[Paper Review] Quantum hydrodynamics of coupled electron-nuclear systems

Rocco Martinazzo, Irène Burghardt|arXiv (Cornell University)|Oct 12, 2023

Quantum, superfluid, helium dynamics7 citations

TL;DR

This paper develops a gauge-invariant quantum hydrodynamics (QHD) framework for coupled electron-nuclear systems using the exact factorization of the wavefunction, enabling exact, non-adiabatic dynamics for both pure and mixed quantum states. It introduces mechanical momentum moments (MMMs) to formulate hydrodynamic equations that are invariant under gauge transformations, unifying and extending mixed quantum-classical approaches and recovering finite-temperature electronic friction from a 'type-e' mixture formulation.

ABSTRACT

The quantum dynamics of electron-nuclear systems is analyzed from the perspective of the exact factorization of the wavefunction, with the aim of defining gauge invariant equations of motion for both the nuclei and the electrons. For pure states this is accomplished with a quantum hydrodynamical description of the nuclear dynamics and electronic density operators tied to the fluid elements. For statistical mixtures of states the exact factorization approach is extended to two limiting situations that we call "type-n" and "type-e" mixtures, depending on whether the nuclei or the electrons are, respectively, in an intrinsically mixed state. In both cases a fully gauge invariant formulation of the dynamics is obtained again in hydrodynamic form with the help of mechanical momentum moments (MMMs). Nuclear MMMs extend in a gauge invariant way the ordinary momentum moments of the Wigner distribution associated with a density matrix of positional variables, electron MMMs are operator-valued and represent a generalization of the (conditional) density operators used for pure states. The theory presented here bridges exact quantum dynamics with several mixed quantum-classical approaches currently in use to tackle non-adiabatic molecular problems, offering a foundation for systematic improvements. It further connects to non-adiabatic theories in condensed-phase systems. As an example, we re-derive the finite-temperature theory of electronic friction of Dou, Miao \& Subotnik (Phys. Rev. Lett. 119, 046001 (2017)) from the dynamics of "type-e" mixtures and discuss possible improvements.

Motivation & Objective

• To eliminate gauge dependence in the exact factorization approach for electron-nuclear dynamics.
• To extend the exact factorization to statistical mixtures of electronic and nuclear states, distinguishing between 'type-n' and 'type-e' mixtures.
• To develop a fully gauge-invariant hydrodynamic formulation for both electrons and nuclei using mechanical momentum moments (MMMs).
• To bridge exact quantum dynamics with existing mixed quantum-classical methods for non-adiabatic processes.
• To recover and generalize the finite-temperature electronic friction theory of Dou, Miao & Subotnik from a quantum hydrodynamic perspective.

Proposed method

• Uses the exact factorization of the electron-nuclear wavefunction to separate electronic and nuclear degrees of freedom exactly.
• Introduces mechanical momentum moments (MMMs) as gauge-invariant generalizations of Wigner momentum moments and conditional density operators.
• Derives hydrodynamic equations for nuclei using fluid elements tied to nuclear positions, ensuring gauge invariance.
• Applies the MMM formalism to both pure states and two types of statistical mixtures: 'type-n' (nuclear mixtures) and 'type-e' (electronic mixtures).
• Derives the finite-temperature friction kernel from the dynamics of 'type-e' mixtures using spectral decomposition and Fermi-Dirac statistics.
• Employs the Lie-Trotter identity and Wigner function formalism to connect the hydrodynamic equations to standard quantum transport and friction theories.

Experimental results

Research questions

• RQ1 How can the exact factorization approach be made fully gauge invariant for electron-nuclear dynamics?
• RQ2 What is the correct hydrodynamic formulation of nuclear and electronic dynamics in the presence of statistical mixtures?
• RQ3 How do mechanical momentum moments (MMMs) generalize standard momentum moments and density operators in a gauge-invariant way?
• RQ4 Can the finite-temperature electronic friction kernel be derived from a quantum hydrodynamic framework of mixed states?
• RQ5 How does the proposed framework unify and improve upon existing mixed quantum-classical and non-adiabatic theories?

Key findings

• The paper successfully formulates a gauge-invariant quantum hydrodynamics for electron-nuclear systems using mechanical momentum moments (MMMs).
• For pure states, the nuclear dynamics is described by a hydrodynamic equation with gauge-invariant momentum moments, while electronic dynamics is governed by operator-valued density matrices.
• For 'type-e' mixtures, the finite-temperature electronic friction kernel is derived as a zero-frequency limit of a spectral function involving Fermi-Dirac distributions and single-particle matrix elements.
• The derived friction kernel matches the form of the Head-Gordon-Tully result in the zero-temperature limit, confirming consistency with established theories.
• The theory provides a systematic foundation for improving mixed quantum-classical dynamics by embedding exact quantum dynamics into a hydrodynamic formalism.
• The gauge invariance of the framework ensures that physical observables are independent of the arbitrary phase choices in the exact factorization, resolving a key ambiguity in prior approaches.