Kleinman symmetry

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

The dispersion coefficients for the mixed-symmetry component 7 5 which describes the deviation from Kleinman symmetry are for methane more than an order of magnitude smaller than coefficients of the same order in the frequencies for 7. Their varations with basis sets and wavefunction models are, however, of comparable absolute size and give rise to very large relative changes for the mixed-symmetry dispersion coefficients. 

For RIKES with circular pump polarization and for IRS, the background interference is surpressed. In the RIKES case, the non-resonant part of x drops out according to Kleinman symmetry 3). In IRS only the imaginary part of x(3) contributes, but all of the probe intensity is admitted to the detector. 

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. 

Kleinman symmetry (index permutation symmetry). Far from resonances of the medium where dispersion is negligible, the susceptibilities become to a good approximation invariant with respect to permutation of all Cartesian indices (without simultaneous permutation of the frequency arguments). This property is called Kleinman symmetry (Kleinman, 1962). It is important in the discussion of the exchange of power between electromagnetic waves in an NLO medium. In many cases approximate validity of Kleinman symmetry can be used effectively to reduce the number of independent tensor components of an NLO susceptibility. 

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, /3 , = holds for the second-order polarizability /3 ,(-2w permutation symmetry in all Cartesian indices (cf. p. 131), generally holds only in the limit w—>0. 

The susceptibility x —2w,o),w,0) is determined in the EFISHG experiment [electric-field-induced second harmonic generation see below (Levine and Bethea, 1974, 1975)]. In order to measure the two independent components and x cxz of this susceptibility, the experiment can be performed under two polarization conditions, the incident IR photons being polarized parallel and perpendicular to the externally applied field (Wortmann et al., 1993). For theoretical treatments see also Andrews and Sherborne (1986), Wagniere (1986) and Andrews (1993). A concentration series finally yields the molar polarizabilities (Kleinman symmetry assumed) through (109),... 

The effective polarizability p was defined in (97). The general result for planar 2 symmetric molecules (42) without the assumption of Kleinman symmetry is shown in (111) and (112),... 

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, = rsi, holds for the second-order polarizability in the second and... 

Table 2 lists the results of calculations of the static first hyperpolarizability. As the perturbation theory expansion shows, for w = 0, Kleinman symmetry holds exactly and the components of the tensor are invariant under permutations of the three co-ordinate indices so that Pz as defined in eqn (4.12) reduces to,... 

Under the assumption of Kleinman symmetry, the hyperpolarizability tensor is fully symmetric and the reduction spectrum simplifies to... 

This expression is only valid for the static field limit, where p (so-called po) is independent of the laser frequency. Following this approach, p is the magnitude of the vectorial hyperpolarizability (p = (p )2 -1- (Py)2 + (P )2with p = p + p,yy + p , aftot assumptiou of the Kleinman symmetry conditions.The frequency-dependent p value is now accessible with the time-dependent density functional theory (TD-DFT). However, and although considerable improvement of this method has been achieved in recent years, the use of TD-DFT for p calculations remains not fully reliable in many cases. 

This approximation is referred to as Kleinman symmetry and is used very often in the description of electro-optic and photorefractive polymers. 

According to Eq. (59), the two independent tensor elements for poled polymers when Kleinman symmetry is valid are given by... 

Far-off from resonances, where there is no energy dissipation in the material, we can use Kleinman symmetry = fij = which reduces the number of independent tensorial components even more ... 

Far off from the resonances, under the assumption that there is no energy dissipation through the nonlinear process, Kleinman symmetry can be used to reduce the number of independent components of yy., [75]. In the case of molecules that belong to the orthorhombic point group, the 21 nonzero components get reduced to only 6 independent components y y, 6y,, and 6y. ... 

Beyond the classical approach, one has to consider the concept of octupolar nonlinearity in order to optimize the NLO response. Thus, it is sometimes convenient to decompose the P tensor into irreducible spherical multipolar components [27, 28]. When Kleinman symmetry applies i.e. under off-resonant conditions, the decomposition for p is as follows. 

One should note that Xsi , Tis > this situation being clearly at variance with that found for linear molecules, where the relation = Xi/ / should be obeyed. Another conclusion is that Kleinman symmetry implying = is not obeyed. As expected, subphthalocyanine bearing the stronger acceptor... 

A frequently-used approximation is to invoke Kleinman symmetry [9] this holds rigorously true for static fields, is reasonably good for low frequencies and is totally unacceptable near a resonance. The Kleinman relations are ... 

As I have said, Sekino and Bartlett [31] were the first to show how to proceed to calculate frequency-dependent hyperpolarizabilities within the TDCPHF approximation. They developed an infinite-order recursive procedure, using density matrices, and, by solving the equations iteratively at each order, could, in principle, calculate any non-linear optical property. Their first application was to H2, FH (the work on FH was analysed in detail in another paper [38]), CH4 and the fluoromethanes. The processes SHG, OR, dc-SHG, dc-OR, IDRI and THG were considered but not all hyperpolarizability components were computed (the assumption of Kleinman symmetry was made). 

This free permutation, known as Kleinman symmetry, is often assumed to hold at low frequencies far from resonance. 

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