active, and the singlet–triplet transitions have electric dipole intensity because of the spin‐orbit mixing of singlet levels with vibrationally excited levels of the triplet state. By identifying four pairs of singlet and triplet levels that perturb each other we can accurately position the singlet and triplet state relative to each other and determine the single–triplet energy splitting. We determine that T0(a 1A1)=3165±20 cm−1 (9.05±0.06 kcal/mol; 0.392±0.003 eV), and Te(a 1A1)=2994±30 cm−1 (8.56±0.09 kcal/mol; 0.371±0.004 eV). A new ab initio calculation of the spin‐orbit matrix element between these two states has been of assista...

The potential surface and stretching frequencies of X̃ 3B1 methylene (CH2) determined from experiment using the Morse oscillator‐rigid bender internal dynamics Hamiltonian

The Morse oscillator‐rigid bender internal dynamics (MORBID) Hamiltonian [P. Jensen, J. Mol. Spectrosc. 128, 478 (1988)] has been used in a fitting to all extant rotation–vibration data for X 3B1 methylene CH2. This fitting leads to an improved determination of the potential energy surface, and in particular to reliable predictions for the stretching frequencies. We predict ν1=2992 cm−1 and ν3=3213 cm−1 for 12CH2, and we hope that the new predictions will encourage the experimental search for these weak fundamentals. In the MORBID approach the rotation–vibration energies are obtained from the potential energy surface in a purely variational calculation, and consequently the present work is an improvement over previous determinations of the CH2 potential energy surface from experiment that used the nonrigid bender formalism [see P. R. Bunker et al., J. Chem. Phys. 85, 3724 (1986), and references therein]; this latter approach treats the stretching vibrations by second order perturbation theory. A fitting ...

The rigid bender and semirigid bender models for the rotation-vibration Hamiltonian

Abstract The rigid bender Hamiltonian used by Bunker and Stone ( J. Mol. Spectrosc. , 41 , 310 (1972)) is reviewed and applied to fit the recently obtained rotation-bending energy levels of the H 2 O molecule. This rotation-bending Hamiltonian treats a triatomic molecule as bending with fixed bond lengths. A simple improvement is made by allowing the bond lengths to vary with the bond angle and the new Hamiltonian is called the semirigib bender Hamiltonian. This Hamiltonian allows for all the important stretch-bend interaction terms from the potential energy and rotation-bending interaction terms from the kinetic energy. As a result the variation of the rotational constants with bending vibrational state is treated better than by the rigid bender. The model is used to fit the rotation-bending energy levels of the H 2 O molecule. While not as good as the nonrigid bender Hamiltonian of Hoy and Bunker ( J. Mol. Spectrosc. , 52 , 439 (1974)) the semirigid bender Hamiltonian is easier to apply to a larger molecule, and this is the reason for introducing it. At the end of the paper the rigid bender model is compared to a more approximate model in which the variation of the bending reduced mass with bending angle is ignored. The approximate model can introduce serious errors but since we know how the bending reduced mass varies with bond angle in the rigid bender model the approximation of the simpler model is unnecessary.

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

Abstract 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 A 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.

The effective rotation-bending Hamiltonian of a triatomic molecule, and its application to extreme centrifugal distortion in the water molecule

Abstract This paper is concerned with the determination of the shape of the potential energy surface of a triatomic molecule over a wide range of values for the bending coordinate, and numerical integration of the wave equation is used in order to relate the shape of the potential surface directly to the rotation-bending energies. The rotation-vibration Hamiltonian that is used was derived by Hougen, Bunker, and Johns [ J. Mol. Spectrosc. 34 , 136 (1970)] in a manner that allowed explicitly for the dependence of the inverse moment of inertia tensor elements on the bending angle. We now treat the stretching vibrational coordinates in this Hamiltonian by Van Vleck perturbation theory to obtain the effective rotation-bending Hamiltonian. We call this Hamiltonian the nonrigid bender Hamiltonian, in contrast to the simpler rigid-bender Hamiltonian used by Bunker and Stone [ J. Mol. Spectrosc. 41 , 310 (1972)]. The Hamiltonian is used to calculate the energies of the higher rotational energy levels ( J ≅ 10) of the v 2 = 0 and 1 states of H 2 O, D 2 O, and HDO from the equilibrium structure and force constants of the water molecule, and a slight refinement of the structure and force field has been possible. The fit is at the level where any further improvement will necessitate considering the breakdown of the Born-Oppenheimer approximation among other higher-order corrections. Many of the ideas developed here can be applied to the treatment of the rotation-vibration problem in any molecule having a large amplitude coordinate, or a coordinate on which a moment of inertia element strongly depends.

