Second-order distortions field

In the expressions (184) and (184b) the second, temperature-dependent term defines the Born effect due to superposition of the two non-linear processes of second-order distortion and reorientation of permanent dipole moments in the electric field. Buckingham et al. determined nonlinear polarizabflities If and c for numerous molecules by Kerr effect measurements in gases as a function of temperature and pressure. It is here convenient to use the virial expansion of the molar Kerr constant, when the first and second virial coefficients Ak and Bk result immediately from equations (177), (178), and (184). Meeten et al. determined nonlinear molecular polarizabilities by measuring K in liquids as a function of temperature. 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

An applied electric field can also change a material s linear susceptibility, and thus its refractive index. This effect is known as the linear electro-optic (LEO) or Pocket s effect, and it can be used to modulate light by changing the voltage applied to a second-order NLO material. The applied voltage anisotropically distorts the electron... 

Second-order optical nonlinearities result from introduction of a cubic term in the potential function for the electron, and third-order optical nonlinearities result from introduction of a quartic term (Figure 18). Two important points relate to the symmetry of this perturbation. First, while negative and positive p both give rise to the same potential and therefore the same physical effects (the only difference being the orientation of the coordinate system), a negative y will lead to a different electron potential than will a positive y. Second, the quartic perturbation has mirror symmetry with respect to a distortion coordinate as a result, both centrosymmetric and noncentrosymmetric materials will exhibit third-order optical nonlinearities. If we reconsider equation 23 for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric), we see that polarization at a given point in space is ... 

Such second-order molecular properties as spin-spin coupling depend upon distortion of electron clouds by additional external perturbations that is, in the NMR experiment they depend upon the electronic motion induced by an applied magnetic field. Theories for such second-order molecular properties require a study of the change in the molecular-orbital wavefunctions, which may be found by using a perturbation method to describe the effects occurring when a magnetic field is applied.8-1065-67... 

All of this assumes that the proton in question is only coupled to other protons that are far away in chemical shift, so that its coupling pattern is simple ( first order or weak coupling ). If it is coupled to nearby peaks, distortions of the peak intensities and more complex patterns can result, and this effect is strongest at lower field strengths ( second order or strong coupling ). To state this more precisely, the 7 coupling in hertz between two spins must be much less than the chemical shift difference in hertz to see a simple first-order pattern. We could write this as... 

The method of symmetric points was used to determine the center of the interference curve. Extensive calculations showed that the line profile should be symmetric about the center frequency. The line center was then corrected for the second order Doppler shift, The Bloch-Siegert and rf Stark shifts, coupling between the rf plates, the residual F=1 hyperfine component, and distortion due to off axis electric fields. A small residual asymmetry in the average quench curve was attributed to a residual variation of the rf electric field across the line and corrected for on the assumption this was the correct explanation. Table 1 shows the measured interval and the corrections for one of the 8 data sets used to determine the final result. 

Equation (1-239) relates the interaction-induced part of the dipole moment of the complex AB to the distortion of the electron density associated with the electrostatic, exchange, induction, and dispersion interactions between the monomers. The polarization contributions to the dipole moment through the second-order of perturbation theory (A/a, A/a, and A/a ) have an appealing, partly classical, partly quantum, physical interpretation. The first-order multipole-expanded polarization contribution (F) is due to the interactions of permanent multipole moments on A with moments induced on B by the external field F, and vice versa. The terms... 

Second-order perturbation theory provides expressions for the polarization or induction energy. This is the attractive energy term arising from the distortions of the charge density of each molecule due to the field arising from the other (undistorted) molecule ... 

An atom or molecule distorts under the action of an external field, the measure of distortion being expressed through a second-order electrical quantity called the (dipole) polarizability a, which we define in terms of a transition moment /q from state i//0 to t/q and an excitation energy s, as ... 

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