Vibrational-rotational energy, formulas

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... 

It should be noted that the spaeings between the experimentally observed peaks in HCl are not eonstant as would be expeeted based on the above P- and R- braneh formulas. This is beeause the moment of inertia appropriate for the v = 1 vibrational level is different than that of the v = 0 level. These effeets of vibration-rotation eoupling ean be modeled by allowing the v = 0 and v = 1 levels to have rotational energies written as... 

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... 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

At this point the first-principles perturbative (FP) approach becomes valuable. The same kinds of perturbative models are used to describe the vibrational-rotational motions as in the SP approach. However, data from electronic structure theory computations or potential energy functions are used to parameterize the formulas instead of spectroscopically obtained data. The FP approach has for example, been pursued by Martin et al. [16-18] and by Isaacson, Truhlar, and co-workers [19-25]. This avenue is especially valuable when spectroscopic data are not available for a molecule of interest. Codes are available that can carry out vibrational perturbation theory computations, using a grid of ab initio data as input SURVIBTM... 

This looks reasonable for smaU. amplitude vibrations only. However, this amplitude becomes larger under rotational excitations. Thus, in principle. Re should increase if J increases and therefore the rotational energy is lower than shown by the formula. 

The Dunham coefficients Yy are related to the spectroscopical parameters as follows 7io = cOe to the fundamental vibrational frequency, Y20 = cOeXe to the anharmonicity constant, Y02 = D to the centrifugal distortion constant, Yn = oie to the vibrational-rotational interaction constant, and Ym = / to the rotational constant. These coefficients can be expressed in terms of different derivatives of U R) at the equilibrium point, r=Re. The derivatives can be either calculated analytically or by using numerical differentiation applied to the PEC points. The numerical differentiation of the total energy of the system, Ecasccsd, point by point is the simplest way to obtain the parameters. In our works we have used the standard five-point numerical differentiation formula. In the comparison of the calculated values with the experimental results we utilize the experimental PECs obtained with the Rydberg-Klein-Rees (RKR) approach [58-60] and with the inverted perturbation approach (IPA) [61,62]. The IPA is method originally intended to improve the RKR potentials. 

The internal partition function for molecules having inversion may be factored, to a good approximation, into overall rotational and vibrational partition functions. Although inversion tunnelling results in a splitting of rotational energy levels, the statistical weights are such that the classical formulae for rotational contributions to thermodynamic functions may be used. The appropriate symmetry number depends on the procedure used to calculate the vibrational partition function. 

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