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The overtone v6 = 2 (~520 cm−1) and combination v3 = v6 = 1 (~627 cm−1) vibrational levels of chloroform CH35Cl3 have been studied by millimeter wave spectroscopy, in the regions 150-330 and 360-660 GHz, corresponding to J = 22-49 and 55-99. Like the v6 = 1 fundamental vibration, the v3 = v6 = 1 combination level has an accidentally small value of C-B-Cζ ~ −0.0029 cm−1. Hence, a global q22 resonance (Δk = Δl = ±2) connects the (k-1, l6 = -1) and (k+1, l6 = +1) sublevels. As a consequence, several resonant q12 (Δk = ±1, Δl = ∓2) crossings occur, which enabled the precise determination of the q12 interaction parameter for this state. Altogether, 1182 rotational transitions in the v3 = v6 = 1 combination level have been measured and fitted, with a standard deviation of 41.2 kHz. For the v6 = 2 (A1+E) overtone level, the combined effects of the q22 and q12 l-type interactions induce resonant crossings between the (k, l6 = 0)/(k+1, l6 = -2) pairs of levels, for a large variety of k values. Altogether, 1463 rotational transitions in the (l6 = 0) A1 sublevel and 2924 rotational transitions in the (l6 = ±2) E sublevels, together with nine perturbation-allowed transitions induced by the q12 (Δk = ±1, Δl = ∓2) interaction, have been measured and fitted, with a global standard deviation of 41.4 kHz. This resulted in the first accurate determination of the molecular constants of this overtone level.

We have recently analyzed the broadband millimeter-wave rotational spectrum of chloroform CH35Cl3 in the v3 = v6 = 1 (E symmetry) and v6 = 2 (A1 + E) excited vibrational states . In the present work, we have used the same spectrum (recorded in the 150 - 330 and 360–660 GHz spectral regions) for a rotational analysis of the v2 = 1 and v3 = 2 A1 states of the same molecule, together with a new analysis of the v5=1 (E) fundamental vibration. For the v2 = 1 state (A1, near 677 cm−1), the body of assigned data comprises 392 rotational transitions. The corresponding frequencies were fitted within an isolated-level model, with a standard deviation of 36.4 kHz, and the obtained parameters are in good agreement with the previous IR work. For the v3 = 2 vibrational state (A1, near 734 cm−1), a total of 351 rotational transitions were assigned. Their fit was improved by inclusion of a second-order Coriolis coupling term with the v5 = 1 fundamental level, providing standard deviation of 40.2 kHz. At the same time, the description of the v5 = 1 vibration (E symmetry, 776 cm−1), studied previously with high-resolution as an isolated level, was also improved. The quantitative interpretation of the newly assigned 612 MMW data and the significantly extended set of IR assignments necessitated the introduction of the second-order Coriolis interaction with v3 = 2, but more importantly of an anharmonic interaction with the v6 = 3 (A1 + A2 + E, near 782 cm−1) level. For this latter level more than 200 rotational transitions, together with several series of rovibrational transitions, could finally be assigned and included in the fit.

This paper describes a novel approach for empirical lower state assignments in complex high resolution ro-vibrational overtone spectra of molecules with low rotational constants and complex intramolecular dynamics. Methanol, CH3OH, was chosen as a representative of such molecules - it is an asymmetric top with two non-hydrogen nuclei and hindered internal rotation leading to dense and disordered rotational structure of vibrational overtone bands. We report the first rotationally resolved methanol spectra of the OH-stretch overtone 2ν1 band using sub-Doppler diode laser spectroscopy in a supersonic jet, and describe how the combination of two temperature analysis (TTA) and analysis by ground state combination differences (GSCDs) is used to reliably identify spectral lines that originate from lowest rotational states. In the first step of the analysis, the TTA was utilized to obtain a set of possible rotational assignments for each spectral line using the line intensity variation between two different temperatures in the supersonic jet (13, and 56 K respectively). Thereafter, the GSCDs were used to confirm specific lower state assignment for those spectral lines that have been identified to have low rotational ground states by the TTA. We show that the TTA pre-selection leads to fast and reliable confirmation by GSCDs and avoids false assignments due to accidental GSCD matches. The procedure yields an important subset of reliably assigned spectral lines in the complex ro-vibrational structure that provides a convenient starting point for subsequent application of traditional spectral analysis techniques.

The absorption spectrum of 13 CH 4 was recorded at two low temperatures in the icosad region near 1.38 µm, using direct absorption tunable diode lasers. Spectra were obtained using a cryogenic cell cooled at liquid nitrogen temperature (80 K) and a supersonic jet providing a 32 K rotational temperature in the 7173–7367 cm −1 and 7200–7354 cm −1 spectral intervals, respectively. Two lists of 4498 and 339 lines, including absolute line intensities, were constructed from the 80 K and jet spectra, respectively. All the transitions observed in jet conditions were observed at 80 K. From the temperature variation of their line intensities, the corresponding lower state energy values were determined. The 339 derived empirical values of the J rotational quantum number are found close to integer values and are all smaller than 4, as a consequence of the efficient rotational cooling. Six R (0) transitions have been identified providing key information on the origins of the vibrational bands which contribute to the very congested and not yet assigned 13 CH 4 spectrum in the considered region of the icosad.

International audience; Complete 12CH4 spectrum in the lower icosad spectral range (7050-7350 cm‑1) has been measured in supersonic jet at 28 K for the first time using direct absorption tunable diode laser spectrometer. Due to efficient rotational cooling only spectral lines with lower rotational quantum number J < 4 are observed in the spectrum. Spectral lines in the supersonic jet spectra have been matched with those in liquid nitrogen cooled cell spectra (81 K) and two-temperature method for empirical lower state rotational quantum number assignments has been applied to obtain accurate experimental values of the lower state rotational quantum numbers J″. Especially positions of lines with of J″ = 0 are relevant for vibrational band origin determination, as they are unambiguously the R(0) transitions of the respective bands.

