Effective dipole moment of rotational transitions of X 2 Y molecules

Contributions determining the rotational dependence of the effective dipole moment of molecules are calculated for the ground state of H₂S and H₂O molecules. The calculation is carried out in various ordering algorithms of perturbation theory. It is shown that the convergence of the effective dipole moment for the ground state of an H₂O molecule in the polynomial representation is rather slow in the rotational operator J _z (the convergence radius is K*≤17). Nonpolynomial forms of the dipole moment as a function of rotational operators are discussed.     

Symmetry properties of the effective hamiltonians of some nonrigid molecules

Group-theoretical methods are used to identify the nonvanishing parameters and establish relations between them in the effective Hamiltonians which describe rotation and motion with large amplitude. The cases considered are 1) bending vibration in molecules of the type XY₂ and XYZ and 2) inversion vibration in pyramidal molecules of the type XY₃. For the latter type of molecule, the symmetry properties of the parameters in the effective centrifugal Hamiltonian for quasidegenerate inversion states are investigated. The general form of this Hamiltonian and its symmetry properties are established and studied in an arbitrary order in the small parameter.Translated from Izvestiya Vysshikh Uchebnykh Zavednii, Fizika, No. 2, pp. 22–26, February, 1981.     

Critical phenomena in the rotational spectra of H2X molecules

Using the method of generating functions for the effective rotational Hamiltonian of H₂X molecules, an expression for estimating the critical value J _c of the rotational quantum number is obtained, starting with which the formation of four-level clusters in a molecule is possible. The analysis made in the harmonic approximation for rotational operators and for a particular form of generating functions showed the impossibility of clusterization for nonrigid H₂X molecules with strong centrifugal effects.     

Asymptotic behavior of rotational energies of rigid X2Y-type molecules

In this study the asymptotic behavior of rotational energies of simple rigid molecules of X₂Y-type is considered. This behavior is obtained using various forms of an effective rotational Hamiltonian. The results of calculations for highly excited rotational energies of the molecules H₂O and CH₂ are presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 9–12, January, 1990.     

A self-consistent method of constructing an effective rotational Hamiltonian

Methods have become widely used in molecular spectroscopy that involve the construction of effective rotational Hamiltonians [1, 2]. One of these methods has been developed by one of us over several years [3, 4]. A simple basis is provided for this method and it is shown that the scheme for constructing the effective operator for the rotational energy is substantially self-consistent.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 68–72, May, 1981.     

The isotope effect in linear X2Y molecules

We discuss the isotopic relations linking molecular and spectroscopic constants in linear X₂Y molecules. As an example we have examined the isotopic modifications of the CO₂ molecule and of the NH₂ radical.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 54–58, April, 1976.     

The effective rotational hamiltonian for open-shell molecules

A formalism for obtaining the effective rotational hamiltonian for any open-shell molecule is described. This effective hamiltonian is shown to be formed of series of irreducible tensors composed of real and fictitious angular momentum operators. The operations of these tensors may in all cases be restricted to a particular electronic state. Application of such a formalism should be particularly valuable in the interpretation of the high resolution spectra (zero-field microwave, gas-phase electron resonance, etc.) of open-shell molecules, for it follows that the experimental data may always be interpreted without explicit reference to electronic states other than that wherein the spectrum originates. Useful rules are given which determine the relevant orders of tensors in the series and the method is illustrated for the case of a ³Σ diatomic molecule.     

