Hyperpolarizability first-order

If we neglect pure dephasing, the general tensor element of the third order hyperpolarizability relates to those of the first order polarizability tensor according to... 

The polarizability expresses the capacity of a system to be deformed under the action of electric field it is the first-order response. The hyperpolarizabilities govern the non linear processes which appear with the strong fields. These properties of materials perturb the propagation of the light crossing them thus some new phenomenons (like second harmonic and sum frequency generation) appear, which present a growing interest in instrumentation with the lasers development. The necessity of prediction of these observables requires our attention. 

Trans-polyenes H-(-HC=CH-),, -H, trans-polyenynes H-(HC=CH-C=C) -H, cumulenes H2C=(C=C) =CH2 and polyynes H-(C=C) -H have been studied (M=N-1). For eentrosymmetrie molecules, the first order hyperpolarizability p is equal to zero so that non linear effects are of second order nature. Furthermore, (the x axis goes through the middle of the C-C bonds of the polyenes, or is the intemuclear axis in the case of linear molecules) is the most important component of the second order y hyperpolarizability tensor, the other components being negligible. Both y and the mean hyperpolarizability... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

Xjj is the linear polarizability, f the first-order hyperpolarizability (or second-order polarizability), and the second-order hyperpolarizability (or third-order polarizability), which is the main focus of this chapter. 

For the isotropic average of the first-order hyperpolarizability fi only elements with all three spatial indices different from one another are non zero. sgn(o) is the sign of the permutation o(xyz)=ijk of the spatial indices. For the nonlinear polarization in z-direction for example, the two elements (P)zxy= +Pchiral and (P)Zyx= Pchiral are different from zero. Far from any resonance Kleinman symmetry is valid (Pxyz=Pyzx=PZXy=PyxZ=Pzyx=PXZy) and the terms in the numerator of rra/ cancel each other resulting in (/J)p=0 for all averaged tensor elements of the first-order hyperpolarizability. 

Only little has been reported on second-order hyperpolarizabilities yin two-di-mensionally conjugated molecules. Planar systems as e.g. phthalocyanines have been studied for two photon absorption which is proportional to the imaginary part of the nonlinearity y. For planar molecules with a three-fold symmetry, the importance of charge transfer from the periphery to the center of the molecule in order to realize large nonlinearities ywas reported [65]. Off-resonant DFWM experiments revealed promising third-order nonlinearities in two-dimensional phenylethynyl substituted benzene derivatives [66]. Recently, the advantage of two-dimensional conjugation to increase the values of the first-order hyperpolarizability p has also been pointed out [67-69]. 

A further approach to enhance two-photon absorption deals with donor-acceptor substituted compounds [87] (Fig. 28). An optimization can be achieved if the chromophore is designed for maximum values of the first-order hyperpolarizability fi and for two-photon absorption into the lowest excited state Sj. A second possibility is to consider two-photon absorption into the second excited state S2 and to optimize the molecule for large linear absorption at the same... 

The photoinduced susceptibility shown in Equation 11.14a is the sum of two terms one with exp(-2Dt) (relaxation of the first-order parameter A ) decay and the second with exp(-12D ) (relaxation of the third-order pammeter A3) decay. Hence, the first very rapid decay may contain the fast exp(-12D ) contribution. However, as can be seen from Figure 11.14, the relative magnitude of this initial very fast decay does not depend on the optimization of the intensity ratio between the writing beams. So, this first rapid decay may not be due to the decay of the third-order parameter A3. In addition, because the hyperpolarizability P of DRl is different in the ds and in the trans state, the first very rapid decay also contains a contribution connected with the hferime of the metastable ds form, which is due to molecules coming back to the trans form without any net orientation. A better model would have to account for a distribution of diffusion constants for molecules embedded with various free volumes, which may explain the multiexponential behavior of the decay. 

It is possible to differentiate the quantum-mechanical electronic energy beyond first order, and means for doing this are discussed in Section III. The second derivatives are the usual polarizabilities, the third derivatives are the hyperpolarizabilities, and so on. These properties are associated with a power series expansion of the energy in terms of the elements of V. A second-degree polytensor is introduced for handling all the polarizabilities [7]. It is a square matrix whose rows and columns are labeled, in anticanonical order, by the same indices that label the elements of the column array M. For example. 

Working to similar levels of accuracy, Pawlowski et al have calculated the static and frequency-dependent linear polarizability and second hyperpolarizability of the Ne atom using coupled-cluster methods with first order relativistic corrections. Good agreement with recent experimental results is achieved. Klopper et al.s have applied an implementation of the Dalton code that enables... 

Norman and Jensen27 have implemented a method for obtaining second order response functions within the four component (relativistic) time-dependent Hartree-Fock scheme. Results are presented for the first order hyperpolarizabilities for second harmonic generation, />(—2o o),o ) for CsAg and CsAu. A comparison of the results with those of non-relativistic calculations implies that the nonrelativistic results are over-estimated by 18% and 66% respectively. In this method transitions that are weakly-allowed relativistically can lead to divergences in the frequency-dependent response, which would be removed if the finite lifetimes of the excited states could be taken into account. 

N- and N, (9-heterocycles as second-order nonlinear optical chromophores with both large first-order molecular hyperpolarizability and good transparency 03MI108. 

Just as an explicit formula for the linear polarizability is identified from the linear polarization, we are able to retrieve the corresponding formula for the first-order hyperpolarizability from the second-order polarization. We obtain the second-order polarization from Eq. (35) by insertion of the first-order and second-order... 

Before closing the derivation of the first-order hyperpolarizability, we wish to remove the apparent divergences of Eq. (53) in the limit of non-oscillating perturbing... 

The first-order hyperpolarizability has both single and double residues. From Eq. (53), we see that one of the first-order residues becomes... 

In tills very general formula we are using the notation of Bishop [5], where conventionally (linear polarizability), (first-order hyperpolariz-... 

The explicit formula for the first-order hyperpolarizability thus becomes d (kE,E)... 

Figure 8. Dispersion of the first-order hyperpolarizability /3(—Wq. Wi, (-Oi in die two-states model...

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