  In the first part, expressions are given for the {\it sign} of the
topological angle that is acquired upon making a loop around a degeneracy
("conical intersection") point of two molecular energy surfaces. The
expressions involve the partial derivatives (with respect to the nuclear
coordinates) of the matrix elements of the coupling Hamiltonian. Examples are
given of a few studied cases, such as of excited states that have topological
angles with a sign opposite to those in the ground states. In the second part,
the two dimensional (or two parameter) situation that characterizes a conical
intersection (ci) between potential surfaces in a polyatomic molecule is
constructed as a limiting case of the three dimensional Dirac-monopole
situation. For an electron occupying a twofold state, we obtain both the
"magnetic-field" (or curl-field) and the tensorial (or Yang-Mills-) field
(which is the sum of a curl and of a vector- product term). These pseudo-
fields represent the reaction of the electron on the nuclear motion via the
nonadiabatic coupling terms (NACTs). We find that both fields are aligned with
the orthogonal, (so called) seam directions of the ci and are zero everywhere
outside the seam, but they differ as regards the flux that they produce. In a
two-state situation, the fields are representation dependent and the values of,
e.g., the fluxes depend on the state that the electron occupies. The angular
dependence of the NACTs and the fields calculated from a general linearly
coupled model agrees with recently computed results for $C_2 H$ [A.M. Mebel, M.
Baer and S.H. Lin, J.Chem. Phys. {\bf 115} 3673 (2001)]. An
effective-Hamiltonian formalism is proposed for experimentally observing and
distinguishing between the different fields.

Quantum Physics