Rotational constants definitions

Taking into account the fact that the vector of angular velocity is constant, the right hand side of this equation can be simplified and with this purpose in mind we make use of Fig. 2.3e, where the vector R is perpendicular to the axis of rotation. By definition,... 

Similar operational definitions have to be taken into account for every experimental tool of structural chemistry to define the meaning of the observables that it provides6. In microwave spectroscopy, for example, structural information is obtained from the rotational constants... 

In order to assign the Zeeman patterns for the three lowest rotational levels quantitatively, one must determine the spacings between the rotational levels, and the values of g/and gr-In the simplest model which neglects centrifugal distortion, the rotation spacings are simply B0. /(./ + 1) this approximation was used by Brown and Uehara [10], who used the rotational constant B0 = 21295 MHz obtained by Saito [12] from pure microwave rotational spectroscopy (see later in the next chapter). The values of the g-factors were found to be g L = 0.999 82, gr = —(1.35) x 10-4. Note that because of the off-diagonal matrix elements (9.6), the Zeeman matrices (one for each value of Mj) are actually infinite in size and must be truncated at some point to achieve the desired level of accuracy. In subsequent work Miller [14] observed the spectrum of A33 SO in natural abundance 33 S has a nuclear spin of 3/2 and from the hyperfine structure Miller was able to determine the magnetic hyperfine constant a (see below for the definition of this constant). 

From what has been shown in the preceding sections (cf. Eqs. 61 and 73, 83), it is possible to present the molecular structure resulting from both the r -fit method and any of the r()-derived methods in a convenient and easily comparable form, as a structural description in both Cartesian and internal coordinates, and with consistent errors and correlations (for small and larger molecules). A detailed comparison would require a sufficiently large SDS to determine a complete molecular structure, but the requirements are still the least restrictive of all methods presented. The input data must include the covariance matrix of the rotational constants or moments. This matrix may have to be adequately modeled to avoid grossly different weighting of isotopomers which is usually not warranted. The definition of the input data set... 

To be able to utilize this formula a great deal of information concerning molecular parameters is required. To calculate N E) rotational constants and vibrational frequencies of internal motion are required and in many case these are available from spectroscopic studies of the stable molecule. Unfortunately the same cannot be said for the parameters required to calculate G E) because, by definition, the transition state is a very short lived species and is therefore not amenable to spectroscopic analysis. The situation is aggravated still further by the fact that many unimolecular dissociation processes do not have a well defined transition state on the reaction coordinate. It is precisely these difficulties that make ILT an attractive alternative as it does not require a detailed knowledge of transition state properties. 

The equilibrium intemuclear distance Rg is determined by the rotational constant. By definition. 

In the case of oxetanone-3, the coefficients in the rotational constant expansions [Eq. (4.2)] were treated as empirical parameters and the potential function was taken from a previous vibrational study10). Figure 2.5 shows the smooth variation, with a definite curvature, of the B rotational constant with ring-puckering vibrational state. Table 4.2 lists the observed and calculated values of the rotational constants. The smooth variation indicates a single-minimum potential with a definite curvature due to the quartic potential term and the quartic terms in the expansion [Eq. (4.3)]. 

Derivation of molecular parameters from rotational constants is not a new subject, and many discussions of the various methods have previously been presented. One of the more complete of the recent accounts is that by Gordy and Cook.1 Definitions of the different molecular parameters are given by Laurie elsewhere in this volume and have also been given recently by Kuchitsu and Cyvin.2 The fundamental papers on this subject include the series by Herschbach and Laurie3-5 and by Morino, Oka, and co-workers.6-9 The present discussion will concentrate on a description of the computational strategies that may be employed and the problems that occur in practical cases. For the usual reasons of familiarity the examples will be dominated by work done at Michigan State University. 

The principal application of the Kraitchman equations [Eq. (9)1 is for the determination of the atomic coordinates, at, bSi and cs. From a study of the rotational spectrum of the parent and of a species with single isotopic substitution the coordinates of the substituted atom may be determined. These coordinates are referred to as substitution coordinates or rs coordinates. Each new species yields new coordinates, and since all of the coordinates are in the same coordinate system, the calculation of substitution or rs bond distances and bond angles is a simple process. Costain,s demonstrated that there are definite advantages to the use of the Kraitchman equations to obtain molecular parameters. These advantages are sufficient to make the use of Kraitchman s equations the preferred method of structure determination from ground-state rotational constants. 

