Determination of rotation constants

By far the most important use of rotation spectra (including spectra containing rotational fine structure) is in the measurement of rotation constants, which can be used to define moments of inertia and, hence, give 

For a linear molecule, there is only one rotation constant, which can be measured from the pure rotation spectrum, from a vibration-rotation band or from an electronic spectrum. As the fine positions depend to some extent on other parameters as well as Bq, it is necessary to measure more than one fine position to obtain 

For a symmetric top, only B can be determined easily from a pure rotation spectrum or a vibration- -otation band in absorption, and the same problem can arise. This is described in more detail in the on-line supplement for Chapter 7. It is also possible in principle to determine A for a symmetric top if the Raman spectrum is studied with sufficient resolution, but this has been done only for a few simple molecules. In other cases, A has been found from high-resolution IR spectra by making use of a breakdown of the simple rotational selection mles arising from perturbations, or couplings between close-lying levels. However, these occur only by chance in particular systems. 

For an asymmetric top, all three rotation constants can in principle be determined from pure rotation spectra or from vibration-rotation spectra, but the accuracy of some of the constants might be low even though they are derived from very precise frequencies measured in the microwave spectrum. This arises in part because of the problems of centrifugal distortion, but also because the measured line positions could well depend only weakly on one of the rotation constants, which is then correspondingly uncertain. 

The rotation constant for a spherical top can only be determined from the forbidden pure rotation spectrum or from a detailed analysis of the fine stmcture of a vibration-rotation band. An analysis based on the use of combination differences (differences between observed transition energies involving the same upper-state level but different lower-state levels) is most effective in this case. 

---

With a series of small radicals, the analysis of rotation lines has resulted in determination of rotational constants. Some of those which are employed in calculations of molecular geometry (91) are summarized in Table IV. 

The most important use of rotational spectroscopy is the determination of rotation constants. A problem besetting rotational spectroscopy is centrifugal distortion however, it is possible to take account of this if precise data over a range of quantum munbers is available. 

Both CO2-HF and CO2-HCI complexes were first examined in the gas phase by Klemperer and co-workers, using the molecular beam electric resonance technique, with the radio-frequency and microwave spectra of CO2-HF and CO2-HCI indicating nearly linear, hydrogen-bonded structures [35, 36], Accurate determinations of rotational constants allowed the separations between the centers-of-mass of each sub-unit to be obtained, and assuming that neither CO2 nor HX underwent intramolecular change, the O—H bond lengths could be estimated for each complex. A comparison of CO2-HF and CO2-HCI showed the O—H separation in the former to be 10% less than in the latter. Equilibrium structures for CO2-HF and CO2HCI are shown in Fig. 10, and Table 1 lists structural parameters for all known CO2-HX complexes. 

Infrared and Raman spectra contain information about what compound(s) could be present in a sample and about the relative amounts of such constituents, as well as giving insights into the stmctures of compounds. Determination of rotation constants from high-resolution spectra of gases is described in Section 7.3.2, and derivation of force constants is discussed in Section 8.9. Absolute quantitative measurements of concentrations, such as can be obtained by integration of NMR spectra, are not possible for unknown species, because the intensity of each IR or Raman band for each compound is a specific property of that molecule. We have seen that the shapes and positions of bands vary depending on the phase of the sample, but even so the overall profile of absorption (or transmission) of a spectrum can be analyzed as a function of the relative concentrations of the components of the sample, if reference spectra for each of the components are... 

The existence of the A-H...O hydrogen bonds was confirmed by experiment. For example, gas phase measurements of dipole moment and vibrational predissociation lifetimes as well as the determination of rotational constants performed for the F-H...H2 complex [73] confirmed its T-shaped structure. The high-resolution infrared spectra were analyzed for the H2-HF, D2-HF and HD-HF complexes solvated in helium nanodroplets and also for those species the T-shaped structure was confirmed [74—76]. The infrared vibrational predissociation spectra... 

The moment of inertia 1 determines the rotational constant 0 = h /IT, which is the parameter that controls the strength of quantum effects. The other parameter of the model, which is the quadrupolar coupling constant K, can be conveniently taken as the energy and temperature scale. We can thus reduce all quantities related to energies by K, and define, e.g., the dimensionless temperature = k T/K, energy = E/K, and rotational... 

A=5.4+0.4GHz B+C=1445.2 0.5MHz of the per-deuterated acetic acid dimer (CD3COOD)2 were extracted. It was not possible to determine the rotational constants B and C independently due to the symmetric prolate top nature of the dimeric structure (Ray s asymmetry parameter k=-0.965). More detailed consideration of the fs DFWM spectra taken in the gas cell and in a supersonic expansion for the acetic acid dimer are under way in our laboratory and will be presented in a forthcoming publication. 