The paper reports on the first high-resolution study of the weak ν6 fundamental rovibrational band of monoisotopic chloroform HC35Cl3. More than 4500 infrared transitions pertaining to the ν6 band have been assigned, spanning the rotational states J=0–95 and −73≤KΔK≤89. The new Fourier transform infrared data were combined with accurate submillimeter-wave transition frequencies in the v6=1 state measured previously [Journal of Molecular Structure 795; 2006: 157–162] in a simultaneous fit. The existence of resonance crossings due to the Δ k = ± 1 , Δ l = ∓ 2 l-type resonance in the v6=1 state, which generated perturbation-allowed rotational transitions, provided independent values of the C and Cζ6 constants. When we combined them with the wavenumbers of the fundamental ν6 band, we were able to determine accurately the axial rotational constants in the vibrational ground state C0=0.05715783 (20) cm−1, D K 0 =2.759 (63)10−8 cm−1, and even H K 0 =−8.27 (75)10−14 cm−1.

The degenerate combination level ν3 = ν5 = 1 (1669.09 cm−1, E symmetry) was investigated with high-resolution, using a Fourier-transform infrared spectrum recorded in the 1600–1800 cm−1 region, at a resolution of 0.0024 cm−1. A second spectrum, recorded in the 900–1100 cm−1, at a resolution of 0.0024 cm−1, was employed to analyze the ν3 + ν5 − ν3 hot band. The studied combination level has to be treated as a part of a polyad including two other dark states: (ν3 = 1, ν6 = 2, l6 = 0, ∓ 2) and (ν4 = ν6 = 1). The important anharmonic, Coriolis, and α-resonance interactions were extrapolated from the previously studied dyads ν5/2ν6 and ν4/ν3 + ν6. The reproduction thus achieved is quantitative for all the assigned data and provides spectroscopic parameters which are consistent with those of the system of vibrational levels ν5/2ν6.

The degenerate combination level ν 2 = ν 6 = 1 (1611.84 cm −1 , E symmetry) of DCF 3 was investigated with high-resolution using a Fourier-transform infrared spectra. They were recorded in the regions of the combination band between 1550 and 1650 cm −1 and the corresponding hot band ν 2 + ν 6 − ν 6 between 1070 and 1150 cm −1 . The analysis justifies the use of an isolated vibrational level model which yields a fully quantitative reproduction of all assigned experimental data.

The degenerate fundamental vibrational level v5 = 1 (975.51 cm−1, E symmetry) and the overtone vibrational level v6 = 2 (1004.06/1005.98 cm−1, A1 + E symmetry) were reinvestigated with inclusion of their inter-vibrational couplings: the anharmonic Fermi interaction ( Δ K = 0 , Σ t Δ l t = Δ l 5 + Δ l 6 = ± 3 ), the second-order Coriolis resonance ( Δ K = ± 1 , Σ t Δ l t = ± 1 ), and the second-order α-resonance ( Δ K = ∓ 2 , Σ t Δ l t = ± 1 ). This required significant extension of assignments to high J and K rotational states where resonant crossings due to the latter two interactions occur. The newly assigned data from Fourier-transform IR spectroscopy were combined with rotational and laser-sideband rovibrational data pertaining to the v5 = 1 vibrational level, which were used already in a previous analysis, in a fully quantitative simultaneous fit.

The parallel vibration-rotation band ν(4) of methyl isocyanide (CH(3)NC), with a band center at 944.9 cm(-1), was studied by FTIR spectroscopy between 890 and 980 cm(-1) in order to improve the ground-state rotational constants. Such improvement is essential for the scheduled studies of excited vibrational levels and their mutual anharmonic resonances occurring at higher values of the K rotational number. Ground-state combination differences generated from this band, spanning values of J/K from 0 to 85/13, were combined with rotational data from the literature and newly measured rotational transitions, extending the J/K range from 3/0 up to 31/14, and fitted simultaneously with a fully quantitative reproduction of the data. The infrared data of the ν(4) band were analyzed together with rotational data of the ν(4) = 1 level, spanning values of J/K from 4/0 to 14/12. The fit in the approximation of an isolated vibrational state, with the transitions perturbed by weak local resonances excluded, yields reproduction of the data within experimental uncertainties.

International audience; The infrared spectrum of the fluorocarboxyl radical, FCO2, was recorded at high resolution (0.0035 cm−1) in the 600−1400 cm−1 region on a Bruker IFS 120 HR Fourier transform spectrometer of the University of Wuppertal. The analysis of the A-type nu2 band of FCO2 (CF stretching mode) centred at 970.208 cm−1 was performed taking use of the ground state parameters achieved by Kolesnikova et al. [Kolesnikova L, Varga J, Beckers H, Simeckova M, Zelinger Z, Nova Striteska L, Kania P, Willner H, Urban S. J. Chem. Phys. 128, 2008, pp. 224302-1-8]. For the FCO2 radical, the nu2 transitions are, in principle, splitted into two spin - rotation subcomponents corresponding to J= N +/-1/2. However the spin- rotation parameters in the 21 vibrational state have values similar to those of the ground state, and spin-rotation doublings are observable only for the weaker transitions involving medium Ka or Kc values in the P and R branches. The nu2 fundamental band is weakly perturbed by the 2nu5 dark overtone band at 965.4 cm−1 and the 21 and 52 energy levels of FCO2 are coupled through Fermi type resonances. The final energy level calculation was performed accounting both for the spin -rotation interaction within the 21 and 52 vibrational states, and the 21 52 Fermi-type resonances.

