  In order to explain the reason that all the sythesised \ce{AX2MQ6}
chalcogenides compounds are phase-matching while isomorphic chalcogenides
compounds \ce{AX4M5Q12} are not phase-matched materials. The linear optical
property birefringence and non-linear optical property of $R3$ space group
crystals \ce{AX2MQ6} have been calculated to compare with \ce{AX4M5Q12}. Their
Electron Localization Function(ELF) was calculated to show that the acentricity
of the bonds on $sp^3$ hybridization \ce{Q^2-} ions is the main origin of
optical anisotropy. To quantificat the acentricity, a geometric parameter
dihedral angle between tetrahedral undersides and xy-planes was defined. We
find theoretically the birefringence depend on the defined geometric parameter
and ion radius. This relation between birefringence and structure resonalbly
explain the abnormal difference of birefringence of isostructural \ce{AX2MQ6}
and \ce{AX4M5Q12} and useful for exploring new phase-matching IR-NLO crystals.
