  The recently proposed diagrammatic expansion (DE) technique for the full
Gutzwiller wave function (GWF) is applied to the Anderson lattice model (ALM).
This approach allows for a systematic evaluation of the expectation values with
GWF in the finite dimensional systems. It introduces results extending in an
essential manner those obtained by means of standard Gutzwiller Approximation
(GA) scheme which is variationally exact only in infinite dimensions. Within
the DE-GWF approach we discuss principal paramagnetic properties of ALM and
their relevance to heavy fermion systems. We demonstrate the formation of an
effective, narrow $f$-band originating from atomic $f$-electron states and
subsequently interpret this behavior as a mutual intersite $f$-electron
coherence; a combined effect of both the hybridization and the Coulomb
repulsion. Such feature is absent on the level of GA which is equivalent to the
zeroth order of our expansion. Formation of the hybridization- and
electron-concentration-dependent narrow effective $f$-band rationalizes common
assumption of such dispersion of $f$ levels in the phenomenological modeling of
the band structure of CeCoIn$_5$. Moreover, we show that the emerging
$f$-electron coherence leads in a natural manner to three physically distinct
regimes within a single model, that are frequently discussed for 4$f$- or 5$f$-
electron compounds as separate model situations. We identify these regimes as:
(i) mixed-valence regime, (ii) Kondo-insulator border regime, and (iii)
Kondo-lattice limit when the $f$-electron occupancy is very close to the $f$
electrons half-filling, $\langle\hat n_{f}\rangle\rightarrow1$. The non-Landau
features of emerging correlated quantum liquid state are stressed.
