Dynamical pseudopotentials

This paper introduces a framework for dynamical, energy-dependent pseudopotentials that utilize a sum-over-poles representation to accurately reproduce all-electron scattering across extended energy ranges while enabling a unified treatment of atoms and solids within many-body total energy functionals.

The Gist

Imagine you are trying to simulate a complex chemical reaction on a computer. To do this accurately, you need to model every single electron in every atom involved. However, atoms have two types of electrons: the "core" electrons, which are tightly glued to the nucleus and rarely move, and the "valence" electrons, which are on the outside and do all the interesting chemical work.

Calculating the behavior of every single core electron is like trying to count every grain of sand on a beach just to measure the shape of a single dune. It's computationally impossible for large systems.

The Old Solution: The "Static Mask"
For decades, scientists have used a trick called a pseudopotential. Think of this as a "mask" or a "filter" that hides the core electrons. Instead of calculating the messy, complex core, the computer replaces it with a smooth, simplified potential (a force field) that only acts on the valence electrons.

However, traditional masks are static. They are designed to work perfectly at one specific energy level (like a key cut for one specific lock). If you try to use them to study excited states (where electrons have more energy) or high-energy collisions, the mask doesn't fit well anymore. To make it work, scientists often have to make the mask "harder" (more detailed), which slows down the computer, or they have to use complex, unstable workarounds.

The New Solution: The "Smart, Shapeshifting Mask"
This paper introduces a new kind of pseudopotential: a Dynamical Pseudopotential.

Here is the core idea using a simple analogy:

1. The "Bath" Analogy

Imagine the valence electrons are a swimmer in a pool. The core electrons are the water molecules they are displacing.

• Old Way: You replace the water with a rigid, static wall. The swimmer can move, but the wall never changes shape. If the swimmer moves fast (high energy), the wall feels wrong.
• New Way: The authors treat the core electrons as an "auxiliary bath" (like a flexible, responsive fluid) that is coupled to the swimmer. The "mask" isn't a wall; it's a dynamical force that changes depending on how fast the swimmer is moving (their energy).

2. The "Sum-of-Poles" Trick

The biggest challenge with making a mask that changes with energy is that it usually requires a massive amount of data, leading to computer crashes (mathematical "ill-conditioning").

The authors solved this using a Sum-over-Poles representation.

• Analogy: Imagine you want to describe a complex, wiggly curve. Usually, you might need 100 different points to draw it accurately.
• The Innovation: The authors found a way to describe that same wiggly curve using just a few "poles" (like anchor points) and a clever mathematical formula.
• The Result: They can now match the behavior of the real atom (all-electron) at many different energy levels simultaneously, using very few "projectors" (mathematical tools). It's like having a single key that can open 7 different locks perfectly, whereas before you needed 7 different keys, and trying to combine them often broke the lock mechanism.

3. The "Universal Translator"

The paper claims this new method unifies three different worlds that were previously treated separately:

1. The All-Electron Atom (the real, messy thing).
2. The Pseudo-Atom (the simplified model).
3. The Solid (the material made of many atoms).

By treating the core electrons as a dynamic "bath," the math naturally flows from the single atom to the solid material without needing different rules for each. This is crucial for advanced theories (like GW or DMFT) that study how electrons interact over time, which static masks struggle to handle.

What They Actually Proved

The authors didn't just propose a theory; they built it and tested it:

• The Test: They applied this to Copper (Cu) and Erbium (Er) atoms.
• The Result: They created a pseudopotential that could accurately mimic the real atom's behavior across a huge range of energies (up to 60 Ry, which is very high).
• The Efficiency: They managed to reproduce the accuracy of using 7 different reference energies using only 3 mathematical "basis states." In the old methods, using 7 references would have required 7 states, often causing the math to break down due to redundancy.
• The Smoothness: They showed that the resulting "pseudo-orbitals" (the shapes of the electron clouds) are very smooth, meaning computers can simulate them much faster than the real, jagged all-electron versions.

Summary

In short, this paper replaces the old, rigid "one-size-fits-all" mask for atoms with a smart, energy-responsive filter. By treating the hidden core electrons as a dynamic partner rather than a static wall, and using a clever mathematical shortcut (sum-over-poles), they created a tool that is accurate across a wide range of energies, stable, and ready to be used in the most advanced theories of how materials behave.

Technical Summary

Technical Summary: Dynamical Pseudopotentials

Problem Statement
Pseudopotential theory is a cornerstone of modern electronic-structure calculations, enabling efficient plane-wave density-functional theory (DFT) by replacing chemically inert core electrons with an effective potential acting on valence states. However, conventional pseudopotentials are inherently static (energy-independent). This limitation becomes critical in excited-state calculations and many-body perturbation theories (such as GW, dynamical mean-field theory, and dynamical Hubbard functionals), where the valence electrons are described by frequency-dependent self-energies.

