We present a scheme for calculating the electronic structure of disordered alloys, self-consistent in the local-density-approximation sense. It is based on expanding the one-electron Green's function in the basis of modified atomic orbitals [H. Eschrig, Optimized LCAO Method and the Electronic Structure of Extended Systems (Springer, Berlin, 1989)]. The two-terminal approximation introduced for the Hamiltonian and the overlap matrix permits us to treat both the diagonal and off-diagonal disorder using an extension of the Blackman-Esterling-Berk form of the coherent-potential approximation (CPA) [Phys. Rev. B 4, 2412 (1971)] to a nonorthogonal basis set. Calculations using the scalar relativistic density functional for the magnetic binary transition-metal alloys Fe-Co, Fe-Pt, Co-Pt, and for the ternary alloy Al-Fe-Mn give results comparing well with experimental data and calculations based on the Korringa-Kohn-Rostoker (KKR)-CPA and linear muffin-tin orbital-CPA techniques.

Triangular Heisenberg antiferromagnets are prototypes of geometric frustration, even if for nearest-neighbor interactions quantum fluctuations are not usually strong enough to destroy magnetic ordering: stronger frustration is required to stabilize a spin-liquid phase. On the basis of static magnetization and electron spin resonance measurements, we demonstrate the emergence of ${\tilde{j}}_{\text{eff}}=\frac{1}{2}$ moments in the triangular-lattice magnet ${\mathrm{Na}}_2\mathrm{Ba}\mathrm{Co}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$. These moments are subject to an extra source of frustration that causes magnetic correlations to set in far above both the magnetic ordering and Weiss temperatures. Corroborating the ${\tilde{j}}_{\text{eff}}=\frac{1}{2}$ ground state, theory identifies ferromagnetic Kitaev exchange anisotropy as an additional frustrating agent, altogether putting forward ${\mathrm{Na}}_2\mathrm{Ba}\mathrm{Co}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ as a promising Kitaev spin-liquid material.

The Blackman-Esterling-Berk (BEB) theory, generalized to nonorthogonal basis sets and complicated elementary cells [K. Koepernik, B. Velický, R. Hayn, and H. Eschrig, Phys. Rev. B 55, 5717 (1997)], is inspected in great detail with respect to its analytic properties and numerical accuracy. We give a proof of the single impurity limit fulfilled by BEB theory, thus clarifying some critical points in the derivation of the theory. We undertake extensive numerical tests for the BEB coherent-potential approximation (CPA) to check the range of applicability of the included single-site approximation. The BEB CPA turns out to give good results in a wide range of energy parameters.