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.     

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.     

Rotational energy surface and quasiclassical analysis of the rotational energy level cluster formation in the ground vibrational state of PH3

We report and substantiate a method for constructing the rotational energy surface (RES) of a molecule as a pure classical object. For an arbitrary molecule we start from the potential energy surface rather than from a conventional “effective Hamiltonian”. The method is used for constructing the RES of the PH₃ molecule in its ground vibrational state. We have used an ab initio potential energy surface [D. Wang, Q. Shi, Q.-S. Zhu, J. Chem. Phys. 112 (2000) 9624-9631; S.N. Yurchenko, M. Carvajal, P. Jensen, F. Herregodts, T.R. Huet, Chem. Phys. 290 (2003) 59-67.]. The shape of the RES is shown not to change for J from 0 to 120. The procedure of quasiclassical quantization of the RES was also undertaken, yielding a set of quasiclassical critical values of the angular momentum. The results explain the structure of quantum rotational energy levels obtained by variational calculations [S.N. Yurchenko, W. Thiel, S. Patchkovskii, P. Jensen, Phys. Chem. Chem. Phys. 7 (2005) 573-582].     

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.     

A theoretical study of FeNC in the Δ electronic ground state

We report an ab initio calculation, at the MR-SDCI + Q + E_rel/[Roos ANO (Fe), aug-cc-pVQZ (C, N)] level of theory, of the potential energy surface for ⁶Δ_i FeNC. From the ab initio results, we have computed values for the standard spectroscopic parameters of FeN¹²C and FeN¹³C. Analytical representations of the potential energy surfaces have been fitted through the ab initio points, and the resulting functions have been used for directly solving the rotation-vibration Schrödinger equation by means of the MORBID program and by means of an adiabatic-separation method. For ⁶Δ_i FeNC, our ab initio calculations show that the equilibrium structure is linear with r_e (Fe-N) = 1.9354 Å and r_e (N-C) = 1.1823 Å. We find that the bending potential is very shallow, and the MORBID calculations show that the zero-point averaged structure is bent with the expectation values 〈r (Fe-N)〉 = 1.9672 Å, 〈r(N-C)〉 = 1.1866 Å, and . The experimentally derived bond length r₀ (N-C) = 1.03(8) Å reported for ⁶Δ_i FeNC by Lie and Dagdigian [J. Chem. Phys. 114 (2001) 2137-2143] is much shorter than the corresponding ab initio r_e-value and the averaged value from MORBID. Our calculations suggest that this discrepancy is caused by the inadequate treatment of the large-amplitude bending motion of ⁶Δ_i FeNC. It would appear that for floppy triatomic molecules such as FeNC, r₀-values have little physical meaning, at least when they are determined with the effects of the large-amplitude bending motion being ignored, i.e., under the assumption that the r₀ structure is linear.     

Spectroscopically determined potential energy surfaces of the H2O, H2O, and H2O isotopologues of water

Adiabatic potential energy surfaces (PESs) for three major isotopologues of water, H₂¹⁶O, H₂¹⁷O, and H₂¹⁸O, are constructed by fitting to observed vibration-rotation energy levels of the system using the nuclear motion program DVR3D employing an exact kinetic energy operator. Extensive tests show that the mass-dependent ab initio surfaces due to Polyansky et al. [O.L. Polyansky, A.G. Császár, S.V. Shirin, N.F. Zobov, P. Barletta, J. Tennyson, D.W. Schwenke, P.J. Knowles, Science 299 (2003) 539-542.] provide an excellent starting point for the fits. The refinements are performed using a mass-independent morphing function, which smoothly distorts the original adiabatic ab initio PESs. The best overall fit is based on 1788 experimental energy levels with the rotational quantum number J = 0, 2, and 5. It reproduces these levels with a standard deviation of 0.079 cm⁻¹ and gives, when explicit allowance is made for nonadiabatic rotational effects, excellent predictions for levels up to J = 40. Theoretical linelists for all three isotopologues of water involved in the PES construction were calculated up to 26 000 cm⁻¹ with energy levels up to J = 10. These linelists should make an excellent starting point for spectroscopic modelling and analysis.     

