Explanation of the diabatic basis In Curl Wittig's paper, (J. Phys. Chem. B 2005, 109,8428-8430) on the Landau Zener formula, he defines something called the "diabatic basis" for a wavefunction describing a two state system? What is the definition of the diabatic basis? Is it not simply the same as the mass eigenbasis?
 A: Recently, I have also came across this definition. Somehow, the concept is poorly explained in recent literature. Therefore, I went back to the original Zener's paper: 

Proc. R. Soc. Lond. A 1932 137 696-702; DOI: 10.1098/rspa.1932.0165.
  Published 1 September 1932

Which can be found here: http://rspa.royalsocietypublishing.org/content/137/833/696
What I found is that if the Hamiltonian of the system is
$$
H = \begin{pmatrix}
\epsilon_{11}(t) & \epsilon_{12} \\
\epsilon_{12} & \epsilon_{22}(t)
\end{pmatrix}
$$
then the diabatic basis is composed of the two eigenstates in the limit of $\epsilon_{12} \rightarrow 0$. Now, since in the Landau-Zener problem you know that $\epsilon_{11} - \epsilon_{22} = \alpha t$, you can easily see why the diabatic basis crosses at $t = 0$ and you get the energy vs. time figure with a crossing point:

The black solid and dashed lines represent the diabatic basis.
Note: The figure shows plot of Energy vs. Magnetic field. This is because in problems where Landau-Zener formula is used you are dealing with time dependent field. Hence you could just as easily plot Energy vs time since Magnetic field is a function of time.
In a problem where you are using the Landau-Zener formula, the system has eigenstates which are equivalent to the diabatic basis only far away from the crossing point. Landau and Zener introduced the diabatic basis to simplify the problem - real eigenstates are not equivalent to the diabatic basis. Therefore, the Landau-Zener formula is not applicable when $\epsilon_{12} >> 0$.
Finally, Landau and Zener used these idealized eigenstates to calculate the probability that a system in the ground state gets excited (in the figure this would mean a jump from the red state to the blue one). This is what the Landau-Zener formula is used for.
