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).     

Microwave l-type resonance transitions of the v4 = 1 state in PF3: Detailed interactions and molecular structure

Observation of the direct l-type resonance transitions in the microwave spectrum of the v₄ = 1 state of PF₃ has been extended to J = 36. The w-type interaction, (Δl = 0, ΔK = 6), has been found from measurements on the “forbidden” Stark trasitions in the $\text{K}\text{= 3}$ series. Also in this series a close accidental degeneracy was found between J = 30, $\text{K}\text{= 3}$ and 0, leading to new zero-field “forbidden” transitions through the r-type interaction (Δl = 2, ΔK = ?1) and to the determination of the C rotational constant. Nine spectroscopic parameters were determined using 140 observed frequencies including two “forbidden” trasitions. After suitable correction the B and C constants were used to determine the r₀, r_z, and r_e structures for PF₃. The equilibrium structure is estimated to be P-F = 1.561 ± 0.001 Å and ∠FPF = 97.7 ± 0.2°.     

Forbidden vibrational-rotational transitions and Herman-Wallis factors in fundamental IR bands of symmetric-top molecules

Within the theory of coupled schemes of ordering of vibrational-rotational interactions, the operator of the effective dipole moment of single-quantum vibrational transitions is represented in the form of an infinite series in vibrational (normal coordinates and conjugate momenta) or rotational variables (components of the total angular momentum). Mechanisms of activation of infrared-inactive totally symmetric vibrations in molecules of the D _2a , D _3h , C _3h , D _n (n ≥ 3), S ₄, T, T _a , and O symmetries and forbidden vibrational-rotational transitions in IR bands of active vibrations have been studied. The group-theoretic analysis of tensor parameters in higher-order effective dipole moments of single-quantum vibrational transitions in axially symmetric molecules has been performed. The strengths of allowed transitions and forbidden transitions in fundamental and hot IR bands of axially symmetric molecules are calculated with allowance for the Herman-Wallis factors. For effective dipole moments of multiquantum transitions in molecules, models are developed in the form of infinite series in rotational variables and in the form of Padé approximants.     

Δk = ±3 “forbidden” infrared transitions in the ν2-band of NH3

Two Δk = ±3 “forbidden” vibration-rotation transitions in the ν₂-band of NH₃ have been measured by using infrared-microwave two-photon spectroscopy and laser Stark spectroscopy. Combining these results with Rao's recent measurement of the band, we have obtained the C₀ rotational constant of 6.2280 ± 0.0008 cm^?1.     

Forbidden pure rotational Raman spectra of C3v and Td molecules

Some of the pure rotational Raman transitions of polyatomic molecules that are forbidden according to rigid rotor selection rules may acquire intensity by a centrifugal distortion mechanism. These are analogous to forbidden rotational electric dipole transitions which recently have been studied extensively (1–6). A simple technique to obtain the intensity factors leading to the line strengths of such Raman transitions is presented. The intensity factors $\text{b}{}_{\mathrm{J\prime k\prime }}{}^{\mathrm{Jk}}$ and $\text{b}{}_{\mathrm{J\prime }}{}^{\mathrm{J}}$ of the Q, R, and S branches of molecules with C_3v and T_d symmetry are obtained. For C_3v molecules, in addition to the usual selection rules for J, those for k are found to be Δk = ±3 and ±6. There is a modification of intensity of the normally allowed Δk = 0 transitions. T_d molecules, which normally do not have pure rotational spectra, now give rise to weak Raman transitions due to centrifugal distortion.     

The ν2 and ν4 bands of AsH3

The infrared absorption of arsine, AsH₃, between 750 and 1200 cm^?1 has been recorded at a resolution of 0.006 cm^?1. Altogether 2419 transitions, including nearly 700 “perturbation allowed” transitions with Δ∥k ? l∥ = ±3, ±6, and ±9, have been assigned to the ν₂(A₁) and ν₄(E) bands. Splitting of the transitions for K″ = 3, 6, and 9 was also observed. To fit the rotational pattern of the v₂ = 1 and v₄ = 1 vibrational states up to J = 21, all the experimental data were analyzed simultaneously on the basis of a rovibrational Hamiltonian which took into account the Coriolis interaction between ν₂ and ν₄ and also included several essential resonances within them. The derived set of 38 significant spectroscopic parameters reproduced the 2328 transition wavenumbers retained in the final fit within the accuracy of the experimental measurements.     

Pure rotational spectrum of cyanopropyne in the v12 = 1 vibrational state

The micoowave and millimeter-wave spectrum of cyanopropyne, CH₃CCCN, in the lowest excited vibrational state, v₁₂ = 1 (E), has been observed in the frequency range from 8 to 220 GHz. The measurements up to the J = 53−52 transitions allowed us to determine the Coriolis coupling term and the l-type interaction term very precisely in addition to some of the sextic centrifugal terms. The contribution of the r-type interaction (Δk = ±1, Δl = ±2) has been found to be correlated with that of η_J, a centrifugal correction term to the first-order Coriolis interaction.     

