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Non-adiabatic Ehrenfest dynamics with norm-conserving and ultra-soft pseudo-potentials with nuclear velocity corrections on the atomic orbitals within the Projector Augmented Wave Method framework

This paper derives a Galilean-invariant first-principles Ehrenfest molecular dynamics framework within the Projector-Augmented-Wave method that incorporates nuclear-velocity-dependent phases on atomic orbitals to eliminate spurious non-adiabatic couplings for both norm-conserving and ultra-soft pseudo-potentials.

Original authors: Paolo Fachin, Francesco Macheda, Paolo Barone, Francesco Mauri

Published 2026-06-05

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CC BY 4.0

Original authors: Paolo Fachin, Francesco Macheda, Paolo Barone, Francesco Mauri

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 you are trying to predict how a crowd of people (electrons) moves around a group of dancers (atomic nuclei) in a room. In the world of quantum physics, we use complex math to simulate this dance. Usually, scientists assume the dancers are standing still while the crowd moves around them. But in reality, the dancers are constantly moving, spinning, and changing positions.

This paper tackles a specific problem that happens when we try to simulate what happens when those dancers start moving.

The Problem: The "Ghost" Moves

When scientists simulate atoms moving, they often use a shortcut called a "pseudo-potential." Think of this like using a simplified map instead of a detailed satellite photo. It saves a lot of computing power.

However, the old way of using these maps had a glitch. When the "dancers" (nuclei) moved at a constant speed, the simulation would sometimes incorrectly show the "crowd" (electrons) suddenly jumping to new energy states or changing their behavior.

The paper calls this a violation of Galilean invariance. In everyday terms, this is like saying that if you are on a train moving at a steady speed, the coffee in your cup should stay still relative to you. But the old simulation said the coffee would suddenly slosh over just because the train was moving. That doesn't make sense in the real world, but the math was broken, creating "ghost" movements that shouldn't exist.

The Solution: The "Moving Walkway"

The authors fixed this by changing how they describe the electrons.

In the old method, they treated the electrons as if they were glued to the dancers' positions. If a dancer moved, the electron's "home" just shifted rigidly to the new spot.

In this new method, the authors added a special "speed factor" to the electrons. Imagine the electrons aren't just sitting on the dancers; they are riding on a moving walkway that travels at the exact same speed as the dancer.

The Phase Shift: They added a mathematical "phase" (a kind of timing adjustment) that depends on how fast the nucleus is moving.
The Result: Now, when the nucleus moves, the electron moves with it perfectly, just like a passenger on a moving walkway. This removes the "ghost" movements. The simulation now respects the rule that constant motion shouldn't cause sudden, unexplained changes in the system.

The Two Types of Maps

The paper looks at two different ways of making these simplified maps (pseudo-potentials):

Norm-Conserving (The Standard Map): This is the simpler version. The authors found that adding the "moving walkway" speed factor fixed the problem completely. The math became clean, and the "ghost" forces disappeared.
Ultra-Soft (The Flexible Map): This is a more complex, flexible version used for heavier atoms. Here, the fix was trickier. The authors discovered that not only did they need to account for the speed of the nucleus, but they also had to account for the acceleration (how quickly the nucleus is speeding up or slowing down).
- They found that if a nucleus is accelerating, it creates a tiny "push" on the electrons (like the feeling of being pushed back in your seat when a car accelerates).
- The old math ignored this push. The new math includes it, ensuring the simulation remains accurate even when the atoms are speeding up or slowing down.

Why This Matters (According to the Paper)

The authors didn't just fix a bug; they restored the fundamental laws of physics to their simulations.

No More Paradoxes: They proved that if you move an entire system at a constant speed, the electrons shouldn't suddenly jump to new states. Their new method ensures this doesn't happen.
Better Accuracy: By including these speed and acceleration adjustments, the "simplified map" (pseudo-potential) now behaves exactly like the "detailed satellite photo" (all-electron calculation), but without needing as much computer power.

The Bottom Line

This paper provides a new set of rules for simulating moving atoms. It's like upgrading the software of a video game so that when characters run, the physics engine doesn't glitch out. By adding a "speed adjustment" to the electrons and accounting for "acceleration pushes," the authors ensure that their simulations of how atoms and electrons interact are physically correct, whether the atoms are cruising at a steady speed or speeding up and slowing down.

Technical Summary: Non-Adiabatic Ehrenfest Dynamics with Nuclear Velocity Corrections in the PAW Framework

Problem Statement
Standard first-principles non-adiabatic molecular dynamics, particularly within the Ehrenfest framework, often suffer from a violation of Galilean invariance. This arises because effective electronic Hamiltonians are typically constructed using atomic-like orbitals that are rigidly translated with the nuclei, neglecting the nuclear-velocity-dependent phases (known as electron-translation factors or ETFs). When nuclei move, this omission leads to spurious non-adiabatic couplings, manifesting paradoxically as electronic transitions induced by the constant velocity motion of an atom. Furthermore, this approach breaks sum rules relating Born effective charges and interatomic force constants to frequency-dependent electromagnetic susceptibilities. While these issues have been addressed in all-electron calculations and specific contexts like vibrational circular dichroism (VCD), current implementations of first-principles non-adiabatic molecular dynamics using pseudo-potentials generally neglect these velocity-dependent phases.

