Non-Bloch electron states in perfect cubic crystals

Abstract

The formalism allowing for the calculation of the energy and electron density of an entire band of electron states in a crystal from the standing linearwombination-Of-atomic-Orbitals wave functions has been presented.The calculations for an fcc lattice were carried out with a maximum basis of 10 products of the cubic harmonics and spherical Bessel functions and with a maximum basis of 14 products for the sc lattice. A good agreement for the results of the energy with Bloch's theory has been obtained for the critical point as well as for the band limit, whereas the density of states in the energy interval between the critical point and the limit of the band displays sharp oscillations above the Bloch's density of states. %'ith increasing basis the oscillations have a tendency to attenuate. The charge per atom introduced by the entire band of states is equal to about 1.11 for the fcc lattice and slightly over 1.5 for the sc lattice, whereas the charge in Bloch's band is equal to 1 in both cases.

Thomas-Reiche-Kuhn Sum Rule for Electron Transitions in Solids

“…This is so because any coefficient function given in the expression (31) is also an eigenfunction of the Laplacian operator with the eigenvalue equal to κ2 , see [21][22][23][24][25]. The dependence on λ of the expressions (32)-(32b) can be exhibited when higher powers of κ than the second one are taken into account.…”

“…(4), can be represented as the eigenvalues of W according to Eq. (22). Consequently, the component z of the polarization vector in the ordinary space should be replaced by the component Z in the space of the lattice vector.…”

“…Details of the calculations of Α (11) and EC done, for example, for the electron wave functions in cubic lattices in the case when these functions satisfy the standing-like boundary conditions, are given in Refs. [21][22][23][24][25][26].…”

“…The dependence on λ of the expressions (32)-(32b) can be exhibited when higher powers of κ than the second one are taken into account. This dependence is connected with the fact that any of the expressions AΓ1,γ(κ , .R) has its eigenvalue ΕΓ1,λ(κ), so for a large enough κ we have different energies labelled by λ, see [21,22] and Sec. 6.…”

“…In particular, a site located at the origin Ο of the coordinate system can be taken to calculations as representing other sites of a crystal. In this case solutions (31) belonging only to Γ 1 are of importance because solely the solutions of this kind do not vanish at Ο (see [21,22] and Sec. 7).…”