The Δk = ±2 perturbation-allowed ν4 band and inversion-rotation energy levels of 15NH3

More than 800 Δk = ±2 and 60 Δk = ±3 forbidden transitions to the ν₄ and 2ν₂ vibrational levels, respectively, have been assigned in the Fourier transform spectra of ¹⁵NH₃, recorded with a pathlength of 96 m. Combination differences derived from these transitions provide information on the spacing between the ground state energy levels with different rotational quantum numbers K in the interval from 0 to 16. These data along with wavenumbers of all the available allowed transitions pertaining to the ground and ν₂ states have been subjected to a simultaneous least-squares analysis using two different parametrization models to obtain precise values of the inversion-rotation energy levels.     

The study of microwave pressure lineshifts

The results of experimental investigations of pressure shifts and broadening of spectral lines of polar molecules performed in the submillimeter region by a microwave spectrometer RAD are reported. About 30 measurements were made of self-shift and foreign gas shift parameters of the lines of NH₃, PH₃, AsH₃, and H₂O molecules including lineshifts in excited vibrational states, lineshifts of transitions connected by common levels, lines with various values of J and K quantum numbers, and the “forbidden” |ΔK| = 3 lines. On the basis of the data obtained in this work and data available in the literature, new experimental dependences of lineshifts on molecular parameters are found. The results are well described by a simple “Stark effect” model of lineshifts. Some new directions of investigations are outlined.     

Forbidden rotational transitions in symmetric top molecules

The intensities of the forbidden Δk = ±6 and ±4 transitions in symmetric top molecules arising from rotational intensity borrowing are reported. A perturbation theoretical approach has been used to study the mixing of wavefunctions produced by the additional splitting term in the rotational Hamiltonian of C_3v or C_4v-type molecules. Numerical values are presented for PH₃, PD₃, and BrF₅ molecules. The importance of these results in structural analysis and astrophysical studies is discussed.     

Microwave spectrum of the C4v molecule IF5 in the excited vibrational states v5(B1) = 1 and v9(E) = 1: Analysis of the ν5-ν9 Coriolis resonance

Measurements of the microwave spectrum of the C_4v molecule IF₅ in the excited vibrational states v₅(B₁) = 1 and v₉(E) = 1 are reported for the transitions J4 → 5, 5 → 6, 6 → 7, 8 → 9, and 9 → 10 (27–55 GHz). The Coriolis resonance interaction between these two states is analyzed by diagonalization of Hamiltonian matrices of dimension 3 × (2J + 1) in which all (Δl,Δk) = (±2, ±2)(q⁺), (±2, ±2)(q^?), and (0, ±4)(R₆) interactions are included as off-diagonal terms in addition to the v₅ = 1 ? v₉ = 1, l₉ = ±1(R₅₉) Coriolis interaction. In the v₉ = 1 state spectra, the $\text{B}{}_1\text{B}{}_2$l-doubling of the kl = ?1 transitions and $\text{A}{}_1\text{A}{}_2$ splittings of the kl = ?3 transitions and $\text{B}{}_1\text{B}{}_2$ splittings of the kl = +3 transitions, all enhanced by the Coriolis resonance, have been observed and measured. Least-squares refined rovibrational parameters for the v₅ = 1 and v₉ = 1 states are reported and a preliminary value for the rotational constant C₉ has been obtained.     

Spin forbidden transitions in the Ka = 0 subband of NO2 in the λ = 592.5 nm region

In a high resolution laser excitation spectrum of NO₂, lines were recorded which do not follow the selection rule ΔN = ΔJ = ΔF of “spin allowed” transitions. Line positions and intensities of these “spin forbidden” lines were investigated for all rotational lines up to N″ = 12 of the K_a = 0 subband around λ = 592.5 nm. While the observed line intensities of “spin allowed” transitions can be well described by the J-coupling scheme, neither the J- nor the G-coupling scheme sufficiently describes the “spin forbidden” transitions. The observations can be fitted satisfactorily by perturbation theory, in which the Fermi interaction in ²A₁ is treated as the perturber. This looks similar to a superposition of J and G scheme in the ²A₁ ground state.     

Variation Angulaire,Transitions Interdites Dans Le Spectre De R.P.E. Des Ions Mn2+ Dans L'alumine α

The spin Hamiltonian with trigonal symmetry for Mn²⁺ in Al₂O₃ has been derived. The line positions have been calculated using perturbation theory up to third order. Three groups of forbidden transitions ΔM = ± 1 Δm = ± 1 have been investigated. Q′ and γ have been deduced from for forbidden hyperfine doublets. The two evaluations of the spin Hamiltonian parameters from allowed (Δm = 0) and forbidden lines (Δm = 1) are discussed.     

Far infrared spectrum and spectroscopic constants of AsH3 in the ground state

The far ir spectrum of arsine, AsH₃, was recorded in the range 25–100 cm^?1 with a resolution of approximately 0.004 cm^?1. ΔJ = +1, ΔK = 0 rotational transitions were measured and assigned up to J″ = 12. These transitions, together with the presently available microwave and submillimeter-wave data and ground state combination differences, were analyzed on the basis of a rotational Hamiltonian which includes Δk = ±3 and Δk = ±6 interaction terms. The derived ground state molecular parameters reproduced the transition frequencies of both allowed and “perturbation allowed” transitions within the accuracy of the measurements. The equilibrium structure was determined for the AsH₃ molecule.     

