Two-level model

Figure 12.9 The excitation-power dependence of the emission count rate of single DMPBI nanocrystals (dots), and a saturation curve calculated from a two-level model (solid line). One count rate value to one laser power was calculated as an average of 30 nanocrystals. S. Masuo, A. Masuhara, T. Akashi, M. Muranushi,...

MAP has a non-centrosymmetric structure (24) P2- leading to four non-vanishing crystalline nonlinear coefficients. According to the previously defined two-level model, there are also four non-vanishing molecular nonlinearity coefficients which are linearly related to the crystalline coefficients. (3)... 

Molecular hyperpolarizabilities p of the azobenzene-linked amphiphiles were estimated using the following equation derived from the two-level model [3] ... 

The simplest description of compoimds of this type is obtained limiting the siun-over-state equations to terms depending only on the properties of the groimd state g and the first excited state e (two-state model) [93]. This is analogous to the two-level model introduced to describe other nonlinear optical properties, for example the nonlinear polarizability pS - co coi,co2) [ 104]. In the case of 2PA, this two-state, or dipolar, contribution to the cross section is, on resonance ... 

This is referred to as the 2-level model (6) and was used in the past to understand trends in p with structural modifications of the molecule. With the computational methods and experimental accuracy available today this method is viewed as inadequate although it does serve to illustrate the essential features of molecular structure that control p. In the two level model, px (the component of the tensor along the charge transfer axis) is given by... 

Table I. Measured and Calculated Values of ft Using Two-Level Model...

A simple two level model was proposed for BDN (20) to explain the origin of the nonlinearity obtained by tuning onto the resonance in that material. A similar model may explain the results observed here, although in the model proposed in reference 20 a high quantum yield for fluorescence is important, whereas these materials do not fluoresce (9). 

An attempt is made to fit quadratic electrooptic (QEO) results to a two-level model for the microscopic third order susceptibility, y. 

It has been known experimentally(4) that much of the second order susceptibility generally arises from the lowest singlet excited state. For any particular molecule, the recently introduced Missing States Analysis (MSA)(5,6). can show, via calculation, to what extent P is dominated by the first excited state. For instance, the P of p-nitroaniline has been shown by MSA to be heavily dominated by the first excited state, at least with a PPP (Pariser-Pople-Parr) Hamiltonian and standard basis. The result of these findings is that one can often approximate Equation (4) by including only one excited state in the sum, there-by arriving at a two-level model ... 

For the linear electrooptic effect (EO), the two-level model only differs in dispersion, with the dispersion factor, F q (co), given by... 

In general, the optimization of organic molecules for third order nonlinear optical applications has enjoyed much less success than for second order optical nonlinearities. The major reason for this has been the questionable validity of the two-level model for y, and the difficult assessment of the contribution of two-photon states for the more acceptable three-level model. 

Answering this question involves an examination of Equation (12). The largest QEO nonlinearities are available when the QEO probe frequency, co, is close to resonance. This is the restriction that can potentially allow the two-level model to be applicable. Note that the two-level model depends on the competition between yc and yn, which have opposite sign. It is important to note that under centrosymmetry, yc should be the only contribution to yqeo> while the yqeo for noncentrosymmetric molecules consists of yc+yn. A first order correction to this model will require the additional term yfp. It is important to judge what effect this term may have on this speculation and the potential interpretation. In general, it should be to lower (However, note as in the case of frans-octatetraene,... 

Transition moments, ioi> to the first excited state can be calculated from the integrated absorption of the linear electronic spectrum. This can be used to calculate -Af poi n the first term (defined here as yc) of the two-level model. The second term (Af p.oiAjioi m) from Equation (12) (defined here yn) involves Ajioi, which can be determined directly from solvatochromism( 19), or from a two-level analysis of a molecular EFISH measurement of p. 

With the permutation-symmetry-corrected results(17) it appears that we cannot successfully fit our QEO data to a two-level model, at least for centrosymmetric structures. We have only one data point (9) to make this judgement, however, and... 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

In two-level models with phonon assisted tunneling, there appears coupling of both phonon modes mediated by Rabi oscillations. Therefore, also the interlevel... 

Fig. 23 Stability diagram (the contour plot of the differential conductance) calculated by our Ansatz for the two level model with parameters as in Fig. 22. The latter is indicated with a dash line at VR = 1.0 V.

Fig. 4.3 A simplified two-level model for photoabsorption (left). The two levels are represented by the corresponding wavefunctions (g, ground state x, excited state) with the matrix element (x ji g) giving the transition probability between the ground and excited state. In the right part a simple three-state model for a two-site situation represents the mixing between the zero order states (neutral, excited and charge-transfer) to form the spectroscopic states...

Design of Second-order Chromophores the Two-level Model... 

It should be noted that although /3 at a given frequency is different for different NLO phenomena, both tend to the same zero-frequency limit, /30. Figure 11.3 compares the predictions of the two-level model for the frequency dispersion of the electrooptic effect with that for frequency doubling. While dispersion effects complicate comparison of the intrinsic nonlinearity of various chromophores... 

This technique was first considered theoretically for molecules by Daily (18, 19) in important papers in 1976 and 1977. In these papers Daily extended the two level model used in atomic systems to molecules and attempted to define the range of utility of the model. Baronavski and McDonald reconsidered the model and made the first experimental measurements in flames under saturated conditions to test the model (20, 21) for C, in acetylene flames. During the same year workers at United Technologies began experimental evaluation of the technique (22) and in the last two... 

Figure 23. Schematic of the two-level model used to treat saturation spectroscopy data. See the text for details.

These evaluations are made within the context of the two level model and the steady state approximation. The steady-state approximation is probably valid for this experiment. C is not (electronically, vibronically, or rotationally) a two level system. Other groups, particularly Daily (23) and Berg and Shackleford (18) have developed expressions which allow for the inclusion of more levels and provide for incomplete relaxation. Lucht and Laurendeau (28) have carefully considered the effect of rotational equilibration. There is not time here to discuss these models in detail. The theoretical models which include specifically more than two electronic levels require experimental measurements independently of the radiation coupling the various levels. We have not found a system experimentally tractible for testing the three electronic level model. 

---