Current approaches to improve transferability, such as multiprojector constructions (e.g., Vanderbilt's ultrasoft pseudopotentials), face significant hurdles when targeting broad energy ranges. Increasing the number of reference energies to match scattering properties at higher energies typically requires an equal number of projectors. This leads to near-linear dependencies among projectors, resulting in numerical ill-conditioning and instabilities. Furthermore, static pseudopotentials create a formal disconnect between the treatment of core and valence electrons in theories employing dynamical self-energies, preventing a unified description of the all-electron atom, the pseudo-atom, and the solid within the same framework.

Methodology
The authors reformulate pseudopotential theory as a dynamical embedding problem. By tracing out the core degrees of freedom, they introduce an auxiliary bath coupled to the valence electrons, generating an energy-dependent (dynamical) pseudopotential $v_{PS}(\omega)$.

1. Generalized Norm-Conservation: The authors derive a generalized norm-conservation condition for energy-dependent potentials. They show that the energy derivative of the pseudopotential, $\partial_\omega v_{PS}(\omega)$, naturally acts as an augmentation charge, analogous to the loss of quasiparticle spectral weight in interacting theories. This generalizes the Hamann-Schlüter-Chiang (HSC) condition and recovers the ultrasoft pseudopotential (USPP) formalism as a specific case where the potential has a linear energy dependence.
2. Sum-Over-Poles (SOP) Representation: To construct these potentials practically, the authors employ a sum-over-poles representation:
   $$v_{PS}(\omega) = \sum_{\alpha\beta} \sum_s |b_\alpha\rangle \frac{\Gamma^s_{\alpha\beta}}{\omega - \Omega_s} \langle b_\beta|$$
   where $\Omega_s$ are scalar poles and $\Gamma^s_{\alpha\beta}$ are matrix-valued residues. A crucial innovation is the disentanglement of the number of reference energies from the number of nonlocal projectors (basis functions $|b_\alpha\rangle$).
3. Variational Basis Construction: The authors introduce a variational functional to optimize the localized basis set $\{|b_\alpha\rangle\}$. This allows for a controlled truncation of the basis, retaining only the most significant eigenvectors of the overlap matrix of the projectors. This strategy ensures that a small number of projectors can accurately reproduce scattering properties at a large number of reference energies, avoiding the numerical instabilities associated with multiprojector schemes.
4. Total Energy Formulation: The framework is embedded within a stationary total-energy formulation using Green's function functionals (specifically a Klein functional). This allows for the self-consistent screening of the ionic pseudopotential by dynamical augmentation charges, ensuring consistency with many-body perturbation theories.

Key Results

• Scattering Accuracy: The authors demonstrate the transferability of the dynamical pseudopotentials for Copper (Cu) $d$-electrons and Erbium (Er) $f$-electrons. Using up to seven reference energies, the dynamical pseudopotentials reproduce the all-electron logarithmic derivatives accurately over energy ranges extending up to 60 Ry (well beyond the typical 30 Ry requirement for excited states).
• Efficiency and Stability: In the test cases, the method required only three basis states (projectors) to represent seven reference energies, highlighting the near-linear dependence of the projectors and the effectiveness of the variational truncation. This contrasts with conventional multiprojector schemes where the number of projectors would equal the number of reference energies.
• Interpolation: The dynamical pseudopotentials exhibit excellent interpolation properties, reconstructing pseudo-orbitals at energies not included in the reference set (e.g., 10 Ry) with accuracy comparable to explicit polynomial pseudization.
• Smoothness: The pseudo-orbitals remain smooth, reducing the required plane-wave cutoff for bound states by approximately one order of magnitude compared to all-electron calculations. For unbound states, the cutoff requirements scale with the energy eigenvalue, similar to standard methods.

Significance and Claims
The paper claims to provide the first consistent and unified treatment of the all-electron atom, the pseudo-atom, and the solid within the same electronic-structure theory. By formulating pseudopotentials as dynamical embedding potentials, the work:

1. Generalizes Norm-Conservation: It establishes a rigorous link between energy-dependent potentials and augmentation charges, placing pseudopotentials on the same formal footing as frequency-dependent self-energies used in correlated-electron methods.
2. Overcomes Transferability Limits: The sum-over-poles approach, by decoupling the number of reference energies from the number of projectors, enables high-accuracy scattering reproduction over extended energy windows without the numerical instabilities of current multiprojector schemes.
3. Enables Many-Body Applications: The framework naturally integrates with Green's function functionals, offering a pathway to employ pseudopotentials in advanced many-body perturbation theories (GW, DMFT) where static approximations are insufficient.

The authors emphasize that while the energy dependence solves transferability issues over broad ranges, it does not inherently solve the problem of ghost states, which remains related to the separable structure of the potential and must be addressed through standard pseudization adjustments. The computational overhead is described as modest, as each pole introduces only one auxiliary degree of freedom, allowing the nonlinear eigenvalue problem to be mapped onto a static Hamiltonian solvable with standard diagonalization techniques.