  Gutzwiller wavefunction is a physically well motivated trial wavefunction for
describing correlated electron systems. In this work, a new approximation is
introduced to facilitate evaluation of the expectation value of any operator
within the Gutzwiller wavefunction formalism. The basic idea is to make use of
a specially designed average over Gutzwiller wavefunction coefficients expanded
in the many-body Fock space to approximate the ratio of expectation values
between a Gutzwiller wavefunction and its underlying noninteracting
wavefunction. To check with the standard Gutzwiller approximation (GA), we test
its performance on single band systems and find quite interesting properties.
On finite systems, we noticed that it gives superior performance than GA, while
on infinite systems it asymptotically approaches GA. Analytic analysis together
with numerical tests are provided to support this claimed asymptotic behavior.
At the end, possible improvements on the approximation and its generalization
towards multiband systems are illustrated and discussed.
