Vector potential with an angle to the periodic direction

One of the most difficult aspects in Tight Binding models is the incorporation of the magnetic field. And that is because a lot of the things that exist in simple analytical expressions in quantum mechanics, change when we are talking about a Tight Binding model, and especially one derived from First Principles using Density Functional Theory as that discussed here.

One of the problems that could emerge is that there exists an angle between the periodic direction and the direction the magnetic field is applied. This differs from the case of the Peierls phase, defined as,