  We develop an iterative diagonalization scheme in solving a one-body
self-consistent-field equation in the transcorrelated (TC) method using a
plane-wave basis set. Non-Hermiticity in the TC method is well handled with a
block-Davidson algorithm. We verify the required computational cost is
efficiently reduced by our algorithm. In addition, we apply our
plane-wave-basis TC calculation to some simple $sp$-electron systems with deep
core states to elucidate an impact of the pseudopotential approximation to the
calculated band structures. We find a position of the deep valence bands is
improved by an explicit inclusion of core states, but an overall band structure
is consistent with a regular setup that includes core states into the
pseudopotentials. This study offers an important understanding for the future
application of the TC method to strongly correlated solids.
