Geometric phase and gauge connection in polyatomic molecules‡

文献信息

发布日期 2012-02-08
DOI 10.1039/C2CP22974A
影响因子 3.676
作者

Curt Wittig


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摘要

Geometric phase is an interesting topic that is germane to numerous and varied research areas: molecules, optics, quantum computing, quantum Hall effect, graphene, and so on. It exists only when the system of interest interacts with something it perceives as exterior. An isolated system cannot display geometric phase. This article addresses geometric phase in polyatomic molecules from a gauge field theory perspective. Gauge field theory was introduced in electrodynamics by Fock and examined assiduously by Weyl. It yields the gauge field Aμ, particle–field couplings, and the Aharonov–Bohm phase, while Yang–Mills theory, the cornerstone of the standard model of physics, is a template for non-Abelian gauge symmetries. Electronic structure theory, including nonadiabaticity, is a non-Abelian gauge field theory with matrix-valued covariant derivative. Because the wave function of an isolated molecule must be single-valued, its global U(1) symmetry cannot be gauged, i.e., products of nuclear and electron functions such as χnψn are forbidden from undergoing local phase transformation on R, where R denotes nuclear degrees of freedom. On the other hand, the synchronous transformations (first noted by Mead and Truhlar): ψn → ψneiζ and simultaneously χn → χne−iζ, preserve single-valuedness and enable wave functions in each subspace to undergo phase transformation on R. Thus, each subspace is compatible with a U(1) gauge field theory. The central mathematical object is Berry's adiabatic connection i〈n|∇n〉, which serves as a communication link between the two subsystems. It is shown that additions to the connection according to the gauge principle are, in fact, manifestations of the synchronous (eiζ/e−iζ) nature of the ψn and χn phase transformations. Two important U(1) connections are reviewed: qAμ from electrodynamics and Berry's connection. The gauging of SU(2) and SU(3) is reviewed and then used with molecules. The largest gauge group applicable in the immediate vicinity of a two-state intersection is U(2), which factors to U(1) × SU(2). Gauging SU(2) yields three fields, whereas U(1) is not gauged, as the result cannot be brought into registry with electronic structure theory, and there are other problems as well. A parallel with spontaneous symmetry breaking in electroweak theory is noted. Loss of SU(2) symmetry as the energy gap between adiabats increases yields the inter-related U(1) symmetries of the upper and lower adiabats, with spinor character imprinted in the vicinity of the degeneracy.

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来源期刊

Physical Chemistry Chemical Physics

Physical Chemistry Chemical Physics
CiteScore: 5.5
自引率: 10.3%
年发文量: 3036

Physical Chemistry Chemical Physics (PCCP) is an international journal co-owned by 19 physical chemistry and physics societies from around the world. This journal publishes original, cutting-edge research in physical chemistry, chemical physics and biophysical chemistry. To be suitable for publication in PCCP, articles must include significant innovation and/or insight into physical chemistry; this is the most important criterion that reviewers and Editors will judge against when evaluating submissions. The journal has a broad scope and welcomes contributions spanning experiment, theory, computation and data science. Topical coverage includes spectroscopy, dynamics, kinetics, statistical mechanics, thermodynamics, electrochemistry, catalysis, surface science, quantum mechanics, quantum computing and machine learning. Interdisciplinary research areas such as polymers and soft matter, materials, nanoscience, energy, surfaces/interfaces, and biophysical chemistry are welcomed if they demonstrate significant innovation and/or insight into physical chemistry. Joined experimental/theoretical studies are particularly appreciated when complementary and based on up-to-date approaches.

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