Development of the Hamiltonian and transition moment operators of symmetric top molecules using the O (3) ⊃ C∞v ⊃ C3v group chain

We present a development of the Hamiltonian, dipole moment, and polarizability operators for XY₃Z molecules. These rovibrational operators are written with the aid of a tensorial formalism derived from the one already used in Dijon and adapted to the XY₃Z symmetric tops in a recent paper [A. El Hilali, V. Boudon, M. Loëte, J. Mol. Spectrosc. 234 (2005) 166-174]. We use the O (3) ⊃ C_∞v ⊃ C_3v group chain. Expressions for the matrix elements are derived for these operators.     

Symmetry-adapted tensorial formalism to model rovibrational and rovibronic spectra of molecules pertaining to various point groups

We present a short review on the tensorial formalism developed by the Dijon group to solve molecular spectroscopy problems. This approach, originally devoted to the rovibrational spectroscopy of highly symmetrical species (spherical tops) has been recently extended in several directions: quasi-spherical tops, some symmetric and asymmetric tops, and rovibronic spectroscopy of spherical tops in a degenerate electronic state. Despite its apparent complexity (heavy notations, quite complex mathematical tools), these group theoretical tensorial methods have a great advantage of flexibility: a systematic expansion of effective terms for any rovibrational/rovibronic problem up to a given order is automatically generated. Inclusion of all possible interaction terms for any polyad scheme is therefore easy. This makes such an approach suitable for many types of molecular problems, not only the most symmetric ones.     

Spectroscopy of X2Y4 (D2h) molecules: tensorial formalism adapted to the O(3)⊃D2h chain, Hamiltonian and transition moment operators

A tensorial formalism adapted to the case of the X₂Y₄ molecules with D_2h symmetry has been developed in the same way as in the previous works on XY₄ (T_d) and XY₆ (O_h) spherical tops and XY₅Z (C_4v) symmetric tops. Here, we use the O(3)⊃D_2h group chain. All the coupling coefficients and formulas for the computation of matrix elements are given for this chain and used in the case of the Hamiltonian and transition moment operators.     

Spectroscopy of XY2Z2 (C2v) Molecules: A Tensorial Formalism Adapted to the O(3)⊃Td⊃C2v Chain. Application to the Ground State of SO2F2

A tensorial formalism adapted to the case of quasi-spherical XY₂Z₂ asymmetric tops such as SO₂F₂ has been developed as an extension of the usual one for the tetrahedral molecules. We use the O(3)⊃T_d⊃C_2v group chain. All the coupling coefficients and formulas for the computation of matrix elements are given for this chain. Such relations are then deduced in the C_2v group itself. We also present a development of the Hamiltonian, dipole moment, and polarizability operators for the molecules under consideration using this formalism. These operators are involved in the calculation of the energies and intensities of rovibrational transitions and are essential for spectrum simulations. Expressions for the matrix elements are derived for these operators. A first application to the ground state of SO₂F₂ is presented. Programs for spectrum simulation and fit using these methods are freely available at the URL http://www.u-bourgogne.fr/LPUB/c2vTDS.html.     

Tensorial development of the rovibronic Hamiltonian and dipole moment operators for XY3Z molecules with a degenerate electronic state: Preliminary application to the CH3O radical

We present a development of the Hamiltonian and transition moment operators of XY₃Z (C_3v) symmetric tops molecules in a degenerate electronic state with the aid of a tensorial formalism developed in a recent paper [A. El Hilali, V. Boudon, M. Loëte, J. Mol. Spectrosc. 239 (2006) 41-50]. Electronic operators are defined from group theory properties. They provide a new approach to build an effective rovibronic Hamiltonian as well as an effective dipole moment operator for rovibronic transition of XY₃Z molecules. This model is studied qualitatively thanks to the tensorial algebra properties. Expressions of the matrix elements are derived for these operators. A first simple application to the ground electronic state of CH₃O is proposed as an illustrative example.     

