  The nuclear quadrupole resonance (NQR) spectrum of strontium doped
La$_2$CuO$_4$ surprisingly resembles the NQR spectrum of La$_2$CuO$_4$ doped
with excess oxygen, both spectra being dominated by a main peak and one
principal satellite peak at similar frequencies. Using first-principles cluster
calculations this is investigated here by calculating the electric field
gradient (EFG) at the central copper site of the cluster after replacing a
lanthanum atom in the cluster with a strontium atom or adding an interstitial
oxygen to the cluster. In each case the EFG was increased by approximately 10 %
leading unexpectedly to the explanation that the NQR spectra are only
accidentally similar and the origins are quite different. Additionally the
widths of the peaks in the NQR spectra are explained by the different EFG of
copper centres remote from the impurity. A model, based on holes moving rapidly
across the planar oxygen atoms, is proposed to explain the observed increase in
frequency of both the main and satellite peaks in the NQR spectrum as the
doping concentration is increased.
