Time-dependent electron transfer and energy dissipation in condensed media

This paper employs a time-dependent Newns-Anderson-Schmickler model with Keldysh Green's functions and semiclassical trajectories to demonstrate how adsorbate motion and solvent coupling non-adiabatically suppress electron transfer while facilitating energy dissipation into electron-hole pairs, ultimately deriving an analytical expression for the average energy transfer rate in the slow-motion limit.

The Gist

The Big Picture: A Ball Rolling Down a Hill in a Crowd

Imagine you are watching a game of pinball, but instead of a metal ball, it's a tiny, charged particle (an adsorbate, like a proton) rolling toward a giant, flat, electrically charged trampoline (the metal electrode).

Usually, scientists study what happens when this ball rolls very slowly, or when the trampoline is in a vacuum. But in the real world (like in a battery or a biological cell), this ball is rolling through a thick, sticky liquid (the solvent or water), and it's moving at a decent speed.

This paper asks: What happens to the electrons when this ball zooms past the trampoline while swimming through sticky liquid?

The Main Characters

1. The Adsorbate (The Ball): A tiny particle trying to land on the metal surface. It has its own energy and speed.
2. The Metal Electrode (The Trampoline): A sea of free-moving electrons. When the ball gets close, it can "steal" an electron from the trampoline or "give" one back.
3. The Solvent (The Sticky Liquid): The water or chemical soup surrounding the system. It's made of vibrating molecules that act like a crowd of people jiggling around.
4. The Electron Transfer (The Handshake): The moment the ball grabs an electron from the trampoline.

The Problem: The "Too Fast" Problem

In old theories (called Adiabatic theories), scientists assumed the ball moved so slowly that the electrons on the trampoline had plenty of time to adjust. It was like a slow-motion handshake: the ball approaches, the electrons rearrange themselves perfectly to welcome it, and the handshake happens smoothly.

But in reality, the ball is moving!
When the ball moves fast, the electrons on the trampoline get confused. They can't rearrange themselves fast enough to keep up. This is called Non-Adiabatic behavior. It's like trying to shake hands with someone who is sprinting past you; you might miss, or the handshake might be clumsy and awkward.

The New Discovery: The "Sticky" Effect

The authors of this paper used advanced math (Green's functions and Keldysh formalism—think of these as super-precise GPS trackers for electrons) to simulate this scenario. They found two main things:

1. The "Slippery" Handshake (Reduced Electron Transfer)

Because the ball is moving, it often misses the perfect moment to grab an electron.

• Analogy: Imagine trying to catch a fish in a river. If you stand still and wait, you might catch one. But if you are running along the bank, it's much harder to grab the fish.
• Result: The faster the ball moves, the less likely it is to successfully transfer an electron. The "handshake" fails more often.

2. The "Friction" of the Crowd (Energy Dissipation)

When the ball moves past the trampoline, it disturbs the electrons. Even if it doesn't grab one, it bumps into them, creating little ripples (electron-hole pairs).

• Analogy: Imagine a speedboat moving through a crowded pool. Even if the boat doesn't touch the swimmers, its wake pushes them around. The swimmers push back, slowing the boat down. This is Electronic Friction.
• The Solvent's Role: The "sticky liquid" (solvent) makes things even more complicated. It acts like a cushion or a shock absorber.
  • Strong Solvent Coupling: If the liquid is very "sticky" (strong coupling), it absorbs some of the energy, making the ball move slower and reducing the chance of it grabbing an electron.
  • Weak Solvent Coupling: If the liquid is thin, the ball moves faster, loses more energy to the trampoline's electrons, and is more likely to get "stuck" (adsorbed) because it loses its speed.

The "Sticking" Probability

The ultimate goal of this research is to understand Sticking Probability: Will the ball land and stay, or will it bounce off?

• The Rule: To stick, the ball needs to lose its kinetic energy (speed).
• The Finding:
  • If the ball moves slowly, it has time to adjust, grab an electron, and stick easily.
  • If the ball moves fast, it zips past, doesn't grab an electron, and bounces off.
  • The Twist: The "sticky liquid" (solvent) changes the rules. If the liquid is very active (high reorganization energy), it actually prevents the ball from sticking because it shields the interaction. If the liquid is less active, the ball loses more energy to the metal and is more likely to stick.

Why Does This Matter?

This isn't just about pinball. This physics happens everywhere:

• Batteries: When charging a battery, ions move into the electrode. If they move too fast or the liquid electrolyte interferes, the battery might not charge efficiently.
• Fuel Cells: Converting hydrogen into electricity relies on protons landing on a metal surface. Understanding how fast they move and how the water around them affects them helps us build better fuel cells.
• Corrosion: Rusting is essentially metal atoms reacting with their environment. Knowing how electrons move helps us prevent it.

The Takeaway

The paper tells us that speed matters. You can't just look at a chemical reaction as a static picture; you have to watch the movie. When a particle moves quickly through a liquid toward a metal, the electrons get "distracted," the handshake gets clumsy, and the energy gets dissipated as heat (friction) rather than a successful chemical bond.

By understanding this "electronic friction," scientists can design better materials for energy storage and conversion, ensuring that the "balls" land exactly where we want them to.

Technical Summary

1. Problem Statement

The paper addresses the fundamental challenge of understanding non-adiabatic electron transfer (ET) and energy dissipation in electrochemical systems where an adsorbate (e.g., a proton or atom) moves dynamically near a metal electrode immersed in a solvent.

