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Journal of Physics B Atomic Molecular and Optical Physics

The Berry phase revisited: application to Born–Oppenheimer molecular systems

Tamás Vértesi, Ágnes Vibók, Gábor J. Halász, Michael Baer

Published Nov 23, 2004• 13 citations

Abstract

In this paper, we study the implementation of the Berry approach as expressed in his seminal publication (1984 Proc. R. Soc. Lond. A 392 45) within Born–Oppenheimer molecular systems (1927 Ann. Phys., Lpz. 84 457). This was done with the purpose of revealing the relation between the various time-dependent magnitudes in the adiabatic limit and their time-independent counterparts. The two main results are: (1) in the adiabatic limit, a single Born–Oppenheimer state becomes an eigenstate of the system during the excursion of a system along a given contour, and (2) at the end of a closed contour, the topological phases associated with the states that moved adiabatically along a closed contour are well presented by the diagonal D -matrix as obtained from the time-independent treatment. The theoretical study is supported by a detailed numerical study carried out for two ab initio molecular systems, namely the (Na, H 2 ) and the (H, H 2 ) systems.

DOI: 10.1088/0953-4075/37/23/003

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In this paper, we study the implementation of the Berry approach as expressed in his seminal publication (1984 Proc. R. Soc. Lond. A 392 45) within Born–Oppenheimer molecular systems (1927 Ann. Phys., Lpz. 84 457). This was done with the purpose of revealing the relation between the various time-dependent magnitudes in the adiabatic limit and their time-independent counterparts. The two main results are: (1) in the adiabatic limit, a single Born–Oppenheimer state becomes an eigenstate of the system during the excursion of a system along a given contour, and (2) at the end of a closed contour, the topological phases associated with the states that moved adiabatically along a closed contour are well presented by the diagonal D -matrix as obtained from the time-independent treatment. The theoretical study is supported by a detailed numerical study carried out for two ab initio molecular systems, namely the (Na, H 2 ) and the (H, H 2 ) systems.

By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal terms is derived. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian in the present formulation. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial for any finite time interval T. The topological interpretation of Berry's phase such as the topological proof of phase-change rule thus fails in the practical Born–Oppenheimer approximation, where a large but finite ratio of two time scales is involved.

Two types of phases are discussed in this article: (1) The topological phase as introduced by Berry [Proc. R. Soc. London, Ser. A 392, 45(1984)] and Aharonov and Anandan [Phys. Rev. Lett. 58, 1593 (1987)] and (2) the Longuet–Higgins phase [Proc. R. Soc. London, Ser. A 344, 147 (1975)]. The two types of phases have a common origin, namely the multivaluedness of the electronic adiabatic basis, a phenomenon associated with the existence of a degeneracy in configuration space. It will be shown, by studying an electronic model Hamiltonian that arises from a two-state approximation to the Mathieu equation, that the two phases differ from each other substantially, coinciding only in the adiabatic limit upon completion of a cycle.

Whenever a quantum chemist extracts the eigenstate of an electronic Hamiltonian, he makes, consciously or not, a decision concerning the phase of the wave function. This is done for each calculated state at each nuclear position. Thus he defines a Born–Oppenheimer (BO) frame of reference. There is no absolute phase just as there is no absolute position or time in mechanics. This leads naturally to the question: What are the quantities which do not depend on the arbitrary phases, i.e., what are the BO invariants? In this article we identify BO invariants with respect to an arbitrary path in nuclear configuration space. We identify invariant electronic states along these paths and their Aharonov–Anandan geometric phases. For closed loops not passing through electronic energy degeneracies these invariant states are the BO adiabatic wave functions and the phases are the Berry phases. The results establish rigorous relations between the full nonadiabatic couplings matrix and the geometric phases.

We reconsider the time-dependent Born-Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the rst time when a band crossing is encountered. The adiabatic decoupling leads to an eective Schrodinger equation for the nuclei, including contributions from the Berry connection. 1

We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.