[Paper Review] Signed Phases and Fields Associated with Degeneracies

R. Englman, Asher Yahalom|ArXiv.org|Jun 25, 2004

Elasticity and Wave Propagation6 references3 citations

TL;DR

This paper derives the sign of geometric phases acquired when encircling conical intersections in molecular systems, showing it depends on the partial derivatives of the coupling Hamiltonian. It introduces both a pseudo-magnetic field and a Yang-Mills tensorial field—state-dependent and aligned with seam directions—using a 2D limit of the 3D Dirac monopole model, and proposes an effective Hamiltonian formalism to experimentally distinguish these fields via vibrational coupling effects.

ABSTRACT

In the first part, expressions are given for the {\it sign} of the topological angle that is acquired upon making a loop around a degeneracy ("conical intersection") point of two molecular energy surfaces. The expressions involve the partial derivatives (with respect to the nuclear coordinates) of the matrix elements of the coupling Hamiltonian. Examples are given of a few studied cases, such as of excited states that have topological angles with a sign opposite to those in the ground states. In the second part, the two dimensional (or two parameter) situation that characterizes a conical intersection (ci) between potential surfaces in a polyatomic molecule is constructed as a limiting case of the three dimensional Dirac-monopole situation. For an electron occupying a twofold state, we obtain both the "magnetic-field" (or curl-field) and the tensorial (or Yang-Mills-) field (which is the sum of a curl and of a vector- product term). These pseudo- fields represent the reaction of the electron on the nuclear motion via the nonadiabatic coupling terms (NACTs). We find that both fields are aligned with the orthogonal, (so called) seam directions of the ci and are zero everywhere outside the seam, but they differ as regards the flux that they produce. In a two-state situation, the fields are representation dependent and the values of, e.g., the fluxes depend on the state that the electron occupies. The angular dependence of the NACTs and the fields calculated from a general linearly coupled model agrees with recently computed results for $C_2 H$ [A.M. Mebel, M. Baer and S.H. Lin, J.Chem. Phys. {\bf 115} 3673 (2001)]. An effective-Hamiltonian formalism is proposed for experimentally observing and distinguishing between the different fields.

Motivation & Objective

• To determine the sign of the geometric phase acquired during a loop around a conical intersection in molecular potential energy surfaces.
• To model the nonadiabatic coupling between electronic and nuclear degrees of freedom using a 2D limit of the 3D Dirac monopole formalism.
• To introduce and characterize two distinct pseudo-fields—magnetic and Yang-Mills—arising from nonadiabatic coupling terms (NACTs).
• To propose an effective Hamiltonian formalism capable of distinguishing between the magnetic and Yang-Mills fields experimentally.

Proposed method

• Derives the phase sign using Cartesian, real representations of the coupling Hamiltonian, focusing on partial derivatives of matrix elements near degeneracy points.
• Uses a linearly coupled model near conical intersections to compute the curl-field (pseudo-magnetic field) and the tensorial Yang-Mills field as functions of nuclear coordinates.
• Shows both fields are orthogonal to the seam directions and vanish outside the seam, with fluxes dependent on the electronic state occupation.
• Applies symmetry-based truncation to construct an effective Hamiltonian that includes NACT-induced fields via pseudo-scalar and tensorial operators.
• Introduces empirical coefficients in the effective Hamiltonian to model residual perturbations from states outside the degenerate doublet.
• Proposes experimental verification through vibrational level dependence of the effective Hamiltonian, distinguishing between magnetic and Yang-Mills field contributions.

Experimental results

Research questions

• RQ1 What determines the sign of the geometric phase acquired when encircling a conical intersection in a molecular system?
• RQ2 How do the pseudo-magnetic and Yang-Mills fields arising from nonadiabatic coupling behave in a two-state molecular system?
• RQ3 In what way are the magnetic and Yang-Mills fields dependent on the electronic state or representation?
• RQ4 Can the effective Hamiltonian formalism distinguish experimentally between the magnetic and Yang-Mills fields?
• RQ5 How do the fields' fluxes and spatial distributions relate to the geometry of the conical intersection seam?

Key findings

• The sign of the geometric phase depends on the sign of the determinant $A_X B_Y - B_X A_Y$ in the Cartesian, real representation, with positive or negative values indicating phase increase or decrease with circling angle.
• The pseudo-magnetic field and Yang-Mills tensorial field are both aligned with the orthogonal seam directions and vanish outside the seam, but produce different fluxes depending on the electronic state.
• The fields are representation-dependent, and their fluxes vary with the electronic state occupied, indicating a non-universal behavior across different electronic configurations.
• The effective Hamiltonian formalism includes terms with coefficients $C_1$, $C_2$, etc., which are empirically determined and allow for the inclusion of virtual excitations beyond the degenerate doublet.
• The model's predictions for $C_2H$ match recent ab initio results for nonadiabatic coupling angular dependence, validating the theoretical framework.
• The proposed effective Hamiltonian provides a viable experimental pathway to differentiate between the magnetic and Yang-Mills fields through vibrational state dependence.