The high-resolution infrared spectrum of deuterated fluoroform (DCF3) was studied in the 700 and 1200 cm−1 regions, with the aim of assigning and analyzing the ν4 CF3 asymmetric stretching vibration. The Fermi-type anharmonic coupling between the ν4 = 1 and ν3 = ν6 = 1 rovibrational levels, already mentioned in an early work of Ruoff et al. [Spectrochimica Acta Part A 31A (1975) 1099–1100], was studied here for the first time under high resolution. Assignments in the ν3 + ν6/ν4 band system were confirmed and extended by the identification of the ν3 + ν6 − ν6 and ν4–ν6 bands in the 700 cm−1 region, the latter being enhanced near the Fermi crossings of the studied levels. Data from both the hot and difference bands were included in the analysis. The close separation of the studied vibrational levels of about 14.8 cm−1 produces a large variety of resonance crossings which involve levels with Δ k = 0 , … , 6 . Besides the Fermi ( Δ k = 0 , Δ l = 0 ) and Coriolis ( Δ k = ± 1 , Δ l = ∓ 2 ) resonances, they were accounted for by inclusion of additional higher-order ( Δ k = ± 2 , Δ l = ± 2 and Δ k = ± 3 , Δ l = 0 ) interaction terms between the vibrational states. The least-squares fit of more that 16,000 vibration-rotation transitions provides a quantitative reproduction of data in all bands.

Having in mind their potential quantum physical applications, we classify all geometric hyperplanes of the near hexagon that is a direct product of a line of size three and the generalized quadrangle of order two. There are eight different kinds of them, totalling to 1,023 = 210 − 1 = |PG(9, 2)|, and they form two distinct families intricately related with the points and lines of the Veldkamp space of the quadrangle in question.

The study of vibration resonance physics in propyne is based on experimental measurements of about 600 new rotational transitions between 495–590 and 700–760 GHz in excited vibrational levels v 5 = 1, v 8 = 1, v 10 = 3 and v 9 = v 10 = 1 with vibrational energies around 1000 cm −1 . The limits to the assignments and analysis were imposed by as yet unresolved anharmonic resonances with the states of the next higher polyad of levels lying above 1200 cm −1 , which affect the rotational states involved in transitions that would be measurable with non-vanishing intensities. Vibration–rotation spectra pertaining to the levels in question were studied in the regions 880–1150 cm −1 (the ν 5 and ν 8 fundamental bands), 550–750 cm −1 (the v 9 = v 10 = 1 ← v 10 = 1 hot bands) and 250–400 cm −1 (the v 10 = 3 ← v 10 = 2 “superhot” bands). A simultaneous least-squares fit of both types of data provides their reliable but in the case of accurate rotational data not always fully quantitative reproduction.

The rotational spectrum of NSF3 in the ground and v5 = 1 vibrational states has been investigated in the centimeter- and millimeter-wave ranges. R-branch (J + 1 (kl = +1, A2) with J (kl = -2, A-), (kl = +4, A+) (kl = +1, A-), (kl = +2) (kl = -1), (kl = +3) (kl = 0), (kl = +2) (kl = -3), and (kl = +3) (kl = -3). The various aspects of the regional resonances are discussed in detail. An accidental near-degeneracy of the kl = 0 and kl = -4 levels at J = 26/27 led to the observation of perturbation-allowed transitions following the selection rule delta(k-l) = +/-6 with (kl = +2) (kl = -4). A corresponding near-degeneracy between kl = -1 and kl = -3 levels at J = 30/31 led to the detection of similar transitions, but with (kl = +3) (kl = -3). In the range 230-480 GHz, R-branch rotational transitions have been measured by absorption spectroscopy up to J = 49 in the ground-state and up to J = 50 in the v5 = 1 vibrational state. The transition frequencies have been analyzed using various reduced forms of the effective Hamiltonians. The data for the v5 = 1 vibrational state have been fitted successfully using two models up to seventh order with delta k = +/-3 interaction parameters constrained (dt constrained to zero, and epsilon to zero or to the ground-state value). On the other hand, reductions with the (delta k = +/-1, deltal = -/+2) interaction parameter q12 fixed to zero failed to reproduce the experimental data since the parameters defining the reduction transformation do not arise in the correct order of magnitude. The ground-state data have been analyzed including parameters up to fourth order constraining either parameters of the delta k = +/-3 interactions to zero (reduction A), or of the delta k = +/-6 interactions to zero (reduction B). The unitary equivalence of the different parameter sets obtained is demonstrated for both vibrational states.

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type $\widetilde{D}_n$, $4 \leq n \leq 8$, it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG$(3,2)$. Proper labelling of the vertices of the diagram (for $4 \leq n \leq 7$) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG$(3,2)$. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space $W(3,2)$ of the PG$(3,2)$ whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin-Peres magic square. In the case of $n=8$, a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG$(5,2)$, the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG$(3,2)$ are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin-Peres square. Moreover, save for $n=4$, each Veldkamp space is also endowed with some `exceptional' point(s). Interestingly, two such points in the $n=8$ case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix $Y$., Comment: 19 pages, 17 figures

We have developed advanced graphical software for significantly accelerated and verified assignments of rotationally resolved rovibrational molecular spectra. The program combines the traditional approach of Loomis–Wood diagrams, which are used to visualize spectral branches and facilitate their identification, with the power of interactive combination difference checking, which provides an immediate verification of correct assignments of quantum numbers to spectral lines. Assignment procedures and other tools of the program are designed in a fully graphical Windows environment to accelerate the usually tedious assignment work and make the use of the program highly intuitive. The use of an appropriate rovibrational Hamiltonian for fitting the assigned transitions is made easy by the flexible, user-programmed interfaces for exporting assigned data directly in the needed format and importing rovibrational energies back to the assignment program. The program is designed for the MS Windows operating systems and is available upon request with a full documentation on http://www.lww.amu.edu.pl .

