Bender Hamiltonian Research Articles

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

Abstract 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 v 2 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 C 3 from an ab initio potential surface, and of H 2 O from ab initio and experimental potential surfaces.

An ab initio study of the rotation—vibration energy levels of GeH 2 in the ā 3B 1 state

Thirty-five points on the potential energy surface of the ā 3B 1 first excited state for the GeH 2 radical have been calculated using the (ab initio) MRD CI technique. Thirteen parameters in an analytic expression for the potential have been adjusted (by least-squares optimization) so that the surface fits these points. The rotation-vibration energy levels of GeH 2. GeD 2 and GeHD have been calculated using the non-rigid bender Hamiltonian, and we determine for GeH 2 that ν 1 = 1991 cm -1. ν 2 = 763 cm -1, and ν 3 = 2012 cm -1. The equilibrium structure is found to be r e = 1.545 Å and α z = 119.8°, and the singlet-triplet splitting is calculated to be 22.8 kcal/mole (7975 cm -1).

An ab initio study of the rotation—vibration energy levels of GeH 2 in the X̃ 1A 1 state

Thirty-seven points on the potential-energy surface of the X̃ 1A 1 ground electronic state for the GeH 2 radical have been calculated using the (ab initio) MRD Cl technique. Twelve parameters in an analytic expression for the potential have been adjusted (by least-squares optimization) so that the surface fits these points. The rotation—vibration energy levels of GeH 2 and GeD 2 have been calculated using the non-rigid bender Hamiltonian, and we determine for GeH 2 that ν 1 = 1857 cm −1, ν 2 = 923 cm −1 and ν 3 = 1866 cm −1, in good agreement with the values obtained from a matrix-isolation spectrum. The equilibrium structure is found to be r e = 1.591 A and α e = 91.4°.

A theoretical study of the rotation-vibration energy levels and dipole moment functions of CCN+, CNC+, and C3

Rotation-vibration energy levels of the isomers CCN+ and CNC+ have been determined in the following way: A large number of points in the minimum region of the 1Σ+ electronic ground state surfaces of the two ions have been computed by single reference state complete single and double excitations configuration interaction (CI-SD) calculations using canonical SCF orbitals as a molecular orbital basis. The effect of certain higher excitations has also been taken into account by using the Davidson estimate (CI-SDQ). A force constant expansion has been fitted by least squares to the ab initio points, and the resulting potential expression has been used to calculate rotational and vibrational constants using the perturbation theory expressions. In order to assess the reliability of these calculations and, in particular, the flexibility of the Gaussian basis sets used, identical calculations have been performed for the isoelectronic C3 radical. For C3 the rotational and vibrational energy levels obtained by the perturbational approach are compared to those calculated using the nonrigid bender Hamiltonian. The nonrigid bender results at the CI-SDQ level are found to be in good agreement with the available experimental data. From these CI-SDQ calculations the equilibrium structure of the ground state of C3 is obtained as bent (αe = 162°, re = 1.290 Å). Dipole moment functions and vibrational transition moments have also been calculated for all three molecules. The transition moment for the ν3 fundamental of C3 is found to be very large (0.44 Debye).

C 3O 2 as a semirigid bender: The degenerate ν5 state

The semirigid bender Hamiltonian for carbon su☐ide C 3O 2 [ 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 ν 7 CCC bending mode) superimposed on a high-frequency vibrational state involving excited quanta of the CCO bending modes ν 5 and ν 6. The extended model is used to fit CCC bending and rotation energy level separations for 12C 3 16O 2 superimposed on the ν 5 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 ν 5 state, but estimates of the potential energy parameters are obtained by determining them in two limiting cases.

An extended ab initio study of the rotation-vibration spectrum of HOC +

Configuration interaction calculations have been performed in order to investigate the bending potential of the molecular ion HOC + in detail. It is found that the bending potential has its minimum at the linear configuration and that it is very shallow. The ab initio points on the electronic ground state surface of HOC + were combined with previously calculated points to determine an improved force field. This force field was used in the second-order rotation-vibration perturbation Hamiltonian, as well as in the semirigid bender Hamiltonian, to evaluate rotation and vibration frequencies of HOC + and of some of its isotopes. The ν 0( J = 1−0) rotational transition frequency of the DOC + isotope is predicted to be 76 200 ± 40 MHz.

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 ν 5 HCN bending mode) superimposed on a high frequency vibrational state involving excited quanta of the ν 4 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 ν 4 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 ν 4 fundamental level. It is found that the effective HCN bending potential in the ν 4 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.

The rotational spectrum of the CD2 radical studied by far infrared laser magnetic resonance spectroscopya)

We report the detection of 17 pure rotation transitions in the ground vibronic state of the CD2 radical using far infrared laser magnetic resonance spectroscopy. Fitting the data using an effective rotational Hamiltonian yields values for the three rotational constants, seven centrifugal distortion constants, the three electronic spin-rotation, and two electronic spin–spin parameters. We also fit this data, and CD2 ν2 band data (published separately), using the semirigid bender Hamiltonian and obtain the effective bending potential function for CD2. Combining this with previous CH2 results enables us to predict the rotation bending energy levels of CHD. We also report here the detection of two further rotational transitions in the ν1 excited vibrational state of CH2.

