Basis states and collective excitations of nonlocal three-spin polarons over the singlet phase of the CuO$_2$ plane

An orthonormalized basis describing the single-hole states of the CuO$_2$ plane has been constructed exactly taking into account the $p$–$d$ exchange coupling between the spins of a hole and the two nearest copper ions. The introduced basis makes it possible to include the effects caused by the dependence of the Heisenberg interaction between the nearest copper ions on the occupation of the intermediate oxygen ion by the hole in the calculation of the spectrum of the collective excitations of nonlocal three-spin polarons. The inclusion of this correlation, as well as the nonlocality of the $p$–$d$ exchange coupling, is the main difference of the developed approach to the description of the energy structure from the calculations with the use of the Zhang–Rice singlet.