Methodology
The authors derive a first-principles Ehrenfest molecular dynamics formalism within the Projector Augmented Wave (PAW) method framework, explicitly incorporating nuclear velocity-dependent phases into the atomic orbital basis. The methodology proceeds as follows:

Velocity-Including Atomic Orbitals: The authors establish a localized atomic orbital basis for moving nuclei. By solving the Schrödinger equation for an isolated atom with a moving nucleus, they derive that the atomic orbitals must include a phase factor $e^{i\alpha_s(\hat{r})}$ , where $\alpha_s(\hat{r}) = \frac{m}{\hbar}\dot{R}_s(t) \cdot (\hat{r} - R_s(t))$ . This phase accounts for the instantaneous nuclear velocity $\dot{R}_s(t)$ .
PAW Transformation Generalization: The standard PAW transformation operator $\hat{T}$ , which maps pseudo-wavefunctions to all-electron wavefunctions, is generalized to $\hat{T}_{\dot{R}}$ . This new operator substitutes the static atomic orbitals with the velocity-including ones derived above.
Lagrangian Formalism: The authors employ a semiclassical Ehrenfest Lagrangian formalism. Crucially, because the Lagrangian depends on nuclear acceleration (arising from the time derivative of the velocity-dependent phases), the equations of motion are derived using the generalized Euler-Lagrange equations for higher-order derivatives.
Derivation of Effective Hamiltonians: The effective Hamiltonian is constructed by applying the generalized PAW transformation to the all-electron Hamiltonian. The authors carefully compute the temporal derivatives of the transformation operator $\hat{T}_{\dot{R}}$ , which introduce additional terms dependent on nuclear velocity and acceleration.

Key Contributions and Results
The paper provides explicit derivations for both norm-conserving and ultra-soft pseudo-potentials:

Norm-Conserving Case: The derived effective Hamiltonian features Peierls-like phases in the non-local part of the potential, dependent on the nuclear velocity ( $e^{i\frac{m}{\hbar}\dot{R}_s \cdot \hat{r}} v^{nl}_s e^{-i\frac{m}{\hbar}\dot{R}_s \cdot \hat{r}}$ ). In this case, the nuclear acceleration terms vanish or are negligible, and the result aligns with previous findings for norm-conserving potentials (Ref. [14]). The spurious couplings present in rigidly translated orbital schemes are removed, restoring Galilean invariance.
Ultra-Soft Case: The authors identify and derive additional terms that are specific to ultra-soft pseudo-potentials. These terms depend on both nuclear velocity and nuclear acceleration. They arise from the temporal derivatives of the PAW transformation and the augmentation charges ( $Q_{ij}$ ). Specifically, the Hamiltonian includes terms involving the commutator of the position operator with the non-local potential and terms related to the acceleration-dependent force ( $m\ddot{R}_s \cdot \hat{r}$ ). These contributions were previously overlooked in the literature.
Equations of Motion: The paper presents the complete set of equations for the electronic dynamics (Schrödinger equation) and nuclear dynamics (forces). The nuclear forces include standard Hellmann-Feynman terms, velocity-dependent corrections, and new acceleration-dependent terms.
Conserved Energy: A conserved energy expression is derived using the Ostrogradsky construction, appropriate for Lagrangians depending on acceleration. This expression includes corrections related to the derivatives of the Hamiltonian and the overlap matrix with respect to nuclear velocity and acceleration.

Significance and Claims
The authors claim that their work enables a Galilean-invariant description of non-adiabatic Ehrenfest molecular dynamics using pseudo-potentials. By including nuclear velocity-dependent phases in the atomic orbital basis, the method:

Removes Spurious Couplings: It eliminates artificial electronic transitions that occur when the entire system undergoes a global translation at constant velocity, a flaw inherent in standard rigid-translation approaches.
Restores Physical Consistency: It ensures that the pseudo-potential Hamiltonian correctly reproduces the properties of the all-electron Hamiltonian for moving nuclei, thereby restoring frequency-dependent sum rules and correct vibrational responses.
Generalizes to Ultra-Soft Potentials: The work extends the understanding of velocity-dependent effects beyond norm-conserving potentials, highlighting that nuclear acceleration terms are non-negligible in the ultra-soft case and must be included for accurate dynamical vibrational responses (e.g., force constant matrices and Born effective charges).

The paper concludes that while the local part of the Hamiltonian remains unaffected, the inclusion of these phases in the non-local parts and the augmentation regions is essential for a physically correct description of non-adiabatic processes in condensed matter systems using the PAW method.