  We reconsider the procedure of calculation of fermion-boson vertices and
numerical solution of Bethe-Salpeter equations, used in non-local extensions of
dynamical mean-field theory. Because of the frequency dependence of vertices,
finite frequency box for matrix inversions is typically used, which often
requires some treatment of asymptotic behaviour of vertices. Recently [Phys.
Rev. B 83, 085102 (2011); 97, 235140 (2018)] it was proposed to split the
considered frequency box into smaller and larger one; in the smaller frequency
box the numerically exact vertices are used, while beyond this box asymptotics
of vertices are applied. Yet, this method requires numerical treatment of
vertex asymptotics (including corresponding matrix manipulations) in the larger
frequency box and/or knowing fermion-boson vertices, which may be not
convenient for numerical calculations. In the present paper we derive the
formulae which treat analytically contribution of vertices beyond chosen
frequency box, such that only numerical operations with vertices in the chosen
small frequency box are required. The method is tested on the Hubbard model and
can be used in a broad range of applications of non-local extensions of
dynamical mean-field theory.

Strongly Correlated Electrons