Rotational spectrum of the v11 = 1 and v14 = 1 vibrational states of CH3CCCCH

The pure rotational spectra of the v₁₁ = 1 and v₁₄ = 1 vibrational states of the main isotopic species of methyldiacetylene have been recorded and assigned in the 80-400 GHz frequency range, spanning the quantum numbers 19 ? J ? 95 and 0 ? K ? 15. The present study allows us to provide accurate rotational, centrifugal distortion and vibration-rotation interaction constants. The experimental investigation has been strongly supported by quantum-chemical computations at the second-order Møller-Plesset theory (MP2) in conjunction with a triple-zeta quality basis set.     

The v6 = 1 and v6 = 2 vibrational states of DCF3

The high-resolution infrared spectra of DCF₃ were reinvestigated in the ν₆ fundamental band region near 500 cm⁻¹ and around 1000 cm⁻¹ with the aim to assign and analyze the overtone level of the asymmetric CF₃ bending vibration v₆ = 2.The present paper reports on the first study of both its sublevels (A₁ and E corresponding to l = 0 and ±2, respectively) through the high-resolution analysis of the overtone band and the hot and bands.The well-known “loop method”, applied to and , yielded ground state energy differences Δ(K, J) = E₀(K, J) − E₀(K − 3,J) for the range of K = 6 to 30.In the final fitting of molecular parameters, we used the strategy of fitting all upper state data together with the ground state rotational transitions.This is equivalent to that calculating separately the and coefficients of the K-dependent part of the ground state energy terms from the combination loops.All rotational constants of the ground state up to sextic order could be refined in the calculation.This led to a very accurate determination of C₀ = 0.18924413(25) cm⁻¹, , and also .In the course of analyzing simultaneously the overtone band together with the and ν₆ bands, the original assignment of the fundamental ν₆ band [Bürger et al., J. Mol. Spectrosc. 182 (1997) 34-49] was found to be incompatible with the present one. Assignments of the (k + 1, l₆ = +1)/(k − 1,l₆ = −1) levels had to be interchanged, which changed the value of Cζ₆ = −0.14198768(26) cm⁻¹ and the sign of the combination of constants C − B − Cζ in the v₆ = 1 level to a negative value.     

The ν1 + 3ν2 + 3ν3 and 4ν1 + ν2 + ν3 bands of ozone by CW-cavity ring down spectroscopy between 5900 and 5960 cm

The absorption spectrum of ozone, ¹⁶O₃, has been recorded in the 5903-5960 cm⁻¹ region by high sensitivity CW-cavity ring down spectroscopy (α_min ∼ 5 × 10⁻¹⁰ cm⁻¹). The ν₁ + 3ν₂ + 3ν₃ and 4ν₁ + ν₂ + ν₃ A-type bands centred at 5919.15 and 5947.07 cm⁻¹ were newly observed. A set of 173 and 168 energy levels could be experimentally determined for the (1 3 3) and (4 1 1) states, respectively. Except for a few K_a = 5 levels of the (4 1 1) state, the rotational structure of the two states was found mostly unperturbed. The spectroscopic parameters were determined from a fit of the corresponding line positions by considering the (1 3 3) and (4 1 1) states as isolated. The determined effective Hamiltonian and transition moment operators were used to generate a list of 785 transitions given as Supplementary Material.     

High-resolution rotational analysis of HDS: 2ν3, ν2 + 2ν3, 3ν3, and ν2 + 3ν3 bands

High-resolution Fourier transform spectrum of the HD³²S molecule was studied in the region of 5000-9000 cm⁻¹. More than 1600 observed transitions yielded 239, 264, 131, and 116 upper state ro-vibrational energies of the states (002), (012), (003), and (013), respectively. With a Watson-type effective Hamiltonian model, the ro-vibrational parameters of these four upper states were determined by a least-square fitting which can reproduce the ro-vibrational energies close to the experimental accuracy. The relative linestrengths are also discussed.     

