  For a given quantum impurity model, Wilson's numerical renormalization group
(NRG) naturally defines a NRG Hamiltonian whose exact eigenstates and
eigenenergies are obtainable. We give exact expressions for the free energy,
static, as well as dynamical quantities of the NRG Hamiltonian. The dynamical
spectral function from this approach contains full excitations including intra-
and inter-shell excitations. For the spin-boson model, we compare the spectral
function obtained from the present method and the full density matrix (FDM)
method, showing that while both guarantee rigorous sum rule, the full
excitation approach avoids the causality problem of FDM method.
