Non-linear susceptibilities for use in describing the second-harmonic generation (SHG) recently observed in the hexagonal manganites RMnO3 (R = Sc, Y, Ho, Er, Tm, Yb, Lu) below the Néel temperature TN are derived. Their explicit expressions show that they should give rise to quite different spectra according to whether the magnetic ordering is P6'3cm' or P6'3c'm. The excited states around 2.45 eV are treated as excitons in the antiferromagnetic phase. The calculated SHG spectra are compared with experiments and it is found that they can be used to explain the observed features and spectra satisfactorily.

We present a microscopic model to understand the magnetoelectric and second-harmonic spectra of Cr 2 O 3 and the interference effect between the magnetic ( m ) and electric ( e ) second-order optical polarizations. The spin-orbit interaction and the crystalline fields of correct symmetry around the Cr 3ϩ ion, i.e., the axial and twisted crystalline field of C 3 symmetry, are treated as perturbation on the trigonal states of the d 3 system of Cr 3ϩ . By these treatments we attempt to describe the observed phenomena associated with the transitions between 4 A 2g and 4 T 2g states in Cr 2 O 3 . It is possible to reproduce the observed spectra of the polarization rotation and ellipticity, and the second-harmonic generation spectra described by the nonlinear susceptibilities m and e . The estimated theoretical magnitudes of these quantities are in reasonable agreement with the observed magnitudes.

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We calculated the Madelung energies of both the ground state and excited states in tetrathiafulvalenep-chloranil by taking into account intramolecular charge distribution. The distribution is found to be significant in the neutral-ionic transition. In the ionic phase, the Madelung energy depends more strongly on the intermolecular distance perpendicular to the -stacking chains than on that along the chains. This indicates that simple single-chain models neglecting interchain electrostatic coupling are not adequate. The gain of the Madelung energy due to dimerization is concluded to be small compared with the other structural changes. We also calculated the formation energy of excited state domains, which appear in the primary process of the phase transition. A one-dimensional excited domain has the smallest energy among the possible domains with the same number of molecules when the domain is small, and the energy per molecule is considerably reduced in increasing the domain size. It is consistent with the experimental suggestion that a large number of excitations were generated by only one photon.

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