  Positive-parity spin-1/2 excitations of the nucleon are explored in lattice
QCD. The variational method is used in this investigation and several
correlation matrices are employed. As our focus is on the utility and
methodology of the variational approach, we work in the quenched approximation
to QCD. Various sweeps of Gaussian fermion-field smearing is applied at the
source and at the sink of $\chi_{1}\bar\chi_{1}$ and $\chi_{1}\chi_{2}$
correlation functions to obtain a large basis of operators. Using several
different approaches for constructing basis interpolators, we demonstrate how
improving the basis can split what otherwise might be interpreted as a single
state into multiple eigenstates. Consistency of the extracted excited energy
states are explored over various dimensions of the correlation matrices. The
use of large correlation matrices is emphasized for the reliable extraction of
the excited eigenstates of QCD.
