  We use the Van Vleck-Primas perturbation theory to study the problem of
parallel transport of the eigenvectors of a parameter-dependent Hamiltonian.
The perturbative approach allows us to define a non-Abelian connection
$\mathcal{A}$ that generates parallel translation via unitary transformation of
the eigenvectors. It is shown that the connection obtained via the perturbative
approach is an average of the Maurer-Cartan 1-form of the one-parameter
subgroup generated by the Hamiltonian. We use the Yang-Mills curvature and the
non-Abelian Stokes' theorem to show that the holonomy of the connection
$\mathcal{A}$ is related to the Berry phase.
