why pseudo wave functions can be used to calculate berry connection Berry connection plays a very important role in topological insulators. Berry connection $A(k)$ is defined to be $i\langle u(k)|\nabla_k|u(k)\rangle$, where $|u(k)\rangle$ is the periodic part of Bloch wave function.
I have seen a lot of ab initio computations of berry connection related quantities in literature. However, they all use pseudo wave functions to compute berry connection. To be more clear, they use ab initio computing softwares such as vasp to obtain wave functions. But most ab initio computing softwares are using pseudo potentials so the obtained wave functions are not real ones. It seems berry connection computed this way can hardly reproduce real material properties, but researchers are still doing this.
Does this practice have any theoretical support?
 A: Within projected augmented wave (PAW) method it is incorrect to use the smooth pseudo wave function directly to obtain integrals. The same applies for ultra-soft pseudo potentials, but I will proceed discussing about PAW. The results can indeed be wrong. For example, for coin-metals, the d-band smooth wave functions $\tilde{\Psi}$ can have norms of 0.2-0.4, which will cause serious errors in some Coulomb integrals, unless they are accounted for.
Within PAW formalism, these so called PAW-correction terms are exactly defined. It means that they are exact up to frozen core approximation, finite set of partial waves within the augmentation sphere and finite plane-wave cut-off, since PAW method has a variational partial wave basis within the augmentation sphere and all-electron wave functions are defined as functions of the pseudo wave functions.
In VASP manual it says
VASP (4.6 and higher) is able to calculate the macroscopic electronic
polarization of an insulating system through the evaluation of the Berry
phase expressions of the "Modern theory of polarization" [85], as modified
for the application to USPP's and PAW datasets [86].

And the reference 86, points to an article which has these corrections defined:
http://arxiv.org/abs/cond-mat/9801177
They are usually called corrections within PAW method too, even it actually means that one calculates some integrals exactly, and whatever is the difference between just evaluating the quantity from pseudo wave function and the all-electron wave function is called the correction.
So the answer to your question: "Why pseudo wave functions can be used to calculate X?" is 
In PAW formalism: "Because the exists a linear mapping between the all-electron wave function and the pseudo-wave function, which can be used to obtain numerically attractive exact expressions for X."
For ultra-soft pseudo potentials, one could say that there exists a correction, but the quality is usually rather the same.
But be careful, in some codes, some of these terms might actually be neglected. However, it appears that in the example you gave, this is not the case. 
