Transformation of the effective rotational Hamiltonian of nonrigid X 2 Y molecules

The transformation of the effective rotational Hamiltonian H of nonrigid X ₂ Y molecules to the form having a minimum number of diagonals in the basis of rotational functions of a symmetric top is discussed. Such a transformation is a generalization of the reduction transformation performed for the polynomial effective Hamiltonian H. It is shown that in the general case the transformation substantially changes the form of the initial Hamiltonian, which restricts the region of applicability (J<J*) of the reduced Hamiltonian represented in a class of elementary functions in terms of angular momentum operators. The values of the rotational quantum number J* are estimated for the (000) ground and (010) vibrational states of the H₂O molecule.     

Effective Hamiltonian for degenerate vibrational states in symmetric top molecules

A mixed matrix-operator form of the effective rotational Hamiltonian has been discussed for the degenerate vibrational states of symmetric top molecules. In this scheme, a rotational contact transformation can be applied to the effective Hamiltonian such that the operators of the “2, +2,” “2, −2,” and “2, −1” l-type interactions as well as the operators of the Δk = ±3 and ±4 interactions are eliminated from the first-order terms of the expansion of the rotational Hamiltonian in terms of the small parameter λ. The results have been used to discuss the correlation between various interaction parameters in the effective rotational Hamiltonian for the doubly degenerate fundamental vibrational levels of semirigid symmetric top molecules. For example, for C_3v or D₃ molecules, the parameter of the “2, −1” interaction is correlated with other parameters and cannot be determined separately by fitting the experimental data (unless there are certain accidental resonances between vibrational-rotational levels).     

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.     

Quantum Mechanical Virial Theorem in Systems with Translational and Rotational Symmetry

Generalized virial theorem for quantum mechanical nonrelativistic and relativistic systems with translational and rotational symmetry is derived in the form of the commutator between the generator of dilations G and the Hamiltonian H. If the conditions of translational and rotational symmetry together with the additional conditions of the theorem are satisfied, the matrix elements of the commutator [G,H] are equal to zero on the subspace of the Hilbert space. Normalized simultaneous eigenvectors of the particular set of commuting operators which contains H, J ², J _z and additional operators form an orthonormal basis in this subspace. It is expected that the theorem is relevant for a large number of quantum mechanical N-particle systems with translational and rotational symmetry.     

Calculated energy levels and intensities for the ν1 and 2ν2 bands of HDO

A Hamiltonian taking explicitly into account both Fermi and Coriolis interactions has been set up for triatomic molecules of symmetry C_s and used to reproduce, very satisfactorily, the available rotational energy levels of the {(100), (020)} interacting states of HDO, providing us with realistic wavefunctions as well as precise rotational constants and vibrational energies. Then, to calculate line intensities, these wavefunctions were used together with suitably chosen transition moment operators expanded up to degree 2 in $\text{J}$ and having the correct symmetry in the C_s group, leading to hybrid bands of both A and B type., Using this formalism, it has been possible to determine, from the fit of the existing experimental intensities, the coefficients appearing in the expansions of the transition moment operators of the 2ν₂ and ν₁ bands of HDO. In this way, we have improved upon the F-factor formalism which needs much more parameters to reproduce the line intensities with the same precision. Finally, using the transition moments as well as the wavefunctions and energy levels deduced from the diagonalization of the Hamiltonian matrix, we have calculated the whole spectrum of the ν₁ and 2ν₂ bands of HDO.     

Solution of a one-dimensional torsion Schrödinger equation with a general periodic potential

Relations are found for the calculation of eigenvalues and eigenfunctions of the Hamiltonian of internal rotational motion in molecules in the basis of plane waves. The dependences of kinematic coefficient F(φ) and potential V(φ) on dihedral angle φ are represented by Fourier series for both symmetric and asymmetric functions, as well as for general periodic functions. If a molecule has symmetry elements, the found solution transforms to that previously known.     

Algebraic realization of the quantum rotor — odd-A nuclei

An algebraic realization of the quantum rotor for non-zero spin values (integer as well as half-integer) is established by constructing a model Hamiltonian out of rotationally invariant functions of the generators ofSU(3). The eigenvalues of this Hamiltonian in the leading normal-SU(3) symmetry for²⁵Mg and the so-called leading pseudo-SU(3) symmetries for¹⁵⁹Dy and¹⁶⁵Er are compared with the corresponding rotor results. For spinfree systems the internal symmetry group of the rotor and itsSU(3) realization are known to be D₂, the Vierergruppe. This symmetry extends to integral spin values, while for half-integer spins the rotor and itsSU (3) realization are shown to display an internal quaternion group symmetry. The theory points to a microscopic (many-particle shell-model) picture of nuclear rotational motion with spin degrees of freedom taken fully into account. An algebraic realization of the many-particle Nilsson model for odd-A nuclei, with the orbit-orbit and spin-orbit terms included, is given and applied to²³Na.     

