Non-resonant Term

A major difference between SFG and other surface vibrational spectroscopy techniques is the presence of a non-resonant background, because, in part, of the metal substrate. This background is usually treated as independent of the frequency and characterized as a constant ( nr). although this treatment is not always possible. In electrochemical systems, /nr is not usually independent of the applied potential. This is because of potential-dependent changes in the electronic state 

One method developed to minimize the contribution of /nr to the nonlinear spectrum is the Difference Frequency Generation (DFG) approach [13]. Since /nr is dependent on the electronic states of the metal, shifting the output frequency to lower energy avoids exciting the resonance modes in the metal. Although DFG signal is susceptible to contributions of fluorescence from the sample, it has been shown to help in the analysis of electrochemical interfaces [22]. 

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Figure 5 displays the three partial intensities. Here the upper curve (circles) correspond to the SAXS intensity measured by a conventional SAXS experiment far below the edge. The lowermost curve (triangles) is the self-term of Eq. (12) and the curve in between marks the cross-term (squares). As expected from previous model calculations, the intensities exhibit a very similar dependence on q [17], Note that the self-term which is much smaller than the non-resonant term or the cross term can be obtained up to q = 2.5 nm-1. As mentioned above, this term provides the most valuable information of the ASAXS experiment. It refers to the scattering intensity that would result from a system in which the macroion is totally matched. 

The second terms in Eqs. (1.167) and (1.168) are rapidly vibrating non-resonant terms having minor contribution on average, so they are neglected in further calculations. This is called the rotating wave approximation. By solving these simultaneous equations with the initial condition of T0 = T, for / = 0, we obtain ... 

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first term is the resonant term and the second term is the non-resonant term. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD formula is sometimes called the sum-over-states formula, since formally it requires a sum over all intermediate states j, each intermediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states v]/ - and v]/ The KHD formula is fully equivalent to the time domain formula, equation IAl.6.92). and can be derived from the latter in a straightforward way. However, the time domain formula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

Here, the exponents, 9 and 0, describe the phase of the resonant term and the non-resonant term, respectively. The phase difference appears in the cross-product of the two terms [20, 23]. Therefore, they can interfere either constructively or destrac-tively with each other. The phase of the non-Hnear signal is related to the direction of the oscillating dipole [24]. For example, the phase factor, is -i-l or -1 for up or down orientations. Often the resonant and non-resonant susceptibilities will interfere and give rise to a more compHcated Hne shape in the spectrum As an example, Fig. 5.3 A shows the variation in peak shape for a resonant peak with phase 9 = —nil (on resonance), as the non-resonant phase varies from [Pg.168]

The term resonance has also been applied in valency. The general idea of resonance in this sense is that if the valency electrons in a molecule are capable of several alternative arrangements which differ by only a small amount in energy and have no geometrical differences, then the actual arrangement will be a hybrid of these various alternatives. See mesomerism. The stabilization of such a system over the non-resonating forms is the resonance energy. 

As it concerns the band in the UV region (at 315 nm in the present case), Benesi and Hildebrand [5] assigned this absorption to a charge-transfer transition, where the phenyl ring acts as an electron donor (D) and the iodine as an electron acceptor. The interaction can be described in resonance terms as D-I2 <-> D+I2", the band being assigned to the transition from the ground non polar state to the excited polar state. 

The RWA in Eq. (4.183) breaks down if the transfer time is similar to or less than the inverse energy separation of the qubit, t < This is a common case for qubits whose resonance frequency is in the microwave (GHz) or radio frequency (RE MHz) range. In such cases, the optimization process must take into account both the dephasing due to the bath as in Eqs. (4.189)-(4.191) and the error due to the non-RWA terms when minimizing the infidelity. 

Measurement of non-linear optical properties [580] also provides a means for characterizing size-quantized semiconductor particles. Third-order optical non-linearity of size-quantized semiconductor particles has been discussed in terms of resonant and non-resonant contributions [11]. Resonant non-linearity is expected to increase with decreasing particle size and increasing absorption coefficients. 

For cyclobutadiene, keeping the K0 term as necessary to describe the background exerted by a skeleton, the average of the two Kekule structures is K0-Ja-Jb. Correspondingly, for benzene, confined also to Kekule structures, the non-resonant part can be conventionally defined as K0-3Ja/2-3Jb/2. In this way we estimate a resonance stabilization for benzene of — 8.56 kcal/mol, comparable with the CASVB calculation [22] giving — 7.4 kcal/mol. Here one should note that with respect to the adopted definition and the method of estimation, the resonance energy is disputed in a very large interval (— 5 to — 95 kcal/mol) [23]. The representation in... 

The only non-trivial term in Eq. (3) is the quantity E which is usually identified with the HMO total jc-electron energy. This can be achieved by formally setting a = 0 and P = 1 and then we speak about total ji-electron energy expressed in the units of the resonance integral p. 

Since the non-linear susceptibility is generally complex, each resonant term in the summation is associated with a relative phase, y , which describes the interference between overlapping vibrational modes. The resonant macroscopic susceptibility associated with a particular vibrational mode v, Xr, is related to the microscopic susceptibility also called the molecular hyperpolarizability, fiy, in the following way... 

Here Xo (2w), x (wiiw2) and Xo (O) are the non-resonant values of the hyperpolarisabilities. Thus second harmonic generation is resonantly enhanced at both the fundamental and the harmonic of the optical transition, sum and difference frequency generation at the fundamentals and the sum and difference frequencies, and the rarely observed optical rectification only at the fundamental frequency. The term 3 in the expansion gives rise to effects such as third harmonic generation, x(3) -3oj oj, oj,u>), electric field induced second harmonic generation, x(3) (- 2w 0,w, oj), the optical Kerr effect, x(3) (-oj oj, oj, -cj), etc. that will display resonances at oj, 2oj and 3u>. 

The electromagnetic wave is not absorbed, but deflected. The electromagnetic wave is almost unchanged by this interaction as is the energy content of the molecule this mode of interaction is termed non-resonant or dispersive. 

The above equations tell us, that the ratio between f disperion and flourescence is proportional to the non-resonant amplitude. Fig. (10) shows that in macromolecular systems the f term may easily exceed fluorescence. 

The connection between the non-resonant nonlinear optical (NLO) response to optical pumping well below the absorption edge and the resonant NLO response following absorption and photo-excitation has been discussed in terms of contributions from virtual soliton pairs enabled by nonlinear zero-point fluctuations in the ground state [82,221,222]. Because of the nonlinear zero-point fluctuations, there are finite matrix elements connecting the ground state with the relaxed state following the creation of a soliton-antisoliton (S-AS ) pair. This... 

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