Hyperpolarizability units

Table 4 Comparison of various ab initio results and experimental estimates for the dispersion coefficients of the electronic hyperpolarizabilities 7jj and 7 of methane. (All results in atomic units. Results for the dispersion coefficients refer to single point calculations at the equilibrium geometry. Where available, dispersion coefficients for the vibrational average are given in parentheses.)...

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

Before closing this section, it is worth mentioning that the hyperpolarizability tensors are complex quantities usually given in the old cgs system of units of esu (electrostatic units). The transformation into the International System is readily obtained with the relationship ... 

The computed 6-31G values for the longitudinal polarizability and longitudinal hyperpolarizability per unit cell are given in Table 7. It can be observed that the longitudinal polarizability and longitudinal hyperpolarizability increase with the chain length. However, the rate of variation of these magnitudes decreases with N, in such a way that ay/N... 

TABLE 7. Static longitudinal polarizabilities oil (in a.u.) and longitudinal hyperpolarizabilities yl (in 104 a.u.) per unit for linear C2nH2 +2 polyenes"... 

The ethynyl-linked complexes 105 were prepared and explored as potential building blocks for nonlinear optical (NLO) materials.129 Spectroscopic and cyclic voltammetry data indicate a small but real interaction between the ferrocenyl donor group and the borabenzene unit, increasing in the order RuHyper-Rayleigh scattering revealed small values for the first hyperpolarizability / , which increases in the same order. 

It should be noted that polarizabilities of various orders can be defined in an alternative way in the SI system of units to that discussed previously. A quantity having the dimension of volume a = a/47re0 can be considered to be an SI analogue of the cgs polarizability. Analogously, y = -y/47re0 (or y = y/eo) can be used as the third-order hyperpolarizability in the SI system, with y having the units of m5 V 3. The presence or absence of the factor of An in the definition of the hyperpolarizability is, unfortunately, not always obvious in literature data. 

Figure 5. Effect of complex solute hyperpolarizability on the concentration dependence of the THG intensity. For all curves 72 /71 = 16. Solid line 72"/Y2 = 0 dashed line 727y2 = 0.25 dotted line 72772 = 0.5 dot-dashed line 72772 = 1.0. THG intensity is in arbitrary units.

The molecular hyperpolarizabilities are / , 7, and a is the molecular polarizability. Typical values of / are 10 30 esu (esu units mean that the dimensions are in CGS units and the charge is in electrostatic units, thus / in esu means / in units of cmzesuz /erg2) [1-4]. For an electric field typical of Q-switched laser light, 104 statvolts/cm, the contribution to - //(0) from /3S2 is 10 4 debye. These polarizations are infinitesimal on the scale of our usual chemical thinking. Yet, these small polarizations are responsible for the exotic effects described throughout this volume. The perturbation theory approach used to describe these properties is justified by the fact that so little charge actually moves. 

The prototypical organic material with high P (all values of P quoted here are for 1.06 1 light unless noted) is p-nitroaniline (PNA). In electrostatic units (esu), P for PNA is 34.5 x 10 30 esu. (37) It has a dipole moment of 6.8 D. Extension of the chromophore by insertion of a single double bond to produce 4-amino-4 -nitrostilbene increases P dramatically 248 x 10"30 esu. (38) However, both crystalline materials are centrosymmetric, and so no SHG is possible. Most achiral molecules crystallize in centrosymmetric space groups, but some do not. Several materials have been discovered during the last few years which have combinations of hyperpolarizability, crystal growth and habit, and linear spectral properties which make them candiates for useful NLO materials. As will be obvious from the comparison, considerable compromise must be made to yield a practical material. 

Another issue under investigation is the evolution of the second-order hyperpolarizability upon elongation of the 7r-electron path in a molecular backbone. Measurements and quantum-chemical calculations provide an empirical power law of the second-order hyperpolarizability y versus conjugation length (or the number of monomer units n) in short oligomers. 

The THG experiments of the pure PTA (TMS) series show a power law increase of the second-order hyperpolarizability with an exponent a=2.52 0.10 for the oligomers n=1 to 6. The polydisperse PTA polymer samples exhibit a constant second-order hyperpolarizability per monomer unit of y/ =4.1xl0 48 m5/V2 (300x10 36 esu) indicating an only linear increase for longer PTAs. The... 

Fig. 26. Power law scaling of the second-order hyperpolarizabilities as a function of the number of monomer units for the pure (TES) PTAs. Data from THG, DFWM, and quantum chemical calculations (VEH/SOS) are given and follow the same general trend with the power law exponents close together...

The nonlinearities measured by DFWM reveal a smooth saturation of the second-order hyperpolarizability around ten monomer units as well (Fig. 26). Below the saturation, the experiments show a power law dependence of the second-order hyperpolarizabilities with an exponent a=2.64 0.20. Comparing the exponents for DFWM and THG, the difference is small and within the experimental error. The absolute values of the second-order hyperpolarizabilities are similar with an increasing deviation for longer PTAs. 

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