Td_Group_Adapted_Dou.zip 

Point Groups typical to cubic crystals are Tetrahedral (Td) in Zinc-blende and Octahedral (Oh = i x Td) in Diamond.  Symmetry operations give rise to the widespread application of Group Theory most notably to generate basis functions which transform according to irreducible representations.  Much work has been accomplished using Single Group basis, compatible with integer |L,mL> eigenstates of SO(3) however the incorporation of spin may be done so through either the adapted double group |L,mL> x {↑,↓} or the double group |J,mJ> [1].

In this maple worksheet the transformation of adapted double group into double group basis functions is done from the point of view of a unitary transformation of representation matrices [2].  The key result is the difference between the unitary matrix transforming Γ₄^- x Γ₆⁺ into Γ₈^- + Γ_6^- and Γ₅⁺ x Γ₆⁺ into Γ₈⁺ + Γ₇⁺ [3].  Furthermore the realisation of factors √3/5 and √2/5 in d_3/2 and d_5/2 adapted double group basis bridges the gap with the double group approach.

This work was done at Imperial College London 2011 by Robert Ward - robert.ward04@imperial.ac.uk

[1] Onodera & Okazaki - Journ. Phys. Soc. Jap. (21) 11 2400-2408 1966

[2] Cracknell & Joshua - Proc. Camb. Phil. Soc. (67) 647-656 1970

[3] EWZ - Phys Rev B (83) 165210 1-23 2011