Velocity matrix and non-local pseudo potentials It is known that velocity of bloch wave functions are related to band energy derivatives:
$$v(k)=\frac{1}{\hbar}\frac{\partial \epsilon}{\partial k}$$
However, in the following paper, it is given that the above relation is not true if I am using ab initio methods with non-local pseudo potentials.
http://link.aps.org/doi/10.1103/PhysRevB.47.1651
(the second to last paragraph in page 1653)
However, I cannot find any reference about this. Can anyone explain this to me or guide me to a proper reference?
 A: The justification of interpreting the energy band gradient with the velocity comes from semiclassics, and it turns out that the formula you give is correct only in certain circumstances and only to leading order. To attribute these terms to multiparticle effects is misleading, they are due to neglecting transitions to other bands, and they occur even if you make the single electron approximation. 
Depending on the situation, you obtain additional terms involving the Berry curvature (aka the “anomalous velocity term”) or what some authors dub the “piezocurvature” (which is what King-Smith and Vanderbilt's paper is about). Mathematically speaking, these additional terms are due to geometric (or Berry) phases. 
While the notion of Berry curvature came up in the 1980s, the first results are due to Blount in the early 1960s (see e. g. [1,2]). One standard technique to justify semiclassical limits is the one by Sundaram and Niu by means of a variational principle [3]. If you are more mathematically minded, have a look at the work by Panati, Spohn and Teufel [4] as well as references therein, and Panati, Spohn and Sparber [5]. The latter deals with the polarization, and a justification of the King-Smith-Vanderbilt formula. 
[1] Blount, E.I.: Formalisms of band theory. In: Solid State Physics 13, New York: Academic Press, 1962, pp. 305–373 
[2] Blount, E.I.: Bloch electrons in a magnetic field. Phys. Rev. 126, 1636–1653 (1962)
[3] Sundaram, G., Niu, Q.: Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915–14925 (1999)
[4] Panati, G., Spohn, H. and Teufel, S.: Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond. Commun. Math. Phys. 242, 547–578 (2003)
[5] Panati, G., Sparber, C. and Teufel, S.: Geometric Currents in Piezoelectricity. Arch. Rational Mech. Anal. 191, 387–422 (2009)