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.     

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.     

Optimized semiempirical potential energy surface for H<Subscript>2</Subscript><Superscript>16</Superscript>O up to 26000 cm<Superscript>−1</Superscript>

A semiempirical potential energy surface is obtained for the major isotopologue of the water molecule H₂ ¹⁶O that allows the vibration-rotation energy levels in the range of 0–26000 cm⁻¹ to be calculated with an accuracy almost equal to the average experimental accuracy of measurements in the infrared and visible ranges. Variational calculations using this potential energy surface reproduce the experimental energy values of more than 1500 vibration-rotation levels of H₂ ¹⁶O with the total angular momentum quantum number J = 0, 2, and 5 in the indicated range with a standard deviation of 0.022 cm⁻¹. The potential was obtained by optimizing a starting ab initio surface using a combination of two approaches, i.e., (1) the multiplication of the starting ab initio surface by a morphing function whose parameters were optimized and (2) the optimization of parameters of the ab initio surface using both the experimental values of energy levels and the results of quantum-chemical electronic structure calculations.     

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.     

Accurate ab initio-based DMBE potential energy surface for HLi2(X 2A′) via scaling of the external correlation

A globally accurate potential energy surface is reported for the electronic ground-state HLi₂ by fitting ab initio energies to double many-body expansion formalism. The total 3726 ab initio energies used to map the HLi₂ potential energy surface are calculated using the multi-reference configuration interaction method, with their dynamical correlation being semiempirically corrected by the double many-body expansion-scaled external correlation method. The current potential energy surface generates an excellent fit of the ab initio energies, showing a small root-mean squared derivation of 0.636 ? kcal ? mol^-1. The topographical features of the HLi₂ potential energy surface are examined in detail, which concludes that the H + Li₂(X ? ¹ Σ _g ) → Li + LiH(X ? ¹ Σ) reaction is essentially barrierless and the exothermicity is calculated to be 33.668 ? kcal ? mol^-1, thus corroborates the available experimental and theoretical results.     

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°.     

Variational calculation of low-lying and excited vibrational levels of the water molecule

The variational method of calculating vibrational energy levels for triatomic molecules has been extended to determine the lowest 22 levels for H₂O, using an analytic form for the potential surface developed by Sorbie and Murrell. In addition, complete ab initio calculations of the fundamental frequencies for H₂O have been carried out using extensive configuration interaction forms for the potential. They are predicted to be 3720, 1629, and 3807 cm^?1, to be compared with experimental values of 3656, 1597, and 3754 cm^?1. Also discussed is how the variational method may be extended to allow the calculation of vibrational energy levels for larger molecules.     

An investigation of radiative transition probabilities for the A1Σu+-X1Σg+ bands of Na2

The absolute radiative transition probabilities are calculated for previously observed spontaneous emission from A¹Σ_u⁺ (v′,J′) → X¹Σ_g⁺ (v″, J″ = J′ ± 1) reported in the preceding paper and by Woerdman (Chem. Phys. Lett.53, 219 (1978)). The calculations employ accurate hybrid potential energy curves, based on Rydberg-Klein-Rees (RKR), ab initio and long-range results, and a hybrid transition moment function, based on ab initio calculations. These calculated probabilities are compared with the various experimental results; while overall agreement is reasonable, detailed differences do occur.     

An ab initio investigation of the potential function of PH3

Ninety-three points on the potential energy surface of the ground electronic state of the phosphine molecule (with energies up to 12 000 cm⁻¹ above equilibrium) have been calculated ab initio using second-order many-body perturbation theory. A polynomial potential function (quartic in the in-plane coordinates and containing up to 12th-order terms in the out-of-plane coordinate) has been fitted to these points using the reduced inversion potential and standard polynomials. Using the nonrigid invertor Hamiltonian with this function, rotation-vibration energies have been calculated and found in reasonable agreement with experiment. The results are believed to be helpful in the determination of a realistic experimental potential function of the phosphine molecule over a wide range of values for the out-of-plane motion coordinate.     

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 ν₀(J = 1?0) rotational transition frequency of the DOC⁺ isotope is predicted to be 76 200 ± 40 MHz.     

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.