  The Gutzwiller approximation (GA) for Gutzwiller-projected grand canonical
wave functions with fugacity factors is investigated in detail. Our systems in
general contain inhomogeneity and local magnetic moments. In deriving
renormalization formulae, we also derive or estimate terms of higher powers of
intersite contractions neglected in the conventional GA. We examine several
different constraints, i.e., local/global spin-dependent/independent
particle-number conservation. Out of the four, the local spin-dependent
constraint seems the most promising at present. An improved GA derived from it
agrees with the variational Monte Carlo method better than the conventional GA
does. The corrections to the conventional GA can be interpreted as two-site
correlation including the phase difference of configurations. Furthermore,
projected quasi-particle excited states are orthogonal to each other within the
GA. Using these states, spectral weights are calculated. We show that asymmetry
between electron addition and removal spectra can appear by taking into account
the higher powers of the intersite contractions in the case of the d-wave
superconductors and the Fermi sea; the addition is smaller than the removal.
However, the asymmetry is quite weak especially near the Fermi level. In
contrast, projected s-wave superconductors can have the opposite asymmetry
(addition larger than removal) especially near the Fermi level. In addition,
formulae from the other three constraints are also derived, which may be useful
depending on purposes.
