Two-level approximation

At higher densities, the population factor in Eq. (3.65) ceases to be proportional to the collisional excitation rate, but is rather given (in a two-level approximation) by... 

Using this single laser excitation, the equations have been integrated for a variety of values of I and of ox, Oy, near the nominal ones. The goal of the study is not merely to attain a description in terms of these parameters, but to explore the limits of applicability of more simplified approaches for OH and other molecules. In particular, a two-level approximation is found to be clearly inadequate for OH. More detailed aspects are described in what follows. 

Case 2. Atomic tritium in the Bergkvist two-level approximation (Berg-kvist, 1972) the interval obtained was... 

For dephasing (T2) processes one is concerned with the decay of the vibrational correlation function . In the two-level approximation one may write... 

Using the same notations as in Section 2, the positive difference between the adiabatic potential energies within a two-level approximation varies with 5Q according to... 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

In a similar way it is possible to reduce the SOS expressions defining other nonlinear optical properties. The two-photon transition moment within two-level approximation reads ... 

There were also some attempts to derive the few-states models for second-order hyperpolarizability [23, 36, 48]. In the simplest case, i.e. within the two-level approximation, we may write ... 

Due to this, one can use the two-level approximation both for one- and three-photon absorption. The length of one-photon absorption is... 

This article is devoted to Are methodology of predicting the direction of the changes of molecular (hyper)polarizabilities values as a function of tire solvent polarity. Since the environmental effect on the two-photon absorption (TPA) is still poorly understood, we will consider the two-level approximation to describe the influence of the solvent effects on TPA from the ground to the CT excited state of the D-tt-A type chromophores. Only electronic contributions will be taken into account. In contrast to the TPA process, the substantial progress in theoretical description of the solvent influence on the vibrational (hyper)polarizabilities has been observed recently [64-67],... 

In the present contribution we will discuss the direction of the changes of the NLO response and the solvatochromic behavior as a function of solvent polarity of the D-tt-A chromophores. The best starting point for these considerations seems to be the simple two-state model for the first-order hyperpolarizability (/ ) [8]. To avoid the extreme complexity of the sum-over-states (SOS) expression [101], Oudar and Chemla proposed the relation between the dominant component of 13 along the molecular axis (let it be the x-axis) and the spectroscopic parameters of the low-lying CT transition [8]. The use of the two-level approximation in the static case (ru = 0.0) has lead to the following expression for the static component of the first-order hyperpolarizability tensor ... 

From the results in the last section it is clear that for particular applied radiative frequencies or frequency multiples, close to resonance with particular molecular states, each molecular tensor will be dominated by certain terms in the summation of states as a result of their diminished denominators—a principle that also applies to all other multiphoton interactions. This invites the possibility of excluding, in the sum over molecular states, certain states that much less significantly contribute. Then it is expedient to replace the infinite sum over all molecular states by a sum over a finite set—this is the technique employed by computational molecular modelers, their results often producing excellent theoretical data. In the pursuit of analytical results for near-resonance behavior, it is often defensible to further limit the sum over states and consider just the ground and one electronically excited state. Indeed, the literature is replete with calculations based on two-level approximations to simplify the optical properties of complex molecular systems. On the other hand, the coherence features that arise through adoption of the celebrated Bloch equations are limited to exact two-level systems and are rarely applicable to the optical response of complex molecular media. 

The expression for the perturbation operator leading to reaction was discussed in connection with the nonorthogonality of the electronic wave functions of the initial and final states. For the first time, this problem was investigated in a two-level approximation in 1964 by Kuznetsov (see also Refs. 31, 35, and 36). The final results for the probability of electron transfer between complex ions obtained in Ref. 53 represent a limiting case of the expressions derived in Refs. 49 and 57. Deviation of some general relations in Ref. 53 from those derived earlier (see, e.g.. Ref. 49) is due to a certain mathematical inaccuracy in Ref. 53 (see Ref. 58). 

The Dunham coefficients of Equation 16.11 are determined by fitting the measured spectra. The sensitivity coefficients Ki can be found by making use of Equations 16.12 and 16.13. Some rovibrational levels of different electronic excited states lie very close to each other. Eor such levels, additional nonadiabatic corrections can be included within the two-level approximation [57]. 

Within the framework of two-level approximation at excited nucleus consideration, the general Hamiltonian operator of... 

The thermal average nonradiative decay probability from the manifold rps to the manifold r J( in the two-level approximation is given by the Golden Rule expression... 

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