The high-resolution infrared spectra of trifluorophosphine (PF3) were reinvestigated in the ν4 fundamental region near 350 cm−1, and around 690 cm−1, with the aim to provide a necessary reassignment of the 2±2 sublevel of the v4 = 2 overtone level. The present paper reports on the first complete study of both sublevels of v4 = 2 (of A1 and E symmetry, corresponding to l4 = 0 and ±2, respectively), through the high-resolution analysis of the overtone 2 ν 4 0 band and the 2 ν 4 ± 2 - ν 4 ± 1 hot band. The assignments of the latter were corrected and extended, spanning the rotational states J ⩽ 82 and −80 ⩽ K″ · ΔK ⩽ 48. These new infrared assignments in the v4 = 2 state were combined with accurate infrared, radiofrequency, centimeter-, millimeter- and submillimeter-wave data of the v4 = 1 level (Thiessen et al., 2000), together with rotational data in the ground vibrational state (Cotti et al., 1995), in a simultaneous fit. The existence of resonance crossings due to a Δk = ±1, Δl = ∓2 l-type resonance in the v4 = 1 state, which generated perturbation-allowed transitions, provided independent values of the C4 and Cζ4 constants. Combining these rotational transitions with the wavenumbers of the ν4 fundamental band enabled us to determine accurately the C0 axial ground state constant. Moreover, the assignment of a few, very weak rRK transitions in the 2 ν 4 - 2 overtone band and their inclusion in the global least-squares fit allowed also the first accurate experimental determination of D K 0 . The obtained results are (in cm−1): C0 = 0.159970241(29) and D K 0 =1.80457(49) × 10−7. Quadratic, cubic, and semidiagonal quartic force fields of PF3 have been calculated at the CCSD(T) level of theory employing a variety of large correlation-consistent basis sets. These force fields have been used to evaluate spectroscopic constants which are generally found to be in very good agreement with experiment. The present best estimate of the re structure of PF3 based on an explicitly correlated coupled-cluster approach (CCSD(T)-F12b) is almost identical with its latest experimental counterpart.

The two lowest vibrational levels of chloroform, HC 35 Cl 3 , v 6 =1 and v 3 =1, have been studied by submillimeter wave spectroscopy in the regions 255–350 and 530–635 GHz, corresponding to the ranges of J =38–53 and 80–96. The v 6 =1 level has an accidentally small value of ( C − B − Cζ )∼−0.0029 cm −1 , which lies in the origin of a global q 22 resonance (Δ k =±2, Δ l =±2). As a consequence of this resonance, the levels of the v 6 =1 vibrational state exhibit several resonant crossings due to the q 12 l -type interaction (Δ k =±1, Δ l =∓2), which induce measurable perturbation-allowed rotational transitions. Analysis of these data enabled an independent determination of the C 6 and Cζ 6 constants, which is not a typical case in purely rotational studies of symmetric top molecules. In the spectra of the v 3 =1 level, the splittings of K =3 transitions have been resolved, making the determination of the h 3 splitting term possible. Altogether there have been measured almost 800 rotational transitions in the v 6 =1 vibrational state and more than 250 transitions in the v 3 =1 vibrational state. These have been reproduced quantitatively with standard deviations of 43 and 32 kHz in the v 6 =1 and v 3 =1 levels, respectively, leading to considerable improvement of determination of molecular constants of the studied levels.

The direct l -type resonance spectrum of CF 3 CCH in the vibrational state ν 10 = 2 has been measured by means of waveguide microwave Fourier transform spectroscopy in the range 8–26 GHz. Two types of direct l -type resonance transitions induced by the (Δ k = ±2, Δ l = ±2) interaction could be observed: 262 transitions following the Δ J = 0, Δ k = Δ l = 2 selection rule covering values of J = 17–64 and G = ∣ k − l ∣ from 2 to 15, and 75 transitions following the Δ J = 0, Δ k = Δ l = 4 selection rule covering values of J = 44–70 and G up to 3. The strong (2, 2) resonance furthermore allowed the observation of A 1 –A 2 splittings of the k = l = ±2 states from J = 63–70. The transitions with G = 3 showed splittings due to the (4, −2) and (0, 6) interactions. The corresponding energy level systems and part of the Hamiltonian matrix are discussed. Strong perturbations due to Δ( k − l ) = 3 interactions coupling the states k = ±1, l = ±2 and k = ±4, l = ±2 made possible the observation of perturbation-allowed transitions with selection rule k = ±1, l =± 2 ↔ k = 0, l = ±2. Additionally, the J = 2–1 and 3–2 rotational transitions have been measured. A multiple fitting analysis has been performed in which the experimental data have been fitted using five reduced forms of the effective Hamiltonian as proposed by Sarka and Harder [J. Mol. Spectrosc. 197 (1999) 254]. Parameters up to sixth order have been determined including the axial rotational constant A for both values of ∣ l ∣ and the unitary equivalence of the determined parameter sets has been demonstrated.

Fourier-transform infrared (IR) spectra between 1235 and 1680 cm−1 and millimeter wave spectra between 198 and 455 GHz of monoisotopic 13CH3 37Cl have been recorded and analysed simultaneously, taking into account all Coriolis, α-resonance, and l-type interactions in the polyad of the ν2=1, ν5=1 and ν3=2 levels. Several α-resonances (Δ k = ±2,Δ l = ∓ 1) generating perturbation-allowed transitions have been assigned both in the rovibrational and rotational spectra. This enabled us to determine accurately and independently A 0= 5.2057128 (72) cm−1, = 8.3968 (105)×10−5 cm−1, and even = 3.95 (38)× 10−9 cm−1. With 52 upper state parameters varied, more than 5300 IR wavenumbers and 162 rotational frequencies pertaining to the excited vibrational states were fitted within their experimental accuracy. The textbook example of a first-order Coriolis resonant crossing between the A and E fundamental vibrational levels (ν2=1 and ν5=1) is discussed in detail with implications for the structures of the corresponding pa...

