  In order to increase the accuracy of the linearized augmented plane wave
method (LAPW) we present a new approach where the plane wave basis function is
augmented by two different atomic radial components constructed at two
different linearization energies corresponding to two different electron bands
(or energy windows). We demonstrate that this case can be reduced to the
standard treatment within the LAPW paradigm where the usual basis set is
enriched by the basis functions of the tight binding type, which go to zero
with zero derivative at the sphere boundary. We show that the task is closely
related with the problem of extended core states which is currently solved by
applying the LAPW method with local orbitals (LAPW+LO). In comparison with
LAPW+LO, the number of supplemented basis functions in our approach is doubled,
which opens up a new channel for the extension of the LAPW and LAPW+LO basis
sets. The appearance of new supplemented basis functions absent in the LAPW+LO
treatment is closely related with the existence of the $\dot{u}_l-$component in
the canonical LAPW method. We discuss properties of additional tight binding
basis functions and apply the extended basis set for computation of electron
energy bands of lanthanum (face and body centered structures) and hexagonal
close packed lattice of cadmium. We demonstrate that the new treatment gives
lower total energies in comparison with both canonical LAPW and LAPW+LO, with
the energy difference more pronounced for intermediate and poor LAPW basis
sets.
