Perturbation expansion, vibrational energy

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 ... 

At higher levels of excitation anharmonicity has to be included to obtain accurate energy levels. Perturbation theory has been used to derive the following expression, often called a Dunham expansion (Hirst, 1985), for polyatomic anharmonic vibrational energy levels, which is similar to the Morse energy level expression Eq. (2.59), for a diatomic molecule ... 

From this derivation, it follows that the expansion characterized by Eqs. (51) and (54) is valid if the two expansions—the perturbation and Taylor series expansion—converge rapidly. This will be the case if (0) — (0) is large, typically much larger than a vibrational energy quantum cu,-, and the amplitudes Q are small, i.e., if only low vibrational quantum numbers are involved. 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

The dependence of rag on the internal coordinates is not restricted by requirements other than the center of mass conditions (2.4) and that Eq. (2.6) is invertible. In expressing the rag functions we may therefore also consider how the final Hamiltonian is influenced, so that we obtain an operator of optimum suitability characterized by e.g. rapid convergence of the perturbing terms. In this respect there are two particular concerns, the vibration-rotation interaction and the potential energy expansion. 

Figure 6 The differences between energies of bend overtones in H O with 7=2 and K = 2 and the energies of the corresponding 7 = 0 states are plotted as a function of the number of quanta in the bend. The results of variational, eighth, sixth, fourth, and second-order perturbative calculations are denoted by, , O, A, and 0 respectively. The perturbative results model the rotation-vibration coupling as a polynomial expansion in the bend quantum number. Given the actual form of this coupling, as obtained from the variational calculation, the slow convergence of CVPT is not surprising. (From Ref. 49.)...

The treatment of the vibrational NLO properties in the previous sections employed either the Bishop-Kirtman perturbational theory (BKPT) or the finite field-nuclear relaxation (FF-NR) approach. These approaches may fail for molecules containing large amplitude anharmonic motions, as indeed was suspected to happen in Li C6o-In such cases a more recently proposed variational method, based on analytical response theory [78, 79], would in principle be applicable, but is computationally extremely expensive, as it requires an accurate numerical description of the potential energy surface (PES), at least if the anharmonicity is so large that a power series expansion of the PES is inadequate [80]. 

The anharmonic terms, i.e. the cubic and higher terms in the displacement expansion of the intermolecular potential and the rotational kinetic energy terms, which are neglected in the harmonic Hamiltonian, can be considered as perturbations. They affect the vibrational excitations of the crystal in two ways they shift the excitation frequencies and they lead to finite lifetimes of the excited states, which are visible as spectral line broadening. By means of anharmonic perturbation theory based on a Green s function approach [64, 65] it is possible to calculate the frequency shifts, as well as the line widths. 

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