More than 700 new pure rotational transitions (Δν=0, ΔJ=1, Δk=0) pertaining to the vibrational states ν 10=1, 2 and ν 9=l have been recorded between 85 and 922GHz corresponding to 4≤J″≤53. In the case of ν 10=1, high-K transitions could be recorded up to Kl 10=−17 and +19, respectively, only limited by the vanishing intensities of these lines involving highly excited states. Similarly, the ν 9=1+1 subbranch could be followed up to K=18. Investigations of the other subbranches were limited to K values below those severely perturbed by Fermi interactions with vibrational states of the next tetrad approximately 350cm−1 higher in vibrational energy. Moreover, perturbations in the K=8 stack of ν 5=1 could be traced to a intervibrational q 12 resonance with ν 10=2−2, K=9. Several highly perturbed transitions, including transitions between the two vibrational states, have been recorded in the submillimetre range and fit to within experimental uncertainties. The intervibrational transitions also permitte...

Let $S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2)$ be a Segre variety that is $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of $S_{(N)}$, let us call the triple $\{H', H'', \overline{H' \Delta H''}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2 \leq N \leq 4$, that the properties of geometric hyperplanes of $S_{(N)}$ are fully encoded in the properties of Veldkamp {\it lines} of $S_{(N-1)}$. Using this property, a complete classification of all types of geometric hyperplanes of $S_{(4)}$ is provided. Employing the fact that, for $2 \leq N \leq 4$, the (ordinary part of) Veldkamp space of $S_{(N)}$ is $PG(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric $\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2)$ that contains the $S_{(N)}$ and is invariant under its stabilizer group; in the $N=4$ case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type $LG(4,8)$, to the set of 2295 maximal subspaces of the symplectic polar space $\mathcal{W}(7,2)$., Comment: 16 pages, 8 figures and 7 tables

Almost 300 new rotational transitions within the fundamental vibrational level v 10 =1 of propyne have been measured in selected regions between 495 and 925 GHz spanning the quantum numbers 28≤ J ≤54 and 0≤ K ≤16. The accuracies are mostly between 10 and 20 kHz. In addition, the J ″=4 and 5 transitions near 85 and 103 GHz have been remeasured. Simultaneous analyses with refined rovibrational data have been performed, showing that even this lowest and seemingly isolated vibrational level needs a global treatment when high K transitions are involved. The global model with the v 10 =1 level coupled to the next higher cluster of levels, v 10 =2/ v 9 =1, by Fermi and Coriolis resonances is necessary for a quantitative reproduction of both the rovibrational and rotational data within their experimental uncertainties. Included are also improved ground state spectroscopic parameters from a fit of previous pure rotational data and Δ k =3 ground state combination loops as well as additional data obtained in course of the present study.

The υ6=2 vibrational state of the main isotopomer of trifluorosilane, 28SiHF3, has been investigated in the centimeter- and millimeter-wave ranges. Rotational spectra following the Δ J=1, Δk=Δ l=0 selection rule have been measured up to J=24 and K=23 and for both values of ∣l∣. Two types of direct l-type resonance transitions induced by the (Δ l=Δk=±2) interaction could be observed by means of waveguide Fourier transform microwave spectroscopy in the range 8–26 GHz: 252 transitions following the Δ J=0, Δl=Δk=±2 selection rule covering values of J=7–39 and G=∣k-l∣ from 1 to 18 and 90 transitions following the Δ J=0, Δl=Δ k=±4 selection rule covering values of J=17–52 and G from 1 to 4. Due to the strong (2,2) resonance, 18 A1–A2 splittings of the k=l=±2 states from J=36–53 could also be observed. Accidental near-degeneracies lead to strong perturbations due to Δ (k-l)=±3 interactions, enabling the observation of perturbation-allowed transitions with selection rules k=±3(l=±2)↔±4(±2), ±2(±2), A+↔ ±1(∓2), A− and ± 1(0)↔± 6(±2). In a multiple-fit analysis the experimental data have been refined using five reduced forms of the effective Hamiltonian as proposed by Sarka and Harder [J. Mol. Spectrosc.197, 254–261 (1999)]. Parameters up to seventh order have been determined including the axial rotational constant C for both values of ∣l∣ and the vibrational separation of the ∣l∣=0 and 2 states. The unitary equivalence of the determined parameter sets has been demonstrated up to fifth order. Differences of the rotational constants in the various parameter sets have been explained by the theory of reduction. Sign relations of the fitted parameters and general features of the direct l-type resonance spectrum in a υt=2 level are discussed.

The ν 1 (A 1 , 1578.31 cm −1 )/ν 4 ( E , 1615.17 cm −1 ) Si–D stretching dyad of D 3 SiF has been studied by FTIR spectroscopy with a resolution of 2.4×10 −3 cm −1 . Only weak interactions of Coriolis (Δ K =±1, Δl=±1) and α resonance (Δ K =±2, Δl=∓1) type between ν 1 and ν 4 , and of l (2,−4) type within ν 4 , were revealed. However, the v 1 =1 and v 4 =1 levels were found to be severely perturbed by the v 3 = v 5 =1 ( E , 1590.37 cm −1 ) and v 2 = v 3 =1 ( A 1 , 1604.25 cm −1 ) states. These perturbations are observable only near level crossings involving strong Coriolis and α interactions. The energy structure within these perturbers is severely complicated by strong Coriolis and α resonances and by l (2, 2), l (2,−1), and l (2,−4) interactions as already revealed by the ν 2 ( A 1 , 710.16 cm −1 ) and ν 5 ( E , 701.72 cm −1 ) fundamentals. Interactions of the perturbing states with the ν 1 /ν 4 dyad are particularly evident in local crossings. In total, 12 transitions belonging to the dark states and 68 perturbation-allowed transitions within the ν 1 /ν 4 dyad have been detected among the more than 5000 transitions that have been assigned for the ν 1 /ν 4 dyad, with J max and K max of 50 and 30, respectively. Altogether about 85% of the assigned transitions were fitted with a standard deviation of 0.221×10 −3 cm −1 , leading to 61 parameters of the interacting polyad.

