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The Journal of Chemical Physics

Time-dependent and time-independent approaches to study effects of degenerate electronic states

Michael Baer, Asher Yahalom A, R. Englman

Published Oct 22, 1998• 33 citations
Publisher

Abstract

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.

DOI
10.1063/1.477306

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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.
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.
The level crossing problem and associated geometric terms are neatly formulated by using the second quantization technique both in the operator and path integral formulations. The analysis of geometric phases is then reduced to the familiar diagonalization of the Hamiltonian. If one diagonalizes the Hamiltonian in one specific limit, one recovers the conventional formula for geometric phases. On the other hand, if one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval T. The topological proof of the Longuet-Higgins' phase-change rule, for example, thus fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and T is identified with the period of the slower system.
Berry's (1984) formulation of the topological phase for periodic Hamiltonians is extended to the case of non-adiabatic evolution by proper choice of initial states. This restores a full parallel with the non-adiabatic formalism of Aharonov and Anandan (1987) for periodic orbits in a projective Hilbert space. The Berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by Aharonov and Anandan and by Page (1987).