The application of the nonrigid bender Hamiltonian to a quasilinear molecule

The form of the nonrigid bender has changes that here we do render. We add, nicely paired, a term to J 2 and regroup factors that are singular. As a result, the nonrigid bender Hamiltonian can now be set up using only Van Vleck perturbation theory, for any triatomic molecule (linear, quasi-linear, or bent). It can be used to calculate the rotation-vibration energies of the molecule to high J(⋍10) from the bending potential energy function and the stretch and stretch-bend force constants.

The equilibrium geometry, potential function, and rotation-vibration energies of CH2 in the X̃ 3B1 ground state

By fitting the eigenvalues of the nonrigid bender Hamiltonian to the 26 available experimental rotation-vibration energy level splittings of the 3B1 ground electronic state of the methylene radical CH2 we determine the equilibrium geometry and potential function; the data only involve rotational transitions and transitions in the ν2 bending fundamental band. From the fit we determine that the equilibrium bond length is 1.075 Å, the equilibrium bond angle is 133.9° and the height of the barrier to linearity is 1878 cm−1. We are able to use the nonrigid bender Hamiltonian, with this potential, to predict all the lower rotation-vibration energy levels, including the ν1 and ν3 band origins, of CH2 and its isotopes.

The laser magnetic resonance spectrum of the ν2 band of the methylene radical CH2

Ten rotation–vibration transitions in the ν2 (bending) fundamental band of CH2 in its 3B1 ground electronic state have been measured and assigned using the technique of intracavity CO2 laser magnetic resonance (LMR) spectroscopy in the region 880–1090 cm−1. Using the present data, and the microwave and far-infrared LMR measurements of pure rotational transitions, an effective bending potential energy curve has been derived for CH2 using the semirigid bender Hamiltonian. The bending frequency of CH2 is determined to be ν2=963.0987(4) cm−1, which is considerably lower than previous theoretical and experimental estimates.

The inversion potential and rotation-inversion energy levels of H 3O + and CH 3−

The rotation-inversion-vibration energy levels of H 3O + and CH 3 − 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.

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 b 00( ρ) to rigid bender H b 0( ρ) [P. R. Bunker and J. M. R. Stone, J. Mol. Spectrosc. 41, 310–332 (1972)] to semirigid bender H b 0( r, ρ) [P. R. Bunker and P. M. Landsberg, J. Mol. Spectrosc. 67, 374–385 (1977)] Hamiltonian. With H b 0( r, ρ), the ab initio calculated bending frequency ν 2 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 ν 2 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 v 2 ≤ 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. Acta 47, 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 rigid bender and semirigid bender models for the rotation-vibration Hamiltonian

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 2O 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 2O 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.

Vibration-inversion-rotation spectra of ammonia: Centrifugal distortion, Coriolis interactions, and force field in 14NH 3, 15NH 3, 14ND 3, and 14NT 3

An effective inversion-rotation Hamiltonian has been developed for NH 3 which avoids the necessity of having to include high powers of the inversion motion coordinate in the Taylor expansions of the potential energy and the inverse moment of inertia tensor. This nonrigid bender Hamiltonian describes the centrifugal distortion and the Coriolis interactions in the ground and excited inversion states. It also describes the inversion doublings in the ground and excited vibration-inversion states of ammonia. A least-squares procedure that includes the numerical integration of the Schrödinger wave equation has been used to determine the harmonic force field and the double-minimum inversion potential function for ( 14NH 3, 15NH 3) and for ( 14ND 3 and 14NT 3). The anomalous rotational dependence of the inversion doublings in the (± l) components of the ∥ v 4 = 1〉 state of 14NH 3 has been explained by the Coriolis interactions between ∥ v 2=1〉, ∥ v 4 = 1〉, ∥ v 2 = 2〉, ∥ v 2 = 1, v 4 = 1〉, and ∥ v 2 = 3〉 vibration-inversion states.

Anharmonic potential constants and the large amplitude bending vibration in nitrogen dioxide

The anharmonic potential function of the ground electronic state of nitrogen dioxide has been determined within the framework of three different vibrational Hamiltonians. The first of these, which involves a perturbation expansion of the vibrational wave functions in terms of normal coordinate harmonic oscillator wave functions, is the most widely used and generally applicable of the three. It suffers, however, from demonstrably large systematic errors. The other two are vibration–rotation Hamiltonians which allow explicitly for a large amplitude vibration in the bending vibration of a triatomic molecule; they set up the Hamiltonian operator as an explicit function of the bond angle and solve the Schrödinger equation numerically. The more sophisticated of these, the so-called nonrigid bender Hamiltonian, reproduces the spin-free virtual term values to the (0, ν2, 0) manifold of 14NO2 to a standard deviation of 0.026 cm−1 for states with N ≤ 10 and ν2 ≤ 3. It is, moreover, observed to be a more useful tool for extrapolation than is the ordinary parametrized Hamiltonian.The potential function for the bending coordinate is defined by αe = 133.888 ± 0.002°, fαα = 1.61022 ± 0.00005 mdyn Å/rad2, fααα = −2.1172 ± 0.0003 mdyn Å/rad3, and fαααα = 6.0228 ± 0.0020 mdyn Å/rad4. The equilibrium bond length, re, is found to be 1.19464 ± 0.00015 Å.

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

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 2O, D 2O, 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.