The high-resolution infrared spectrum of DCF 3 was investigated in the 2000 cm −1 region, with the aim of assigning and analyzing the overtone bands 2ν 5 ( A 1 + E ), as well as the combination band ν 2 + ν 5 ( E ). The present paper reports on the first high-resolution study of the v 2 = v 5 = 1 and v 5 = 2 levels, through the analysis of the combination ν 2 + ν 5 and the overtone 2 ν 5 ∓ 2 bands respectively. This analysis can be thought of as a necessary step toward the reinvestigation of the complicated vibrational pattern in the CD stretching fundamental region.

High-resolution (ca. 5 x 10(-3) cm(-1)) infrared spectra of monoisotopic samples of F(35)Cl(16)O(3), F(37)Cl(16)O(3), F(35)Cl(18)O(3), and F(37)Cl(18)O(3) have been recorded in the region of the nu(6) fundamental band. This band is centered at 405.6055, 405.5097, 390.5406, and 390.4984 cm(-1), respectively. About 1000 nonzero-weighted transitions were fitted for each isotopomer with an isolated band model, obtaining a s.d. on the residuals close to the experimental precision. Three different reductions of the Hamiltonian have been tested and found almost equally good, although FClO(3) is a nearly spherical top molecule. Each includes the l(2, 2) interaction and either a d(t) DeltaK = +/-3 interaction term (reduction Q) an epsilon(t) DeltaK = +/-3 interaction term (reduction QD), or an l(2, -1) interaction (reduction D). Good agreement between the theoretical predictions for these reductions and the experimental data is observed. Copyright 1999 Academic Press.

A total of 4892 transition wavenumbers of the nu(2), nu(3), nu(5), and nu(6) fundamental bands of H(3)(12)CF and 1469 frequencies of the rotational transitions in the v(2) = 1, v(3) = 1, v(5) = 1, and v(6) = 1 vibrational states have been fitted simultaneously by taking explicitly into account various Coriolis interactions, l-type interactions, and alpha-resonance terms between and within the vibrational states. The standard deviation of the fit using 71 parameters was 8.10 x 10(-5) cm(-1) for the infrared data and 0.198 MHz for the rotational data. This study clearly shows that in order to obtain a really quantitative fit of the highly precise data of vibration-rotational and rotational spectroscopy, it is necessary to fit all the data simultaneously using a variational approach. The deperturbed values of the upper state rotational constants differ significantly from those obtained previously in fitting the nu(3) and nu(6) bands as isolated bands and the nu(2)/nu(5) bands by a variational approach. This is important for the precise determination of the equilibrium structure of methyl fluoride (J. Demaison, J. Breidung, W. Thiel, and D. Papousek, Struct. Chem., in press). Besides this, the results can be used in further studies of the reduced Hamiltonians for more complicated systems of interacting vibrational levels. Copyright 1999 Academic Press.

High-resolution FTIR spectra of the ν 1 bands of both 14 ND 3 and 15 ND 3 species were recorded with the Bruker spectrometer in Giessen. The spectra were assigned using the interactive program combining the Loomis-Wood technique with ground state combination difference checks. New sets of molecular constants of the ν 1 vibrational levels of both isotopomers are presented. The ν 1 rovibrational energies derived from the frequencies of the assigned transitions have made possible calculations of the Raman transition frequencies required for a simulation of the Coherent Anti-Stokes Raman Scattering (CARS) spectra. A satisfactory agreement of the simulated CARS spectra with the experimental ones was achieved.

Absolute line intensities of sixteen rovibrational transitions in the ν4 band of 14NH3 were measured around 1800 cm−1 using a tunable diode laser spectrometer with a sweep modulation of the laser frequency. Individual laser scans over the line region were recorded to disk storage separately and the spectrum accumulation was performed off-line using a computer to reduce a significant line broadening because of a lower accuracy of the hardware on-line accumulation. From integrated absorption coefficients obtained by the numerical analysis of the measured line profile, the effective transition moments of individual rovibrational lines were derived. Using the second order perturbation theory of the transition moment of rovibrational transitions of the symmetric top molecules, the effective vibrational transition moments for the ν4 fundamental band of 14NH3 were estimated.

High-resolution FTIR spectra of monoisotopic D 3 Si 35 Cl have been recorded in the regions 440–590 cm −1 (ν 3 /ν 6 ) and 900–1100 cm −1 (2ν 6 /ν 3 + ν 6 /2ν 3 ). A detailed rovibrational study was done for the 2ν 0 6 , 2ν ∓2 6 , and 2ν 3 overtone bands and for the 2ν ±2 6 –ν ±1 6 , 2ν 0 6 –ν ∓1 6 , and (ν 3 + ν ±1 6 )–ν ±1 6 hot bands. For the first time the interactions occurring between the v 6 = 2, v 3 = v 6 = 1, and v 3 = 2 levels of any trideuterated silyl halide have been analyzed. The (ν 3 + ν ±1 6 )–ν ±1 6 hot band served to obtain accurate energies of the v 3 = v 6 = 1 level, the ν 3 + ν 6 combination band not being detectable on our spectra. The first experimental determination of A 0 and D 0 K for this molecule was undertaken by a well-established method, using the ν 6 , 2ν ±2 6 –ν ±1 6 , and 2ν ∓2 6 bands. Ground state energy differences Δ K ( J ) = E 0 ( J , K ) − E 0 ( J , K − 3) were calculated for K values from 2 to 16. By a least-squares fit of 163 such differences, the A 0 and D 0 K values thus obtained were (in cm −1 ): A 0 = 1.4278230(8) and D 0 K = 5.3916(31) × 10 −6 .

