Kleinman symmetry conditions

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. 

The number of independent components for the case of Kleinman symmetry conditions is given in parentheses. 

The nonlinear susceptibility tensor can then be represented as a 3 X 6 matrix containing 18 elements. In the transparent region, i.e. outside of absorption bands, one may assume the validity of the Kleinman symmetry condition, vv hich states that the indices /, /, k may be freely permuted ... 

Under Kleinman symmetry conditions, dn = dn, and therefore the d tensor takes the form... 

In the case of third harmonic generation, the frequencies of the input beams are the same (coi = CO2 = CO3) and the tripled beam has a frequency CO4 = 3coi. Therefore, for the THG process, Xyu should be invariant when the relative positions ofjkl indices change. Consequently, the number of independent elements of the susceptibility tensor for the THG process I is reduced to just four %xxxx, Xzzzz, Xxxzz, and Xzzxx-Furthermore, when Kleinman symmetry conditions are adopted for experiments performed far from the resonances, one would have x xzz = Xzzxx> resulting in only three independent elements [206]. 

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),... 

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. 

In deriving Eq. (48) one ignores the anisotropy of the linear part of the dielectric constant and one assumes that Kleinman s symmetry is valid. Under these conditions, using group theory to reduce the number of non-vanishing tensor elements, the electro-optic tensor for poled polymers is given by... 

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