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N. Balakrishnan and B. K. Kendrick
1 Introduction
The Born-Oppenheimer approximation is the basis of much of the development
in electronic structure theory, quantum dynamics of nuclear motion and molecular
spectroscopy. The approach exploits the vast difference in timescale for electronic
and nuclear motion due to the small electron/nuclei mass ratio. In this approach, the
Schrödinger equation for electronic motion is solved for various fixed nuclear configurations and the resulting electronic energy as a function of the nuclear degrees of
freedom is called the electronic potential energy surface (PES). The nuclei evolve
under the influence of this electronic PES (or PESs) and subsequent solution of
the nuclear Schrödinger equation yields energy levels for rotational, vibrational and
translational motion of the nuclei. This two-step procedure for quantum chemical
dynamics has been wildly successful for many elementary chemical reactions. However, when there is an electronic degeneracy for certain nuclear configurations, i.e.,
a conical intersection between two electronic PESs, this adiabatic solution of the
Schrödingier equation for nuclear motion breaks down and a fully non-adiabatic
treatment is desirable. Such non-adiabatic treatments which include the coupling
between the ground and excited electronic PESs are computationally challenging
and not practical for the vast majority of chemical reactions.
An important consequence of the electronic degeneracy is that the real-valued
ground state electronic wave function changes sign when the nuclear motion encircles the conical intersection (CI) between two electronic states. The sign change
requires a corresponding sign change on the nuclear motion wave function to keep the
overall wave function single-valued. In other words, the nuclear motion Schrödinger
equation acquires a vector potential as originally pointed out by Mead and
Truhlar [1]. The effect of the vector potential is equivalent to a magnetic solenoid
centered at the conical intersection [2–4]. Flux of this magnetic field through the
surface enclosed by the CI yields a phase shift. This phase shift, due to its geometric origin, is referred to as the geometric phase (GP) or the Berry phase [5]. The
geometric phase resulting from the vector potential is analogous to the AharonovBohm effect [6] and Mead initially referred to this as the molecular Aharonov-Bohm
effect [7]. There have been numerous attempts in the literature to include the geometric phase in both bound state [8–10] and scattering calculations of triatomic systems [11–23]. While bound state studies of alkali metal trimers such as Li 3 [24],
Na 3 [25] and transition metal systems like Cu 3 [26] showed much better agreement
with experimental results when the GP effect is included, experimental verification
of the GP effects in a bimolecular chemical reaction has not been successful yet
[27–33].
Almost all of the experimental studies of GP effects in bimolecular chemical reactions have so far been limited to H or D atom exchange reactions in H+HD/D+HD
systems at energies close to the conical intersection [27–33]. At these high collision energies, many angular momentum partial waves contribute and any small GP
effect present in a partial wave resolved cross section washes out when a summation
over all partial waves is carried out to evaluate the total differential or integral cross
N. Balakrishnan and B. K. Kendrick
1 Introduction
The Born-Oppenheimer approximation is the basis of much of the development
in electronic structure theory, quantum dynamics of nuclear motion and molecular
spectroscopy. The approach exploits the vast difference in timescale for electronic
and nuclear motion due to the small electron/nuclei mass ratio. In this approach, the
Schrödinger equation for electronic motion is solved for various fixed nuclear configurations and the resulting electronic energy as a function of the nuclear degrees of
freedom is called the electronic potential energy surface (PES). The nuclei evolve
under the influence of this electronic PES (or PESs) and subsequent solution of
the nuclear Schrödinger equation yields energy levels for rotational, vibrational and
translational motion of the nuclei. This two-step procedure for quantum chemical
dynamics has been wildly successful for many elementary chemical reactions. However, when there is an electronic degeneracy for certain nuclear configurations, i.e.,
a conical intersection between two electronic PESs, this adiabatic solution of the
Schrödingier equation for nuclear motion breaks down and a fully non-adiabatic
treatment is desirable. Such non-adiabatic treatments which include the coupling
between the ground and excited electronic PESs are computationally challenging
and not practical for the vast majority of chemical reactions.
An important consequence of the electronic degeneracy is that the real-valued
ground state electronic wave function changes sign when the nuclear motion encircles the conical intersection (CI) between two electronic states. The sign change
requires a corresponding sign change on the nuclear motion wave function to keep the
overall wave function single-valued. In other words, the nuclear motion Schrödinger
equation acquires a vector potential as originally pointed out by Mead and
Truhlar [1]. The effect of the vector potential is equivalent to a magnetic solenoid
centered at the conical intersection [2–4]. Flux of this magnetic field through the
surface enclosed by the CI yields a phase shift. This phase shift, due to its geometric origin, is referred to as the geometric phase (GP) or the Berry phase [5]. The
geometric phase resulting from the vector potential is analogous to the AharonovBohm effect [6] and Mead initially referred to this as the molecular Aharonov-Bohm
effect [7]. There have been numerous attempts in the literature to include the geometric phase in both bound state [8–10] and scattering calculations of triatomic systems [11–23]. While bound state studies of alkali metal trimers such as Li 3 [24],
Na 3 [25] and transition metal systems like Cu 3 [26] showed much better agreement
with experimental results when the GP effect is included, experimental verification
of the GP effects in a bimolecular chemical reaction has not been successful yet
[27–33].
Almost all of the experimental studies of GP effects in bimolecular chemical reactions have so far been limited to H or D atom exchange reactions in H+HD/D+HD
systems at energies close to the conical intersection [27–33]. At these high collision energies, many angular momentum partial waves contribute and any small GP
effect present in a partial wave resolved cross section washes out when a summation
over all partial waves is carried out to evaluate the total differential or integral cross