The degenerate combination levels of E symmetry v 3 = v 4 = 1 (1900.21 cm −1 ) and v 3 = 2, v 6 = 1 (1889.30 cm −1 ) were investigated for the first time with high-resolution, using a Fourier-transform infrared spectrum recorded in the 1600–2000 cm −1 region, at a resolution of 0.0024 cm −1 . These combination levels strongly interact through a Fermi-type anharmonic coupling, which plays an essential role in the studied system. Besides the Fermi-interaction, the other important Coriolis (Δ k = ±1, Δ l = ∓2) and higher-order (Δ k = Δ l = ±2) inter-vibrational interactions were also considered, in a similar manner as in the lower ν 3 + ν 6 / ν 4 dyad [P. Pracna et al., J. Mol. Spectrosc. 259 (2010) 1–10]. The fitted wavenumbers of the rovibrational transitions of the ν 3 + ν 4 and 2 ν 3 + ν 6 bands provide a set of molecular parameters which ensure an accurate reproduction of the spectrum. Moreover, these molecular parameters show good agreement with those of the lower system ν 3 + ν 6 / ν 4 .

The microwave and submillimeter wave spectra of propyne between 17 and 358 GHz were measured and the rotational transitions in thev8= 1 excited vibrational state of the CH3rocking vibration were assigned. About 1050 wavenumbers of the ν8vibration–rotation fundamental band and about 600 wavenumbers of the ν5fundamental band of the[formula]stretching vibration were assigned from the infrared spectrum between 910 and 1130 cm−1which was used previously (G. Graneret al., J. Mol. Spectrosc.161,80–101 (1993)) in the analysis of the combinationv9=v10= 1 and thev10= 3 overtone levels (ν9being the[formula]bending and ν10the[formula]bending vibrations). The rovibrational and rotational data corresponding to the two fundamental levels were analyzed simultaneously in least-squares fits using a model which treats together all the vibrational levels in the region around 1000 cm−1with their strong anharmonic and vibration–rotation resonances. The refined parameters reproduce the infrared and submillimeter wave data of thev5= 1 level with standard deviations of 0.32 × 10−3cm−1and 59 kHz, respectively, while for thev8= 1 level the standard deviations were 0.41 × 10−3cm and 290 kHz. The refined parameters of the combination and overtone levels provide reliable predictions for future submillimeter wave studies.

A high-resolution FT infrared spectrum of H3Si79Br (resolution 0.0022 cm−1) in the region 830-1050 cm−1 was used to analyze the fundamental bands ν2 (A1) and ν5 (E). A total of 4392 transition wavenumbers for ν2 and ν5 have been fitted, together with 632 energy values for the ν3 = 2 level, with a standard deviation of the fit of 0.123 × 10−3 cm−1. A model was used taking into account the strong Coriolis x, y resonance between ν2 and ν5, and also three local resonances: (i) an accidental Coriolis-type resonance of ν5 with 2ν3 (crossing of (kl = −13, ν5 = 1) and (K = 14, ν3 = 2) for J′ between 16 and 17); (ii) an l(±2, ∓1) resonance of the (kl = 2, ν5 = 1) and (kl = −3, ν5 = 1) levels, with a crossing for J′ between 58 and 59; and (iii) an accidental l(∓1, ±2) resonance between ν2 (K = 3) and ν5 (kl = 1, A+ and A−). Some "forbidden" transitions, due to this last resonance, were observed and included in the fit. The I-doubling constant q5 and the Coriolis resonance parameters being strongly correlated, these two resonances cannot be treated independently. Hence, q5 was fixed at zero and a value of 0.09314238(58) cm−1 was obtained for Bζ25.

The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7,2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7,2), the seven-dimensional projective space of order two. The quadric is selected in such a way that it contains all 135 symmetric elements of the group. Under such circumstances, the third element on the line defined by any two points of an ovoid is skew-symmetric, as is the nucleus of the conic defined by any three points of an ovoid. Each ovoid thus yields 36/84 elements of the former/latter type, accounting for all 120 skew-symmetric elements of the group. There are a number of notable types of ovoid-associated subgeometries of the group, of which we mention the following: a subset of 12 skew-symmetric elements lying on four mutually skew lines that span the whole ambient space, a subset of 15 symmetric elements that corresponds to two ovoids sharing three points, a subset of 19 symmetric elements generated by two ovoids on a common point, a subset of 27 symmetric elements that can be partitioned into three ovoids in two unique ways, a subset of 27 skew-symmetric elements that exhibits a 15 + 2 x 6 split reminding that exhibited by an elliptic quadric of PG(5,2), and a subset of seven skew-symmetric elements formed by the nuclei of seven conics having two points in common, which is an analogue of a Conwell heptad of PG(5,2). The strategy we employed is completely novel and unique in its nature, as are the results obtained. Such a detailed dissection of the geometry of the group in question may, for example, be crucial in getting further insights into the still-puzzling black-hole-qubit correspondence/analogy., 18 pages, 13 figures; Version 2 - version accepted in J. Phys A: Math. Theor

Millimeter-wave and submillimeter-wave spectra of the rotational transitions in the v 2 = 1 and v 5 = 1 vibrational states of 13 CH 3 F were measured up to 462 GHz. Far-infrared spectra of these transitions were measured in the region 30-65 cm −1 with a resolution of 0.0019 cm −1 using an interferometric spectrometer Bruker IFS 120 HR. Almost 700 lines including 14 lines of the Δ( k − l 5 ) = ±3 perturbation-allowed transitions in the v 5 = 1 state were assigned. Both vibrational states are strongly perturbed, especially by the x , y -Coriolis resonance and "2, −1" l -type interactions. The corresponding spectroscopical parameters were determined by a nonlinear least-squares fit to the experimental data.

