  We introduce in detail our newly developed \textit{ab initio} LDA+Gutzwiller
method, in which the Gutzwiller variational approach is naturally incorporated
with the density functional theory (DFT) through the "Gutzwiller density
functional theory (GDFT)" (which is a generalization of original Kohn-Sham
formalism). This method can be used for ground state determination of electron
systems ranging from weakly correlated metal to strongly correlated insulators
with long-range ordering. We will show that its quality for ground state is as
high as that by dynamic mean field theory (DMFT), and yet it is computationally
much cheaper. In additions, the method is fully variational, the charge-density
self-consistency can be naturally achieved, and the quantities, such as total
energy, linear response, can be accurately obtained similar to LDA-type
calculations. Applications on several typical systems are presented, and the
characteristic aspects of this new method are clarified. The obtained results
using LDA+Gutzwiller are in better agreement with existing experiments,
suggesting significant improvements over LDA or LDA+U.
