Born-Oppenheimer invariants along nuclear configuration paths

Loop matrices D_⊙ and C_⊙ were studied. It was shown that for loops going through electronic energy degeneracies, these matrices have very simple structure they are diagonal and contain only phase factors. The phase factors of the D-matrix describe the single-valuedness of the adiabatic states and those of the C_⊙ involve the invariant phase factors, obtained from the Berry phases.