Bond charge parameters from integrated infrared intensities

A procedure is outlined to analyze the infrared absorption intensities of fundamental vibration bands in terms of “bond charge parameters”. The method is illustrated for some small C_2v- and C_3v-type molecules: SO₂, NF₃, and PF₃. The values obtained for the “bond charge reorganizations” and “effective bond charges” for SO, NF, and PF bonds are discussed. The method offers the possibility of calculating rotational contributions without the use of a reference molecule.     

Millimeter-wave spectrum of the C4v molecule iodine oxygen pentafluoride IOF5: K-doubling of the |k| = 2 transitions in the ground state spectrum

Measurements are reported for the rotational spectrum of the C_4v molecule IOF₅ in the ground vibrational state in the range 30–75 GHz (J7 ← 6 to J17 ← 16). The K-doubling of |k| = 2 transitions due to an off-diagonal centrifugal distortion interaction of the type (Δl, Δk) = (0, ±4) has been observed. The centrifugal distortion constants D_J, D_JK, and R₆ have been determined as 0.139(2) kHz, 0.107(4) kHz, and 21(2) Hz, respectively.     

High-order EPR transitions of Gd3+ in Ce2Zn3(NO3)12·24H2O

The angular variation of the EPR spectrum of Gd³⁺ in Ce₂Zn₃(NO₃)₁₂·24H₂O has been studied. In addition to the allowed fine structure lines some weak low-field lines, identified as forbidden transitions (ΔM = ±2 , ± 3), have been observed.     

Application of the contact transformation method to torsional problems in methyl silane

The effect of the torsional degree of freedom on redundancies in the Hamiltonian and on the dipole operator has been investigated for methyl silane-like molecules. By applying a rotational contact transformation to the torsion-rotation Hamiltonian H_TR for the ground vibrational state, a systematic method is demonstrated for treating the redundancies that relate different terms in H_TR. In general, with this method, the experimentally accessible molecular parameters in the reduced Hamiltonian can be related to the physically significant molecular parameters in the untransformed Hamiltonian. It is shown that H_TR contains a new term which has matrix elements with selection rules (ΔK = ±3), (Δσ = 0), and Δv_T arbitrary, where v_T and σ label the torsional levels and sublevels, respectively. As a result of this term, the distortion dipole constant μ_D which characterizes (ΔK = ±3) matrix elements in C_3v molecules cannot, in systems like CH₃SiH₃, be ascribed entirely to centrifugal distortion but can contain a significant contribution from torsional effects. Furthermore, new transitions can appear in the pure torsional bands which may be strong enough to observe in low barrier molecules. By applying a vibrational contact transformation, the form is derived of the leading torsional terms in the dipole moment expansion. The four dipole distortion constants μ₀^T, μ₂^T, μ_|;^T, and μ_Λ^T which characterize these terms are related to the molecular parameters that enter the Coriolis, centrifugal distortion, and anharmonicity contributions to the vibration-torsion-rotation Hamiltonian.     

Forbidden rotational spectra of polyatomic molecules

Some of the pure rotational transitions of polyatomic molecules that are forbidden according to the rigid rotor selection rules may acquire intensity by a centrifugal distortion mechanism, in which the intensity depends on the dipole derivatives and on the displacements in the normal coordinates produced by centrifugal distortion. The structures of the predicted spectra are discussed for nonpolar molecules belonging to the point groups D_3h, D_n (n > 2), D_2d, and T_d. Some of the calculated R-branch lines of methane are stronger than some of the lines of HD already observed in the same frequency range. For polar molecules extra contributions to the intensity must be considered and are exemplified for the Δk = ±3 transitions of C_3v molecules. These forbidden rotational transitions may have some astrophysical importance.     

12C18O2: High-resolution emission spectra in the 4.5-μm region rovibrational transitions 0v′2v3 → 0v′2(v3 − 1), v2 = l

Emission spectra of gaseous mixtures involving isotopic species of CO₂ excited by a dc discharge were recorded under Doppler-limited resolution, using a high-information Fourier Transform Interferometer, in the region 4–5 μm. In this paper are given the results concerning 34 vibrational transitions (Δv₃ = 1), for ¹²C¹⁸O₂. The band centers and the spectroscopic constants for the 39 vibrational levels involved are reported. They reproduce more than 1000 experimental wavenumbers with a RMS of the order of 2 × 10^?5 cm^?1 for the best vibrational transition and less than 3 × 10^?4 cm^?1 for most of the others. From a weighted simultaneous fit of all the experimental wavenumbers belonging to the Σ-Σ transitions, a set of molecular parameters was computed. A good reproduction of the experimental wavenumbers was obtained for all the vibrational transitions except those involving the level v₃ = 9, our conclusion being that a local vibrational perturbation exists for this level.