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International Journal of Quantum Chemistry

The Berry phase in molecular physics

Arno Böhm, Brian K. Kendrick, Mark E. Loewe

Published Jan 5, 1992• 39 citations
Publisher

Abstract

Abstract The geometric phase of quantum mechanics is introduced, and its physical effects are then studied in the context of molecular physics. By performing the most general Born–Oppenheimer procedure, we show how gauge groups appear in the study of molecules. This method is then applied to the doubly degenerate Λ‐levels of a diatomic molecule. The resulting dynamics for the slow angular motion of the dumbbell is equivalent to that of a Dirac monopole.

DOI
10.1002/qua.560410108

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Abstract The geometric phase of quantum mechanics is introduced, and its physical effects are then studied in the context of molecular physics. By performing the most general Born–Oppenheimer procedure, we show how gauge groups appear in the study of molecules. This method is then applied to the doubly degenerate Λ‐levels of a diatomic molecule. The resulting dynamics for the slow angular motion of the dumbbell is equivalent to that of a Dirac monopole.
A several-particle system is called a molecule in the Born–Oppenheimer approximation. The nonrigidity of molecules involves difficulty in molecular dynamics. Guichardet [A. Guichardet, Ann. Inst. H. Poincaré 40, 329 (1984)] showed recently that the vibration motion cannot in general be separated from the rotation motion, by using the connection theory in differential geometry. The point of his theory is the observation that a center-of-mass system is made into a principal fiber bundle with rotation group as the structure group, and is equipped with a connection by the Eckart condition of rotationless constraint. The base manifold of this bundle is called the internal space. The fact that the connection has nonvanishing curvature gives rise to the nonseparability of vibration from rotation. This is a mathematical meaning of nonrigidity of molecules. As an application of the connection theory due to Guichardet, this paper establishes a gauge theory for nonrigid molecules on the basis of the observation that the vector bundle associated with the principal fiber bundle (the center-of-mass system) provides a setting for quantum mechanics of the ‘‘internal’’ molecular motion. The interest, however, centers on planar triatomic molecules in order to put forward the gauge theory in an explicit manner. The conclusion is this: The internal space of a planar triatomic molecule is diffeomorphic with R3−{0}, and endowed with Dirac’s monopole field which may be interpreted as a Coriolis field induced by the rotation. The angular momentum eigenvalues, which are twice the quantized monopole strengths, assign the complex line bundles over the internal space. The internal states of the molecule are described as the cross sections of the complex line bundle, on which the internal Hamiltonian operator acts in minimally coupling with the monopole field.
The fundamental theory of the geometric phase is summarized in a way suitable for use in molecular systems treated by the Born-Oppenheimer approach. Both Abelian and non-Abelian cases are considered. Applications discussd include the Abelian geometric phase associated with an intersection of two electronic potential-energy surfaces; screening of nuclei by the electrons from an external magnetic field; non-Abelian gauge potentials in molecular systems with Kramers degeneracy; and the coupling between different electronic levels (Born-Oppenheimer breakdown) represented as a gauge potential. Experimental tests for these systems are discussed, as well as a number of experiments on spin systems.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
Abstract The first part of this article attempts to highlight in chronological order some prominent papers that have direct connexion with the Born‐Oppenheimer approximation. This timeline successively points to the role of level crossing, the notions of gauge field and geometrical or Berry phase, early experiments that were accounted for with that concept, and its major role in understanding anomalous transport in solids. The second part analyzes the time evolution of states under the adiabatic approximation when the states are spin degenerate. The geometrical phase thus found entails an electron spin evolution. Then, we examine the case of moleclar dynamics. Starting from the Born‐Oppenheimer approximation, we obtain a spin evolution by replacing the classical nuclear positions by their quantum mechanical counterparts in the terms that describe electrostatic interactions between nuclei and electrons.