The rc and p-Kr coordinates are determined by adjusting coordinates to fit moments of inertia and differences in moments of inertia, respectively. This is again an operational definition, so the only uncertainty is the result of experimental uncertainty in the rotational constants, and this may be treated by standard methods.18 However, the standard deviation between observed rotational constants and those computed from the final structure should not be used in such a calculation. In the comparison shown above the structural parameters of NF2CN are more reliable than those of PF2CN, but the deviation between observed and calculated moments is greater for NF2CN. The standard deviations of the rotational constants are best obtained from the fit of the rotational spectra. These values should be used to estimate the experimental uncertainties in rQ and p-Kr coordinates. The uncertainties in r0 or p-Kr coordinates as estimates of the equilibrium coordinates are very difficult to compute. The Costain rule is probably satisfactory for p-Kr coordinates, but less so for rQ coordinates. 

Second-order effects arising from the product of matrix elements involving J+ L and L+ S operators have the same form as 7J+S. In the case of H2, the second-order effect seems to be smaller than the first-order effect, but in other molecules this second-order effect will be more important than the first-order contribution to the spin-rotation constant. These second-order contributions can be shown to increase in proportion with spin-orbit effects, namely roughly as Z2, but the direct spin-rotation interaction is proportional to the rotational constant. For 2n states, 7 is strongly correlated with Ap, the spin-orbit centrifugal distortion constant [see definition, Eq. (5.6.6)], and direct evaluation from experimental data is difficult. On the other hand, the main second-order contribution to 7 is often due to a neighboring 2E+ state. Table 3.7 compares calculated with deperturbed values of 7 7eff of a 2II state may be deperturbed with respect to 2E+ by... 

An important and frequently encountered result, often mistakenly taken as evidence for pure precession, is that, in the unique perturber, identical potential curve limit, the effective spin-rotation constant of the 2E+ state, jv, is equal to Py. This result is a direct consequence of the second-order perturbation theoretical definition of the contribution of a 2II state to the spin-rotation splitting in a 2E state (see Section 3.5.4) ... 

It is straightforward to establish the relationship between the rotational constants A, B, C in the principal axis system and the constants, 4ram. 5ram. Cram, and Dab in the rho-axis system using the definition of p or by diagonalizing the 3x3 matrix of the RAM rotational constants. In the particular case of an a b) symmetry plane, it gives... 

An analog situation was found in IF [95Mtll], see later in this subvolume, and CIF [77Fab]. There are different sign definitions in use in the literature for the spin-rotation constant, see the introduction to this table. 

Debye-Falkenhagen-type forces do not suffer from the vanishing of the rotationally averaged interaction since they arise from a force due to induction which can track the rotation of the particle. Unlike the Keesom force in which, in principle, the interaction of distributed point charges are considered, induction forces are dependent upon a collective molecular or material property, the static dielectric constant. Definition of the domain of validity for the classical calculation is required and has been given in the case of metals [5.17]. Those authors pointed out that if the surface of the metal were taken at the center of mass of the surface charge distribution, then classical electrostatic calculations would be valid. 

This variety in rotational selection rules, coupled with our natural endowment of molecules with diverse rotational constants, leads to wide variations in the rotational fine structure exhibited by symmetric and near-symmetric tops. For definiteness, we consider a prolate symmetric top whose rotational energy levels are given in Eq. 5.26. Rotational lines will be found at the frequencies... 

It is possible to calculate the internuclear distance in a diatomic molecule by measuring the interlinear spacing in either the P- or the R-branch. If we assume that the equations for a harmonic oscillator— rigid rotor are valid, then this spacing is 2B. The internuclear distance is calculated from the definition of the rotational constant B. If we combine the factors in the definition of the rotational constant B as given by equation (4-10) with the definition of the moment of inertia for diatomic molecules given in equation (4-5), the internuclear distance of a diatomic molecule is given by... 

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