  The evolution of a quantum system is governed by the associated Hamiltonian.
A system defined by a parameter-dependent Hamiltonian acquires a geometric
phase when adiabatically evolved. Such an adiabatic evolution of a system
having non-degenerate quantum states gives the well-studied Berry phase.
Lounguet-Higgins and co-workers discovered a geometric phase when considering
the Jahn-Teller distortion described by the nuclear coordinates traversing a
closed path about the point of intersection of the electronic potential energy
surfaces. Under such a condition, the Born-Oppenheimer wave function undergoes
a sign change corresponding to an introduced global phase of $\pi$ radian. This
change further introduces a multiple valuedness in the wavefunction which may
be removed by adding a vector potential like term in the Hamiltonian for the
nuclear motion giving the Molecular Aharonov Bohm effect. Here, we demonstrate
a scheme to evaluate the introduced global phase for the molecular system
considered by Longuet-Higgins and propose methods as the first principle to do
the same in more complex examples for the molecular Hamiltonian on a quantum
computer.
