Vibrational derivatives, vibration-rotation

Midey A J and Viggiano A A 1998 Rate constants for the reaction of Ar" with O2 and CO as a function of temperature from 300 to 1400 K derivation of rotational and vibrational energy effects J. Chem. Phys. at press... 

Figure B2.3.14. Experimentally derived vibration-rotation populations for the NO produet from the H + NO2 reaetion [43], The fme-stnieture labels and refer to the two ways that the projeetions 2 and A of the eleetron spin and orbital angular momenta along the intemuelear axis of this open-shell ean be eoupled (D =...

The complete vibration-rotation Flamiltonian for acetylene-like tetraatomic molecules has been derived by Handy et al. by hand [155] and using a computer algebra program [156]. (Note that in both of the mentioned papers there are some minor errors, see also [144,157,158]). Handy uses as bending coordinates... 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

It is reasonably well established that the non BO coupling term involving second derivatives of the electronic wavefunction contributes less to the coupling than does the term (-ih-3 rk/dRa) (-ib k 9Ra)/nia having first derivatives of the electronic and vibration-rotation functions. Hence, it is only the latter terms that will be discussed further in this paper. 

Here, ej f are the vibration-rotation energies of the initial (anion) and final (neutral) states, and E denotes the kinetic energy carried away by the ejected electron (e.g., the initial state corresponds to an anion and the final state to a neutral molecule plus an ejected electron). The density of translational energy states of the ejected electron is p(E) = 4 nneL (2meE) /h. We have used the short-hand notation involving P P/p to symbolize the multidimensional derivative operators that arise in the non BO couplings as discussed above ... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

It is well established that the average lengths of CH bonds are consistently 0.003 to 0.004 A longer than the corresponding CD bonds in the ground vibrational state (see Fig. 12.1, its caption, and Section 12.2.3). It remains only to establish the dipole moment derivative, (9p/9r), at the equilibrium bond length. That is available from theoretical calculation or spectroscopic measurement (via precise measurements of IR intensities of vibration-rotation bands). Calculations based on Equation 12.7 yield predicted dipole moment IE s in reasonable agreement with experiment. 

We examine the derivation of information about molecular structure and properties from analysis of pure rotational and vibration-rotational spectral data of diatomic molecular species on the basis of Dunham s algebraic formalism, making comparison with results from alternative approaches. According to an implementation of computational spectrometry, wave-mechanical calculations of molecular electronic structure and properties have already played an important role in spectral reduction through interaction of quantum chemistry and spectral analysis. 

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. 

When Dunham [4,5] presented formula 8 for vibration-rotational terms, he derived a functional V + V2 explicitly because in his JBKW formulation the addend V2 results from exact solution of an integral. In contrast, Dunham assumed a functional K(K+l), equivalent to /(/- -1) in contemporary notation, to contain a quantum number K, now J, for rotational angular momentum. To generate an effective potential energy comprising both internuclear potential... 

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... 

Computational spectrometry, which implies an interaction between quantum chemistry and analysis of molecular spectra to derive accurate information about molecular properties, is needed for the analysis of the pure rotational and vibration-rotational spectra of HeH in four isotopic variants to obtain precise values of equilibrium intemuclear distance and force coefficient. For this purpose, we have calculated the electronic energy, rotational and vibrational g factors, the electric dipolar moment, and adiabatic corrections for both He and H atomic centres for intemuclear distances over a large range 10 °m [0.3, 10]. Based on these results we have generated radial functions for atomic contributions for g g,... 

C. G. Joslin, C. G. Gray, and S. Goldman, Chem. Phys. Lett., 227, 405 (1994). Infrared Rotation and Vibration-Rotation Bands of Endohedral Fullerene Complexes. Helium in C60-Derived Nanotubes. 

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