Abstract：Multi-band k.p theory is often implemented with the one electron Schr？dinger equation without spin(single group) as the un-perturbed system. The effect of spin is taken into account by considering basis functions formed by a direct product between single group eigenstates and spinor states (which give rise to the adapted double group basis after a unitary transformation), with the spin-orbit interaction also treated as a perturbation. The k.p perturbation between these states is calculated using the single group basis functions. This approach leads to a one-to-one link between occurrence of basis states in the single group with those under the double group classification, placing constraints on the adapted double group basis. This talk considers energy eigenstates which form basis of irreducible representations (IRs) of the double group, and derives the direct and remote (L？wdin term) interaction matrices between the states using perturbation theory and symmetry properties of crystal lattice. The use of general double group basis functions removes the constraints placed on the adapted double group basis under the single group formulation. Together with a change of paradigm in constructing atomic site wave functions using hybridised orbitals (rather than atomic orbitals), it allows direct contributions from d and higher orbitals to the valence band with additional interaction matrices permitted by symmetry. A full description of interactions between states of Gamma_8^\pm IRs require two linearly independent matrices and two scaling constants rather than the single matrix and scaling constant under single group consideration. This formulation is developed from both perturbation theory and method of invariant approaches utilising Wigner-Eckart theorem and other group theoretical techniques for calculation of matrix elements. Crystals with diamond lattice is investigated first with results for zincblende lattice obtained under the compatibility relation between the Oh and Td groups. We show that a unitary transformation of the Gamma_8^- basis of the Oh group is required before they can be used in Gamma_8 IR of the Td group. Consequently, existing data and optical transition selection rules shows that the symmetry assignment of the zone centre conduction band edge state should be Gamma_6^- (Gamma_7) in Ge (GaAs and other semiconductors with zincblende lattice) with spin orbit split-off band as origin. In addition to the new interaction matrix between states of Gamma_8^\pm (Gamma_8) IRs, the form of interband Lo \umlaut wdin term between Gamma_8^+ (Gamma_8) and Gamma_7^+ (Gamma_7 ) in the Hamiltonian used in the literature is shown to be incorrect. A linear k term between the degenerate valence band, different from those obtained previously, is shown to exist. It modifies the dispersion and density of state in the vicinity of Gamma point but does not lift the Krammer's degeneracy. When quantum well, wires, and dots are considered, operator ordering in the remote interaction emerges naturally by treating wave vector as operators on the envelope functions. This differs from previous schemes based on single group formulation and a new term, arising from interfacial symmetry breaking, is identified in the valence band Hamiltonian coupling the degenerate heavy hole states.