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Exact-factorization framework for electron-nuclear dynamics in electromagnetic fields

Vladimir U. Nazarov, E. K. U. Gross

TL;DR

The paper extends the Exact Factorization (EF) formalism to systems under time-dependent external electromagnetic fields, deriving coupled EF equations for the nuclear and electronic components that include both physical and Berry-connection vector potentials. For a neutral atom in a uniform magnetic field it proves that the physical magnetic field and the Berry-curvature field exactly cancel in the nuclear equation for eigenstates, yielding free center-of-mass motion, while a residual constant Berry-connection can affect gauge-invariant current. The Harmonium atom in a magnetic field provides a concrete analytic example with explicit expressions for the residual Berry-connection and current, illustrating how pseudo-momentum and Berry-connection couple in a solvable model. These results establish an exact EM-extended EF framework, clarifying nonadiabatic electron-nuclear dynamics in magnetic fields and informing molecular dynamics in external fields.

Abstract

The Exact Factorization (EF) theory aims at the separation of the nuclear and electronic degrees of freedom in the many-body (MB) quantum mechanical problem. Being formally equivalent to the solution of the MB Schrödinger equation, EF sets up a strategy for the construction of efficient approximations in the theory of the correlated electronic-nuclear motion. Here we extend the EF formalism to incorporate the case of a system under the action of an electromagnetic field. An important interplay between the physical magnetic and the Berry-curvature fields is revealed and discussed within the fully non-adiabatic theory. In particular, it is a known property of the Born-Oppenheimer approximation that, for a neutral atom in a uniform magnetic field, the latter is compensated by the Berry-curvature field in the nuclear equation of motion (\citet{Yin-92}). From an intuitive argument that the atom must not be deflected by the Lorentz force from a straight line trajectory, it has been conjectured that the same compensation should occur within the EF theory as well. We give a rigorous proof of this property.

Exact-factorization framework for electron-nuclear dynamics in electromagnetic fields

TL;DR

The paper extends the Exact Factorization (EF) formalism to systems under time-dependent external electromagnetic fields, deriving coupled EF equations for the nuclear and electronic components that include both physical and Berry-connection vector potentials. For a neutral atom in a uniform magnetic field it proves that the physical magnetic field and the Berry-curvature field exactly cancel in the nuclear equation for eigenstates, yielding free center-of-mass motion, while a residual constant Berry-connection can affect gauge-invariant current. The Harmonium atom in a magnetic field provides a concrete analytic example with explicit expressions for the residual Berry-connection and current, illustrating how pseudo-momentum and Berry-connection couple in a solvable model. These results establish an exact EM-extended EF framework, clarifying nonadiabatic electron-nuclear dynamics in magnetic fields and informing molecular dynamics in external fields.

Abstract

The Exact Factorization (EF) theory aims at the separation of the nuclear and electronic degrees of freedom in the many-body (MB) quantum mechanical problem. Being formally equivalent to the solution of the MB Schrödinger equation, EF sets up a strategy for the construction of efficient approximations in the theory of the correlated electronic-nuclear motion. Here we extend the EF formalism to incorporate the case of a system under the action of an electromagnetic field. An important interplay between the physical magnetic and the Berry-curvature fields is revealed and discussed within the fully non-adiabatic theory. In particular, it is a known property of the Born-Oppenheimer approximation that, for a neutral atom in a uniform magnetic field, the latter is compensated by the Berry-curvature field in the nuclear equation of motion (\citet{Yin-92}). From an intuitive argument that the atom must not be deflected by the Lorentz force from a straight line trajectory, it has been conjectured that the same compensation should occur within the EF theory as well. We give a rigorous proof of this property.

Paper Structure

This paper contains 15 sections, 96 equations, 2 figures.

Table of Contents

  1. Introduction: THE CONCEPT OF EXACT FORCES ON THE NUCLEI
  2. Exact factorization formalism in the presence of external electromagnetic fields
  3. Nuclear density and current density
  4. Single neutral atom in uniform magnetic field
  5. Berry-connection $\mathbfcal{A}({\boldsymbol{R}})$
  6. Scalar potential $\epsilon({\boldsymbol{R}})$
  7. Residual Berry-connection vector potential $\mathbfcal{A}_0$
  8. Harmonium atom in magnetic field
  9. Implications for molecular systems
  10. Conclusions
  11. Derivation of equations of motion
  12. Nuclear equation of motion [Eq. (\ref{['EMchi']})]
  13. Electronic equation of motion [Eq. (\ref{['EMPhi']})]
  14. Born-Oppenheimer approximation
  15. Counterexample of a wave-packet violating the Berry-curvature compensation condition

Figures (2)

  • Figure 1: Harmonium atom in magnetic field. Coefficient of proportionality between the residual Berry-connection vector potential $\mathbfcal{A}_0$ and the normal-to-the-magnetic-field component of the pseudo-momentum $\mathbf{K}_\perp$ [Eq. \ref{['A01']}] versus the ratio of masses $M/m$ of the two particles. Results for four magnitudes of the magnetic field, in the units of $m c \omega_0/e$, are shown.
  • Figure 2: Harmonium atom in magnetic field. Coefficient of proportionality between the residual Berry-connection vector potential $\mathbfcal{A}_0$ and the normal-to-the-magnetic-field component of the pseudo-momentum $\mathbf{K}_\perp$ [Eq. \ref{['A01']}] versus the magnitude of the magnetic field, the latter in the units of $m c \omega_0/e$. Results for two values of the ratio of the masses of the particles are shown.