The bending-rotation Hamiltonian for the triatomic molecule and application to HCN and H2O

Abstract In this paper we have extended the work of Hougen, Bunker and Johns ( 1 ), in which the vibration-rotation Hamiltonian of a triatomic molecule was derived in a manner that allowed explicitly for the large amplitude of the bending vibration. In this technique the rotation-bending-stretching Hamiltonian is separated, in zeroth-order, into a rotation-bending part and a stretching part. This proves to be much closer to the physics of the motion than the customary separation into a rotation part and a bending-stretching part since interaction terms are much less important and fewer parameters are required. Our extension to the previous work has been to include overall molecular rotation and we have written a computer program to calculate the rotation-bending energy levels E ( v 2 , J ) using this model. The calculation involves, at most, five parameters (the two zero-point bond lengths, the equilibrium bond angle, the quadratic bending force constant, and the potential energy barrier opposing the straightening of the molecule). We have not explicitly included any vibration-rotation interaction constants since the most important of these are an implicit part of the model. We have fitted the rotation-bending energy levels of HCN and DCN in their bent A electronic state and in their ground electronic state, and the rotation-bending energy levels of H 2 O, D 2 O and HDO in their ground electronic state. The agreement between theory and experiment is very satisfactory in all cases, especially when it is considered how few parameters are used. We have also obtained some new experimental results for the A electronic state of HCN and DCN which is used to further test the theory.

The potential surface of X̃ 3B1 methylene (CH2) and the singlet–triplet splitting

The data in the two immediately preceding papers, when combined with the extant microwave, infrared, and photodetachment spectroscopic data, provide 152 rotation and rotation‐bending energy level separations in X 3B1 methylene (involving 12CH2, 13CH2, and CD2). In the present paper we fit all this data using the two nonrigid bender Hamiltonians NRB1 and NRB2. The more refined model (NRB2) leads to the following results for triplet methylene: re=1.0766±0.0014 A, αe=134.037°±0.045°, and the barrier height to linearity=1931±30 cm−1 (the uncertainties are three times the standard errors). Rotation‐bending energy levels for CH2, CD2, and CHD are calculated for v2≤4 and N≤6. The determination of the rotation‐bending energy levels in CH2 leads to an improved determination of the singlet–triplet splitting T0(a1A1) in methylene as 3156±5 cm−1 (9.023±0.014 kcal/mol, 0.3913±0.0006 eV). Although the rotation‐bending energy levels are accurately predicted it is not possible to predict the stretching frequencies of C...

The breakdown of the Born-Oppenheimer approximation: the effective vibration-rotation hamiltonian for a diatomic molecule

An effective vibration-rotation hamiltonian for the ground electronic state of a diatomic molecule is derived. The contact transformation used to account for the effect of the excited electronic states is complicated by the parametric dependence of the zeroth-order electronic energies on the internuclear distance. The effective vibration-rotation hamiltonian contains an effective internuclear potential and two effective reduced masses, one for the vibrational and one for the rotational kinetic energies. Although the method is applied to 1Σ diatomic molecules it can readily be extended to other states and polyatomic molecules.

The rotational spectrum and hyperfine structure of the methylene radical CH2 studied by far-infrared laser magnetic resonance spectroscopy

Thirteen pure rotational transitions of CH2 in its X 3B1 ground vibronic state have been measured and assigned using the technique of far‐infrared laser magnetic resonance (LMR) spectroscopy. The energy levels thus determined led to the prediction and subsequent detection by microwave spectroscopy of a further rotational transition 404–313, at lower frequency (∼70 GHz). The analysis of these observations yields precise rotational constants as well as spin–spin, spin‐rotation, and hyperfine interaction parameters for gas phase CH2. Its rotational spectrum may enable interstellar CH2 to be detected by radio astronomy. Two rotaional transitions within the v1=1 excited vibrational state have also been identified in the LMR spectrum. Future observations of vibrationally excited CH2 may afford a means of determining the singlet–triplet splitting in methylene, and studies of CD2 and CHD will result in improved structural determinations.