  We analytically investigate the one-excitation spin dynamics in a homogeneous
closed spin-1/2 chain via diagonalization of the one-excitation block of the
XX-Hamiltonian, which allows to derive the analytical expressions for
probability amplitudes describing state transfers between any two spins of a
chain. We analytically investigate the $M$-neighbor approximation ($M\ge 1$) of
spin dynamics with arbitrary initial state and analyze its accuracy using
special integral characteristics defined in terms of the above probability
amplitudes. We find $M$ providing the required accuracy of evolution
approximation for chains of different lengths.
