This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 6409–6432 6409 Cite this: Phys. Chem. Chem. Phys., 2012, 14, 6409–6432 Geometric phase and gauge connection in polyatomic moleculeswz Curt Wittig* Received 19th September 2011, Accepted 14th December 2011 DOI: 10.1039/c2cp22974a 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 A m , 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 w n c n 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): c n - c n e iz and simultaneously w n - w n e iz , 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 ihn|rni, 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 (e iz /e iz ) nature of the c n and w n phase transformations. Two important U(1) connections are reviewed: qA m 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. 1. Introduction The first item of business is to settle on what is meant by the term ‘‘phase’’ in the context of this article. We shall not be concerned with phase transitions, phase diagrams, and other such things. Rather, the phase to be discussed is the one that appears in an exponent, like the x in e ix . Not to be under- estimated, it has been known to vex even the most ardent of bookkeepers. Indeed, it can be subtle to the point of genuine difficulty. The central theme is the geometric phase associated with the parallel transport of electron wave functions on molecular potential energy surfaces. Seminal papers by C. Alden Mead and Donald Truhlar (1979, 1982) 1,2 inspired renewed enthusiasm in a subject that had been unearthed decades earlier, 3–5 but for the most part had lain dormant. The large amount and high quality of the ensuing research has had a significant and lasting impact on chemical and molecular physics. A simple manifestation, certainly one of the most common, arises with the conical intersection of two adiabatic potential energy surfaces (adiabats). The conical shape in the inter- section region is a consequence of the off-diagonal Hamiltonian matrix elements in a diabatic basis being real. It is well known that encircling the intersection (degeneracy point) on either adiabat results in the wave function acquiring a phase whose magnitude is p. 6,7 This follows immediately in the toy model of Department of Chemistry, University of Southern California, Los Angeles, CA 90089, USA. E-mail: wittig@usc.edu w This article is part of a special collection of PCCP Perspective celebrating the International Year of Chemistry. z Electronic supplementary information (ESI) available. See DOI: 10.1039/c2cp22974a PCCP Dynamic Article Links www.rsc.org/pccp PERSPECTIVE Published on 08 February 2012. Downloaded by University of Southern California on 19/09/2015 22:52:49. View Article Online / Journal Homepage / Table of Contents for this issue