🔬 materials science

Memory-dependent electronic friction for nonadiabatic dynamics at metal surfaces

This paper presents a first-principles theoretical formalism for evaluating configuration-dependent electronic friction memory kernels, demonstrating that incorporating these memory effects in nonadiabatic dynamics at metal surfaces significantly alters energy exchange mechanisms and eliminates the need for simplified Markovian friction coefficients.

Original authors: Xuexun Lu, Connor L. Box, Nils Hertl, Reinhard J. Maurer

Published 2026-08-14

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Xuexun Lu, Connor L. Box, Nils Hertl, Reinhard J. Maurer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, energetic molecule bouncing off a giant, metallic trampoline. This isn't just a simple bounce; it's a chaotic dance where the molecule and the metal are constantly exchanging energy. In the world of chemistry and physics, this happens all the time when gases hit metal surfaces, like in car catalytic converters or industrial sensors. The key question scientists have been asking is: how does the molecule slow down?

For decades, the standard explanation has been like imagining the metal as a thick, sticky syrup. As the molecule moves, it drags through this syrup, losing speed and heat in a smooth, predictable way. Scientists call this "electronic friction." It's a useful idea because it treats the metal's electrons as a simple, forgetful bath that instantly reacts to the molecule's movement. If the molecule speeds up, the friction increases immediately; if it slows, the friction drops instantly. This "instant reaction" idea is called the Markovian approximation, and it's been the go-to tool for simulating these collisions for years.

But what if the metal isn't a forgetful syrup? What if it's more like a complex, springy mattress that remembers how you jumped on it a split second ago? What if the electrons take a tiny, finite amount of time to react, and their response depends on how fast you are moving right now, not just where you are? This is the question of "memory." If the metal has a memory, the friction isn't just a simple drag; it's a complex, time-delayed force that can change the outcome of the bounce entirely. Understanding this could explain why some experiments don't match our old, simple models, helping us design better materials and understand how energy moves at the atomic scale.

The Paper's Big Idea: The Metal Has a Memory

In this paper, the authors, led by Xuexun Lu and Reinhard Maurer, decided to stop pretending the metal surface forgets everything instantly. They developed a new way to calculate "memory-dependent electronic friction." Instead of using a simple, constant drag coefficient, they built a mathematical tool that tracks how the metal's electrons react over time, creating a "friction kernel" that remembers the molecule's past movements.

Think of it like this: In the old "Markovian" model, if you push a shopping cart, the friction is the same no matter how fast you push. But in this new "memory" model, the cart is on a floor made of smart foam. If you push it slowly, the foam sinks gently. If you slam it fast, the foam might actually push back differently, or even wiggle in a way that changes how much energy you lose. The authors show that for atoms and molecules bouncing off metal, this "wiggle" and the time it takes for the electrons to respond matter a lot.

What They Found: It Depends on How You Bounce

The team tested their new math on two different scenarios. First, they used a simplified model (the Erpenbeck-Thoss model) representing a single atom hitting a surface. Here, they found that when the atom moves very fast, the memory effect actually reduces the energy loss compared to the old models. It's as if the fast-moving atom "outruns" the metal's ability to react, slipping through the friction before the metal can fully grab hold. In this case, the old "instant reaction" model was actually overestimating how much the atom would slow down.

However, things get much more interesting with a real molecule: Nitric Oxide (NO) hitting a gold surface (Au(111)). This is a famous experiment where the old models have struggled to match reality. The authors ran simulations using their new memory-aware math and found a dramatic shift.

When NO bounces off gold, the memory effects cause the molecule to lose more vibrational energy (shaking apart) but less translational energy (moving forward). In the old "instant friction" models, the molecule would just slide off with a lot of speed left. But with the memory included, the molecule gets "caught" in the friction for a split second longer, transferring more of its shaking energy to the metal electrons.

The authors calculated that for highly excited NO molecules, the memory effect increases the vibrational energy loss significantly. In fact, the old models underestimated this loss so badly that they missed the mark compared to real-world experiments. By including the memory, the new simulations suggest the molecule loses about 2.5 times more energy in its internal vibrations than the simple models predicted. This helps explain why previous experiments showed more energy loss than the "instant friction" theory could account for.