The vibration-rotational bands ν 2 and ν 5 of gaseous fluoromethane 13 CH 3 F were measured in the region 1200-1700 cm −1 at an unapodized resolution of 0.0041 cm −1 using an interferometric spectrometer Bruker IFS 120 HR. Almost 3000 lines were assigned including 381 lines of the Δ k = ±2 perturbation-allowed transitions to the doubly degenerate vibrational state v 5 = 1. Combination differences were generated for the ground state from the allowed and "forbidden" transitions of the ν 2 and ν 5 bands. They were used in a simultaneous fit with the previously measured frequencies of the ground state rotational transitions and the combination differences generated from the ν 3 band (D. Papousek, Z. Papouskova, Yen-Chu Hsu, P. Pracna, S. Klee, M. Winnewisser, and J. Demaison, J. Mol. Spectrosc. 159, 62-68, 1993) to improve the values of the ground state spectroscopical parameters of 13 CH 3 F. The upper state parameters were determined by fitting simultaneously the infrared data with the frequencies of the rotational transitions in the v 2 = 1 and v 5 = 1 levels (preceding paper), using a variational approach with the constrained values of the ground state parameters.

The Fourier transform spectrum of the vibration-rotational bands ν 2 and ν 5 of gaseous fluoromethane 13 CH 3 F was measured at an unapodized resolution 0.0041 cm −1 . Among the assigned lines, 102 belonged to the Δ k = ±2 perturbation-allowed transitions to the doubly degenerate vibrational state v 5 = 1. Combination differences were generated for the ground state of 13 CH 3 F from the wavenumbers of the allowed and forbidden vibration-rotational transitions to the v 5 = 1 level. They were used in a simultaneous least squares fit with the previously measured microwave, submillimeterwave, and far infrared transition frequencies of the pure rotational transitions in the ground state and with the combination differences generated from the allowed transitions to the v 3 = 1 state of 13 CH 3 F (Pracna, Klee, Winnewisser, Papousek, Yen-Chu Hsu, and Hann-Sen Chen, J. Mol. Spectrosc. 156, 217-226 (1992). This made it possible to determine precisely for the first time A 0 and D 0 K for 13 CH 3 F: A 0 = (5.1824246 ± 0.0000015) cm −1 , D 0 K = (70.912 ± 0.033) × 10 −6 cm −1 .

International audience; An improved procedure for accurate determination of empirical lower state rotational quantum numbers from molecular absorption spectra is demonstrated for methane. We combine the high resolution absorption spectra in the 7070–7300 cm−1 frequency range obtained in liquid nitrogen cooled cryogenic cell (T = 81 K) and in supersonic planar jet expansion (TR = 25 K). Empirical lower state energies of 59 transitions are determined from the ratio of the absolute absorption line strengths at 25 and 81 K. The procedure relies on the realistic description of rotational state populations in the supersonic jet expansion where non-equilibrium nuclear spin isomer distributions are generated due to the rapid cooling. The accuracy of the experimental determination of the lower state energies with J ≤ 3 is found to considerably improve the results of the same approach applied to spectra at 296 and 81 K. The 59 transitions with determined lower J values provide a good starting point for the theoretical interpretation of the highly congested icosad region of methane. In particular, the centres of nine vibrational bands are estimated from the transitions with J = 0 lower state rotational quantum number.

It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomorphic to PG(5,2). Since the GQ(2,4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2,2)s, the 63 points of PG(5,2) split into two families; 27 being represented by perp-sets and 36 by GQ(2,2)s. The 651 lines of PG(5,2) are found to fall into four distinct classes: in particular, 45 of them feature only perp-sets, 216 comprise two perp-sets and one GQ(2,2), 270 consist of one perp-set and two GQ(2,2)s and the remaining 120 ones are composed solely of GQ(2,2)s, according as the intersection of two distinct hyperplanes determining the (Veldkamp) line is, respectively, a line, an ovoid, a perp-set and a grid (i. e., GQ(2,1)) of a copy of GQ(2,2). A direct "by-hand" derivation of the above-listed properties is followed by their heuristic justification based on the properties of an elliptic quadric of PG(5,2) and complemented by a proof employing combinatorial properties of a 2-(28, 12, 11)-design and associated Steiner complexes. Surmised relevance of these findings for quantum (information) theory and the so-called black hole analogy is also outlined., Comment: 6 pages, 2 figures, 1 table; v2 - substantially extended, another proof of the isomorphism furnished, a coauthor and 11 references added

It is shown that the $E_{6(6)}$ symmetric entropy formula describing black holes and black strings in D=5 is intimately tied to the geometry of the generalized quadrangle GQ$(2,4)$ with automorphism group the Weyl group $W(E_6)$. The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ$(2,4)$. Different truncations with $15, 11$ and 9 charges are represented by three distinguished subconfigurations of GQ$(2,4)$, well-known to finite geometers; these are the "doily" (i. e. GQ$(2,2)$) with 15, the "perp-set" of a point with 11, and the "grid" (i. e. GQ$(2,1)$) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non- commutative labelling for the points of GQ$(2,4)$. For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares -- objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the $E_{7(7)}$ symmetric entropy formula in D=4 by observing that the structure of GQ$(2,4)$ is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order two, featuring 27 points located on 9 pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the D=5 entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying non-commutative geometric structure based on GQ$(2,4)$., 17 pages, 3 figures, v2 a new paragraph added, typos corrected