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Adiabatic Decoupling and Time-Dependent Born-Oppenheimer Theory

Herbert E. Spohn, Stefan Teufel

Published Jan 1, 2001• 61 citations
OpenAlex

Abstract

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

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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
A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schroedinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications we discuss the Born-Oppenheimer theory to second order and derive the nonperturbative definition of the g-factor of the electron within nonrelativistic quantum electrodynamics.
A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the g factor of the electron within nonrelativistic quantum electrodynamics.
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.
Coherent states are well adapted to understand adiabatic decoupling for multi-components systems like the Dirac operator or systems obtained by reduction in the Born-Oppenheimer approximation [58]. Hagedorn [130] has done an accurate analysis of this using a variant of coherent states. The new fact for systems is that now states are vectors valued in Cm\mathbb C^mCm for m≥2m\ge 2m≥2 and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.