Vibrational energy levels Hamiltonian parameters

Normal coordinates are perfectly adequate for calculating low-lying vibrational energy levels. They have been used extensively by spectroscopists to derive theoretical expressions (in terms of harmonic frequencies and cubic and quartic force constants) for the parameters of effective Hamiltonians, often used to fit experimental spectra. The simplest version of the normal coordinate kinetic energy operator was derived by Watson from the earlier work of Wilson and Howard, and Darling and Dennison. The vibrational part of the Watson kinetic energy operator is. 

Keeping this result in mind, we now proceed to construct the algebraic Hamiltonian in its blocked form. For those families of levels excluding spurious modes, we apply the technique discussed in Section III.C.l. Thus we must fit the algebraic parameters A, A, and (plus N, separately fixed) over a convenient database of experimental levels. A complete list of parameters can be found in Ref. 47. In Table II we report only the fundamental vibrational energies of benzene. 

Here G(vj, v2, v3) is the level energy in wave number units (as far as possible we follow the notation of Herzberg, 1950) and the constants in Equation (0.1) are given in Table 0.1. As usual the vs are the vibrational quantum numbers of S02 and rather high (above 10) values can be reached using the SEP technique. Equation (0.1) provides a fit to the observed levels to within an error below 10 cm 1, which is almost the experimental accuracy. We need, however, to be able to relate the parameters in this expansion directly to a Hamiltonian. The familiar way of doing this proceeds in two steps. First, the electronic problem is solved in the Bom-Oppenheimer approximation, leading to the potential for the... 

The popularity of the SOS methods in calculations of non-linear optical properties of molecules is due to the so-called few-states approximations. The sum-over-states formalism defines the response of a system in terms of the spectroscopic parameters, like excitations energies and transition moments between various excited states. Depending on the level of approximation, those states may be electronic or vibronic or electronic-vibrational-rotational ones. Under the assumption that there are few states which contribute more than others, the summation over the whole spectrum of the Hamiltonian can be reduced to those states. In a very special case, one may include only one excited state which is assumed to dominate the molecular response through the given order in perturbation expansion. The first applications of two-level model to calculations of j3 date from late 1970s [93, 94]. The two-states model for first-order hyperpolarizability with only one excited state included can be written as ... 

In this paper, the effect of the pseudopotential term, arising from the quantum mechanical correction to classical mechanism (F ), on the torsional levels of hydrogen peroxide and deuterium peroxide is evaluated. The V operator, depends on the first and second derivatives with respect to the torsional coordinate of the determinant of the g inertia matrix and on the first derivatives of the B kinetic energy parameter of the vibrational Hamiltonian. V has been determined for each nuclear conformation from the optimized coordinates obtained using MP2/AUG-cc-pVTZ ab initio calculations. 

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