Sub-Doppler and Doppler spectroscopy of DCN isotopomers in the terahertz region: ground and first excited bending states (v1v2v3)=(0 1 0)

The rotational spectra of the deuterium cyanide isotopic species DCN, D¹³CN, DC¹⁵N, and D¹³C¹⁵N were recorded in the vibrational ground and first excited bending state (v₂=1) up to 2 THz. The R-branch transitions from J=3←2 to J=13←12 were measured with sub-Doppler resolution. These very high resolution (∼70 kHz) and precise (±3-10 kHz) saturation dip measurements allowed for resolving the underlying hyperfine structure due to the ¹⁴N nucleus in DCN and D¹³CN for transitions as high as J=10←9. Additional high JR-branch (J=25←24 to J=28←27) transitions around 2 THz and direct l-type (ΔJ=0, J=19 to J=25) transitions from 66 to 118 GHz were recorded in Doppler-limited resolution. For the ground state of D¹³C¹⁵N, the J=1←0 transition was measured for the first time. The transition frequency accuracies for the other deuterated species were significantly improved. These new experimental data, together with the available infrared rovibrational data and previously measured direct l-type transitions, were subjected to a global least squares analysis for each isotopomer. This yielded precise sets of molecular constants for the ground and first excited vibrational states, including the nuclear quadrupole and magnetic spin-rotation coupling constants of the ¹⁴N nucleus for DCN and D¹³CN. The hyperfine structure due to the D, ¹³C, and ¹⁵N nuclei have not been resolved, but led to a broadening of the observed saturation dips.     

Accurate rotational spectroscopy of sulfur dioxide, SO2, in its ground vibrational and first excited bending states, v2 = 0, 1, up to 2 THz

Approximately 150 pure rotational transitions each have been recorded for SO₂, v₂ = 0 and 1, in selected frequency regions up to 2 THz. The J and K_a quantum numbers reach very high values: 92 and 23, respectively, for the ground vibrational state and 81 and 21, respectively, for the first excited bending state. The highest levels accessed are almost 3000 cm⁻¹ above ground. The relative experimental uncertainties Δν/ν are about 10⁻⁸ for several medium to strong, isolated lines, and generally better than 2.5 × 10⁻⁷. Improved spectroscopic parameters have been obtained for both states, particularly for the excited bending state. In fact, the accuracies with which the energy levels of the v₂ = 1 state are known depend essentially only on the accuracy with which the vibrational spacing is known from infrared spectroscopy.     

High-precision frequency measurements of the ν1 + ν3 combination band of C2H2 in the 1.5 μm region

A pair of 1.5 μm semiconductor laser frequency standards have been developed for optical telecommunications use, stabilised to transitions of ¹²C₂H₂ and ¹³C₂H₂, using cavity-enhanced Doppler-free saturation absorption spectroscopy. The absolute frequencies of 41 lines of the ν₁ + ν₃ band of ¹²C₂H₂, covering the spectral region 1520-1545 nm, have been measured by use of a passive optical frequency comb generator, referenced to ¹³C₂H₂ transitions of known frequency. The mean experimental uncertainties (coverage factor k = 1) of the frequency values are 3.0 kHz (type A) and 10 kHz (type B). Improved values of the band origin ν₀, rotational constants B′ and B″, and centrifugal distortion coefficients D′, D″, H′, and H″ are presented.     

Molecular dynamics simulations of CFx (x = 2, 3) molecules at Si3N4 and SiO2 surfaces

Molecular dynamics simulations of the interaction between CF_x (x = 2, 3) molecules and crystalline as well as amorphous Si₃N₄ and SiO₂ surfaces using a density-functional based method are reported. The binding energies of various configurations at the crystalline surfaces were calculated. The effect of hydrogen substitution was studied.     

Orientation of O(3) and SU(2)⊗CI representations in cubic point groups (Oh,Td) for application to molecular spectroscopy

We propose a detailed method for the symmetrization of the standard O(3) or SU(2)⊗C_I basis |j_τ,m〉 (τ=g or u) into the O_h or T_d point group. This is realized by means of an orientation matrix called G. The oriented basis obtained in this way allows matrix element calculations for rovibronic spectroscopic problems concerning octahedral or tetrahedral molecules. Particular attention has been put on careful phase choices. A numerical calculation of all the G matrix elements for both integer and half-integer j values up to 399/2 has been performed. Such high angular momentum values are necessary for the case of heavy molecules with high rotational excitation. To calculate the G coefficients with high precision at high j values we pre-calculated the necessary Wigner functions using symbolic MAPLE software and made then the numerical calculations with quadruple precision. The complete list of these coefficients can be obtained freely at the URL: http://www.u-bourgogne.fr/LPUB/group.html. As an illustration, we also present briefly an application to two typical spectroscopic calculations: the pure rotational levels of SF₆ in its ground vibrational state and the ν₃ band of ReF₆ (an open-shell molecule with an odd number of electrons and a fourfold degenerate electronic ground state).     

