META-POLITICAL NOTE: <PERSON> WOULD DELETE THIS ANSWER, AS INSUFFICIENTLY UNREFERENCED. DO NOT VOTE FOR <PERSON>
You should understand it in 1d first, because it's essentially a 1d phenomenon. <PERSON> and <PERSON> unfortunately have a fundamental misconception about this, and they say wrong things throughout their paper. Here are the false claims:
* They claim that a line of Na atoms in 1d do not have a purely electronic instability if the ions are clamped in place. This is true, but it's implying falsely that someone expected otherwise. The <PERSON> instability is by definition an ion motion, and it obviously can't happen in a strictly periodic lattice (see below). They say the same thing about the realistic DFT simulations of real CDW materials, that the electrons 'even when prompted by being artificially placed in a periodic charge configuration relax to the symmetric state". Again, this shows a confusion about the origin of CDW configurations.
* They falsely claim "In the <PERSON> picture, lattice distortion is a secondary effect that arises in response to an electronically driven charge redistribution that would occur regardless of whether or not the ions subsequently shift from their high symmetry positions." No.
* The claim "In real materials, the electronic and ionic instabilities always occur simultaneously." This is true, but it is exactly what <PERSON>' argued.
* They falsely claim that the wavevectors which produce experimental CDW's are not those which best line up the Fermi surface with itself (this is your question, see further below).
These confusions all come from one core misunderstanding--- they think the electrons organize themselves asymmetricially first, and then the ions follow along. This is incorrect. The ion positions determine the ground state of the electrons, and it is the reduced energy of the electrons in the deformed position that leads the CDW configuration to form.
Having said that, here is...
Peierls 1d argument
Imagine a line of atoms on integer positions sharing their electrons to make a conduction band. You have exactly one electron per atom, and the k-vectors range from $-\pi$ to $\pi$. 70
The electrons are split in spin 50/50, so you get filling of the states in the first band up to exactly $\pm \pi/2$, and that's the Fermi surface (two points). This is called a half-filled band, since the states are filled duplicately until the exact half-way point.
In this situation there is an interesting phenomenon. Imagine that you deform the lattice by shifting all the atoms in the odd positions to the right by $\epsilon$. The new lattice is strictly only periodic with period 2 now, so the k vector range for the deformed lattice is reduced to $[-\pi/2,\pi/2]$. The band from before is now split into two bands of the longer period system, and most importantly, all the electrons are put in the lower band after the deformation, because the original band was half-filled.
The splitting of the old band is accompanied by lowering the energies of the electrons at the edge of the band, the dispersion relation E(k) for the original band, which was linear at $\pi/2$ before, is now split into two hyperbolas, one higher than the other. If the electrons filled up to $\pi/2 + .001$ instead, this wouldn't lower the energy, because the electrons in the lower band would lose, and the electrons in the upper band would gain a nearly equal amount.
But the Fermi surface is exactly at $\pi/2$, so the electronic energy is reduced by this deformation. The shape of the E(k) function is a hyperbola, and it's integral gives $\epsilon^2\log(\epsilon)$ reduction in energy. The $\log(\epsilon)$ is important, because it means that this will always beat the elastic energy cost of the deformation, which goes as $\epsilon^2$, so that this represents a true instability.
Note that the atomic motion is causing the electrons to reconfigure, so the critique in the paper you give is baseless. The 1d lattice is distorted, and this is what makes the electron energy go down. No distortion, no reduction in energy.
The 1d thing becomes an insulator after the band opens, because you opened a band gap, so you can't conduct electricity by giving electrons momentum. But you can still conduct electricity in a strange way, also described by <PERSON>. 969