Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 184-185

Section ABDP and Kleinman symmetries

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: ABDP and Kleinman symmetries

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Intrinsic permutation symmetry, as already discussed, imposes the condition that the nth order susceptibility [\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\omega_2,\ldots,\omega_n)] be invariant under the [n!] permutations of the ([\alpha_i,\omega_i]) pairs as a result of time invariance and causality. Furthermore, the overall permutation symmetry, i.e. the invariance over the [(n+1)!] permutations of the ([\alpha_i,\omega_i]) and ([\mu,-\omega_\sigma]) pairs, may be valid when all the optical frequencies occuring in the susceptibility and combinations of these appearing in the denominators of quantum expressions are far removed from the transitions, making the medium transparent at these frequencies. This property is termed ABDP symmetry, from the initials of the authors of the pioneering article by Armstrong et al. (1962[link]).

Let us consider as an application the quantum expression of the quadratic susceptibility (with damping factors neglected), the derivation of which being beyond the scope of this chapter, but which can be found in nonlinear optics treatises dealing with microscopic interactions, such as in Boyd (1992[link]):[\eqalignno{&\chi^{(2)}_{\mu\alpha\beta}(-\omega_\sigma\semi\omega_1,\omega_2)&\cr&\quad ={Ne^3 \over \varepsilon_o^2\hbar^2}S_T\displaystyle\sum\limits_{abc}\rho_o(a){r_{ab}^{\mu}r_{bc}^\alpha r_{ca}^{\beta}\over (\Omega_{ba}-\omega_1-\omega_2)(\Omega_{ca}-\omega_1)},&\cr&&(}]where N is the number of microscopic units (e.g. molecules in the case of organic crystals) per unit volume, a, b and c are the eigen states of the system, Ωba and Ωca are transition energies, [r_{ab}^\mu] is the μ component of the transition dipole connecting states a and b, and [\rho_o(a)] is the population of level a as given by the corresponding diagonal term of the density operator. ST is the summation operator over the six permutations of the ([\mu, -\omega_\sigma]), ([\alpha, \omega_1]), ([\beta, \omega_2]). Provided all frequencies at the denominator are much smaller than the transition frequencies Ωba and Ωca, the optical frequencies [-\omega_\sigma], [\omega_1], [\omega_2] can be permuted without significant variation of the susceptibility. It follows correspondingly that the susceptibility is invariant with respect to the permutation of Cartesian indices appearing only in the numerator of ([link], regardless of frequency. This property, which can be generalized to higher-order susceptibilities, is known as Kleinman symmetry. Its validity can help reduce the number of non-vanishing terms in the susceptibility, as will be shown later.


Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. (1962). Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939.
Boyd, R. W. (1992). Nonlinear optics. San Diego: Academic Press.

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