The bending triad of the quasi-spherical top molecule SO2F2 in the 550 cm region

The analysis of the ν₃/ν₇/ν₉ bending triad of SO₂F₂ has been recently performed with the Watson’s Hamiltonian up to octic terms employing 79 rovibrational parameters but including only the first order Coriolis interaction terms, fixed to ab initio values [H. Bürger, J. Demaison, F. Hegelund, L. Margulès, I. Merke, J. Mol. Struct. 612 (2002) 133-141]. Since SO₂F₂ is a quasi-spherical top, it can also be considered as derived from the sulfate ion. We have thus developed a new tensorial formalism in the O (3) ⊃ T_d ⊃ C_2v group chain [M. Rotger, V. Boudon, M. Loëte, J. Mol. Spectrosc. 216 (2002) 297-307]. This approach allows a systematic development of rovibrational interactions and makes global analyses easier to perform even for complex polyad systems. We present here an application of this model to the analysis of the bending triad, with the same set of microwave assignments and almost the same set of infrared assignments as in the previous study of Bürger et al. It appears that we need to expand our Hamiltonian to a lower degree than the “classical” one (six instead of eight) when including also the second order Coriolis interactions. Our fit does not include more parameters. Furthermore, all of them are determined and the standard deviation of the rotational transitions is twice smaller. The analysis has been performed thanks to the C_2v TDS program suite, which is freely available at the URL: http://www.u-bourgogne.fr/LPUB/c2vTDS.html.     

Probing Rb MOT in 5P3/2(F = 4) → 5D5/2(F′) transitions. In search for effective Rabi frequency

The aim of this work was to provide a simple justification and applicability limits for the concept of effective Rabi frequency being related to an average atom-field interaction in MOT. We sampled ⁸⁵Rb MOT with a weak probe beam tuned across the 5P_3/2 (F′ = 4) → 5D_5/2(F″ = 3, 4, 5) hyperfine transitions, while the 5S_1/2(F = 3) → 5P_3/2(F′ = 4) transition was driven by the red-detuned trapping beam. The probe absorption spectra were registered for a number of detunings Δ and intensities P of the trapping beam. The Autler-Townes splitting δ of the clearly dominating F′ = 4 → F″ = 5 line was the subject of analysis. The character of the space-dependent interactions of atoms with MOT fields is of a complex nature, which brings the notion of the effective Rabi frequency for MOT into challenge. However, we argue that for the range of the typical values of P and Δ, it is justified to characterize MOT with an effective Rabi frequency Ω_0eff, by using the intuitive formula , where is a mean scaling factor experimentally determined, basing on predictions of a straightforward 3-level model. We postulate that our simple procedure, providing both the value and the applicability limits of the approach, should be repeated with each new implementation of MOT (e.g., with trap beams realignment), which may change conditions experienced by cold atoms.     

DFT study of structural, electronic and vibrational properties of pure (Al2O3)n (n = 9, 10, 12, 15) and Ni-doped (Al2O3)n (n = 9, 10) clusters

The geometrical, electronic and vibrational properties of pure (Al₂O₃)_n (n = 9, 10, 12, 15) clusters and Ni-doped (Al₂O₃)_9-10 clusters are investigated by density functional theory. There are four different Ni-doped (Al₂O₃)₉ clusters and one Ni-doped (Al₂O₃)₁₀ cluster taken into account. Compared with the pure clusters, the Ni-doped (Al₂O₃)_9-10 clusters have narrower HOMO-LUMO energy gaps. The results indicate that the impurity of Ni atom is mainly responsible for the reduction of the HOMO-LUMO energy gap. One characteristic vibration band at about 1030 cm⁻¹ is found in the vibrational frequencies of the Ni-doped (Al₂O₃)_9-10 clusters, which is caused by the asymmetric Al-O-Al stretching vibration. Another band at around 826 cm⁻¹ involving the characteristic vibration of Ni-O bond is in good agreement with experimental results.