• Limitations of Existing Models: Traditional theories often rely on the Born-Oppenheimer approximation (assuming the adsorbate is static or moves infinitely slowly) or time-independent Hamiltonians. These approaches fail to capture the dynamics when the adsorbate moves at finite velocities, particularly in metal electrodes where dynamical processes induce electron-hole (e-h) pair excitations.
• The Gap: There is a lack of comprehensive theoretical frameworks that simultaneously account for:
  1. The time-dependent motion of the adsorbate.
  2. The coupling between the adsorbate and a continuum of electronic states in the metal.
  3. The interaction with a solvent bath (modeled as phonons) which causes reorganization energy.
• Core Question: How does the motion of an adsorbate suppress electron transfer, and how is kinetic energy dissipated into the metal and solvent via non-adiabatic effects?

2. Methodology

The authors employ a rigorous theoretical approach combining quantum dynamics for electrons and phonons with a semiclassical trajectory for the adsorbate.

• Model Hamiltonian: They utilize a time-dependent version of the Newns-Anderson-Schmickler model.
  • System: A moving adsorbate orbital coupled to a metal band (electrode) and a harmonic phonon bath (solvent).
  • Transformation: A non-perturbative Lang-Firsov canonical transformation is applied to eliminate the linear electron-phonon coupling term. This "dresses" the adsorbate electron states and renormalizes the energy level by the reorganization energy ($\lambda$).
• Formalism: The Keldysh non-equilibrium Green's function (NEGF) scheme is used to solve for the time-dependent orbital occupancy and energy transfer rates.
  • The Dyson equation is solved for the lesser Green's function ($G^<$) to obtain the time-dependent occupation number $\langle n_a(t) \rangle$.
  • The Wide-Band Approximation is employed to simplify the metal's density of states.
• Approximations:
  • Trajectory Approximation: The adsorbate's nuclear motion is treated classically with a known trajectory $R(t)$ (typically $z = v|t|$), while electrons and phonons are treated quantum mechanically.
  • Slow-Motion Limit: Analytical expressions for electronic friction and energy transfer are derived by performing Taylor expansions of time-dependent parameters around their adiabatic values.

3. Key Contributions

1. Derivation of Time-Dependent Orbital Occupancy: The authors derived an analytical expression for the time-dependent projected density of states (PDOS) and orbital occupancy that explicitly includes solvent correlation functions and retardation effects caused by finite adsorbate velocity.
2. Non-Adiabatic Suppression Mechanism: They demonstrated that the motion of the adsorbate induces a non-adiabatic suppression of electron transfer. The electron system cannot relax fast enough to the instantaneous ground state as the adsorbate moves, leading to a deviation from the adiabatic occupancy.
3. Analytical Electronic Friction Coefficient: In the slow-motion limit, the authors derived an analytical expression for the electronic friction coefficient ($\eta(z)$). This coefficient quantifies the dissipative force slowing the adsorbate due to e-h pair excitations in the metal.
4. Connection to Marcus Theory: The formalism was shown to reduce to the standard Marcus theory in the limit of a static adsorbate and high temperature ($k_B T \gg \Delta$), validating the approach against established electrochemical theory.
5. Sticking Probability Estimation: By calculating the probability of e-h excitation, the authors estimated the sticking probability of the adsorbate, linking energy dissipation directly to the likelihood of adsorption.

4. Key Results

Numerical calculations were performed for a proton coupled electron transfer (PCET) reaction (Volmer process) on a metal electrode.

• Velocity Dependence:
  • As the adsorbate velocity ($v$) increases, the orbital occupancy deviates significantly from the adiabatic value.
  • Asymmetry: The occupancy becomes highly asymmetric between the incoming and outgoing trajectories. For fast velocities, the orbital filling is delayed (retarded), and the maximum occupancy is reduced.
  • Suppression: At thermal velocities, the electron transfer rate is significantly lower (approx. 4 times) than the adiabatic rate.
• Solvent Reorganization Energy ($\lambda$):
  • Stronger coupling to the solvent (larger $\lambda$) renormalizes the adsorbate energy level further away from the Fermi level ($\epsilon_F$).
  • This shifts the point of Fermi level crossing to distances closer to the metal surface, where the resonance width ($\Delta$) is larger.
  • Counter-intuitive Finding: While strong $\lambda$ usually implies high reorganization energy, in this dynamic context, weak $\lambda$ leads to higher energy loss. This is because weak $\lambda$ allows the Fermi level crossing to occur far from the surface (where $\Delta$ is small), maximizing the time available for non-adiabatic e-h excitation.
• Electronic Friction and Energy Dissipation:
  • The electronic friction coefficient $\eta(z)$ peaks where the adsorbate energy level crosses the Fermi level.
  • Electrode Potential ($\phi$): Making the electrode potential more negative shifts the energy level crossing away from the surface. This broadens the friction profile and increases the total energy dissipation, thereby increasing the probability of the adsorbate "sticking" (being trapped) on the surface.
• Sticking Probability:
  • The probability of adsorption ($S$) is high when the average energy loss exceeds the initial kinetic energy.
  • $S$ is maximized for small $\lambda$ (weak solvent coupling) and negative electrode potentials, as these conditions maximize energy dissipation into the metal.

5. Significance

This work provides a critical bridge between surface physics (metal-adsorbate dynamics) and electrochemistry (solvent effects).

• Beyond Born-Oppenheimer: It rigorously quantifies the breakdown of the Born-Oppenheimer approximation in electrochemical reactions, showing that finite velocity effects cannot be ignored even for "slow" thermal processes.
• Design of Electrochemical Systems: The findings suggest that controlling the solvent environment (reorganization energy) and electrode potential can actively tune the energy dissipation pathways. This is crucial for optimizing reaction rates in fuel cells, batteries, and electrocatalysis where adsorbate dynamics play a role.
• Theoretical Framework: The derived analytical expressions for electronic friction and the connection to e-h excitation probabilities offer a powerful tool for predicting reaction outcomes in dynamic electrochemical environments without resorting to computationally expensive full quantum dynamics simulations.