A further goal is to determine the rotational constants of the different van der Waals states. The effective rotational constants represent a value averaging over the vibrational motion and thus information about the vibra-... 

Krugh and Gold in an early study determined the rotational constants and dipole moment of the parent tetrazole by means of microwave spectroscopy in the gas phase A 10667.3, B 10310.9, C 5240.4 MHz, dipole moment 2.19 D <1974JSP423>. The results of calculations carried out by Fausto and co-workers using the B3LYP/6-31G method are only consistent with the data for the 277-tautomer of tetrazole confirming the prevalence of exactly this form of the compound in the gas phase <2001PCP3541>. 

For determinations of reaction constants, anomeric purity is not necessary, but for the determination of initial rotations, only one anomer can be present. Pure anomers can often be obtained by slow crystallization from a suitable solvent in the presence of nucleating crystals of the desired form and in the absence of crystals of other forms. For measurements in aqueous solution, use of mM potassium hydrogen phthalate as a buffer (pH 4.4) is recommended, to avoid variation in acidity during measurements. Precautions for purification, drying, and use of organic solvents have been described by Lowry and Baker. Anomeric impurities can be removed by lixiviation with a solvent (such as aqueous alcohol) in which both anomers are slightly soluble. 

Willker and Leibfritz (1992a) introduced an extension of the E.COSY principle that yields additional flexibility. In addition to coherence transfer between the active spins i and j, polarization of spin p, which is passive during is transferred to a spin q, which plays the role of the passive spin during t - Hence, in general, the E.COSY triad is opened up. The two- and three-dimensional JHH-TOCSY experiments for the determination of coupling constants of Willker and Leibfritz (1992a) use a combination of homonuclear TOCSY transfer and two BIRD (bilinear rotation decou-... 

The variation of rotational constants with ring-puckering vibrational state is very sensitive to the presence of a barrier at the planar conformation. This is shown for cyclobutanone16 and methylenecyclobutane17) in Fig. 2.6. The presence of a very small barrier, ca. 7.6 cm-1 in the case of cyclobutanone, causes deviation from a smooth variation for the lower levels. In the case of methylenecyclobutane, a very pronounced zig-zag of the rotational constants is observed due to the presence of a 140 cm-1 barrier. The dependence of the rotational constants on vibrational state may be used quantitatively to determine the shape of the potential function as discussed in subsequent sections. 

Model calculations to reproduce the variation of rotational constants with vibrational state are sensitive to the value of to. Thus an approximate value may be determined from them that yields experimental information about the dynamics of the vibration. 

The expectation values represented by the double sums in Eq. (4.3) depend on the potential function in Eq. (3.27). For a given harmonic frequency in the basis set, the matrix elements Zy and Zy are fixed but the t and tjv depend on the value of B in the dimensionless potential of Eq. (3.27). For a single-minimum potential there is a high degree of correlation between the values and the value of B, each of which leads to curvature in the rotational-constant variation with vibrational state1S). Since there are ten adjustable parameters, namely, three coefficients for each of the rotational constants plus one potential constant, B, in the reduced potential, it is necessary to determine the rotational constants in a large number of vibrational states if microwave data alone are used. 

C3H2O3 CH=CHOCO II 0 Vinylene carbonate MW Planar based on variation of rotational constants. No potential function determined. Isotopic species studied 135, 136)... 

MW Double minimum potential function determined from the variation of rotational constants and from analysis of the non-rigid rotor spectra in the v = 0, 1 and 2,3 inversion states. Barrier =241 cm-1 125)... 

Structural parameters are determined usually by fitting tiiem to experimental data by the least-squares method. The exceptions are the traditional method and cases in which the number of rotational constants or... 

According to Rudolph [33], structures determined by fitting structural parameters directly to the experimentally determined rotational constants Bq, the moments of inertia Iq or the planar moments Pq of a series of isotopomers are called rg structures. If the covariances of the observables are transformed correctly, the structures resulting fi-om any of these fits are identical. However, if covariances are not transformed, these structures are not identical nor equivalent (for examples, see [34]). Therefore, one should always specify what kind of covariance matrix is used with a particular fit. Pseudo-Kraitchman (p-Kr) or pseudo-r fits are those in which the structures are fit to differences of rotational constants, moments of inertia or planar moments. The structures called and again are equivalent to each other if the covariances are transformed correctly. On the other hand, the structure is theoretically different fi om the and structures because of a lack of a linear relation between the AB and A/. In practice, structures are very close to other p-Kr structures. The rig structure is obtained when the isotopic moments Iq are fitted to structural parameters and three e parameters. It is... 

---