  In this set of three papers, we present the results of a combined study by
density-functional (LDA) band theory (NMTO) and angle-resolved photoemission
spectroscopy (ARPES) of lithium purple bronze, 2(Li$_{1x}$Mo$_{6}$O$_{17}$).
This material is particularly notable for its unusually robust
quasi-one-dimensional (quasi-1D) behavior. The band structure, in a large
energy window around the Fermi energy, is basically 2D and formed by three Mo
$t_{2g}$-like extended Wannier orbitals (WOs) per cell, each one giving rise to
a 1D band running at a 120$^{\circ }$ angle to the two others. A structural
"dimerization" from $\mathbf{c}/2$ to $\mathbf{c}$ gaps the $xz$ and $yz$ bands
while leaving the $xy$ bands metallic in the gap but resonantly coupled to the
gap edges and, hence, to the two other directions. The resulting complex shape
of the quasi-1D Fermi surface (FS), verified by our ARPES, thus depends
strongly on the Fermi energy position in the gap, implying a great sensitivity
to Li stoichiometry of properties dependent on the FS, such as FS nesting or
superconductivity. The band structure, expressed as a six-band, analytical
tight-binding (TB) Hamiltonian, is verified in detail by the recognition and
application of an ARPES selection rule that enables, for the first time, the
separation in ARPES spectra of the two barely split $xy$ bands and the
observation of their complex split FS. The strong resonances prevent either a
two-band TB model or a related real-space ladder picture from giving a valid
description of the low-energy electronic structure. Down to a temperature of
6$\,$K we find no evidence for a theoretically expected downward
renormalization of perpendicular single particle hopping due to LL fluctuations
in the quasi-1D chains.
