  To treat effects of electron correlations in geometrically frustrated
pyrochlore and checkerboard lattices, an extended single-orbital Hubbard model
with nearest neighbor hopping $\sim t$ and Coulomb repulsion $\sim V$ is
applied. Infinite on-site repulsion, $U\to\infty$, is assumed, thus double
occupancies of sites are forbidden completely in the present study. A
variational Gutzwiller type approach is extended to examine correlations due to
short-range $V-$interaction and a cluster approximation is developed to
evaluate a variational ground state energy of the system. Obtained analytically
in a special case of quarter band filling appropriate to LiV$_2$O$_4$, the
resulting simple expression describes the ground state energy in the regime of
intermediate and strong coupling $V$. Like in the Brinkman-Rice theory based on
the standard Gutzwiller approach to the Hubbard model, the mean value of the
kinetic energy is shown to be reduced strongly as the coupling $V$ approaches a
critical value $V_{c}$. This finding may contribute to explaining the observed
heavy fermion behavior in LiV$_2$O$_4$.