A reduced form of the spin-rotation Hamiltonian for asymmetric-top molecules,with applications to HO2 and NH2

The spin-rotational Hamiltonian for an asymmetric-top molecule in a given vibrational level of an open-shell electronic state may contain more parameters than can be determined from the observed energy levels. This paper describes the reduction of the Hamiltonian by means of a unitary transformation to a form suitable for fitting to observed energies. It is shown that, for molecules of lower than orthorhombic symmetry, there are fewer determinable quadratic spin-rotation parameters than have been used previously. For example, for a molecule belonging to the group C₈, there are four, not five, determinable spin-rotation constants, ?_αβ. Similar indeterminacies exist among the quartic terms of the spin-rotation Hamiltonian. The case of a molecule of orthorhombic symmetry, for which there are six determinable quartic parameters, is considered in detail. The results are applied to the experimental data available on the spin-rotation splittings of the HO₂ and NH₂ radicals in their ground vibrational and electronic states.     

Effective torsion-rotation Hamiltonian for methanol-type molecules

A fourth-order effective Hamiltonian has been derived for the torsion-rotation problem of a methanol-type molecule, i.e., for a C_3v top attached to a C_s frame. First, symmetry considerations based on a frame-fixed axis system are used to determine allowed terms in the Hamiltonian. These terms are then subjected to a contact transformation to remove the indeterminate ones. This procedure is essentially an extension of Watson's method for semirigid molecules to the torsion-rotation problem. It is demonstrated that the Hamiltonian derived in the present work is capable of improving the fittings of the millimeter and submillimeter absorption frequencies of CH₃OH and CH₃SH.     

动力学对称群方法对三原子分子高激发振动态的理论研究

利用动力学对称群方法研究了弯曲三原子分子的振动高激发态能谱-该方法显示：三原子分子的动力学对称性为U₁(4)U₂(4),则三原子分子的Hamiltonian量可写成代数的各元素之和,通过李代数处理而求得分子代数Hamiltonian量的本征值,进而得到分子的振动能谱-并具体计算了O₃分子- 关键词：     

Line frequencies and intensities in the ν3 fundamental band of molecules with an octahedral symmetry

Frequencies and lines intensities in the υ₃ band of molecules with an octahedral symmetry have been calculated by the use of second-order perturbation theory, according to the model of clusters. The calculations are developed by the use of the Hamiltonian of Hecht and the special properties of these molecules. Comparisons are made between the present theoretical calculations and other theoretical and experimental works.     

A frame bundle generalization of multisymplectic momentum mappings

We construct momentum mappings for covariant Hamiltonian field theories using a generalization of symplectic geometry to the bundle L_VY of vertically adapted linear frames over the bundle of field configurations Y. Field momentum observables are vector-valued momentum mappings generated from automorphisms of Y, using the (n + k)-symplectic geometry of L_VY. These momentum observables on L_VY generalize those in covariant multisymplectic geometry and produce conserved field quantities along flows. Three examples illustrate the utility of these momentum mappings: orthogonal symmetry of a Kaluza-Klein theory generates the conservation of field angular momentum, affine reparametrization symmetry in time-evolution mechanics produces a version of the parallel axis theorem of rotational dynamics, and time reparametrization symmetry in time-evolution mechanics gives us an improvement upon a parallel transport law.     