Why the Old Way Wasn't Enough

The paper explicitly argues against the idea that we can just pick one "average" number to represent friction and call it a day. The authors tested two common shortcuts: one that looks at the friction when the molecule is barely moving (zero frequency) and another that averages the friction over a specific range of speeds.

They found that neither of these shortcuts works. The "zero frequency" model (the standard one) was way too weak, predicting too little energy loss. The "average" model was closer in total numbers but completely wrong about how that energy was lost. It guessed the molecule would lose energy by sliding forward, while the memory model showed it was actually losing energy by shaking violently.

The authors emphasize that this isn't just a tiny correction; it changes the direction of the energy flow. The metal's response is "anisotropic," meaning it acts differently depending on whether the molecule is vibrating, rotating, or sliding. The memory effect amplifies this difference, making the friction much more directional than previously thought.

How Sure Are They?

The authors are very confident in their mathematical derivation and the results of their simulations. They didn't just guess; they built the theory from first principles using quantum mechanics and density functional theory (DFT), which is the gold standard for calculating how electrons behave. They ran these calculations on specific models and on real-world trajectories of NO hitting gold.

However, they are careful to note that their current results are based on "Classical Path Approximations" (CPA). This means they calculated the energy loss along a path that the molecule would have taken if there were no friction, and then measured how much energy the friction would have taken. They didn't run the full, messy simulation where the friction actually changes the path of the molecule in real-time (which is computationally very heavy).

So, while they are sure that memory effects exist and that they would increase vibrational energy loss, they suggest that the full picture might be even more complex once you let the friction actually steer the molecule. They propose that future work needs to run these full simulations to see if the memory effect changes the molecule's path, trapping it on the surface or bouncing it off in a new direction.

The Bottom Line

This paper suggests that the metal surface isn't a simple, forgetful sponge. It's a dynamic, responsive partner in the dance. When a molecule hits it, the metal remembers the hit, and that memory changes the outcome. For fast-moving, vibrating molecules like Nitric Oxide, ignoring this memory leads to big mistakes in predicting how much energy is lost. By adding this "memory" back into the equations, the authors show a path toward explaining experimental results that have puzzled scientists for years, revealing that the friction is not just a drag, but a complex, time-delayed conversation between the molecule and the metal.

Technical Summary: Memory-Dependent Electronic Friction for Nonadiabatic Dynamics at Metal Surfaces

Problem Statement
Nonadiabatic energy exchange between adsorbates and metal surface electrons is a critical mechanism in gas-surface dynamics, influencing observables such as chemi-currents, activated diffusion, and inelastic scattering. While Molecular Dynamics with Electronic Friction (MDEF) is a standard approach to model these effects by treating electrons as a dissipative bath, conventional implementations rely on the Markov approximation. This approximation assumes the friction force is instantaneous (memoryless), reducing the generalized Langevin equation (GLE) to a product of a friction tensor and the nuclear velocity. However, this neglects the frequency dependence of the electronic response and the configuration dependence of the system-bath coupling, which can lead to significant inaccuracies, particularly in hyperthermal scattering regimes where vibrational energy loss is often underestimated compared to experimental data.

Methodology
The authors present a theoretical formalism to evaluate tensorial, configuration-dependent electronic friction memory kernels from first principles. The approach is built upon a local linear-response formulation within the quasi-adiabatic limit, where nuclear velocities are small relative to electronic correlation rates.

Theoretical Framework:
- The formalism derives the memory kernel $K_{\mu\nu}(t-t'; x)$ from the time-time correlation function of nonadiabatic coupling.
- By linearizing the electron-nuclear coupling Hamiltonian around a single instantaneous nuclear configuration $x(t)$ , the authors avoid the computationally prohibitive requirement of tracking electronic states across different configurations (two-time kernel dependence). Instead, the kernel depends on the instantaneous configuration but retains non-Markovian character through its frequency dependence.
- The friction kernel in the frequency domain, $K_{\mu\nu}(\omega; x)$ , is expressed via the spectral density of electron-hole pair excitations, derived using Keldysh non-equilibrium Green's function (NEGF) formalism or linear-response theory.
Models and Systems:
- Newns–Anderson Hamiltonian (NAH): The authors first analyze parametrized models for atom and molecule scattering (Erpenbeck-Thoss and Gardner-Habershon-Maurer models). These allow for analytical derivation of memory kernels and investigation of mathematical properties, such as positive semi-definiteness (PSD) conditions.
- First-Principles DFT: The formalism is implemented within the FHI-aims software package using Kohn-Sham Density Functional Theory (DFT). The friction kernel is calculated from Kohn-Sham states and electron-phonon coupling matrix elements for NO scattering on Au(111).
Dynamics Analysis:
- Energy dissipation is quantified using the Classical Path Approximation (CPA). In this scheme, Born-Oppenheimer Molecular Dynamics (BOMD) trajectories are generated, and the work done by the memory-dependent friction force is integrated along these fixed paths. This isolates the effect of memory on energy dissipation without the computational cost of propagating full non-Markovian dynamics with colored noise.

