Method of induced corepresentations and an invariant expression for the energy in the lattice dynamics problem

The method proposed in the preceding paper for inducing irreducible representations (IR) and irreducible corepresentations (ICR) for the local position symmetry groups on representations and corepresentations of the groups Φ and Φ + KΦ is made specific for the case of atomic displacements. The main differences between this scheme and the customary vibration-analysis proposed by E. Wigner for molecules and G. Ya. Lyubarskii for crystals boil down to: 1) The initial local unit vectors in the first position are chosen arbitrarily while the unit vectors in the other positions are obtained by applying the elements of group Φ to the initial unit vectors. The directions of the unit vectors, therefore, depend on the positions, and the construction of the basis vectors of the ICR of the group Φ + KΦ is simplified substantially. 2) Only individual irreducible components of the vector representation of the local group, and not the whole representation at once, are induced. The composition of the vector corepresentation V of the group Φ + KΦ, therefore, can be easily determined from the generalized Frobenius theorem whereas V itself is obtained in partially reduced form. 3) It is shown that in the eigenfrequency problem it is sufficient to find only those parts of the unitary matrix R2 which relate the vectors of the first row of the ICR to the local unit vectors. 4) Initially the energy is written in terms of projections onto specially introduced unit vectors. Using the matrix R2 we transform the energy into a form which contains displacement coordinates symmetrized with respect to the group Φ + KΦ, i.e. projections onto the basis vectors of the ICR of the group Φ + KΦ. The resulting dynamical matrix thus differs significantly from that in the generally accepted theory. The general Hamiltonian is separated into the parts H(F,K), each of which is characterized by an ICR star and is related to only one system of equivalent irreducible corepresentations (F is the type of equivalency and K is the star symbol).

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