Analysis and fit of the Fourier-transform microwave spectrum of the two-top molecule N-methylacetamide

The jet-cooled Fourier-transform microwave spectrum of N-methylacetamide (CH₃NHC(O)CH₃), a molecule containing two methyl tops with relatively low barriers to internal rotation, has been recorded and fit to nearly experimental uncertainty. Measurements were carried out between 10 and 26 GHz, with the nitrogen quadrupole splittings resolved for many transitions. The permutation-inversion group for this molecule is G₁₈ (not isomorphic to any point group), with irreducible representations A₁, A₂, E₁, E₂, E₃, and E₄. One of these symmetry species and the usual three asymmetric rotor quantum numbers J_KaKc were assigned to each torsion-rotation level involved in the observed transitions. F values were assigned to hyperfine components, where . Transitions involving levels of A₁ and A₂ species could be fit to an asymmetric rotor Hamiltonian. The other transitions were first fit separately for each symmetry species using a Pickett-like effective rotational Hamiltonian. Constants from these fits show a number of additive properties which can be correlated with sums and differences of effects involving the two tops. A final global fit to 48 molecular parameters for 839 hyperfine components of 216 torsion-rotation transitions involving 152 torsion-rotation levels was carried out using a newly written two-top computer program, giving a root-mean-square deviation of observed-minus-calculated residuals of 4 kHz. This program was written in the principal axis system of the molecule and uses a free-rotor basis set for each top, a symmetric-top basis set for the rotational functions, and a single-step diagonalization procedure. Such an approach requires quite long computation times, but it is much less prone to subtle programming errors (a consideration felt to be important since checking the new program against precise fits of low-barrier two-top molecules in the literature was not possible). The two internal rotation angles in this molecule correspond to the Ramachandran angles ψ and φ often defined to describe polypeptide folding. Barriers to internal rotation about these two angles were found to be 73 and 79 cm⁻¹, respectively. Top-top coupling in both the kinetic and potential energy part of the Hamiltonian is relatively small in this molecule.     

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.     

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.     

Calculation of the intensities of the vibrational-rotational lines of polyatomic molecules using the formalism of irreducible tensor systems

The theory of irreducible tensor systems is used to obtain general formulas for the dependence of the matrix elements of the transformed vibrational—rotational dipole moment on the rotational quantum numbers. The relations obtained are valid for molecules of arbitrary symmetry and can take into account all possible intramolecular effects and interactions (resonances, vibration—rotation interactions, splitting, etc.). As illustrations we consider molecules of the type XY₄ (symmetry T_d) and XY₂ (symmetry C_2v).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, No. 8, pp. 3–10, August, 1987.     

IBM: Discrete symmetry viewpoint

It is shown that the set of transformations of the s and d boson operators that maintain the IBM-like form of the Hamiltonian comprises a discrete point symmetry group D ₂′. The transformations manifest themselves as a parameter symmetry of the IBM-1 Hamiltonian. The transformations considered are also necessary for constructing the most general IBM-2 Hamiltonian. The properties of the potential energy surfaces arising in connection with these transformations are discussed.     

Ab initio potential energy curves and vibrational levels for the C and D 1Πu states of the hydrogen molecule

Two lowest eigenvalues of the Hamiltonian for the hydrogen molecule corresponding to the ¹Π_u symmetry have been calculated in the Born-Oppenheimer approximation. They represent the C and D states, respectively. Different highly flexible wavefunctions were used for both states and the calculations were performed for a wide range of internuclear distances: 1 ≤ R ≤ 12 a.u. for the C state, and 1 ≤ R ≤ 25 a.u. for the D state. The calculated potential energy curves are more accurate than any previous ab initio results for the above states. The vibrational Schrödinger equation for both states has been solved for H₂, HD, and D₂ using the Numerov method. The resulting energies and rotational constants are compared with the experimental values.     

Analysis of the microwave spectrum of hydrazine

Microwave measurements in the interval from 6 to 133 GHz, consisting of 444 rotational transitions in the vibrational ground state of hydrazine with J ≤ 31 and K_a ≤ 6 were fit to an effective rotational Hamiltonian containing 9 asymmetric rotor constants, 14 NH₂ inversion parameters, and 1 internal rotation parameter, with an overall standard deviation of the fit of 0.40 MHz. This set of parameters contains: (i) the three rotational constants; (ii) tunneling splitting constants for NH₂ inversion at one end of the molecule, for NH₂ inversion at both ends of the molecule, and for internal rotation through the trans barrier; (iii) two K-type doubling constants affecting the K = 1 levels; (iv) an a-type Coriolis interaction with matrix elements linear in K; and (v) various centrifugal distortion corrections to the above parameters. A consistent group theoretical formalism was used to label the energy levels and to select terms in the phenomenological rotational Hamiltonian. The Hamiltonian matrix, which is set up in a tunneling basis set, is of dimension 16×16 and contains only ΔK_a = 0 matrix elements, asymmetric rotor effects being taken into account on the diagonal by terms from a Polo expansion in bⁿ. Hyperfine splittings and barrier heights are not discussed.