Derived factors, rotation

So many factors affect the accuracy of bond distances and bond angles derived from rotational constants that nearly every molecule is a unique case. Nevertheless, there are several key questions that should be answered by every report of structural analysis from rotational spectra. Some of these are as follows ... 

The emission spectrum observed by high resolution spectroscopy for the A - X vibrational bands [4] has been very well reproduced theoretically for several low-lying vibrational quantum numbers and the spectrum for the A - A n vibrational bands has been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrationm and rovibrational transitions appear to be the largest for the X ground state followed by those... 

Similar to the PIP, the Hamiltonian [Eq. (52a)] of a periodic pulse shows an infinite number of effective RF fields with both x and y components of the scaling factors X a and the phases 0na. The periodic pulse, however, acquires a different symmetry as that of the PIP. From Eq. (52c) and = ana, it follows that the scaling factor Xm, is symmetric in respect to the sideband number n, while the phase 6na is anti-symmetric according to Eq. (51c). These symmetries seem to be a coincidence arising from the mathematical derivations. As a matter of fact, they are the intrinsic natures of the periodic pulse. Considering the term f x i)Ix for instance, any Iy component created by the rotating field denoted by a> must be compensated at any time t by its counter-component oj n in order to reserve the amplitude modulated RF field. 

In contrast to PCA which can be considered as a method for basis rotation, factor analysis is based on a statistical model with certain model assumptions. Like PCA, factor analysis also results in dimension reduction, but while the PCs are just derived by optimizing a statistical criterion (spread, variance), the factors are aimed at having a real meaning and an interpretation. Only a very brief introduction is given here a classical book about factor analysis in chemistry is from Malinowski (2002) many other books on factor analysis are available (Basilevsky 1994 Harman 1976 Johnson and Wichem 2002). 

There exists no significant comprehensive fit of spectral data of H2 with which we might here make comparison. Our discussion above demonstrates that, as for GaH above, application of an algorithm based on Dunham s algebraic approach to analysis of vibration-rotational spectral data of H2, especially through implementation of hypervirial perturbation theory [30,72] that allows the term for the vibrational g factor in the hamiltonian in formula 29 to be treated directly in that form, proves extremely powerful to derive values of fitting parameters that not only have intrinsic value in reproducing experimental data of wave numbers of transitions but also relate to other theoretical and experimental quantities. 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

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