Key Contributions

Derivation of Local Memory Kernels: The paper provides a consistent derivation of memory kernels that depend on a single nuclear configuration but retain frequency dependence, enabling practical first-principles calculations.
Mathematical Analysis of PSD: The authors derive exact conditions for the positive semi-definiteness of frequency-dependent friction tensors. They identify a threshold frequency $\omega_*$ beyond which the friction may lose strict PSD properties in certain model Hamiltonians, indicating potential breakdowns of the electronic friction approximation in specific regimes.
Quantification of Memory Effects: The study demonstrates that memory effects are not merely corrections but fundamentally alter the energy dissipation profile. Specifically, they show that memory effects can either increase or decrease energy loss depending on the system configuration and the nature of the scattering (e.g., near state crossings vs. equilibrium geometries).

Results

Model Hamiltonians (ET and GHM):
- In the Erpenbeck-Thoss (1D) model, memory effects lead to a reduction in energy loss for weak coupling strengths, as the trajectory samples high-frequency friction values that are less dissipative than the zero-frequency Markovian limit.
- In the Gardner-Habershon-Maurer (2D NO/Au) model, memory effects lead to an increase in vibrational energy loss and a reduction in translational energy loss. This occurs because the scattering trajectories predominantly sample configurations where the friction spectrum is underdamped and peaks at finite frequencies, rather than at the zero-frequency limit.
- The study highlights that the Markovian approximation can either overestimate or underestimate friction depending on the geometry, particularly near diabatic state crossings where overdamped dynamics prevail.
First-Principles NO/Au(111) Scattering:
- Calculations of the friction tensor for NO on Au(111) reveal strong anisotropy and structured frequency dependence. The intramolecular stretch ( $r$ ) mode probes higher frequencies (peaks around 0.5–2 eV) compared to the surface-normal translation ( $z$ ) mode.
- Comparing three friction models—zero-frequency Markovian (ODF), finite-frequency averaged Markovian ( $\eta_{Avg}$ ), and non-Markovian memory kernels—the non-Markovian treatment yields the highest vibrational energy loss (median 0.42 eV vs. 0.21 eV for ODF).
- While a frequency-averaged Markovian tensor ( $\eta_{Avg}$ ) can approximate the total energy loss, it fails to reproduce the correct partitioning of energy between vibrational and translational modes. The memory kernel correctly redistributes dissipation toward high-frequency intramolecular vibrations, increasing the directional anisotropy of friction.

Significance and Claims
The paper claims that the inclusion of memory-dependent electronic friction is essential for accurately describing nonadiabatic energy dissipation in hyperthermal scattering, particularly for systems like NO on Au(111) where previous Markovian MDEF studies underestimated vibrational energy loss.

The authors assert that their formalism removes the need to define a single effective Markovian friction coefficient from a structured frequency-dependent response, which is often an arbitrary choice. Instead, the memory-dependent evaluation captures the full spectral weight relevant to the specific nuclear motion.

The study concludes that while Markovian MDEF may be insufficient for hyperthermal scattering, the inclusion of memory effects extends the applicability of the electronic friction approach. However, the authors remain modest regarding the ultimate validity, noting that the breakdown of positive semi-definiteness in certain parameter regimes suggests limits to the linear-coupling approximation. They emphasize that while CPA results are promising, fully propagating tensorial, memory-dependent dynamics with colored noise is required to account for dynamical steering effects and fully validate the method against experimental observables.

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