Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

The Berry connection and Born–Oppenheimer method

Arno Böhm, Brian K. Kendrick, Mark E. Loewe, Luis J. Boya

Published Mar 1, 1992• 40 citations
Publisher

Abstract

By performing the most general Born–Oppenheimer procedure, the (non-Abelian) Berry connection for quantum systems in a quantum environment is derived. This method is then applied to the rapid rotation of a particle about a slowly changing axis, as exemplified by the electronic motion of a diatomic molecule. The angular part of the resulting dynamics for the quantum environment is equivalent to that of a monopole.

DOI
10.1063/1.529751

Related papers

The Berry phase in molecular physics

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

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.

International Journal of Quantum Chemistry•Jan 5, 1992•39 citations

The monopole, pure gauge and Born - Oppenheimer equation

Yue-lin Shen

The modified Born - Oppenheimer equation arising from Berry's phase is studied extensively in two physical systems. One is the spin- neutral particle in the spherically symmetric magnetic field, the other is the electron in the cylindrically symmetric magnetic field. The results show the significant topological effects in simple quantum systems.

Journal of Physics A Mathematical and General•Feb 21, 1996•1 citation

Rotation Symmetry and Nonabelian Berry Potential

Hyun Kyu Lee, Mannque Rho C

We study the role of rotational symmetry in the systems where nonabelian Berry potentials emerge as a result of integrating out fast degrees of freedom. The conserved angular momentum is constructed in the presence of a non-abelian Berry potential, which is formulated using Grassmann variables. The modifications on conventional angular momentum are discussed in close analogy with monopole systems. The diatomic molecular system discussed by Zygelman is found to have a similar structure to that of a non-abelian $SU(2)$ charge coupled to a 't Hooft - Polyakov monopole. The abelian limit of the Berry potential in diatomic systems is shown to be the same as $U(1)$ monopole and in a large separation limit, we observe the decoupling associated with a vanishing field tensor.

arXiv:hep-th/9210048•Oct 9, 1992•1 citation

Remarks on the Born-Oppenheimer approximation

José Wudka

The Born-Oppenheimer formulas, as modified by the introduction of the adiabatic connection, are derived from the effective action obtained by integrating out the fast variables. The parallels and differences with Berry's approach are stressed, and the corrections to the approximation are investigated.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 15, 1990•6 citations

Quantum theory of large amplitude collective motion and the Born-Oppenheimer method

Abraham Klein, Niels R. Walet

We study the quantum foundations of a theory of large amplitude collective motion for a Hamiltonian expressed in terms of canonical variables. In previous work the separation into slow and fast (collective and noncollective) variables was carried out without the explicit intervention of the Born-Oppenheimer approach. The addition of the Born-Oppenheimer assumption not only provides support for the results found previously in leading approximation, but also facilitates an extension of the theory to include an approximate description of the fast variables and their interaction with the slow ones. Among other corrections, one encounters the Berry vector and scalar potential. The formalism is illustrated with the aid of some simple examples, where the potentials in question are actually evaluated and where the accuracy of the Born-Oppenheimer approximation is tested. Variational formulations of both Hamiltonian and Lagrangian type are described for the equations of motion for the slow variables.

arXiv:nucl-th/9303017•Physical Review C•Jul 1, 1993•10 citations

A gauge theory for the quantum planar three-body problem

Toshihiro Iwai

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.

Journal of Mathematical Physics•Apr 1, 1987•57 citations
By performing the most general Born–Oppenheimer procedure, the (non-Abelian) Berry connection for quantum systems in a quantum environment is derived. This method is then applied to the rapid rotation of a particle about a slowly changing axis, as exemplified by the electronic motion of a diatomic molecule. The angular part of the resulting dynamics for the quantum environment is equivalent to that of a monopole.
We study the role of rotational symmetry in the systems where nonabelian Berry potentials emerge as a result of integrating out fast degrees of freedom. The conserved angular momentum is constructed in the presence of a non-abelian Berry potential, which is formulated using Grassmann variables. The modifications on conventional angular momentum are discussed in close analogy with monopole systems. The diatomic molecular system discussed by Zygelman is found to have a similar structure to that of a non-abelian SU(2)SU(2)SU(2) charge coupled to a 't Hooft - Polyakov monopole. The abelian limit of the Berry potential in diatomic systems is shown to be the same as U(1)U(1)U(1) monopole and in a large separation limit, we observe the decoupling associated with a vanishing field tensor.
We study the quantum foundations of a theory of large amplitude collective motion for a Hamiltonian expressed in terms of canonical variables. In previous work the separation into slow and fast (collective and noncollective) variables was carried out without the explicit intervention of the Born-Oppenheimer approach. The addition of the Born-Oppenheimer assumption not only provides support for the results found previously in leading approximation, but also facilitates an extension of the theory to include an approximate description of the fast variables and their interaction with the slow ones. Among other corrections, one encounters the Berry vector and scalar potential. The formalism is illustrated with the aid of some simple examples, where the potentials in question are actually evaluated and where the accuracy of the Born-Oppenheimer approximation is tested. Variational formulations of both Hamiltonian and Lagrangian type are described for the equations of motion for the slow variables.
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.