Reduction of the effective rotational Hamiltonian for pyramidal XY3 molecules with a large-amplitude inversion motion

The previously described reduction of the effective rotational Hamiltonian for semirigid molecules of C_3v or D₃ point group symmetry [M. R. Aliev and V. T. Aleksanyan, Opt. Spectrosc.24, 201–206 (1968)] has been extended to nonrigid molecules with a significant inversion splitting of the energy levels (e.g., NH₃, H₃O⁺, CH₃^?, or SiH₃ molecules). Although for semirigid molecules like PH₃ or AsH₃, the parameters α and η′₃ which appear in the terms α[J₊³ + J_?³, J_z]₊ and η′₃(J₊⁶ + J_?⁶) are almost completely correlated, the effects of the inversion splitting and the accidental resonance which can occur between the interacting rotational levels in nonrigid molecules make it possible to determine α and η′₃ separately. The results of fitting the experimental data for ¹⁴NH₃ and ¹⁵NH₃ [?. Urban, Romola D'Cunha, K. Narahari Rao, and D. Papou?ek, Canad. J. Phys.62, 1775–1791 (1984); Romola D'Cunha, ?. Urban, K. Narahari Rao, L. Henry, and A. Valentin, J. Mol. Spectrosc.111, 352–360 (1985)] are in agreement with this conclusion. The possibility of the determination of the sign of η₃ from a simultaneous analysis of the allowed and Δk = ±3 forbidden transitions in semirigid XY₃ molecules has been discussed.     

Deviations from the born-oppenheimer approximation in nonlinear X2Y molecules

The full electron-nuclear operator of a nonlinear molecule is diagonalized on the (nondegenerate) basis of electron states. Corrections to the vibration-rotation Hamiltonian of the Born-Oppenheimer adiabatic approximation are derived from it. Corrections to harmonic frequencies and reciprical moments of inertia, generated by deviations from the Born-Oppenheimer approximation, are obtained for molecules of the H₂O type.     

Mean amplitudes of vibration: Planar X2Y4 molecules

The theory of mean-square amplitude matrices is applied to the ethylene type of molecules. The mean amplitudes of thermal vibrations of atomic distances in these molecules are evaluated at 300K from their vibrational frequencies.One of the authors (S.M.) is grateful to the Council of Scientific and Industrial Research, Government of India, for the award of a Senior Research Fellowship.     

Symmetry coordinates of nonrigid molecules

Motivated by Altmann’s definition of symmetry groups of nonrigid molecules, Wigner’s method of obtaining the symmetry coordinates of a molecule is extended to nonrigid molecules with free internal rotations. The molecule BF₂ CH₃ is exemplified.     

One-dimensional approximation of the effective rotational Hamiltonian of the ground state of the water molecule

A method is presented for summation of the divergent perturbation-theory series used in molecular spectroscopy, by means of a one-dimensional approximation. Application of the proposed method enables one to fit the energy levels of the ground state of the water molecule (Mol. Phys. 32, 499–521 (1976)) to a maximum difference between the observed and the calculated levels of less than 8 cm^?1 for levels with the maximum rotational quantum numbers K = 20 and J = 20 in a model with 24 parameters and terms up to J⁸.     

Centrifugal distortion coefficients of asymmetric-top molecules: Reduction of the octic terms of the rotational Hamiltonian

The rotational Hamiltonian of an asymmetric-top molecule in its standard form, containing terms up to eighth degree in the components of the total angular momentum, is transformed by a unitary transformation with parameters S_pqr to a reduced Hamiltonian so as to avoid the indeterminacies inherent in fitting the complete Hamiltonian to observed energy levels. Expressions are given for the nine determinable combinations of octic constants Θ′_i (i = 1 to 9) which are invariant under the unitary transformation. A method of reduction suitable for energy calculations by matrix diagonalization is considered. The relations between the coefficients of the transformed Hamiltonian, for suitable choice of the parameters S_pqr, and those of the reduced Hamiltonian are given. This enables the determination of the nine octic constants Θ′_i in terms of the experimental constants.     

On elimination of ambiguity of parameters in the effective rotational Pade Hamiltonian operator of the molecule

Different methods are considered for the elimination of ambiguity in the determination of model parameters from experimental data for the effective rotational Pade Hamiltonian operator in the case of a nondegenerate state of an asymmetric top-type molecule.     

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 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 J² 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 $\text{J(?10)}$ from the bending potential energy function and the stretch and stretch-bend force constants.     

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.