Excited-state Properties Beyond the Excitation Energy from Orbital-Optimized Density Functional Calculations I: Dipole Moments of Rydberg States

This study demonstrates that orbital-optimized density functional calculations using plane-wave basis sets provide a superior description of the dipole moments for Rydberg excited states compared to traditional atomic orbital approaches, revealing that while hybrid functionals like PBE0 yield the best agreement with high-level benchmarks, standard augmented basis sets often fail to capture accurate dipole moments even when excitation energies appear converged.

The Gist

The Big Picture: Catching the "Ghost" Electrons

Imagine a molecule as a tiny solar system. Usually, the electrons (the planets) stay close to the nucleus (the sun) in neat, tight orbits. But sometimes, an electron gets a huge energy boost and jumps way, way out into the deep, empty space surrounding the molecule. Scientists call these Rydberg states.

These "ghost" electrons are incredibly hard to study because they are so spread out and diffuse. They are like a faint mist rather than a solid ball. If you try to measure them with the wrong tools, you might miss them entirely or get the wrong shape.

This paper is about a new way of calculating where these ghost electrons are and how they affect the molecule's "electric personality" (called the dipole moment). The researchers found that their new method is much better at describing these fuzzy, far-out electrons than the old, standard methods.

The Problem: The "Fence" vs. The "Open Field"

To simulate these molecules on a computer, scientists have to build a digital cage around them.

• The Old Way (Atomic Orbitals): Imagine trying to map a vast, open field using only a few specific, rigid fences placed right next to the house. You can describe the house perfectly, but as you get further out into the field, your fences stop. If a "ghost electron" wanders into that open space, your rigid fences can't capture it properly. You might think the electron is still near the house, or you might get the direction it's pointing completely wrong.
• The New Way (Plane Waves): Instead of fences, imagine the computer uses a giant, invisible grid that covers the entire open field uniformly. There are no gaps. This allows the computer to see the "ghost electron" clearly, even when it is far away from the molecule.

The paper shows that while the old "fence" method (atomic basis sets) is okay at guessing how much energy it took to jump the electron out, it fails miserably at describing where the electron actually is and which way the molecule is pointing electrically.

The Experiment: Testing the Tools

The researchers tested four small molecules (Water, Formaldehyde, Ammonia, and Methanol). They used their new "Open Field" method (Plane Waves) and compared it to the old "Fence" method (Atomic Orbitals) using different mathematical rules (called functionals).

Key Findings:

1. Energy vs. Direction: The old method was surprisingly good at guessing the energy needed to jump the electron out. However, it was terrible at guessing the dipole moment (the direction and strength of the molecule's electric pull). It's like guessing how fast a car is going, but getting the direction it's driving completely wrong.
2. The "Double Fence" isn't enough: Even when the researchers added more fences (extra diffuse functions) to the old method to try to reach further out, it still couldn't match the "Open Field" method for the most spread-out electrons. The problem wasn't just that the fences were too short; it was that they were stuck in one spot and couldn't bend to fit the shape of the electron cloud.
3. The Best Rules: They tried different mathematical "rulebooks" to see which one worked best with the "Open Field" method.
   • PBE0: This rulebook was the winner. It gave the most accurate results, closest to what we expect from high-level physics.
   • Self-Interaction Correction (SIC): Scientists often try to fix errors in calculations by adding a "correction" to account for electrons repelling themselves. The researchers found that while this correction helps with energy, it actually made the direction of the electric pull worse. It was like trying to fix a crooked picture by adding a heavier frame; it didn't help straighten the picture.

The Conclusion: Why This Matters

The main takeaway is that dipole moments are a stricter test than energy. Just because a computer program gets the energy right doesn't mean it understands the shape or direction of the excited electron.

• The "Ghost" needs a big canvas: To accurately describe these far-out, fuzzy electrons, you need a flexible, grid-like system (Plane Waves) rather than a system of fixed, local fences (Atomic Orbitals).
• Better tools exist: The "Orbital-Optimized" method used here is a powerful tool that handles these tricky states much better than the standard methods used in most chemistry software today.

In short, if you want to know exactly how a molecule behaves when it's excited and its electrons are flying far away, you need to stop using "fences" and start using an "open field" grid to see the whole picture.

Technical Summary

Technical Summary: Excited-State Properties Beyond the Excitation Energy from Orbital-Optimized Density Functional Calculations I: Dipole Moments of Rydberg States

Problem Statement
Rydberg excited states, characterized by highly diffuse electron orbitals, pose significant challenges for electronic structure calculations. While Time-Dependent Density Functional Theory (TDDFT) offers computational efficiency, standard adiabatic implementations often fail to describe Rydberg excitations accurately due to a lack of orbital relaxation and erroneous asymptotic behavior of exchange-correlation potentials. Orbital-optimized (OO) density functional calculations have emerged as a promising alternative, providing state-specific variational optimization that accounts for orbital relaxation. However, previous benchmarks for OO methods have predominantly focused on vertical excitation energies and geometries. Assessments of excited-state dipole moments have been limited, particularly for states above the lowest-energy excitation. Furthermore, the performance of these methods regarding dipole moments is complicated by the basis set requirements: accurately describing diffuse Rydberg tails typically necessitates extensive diffuse functions in atomic orbital (LCAO) basis sets, which can introduce linear dependency issues and high computational costs. It remains unclear whether standard LCAO basis sets, even when augmented, can reliably reproduce the anisotropic density redistribution required for accurate dipole moment predictions in Rydberg states.

Methodology
The authors employ a fully variational orbital-optimized density functional approach using a direct optimization (DO) algorithm to target excited states as stationary points (saddle points) of the energy functional. This method avoids the variational collapse issues common in standard self-consistent field (SCF) algorithms.

• System Selection: Calculations were performed on water (H₂O), formaldehyde (CH₂O), ammonia (NH₃), and methanol (CH₃OH), covering singlet and triplet excited states with excitation energies up to 10 eV.
• Basis Sets: A critical component of the study is the comparison between plane wave (PW) basis sets and LCAO basis sets (aug-cc-pVDZ+sz and d-aug-cc-pVDZ+sz). PWs are utilized to provide a flexible, delocalized representation of diffuse orbitals without confinement effects, serving as a robust reference for the spatial extent of the electron density.
• Functionals: The study evaluates the Generalized Gradient Approximation (PBE), the hybrid functional (PBE0), and PBE with explicit Perdew-Zunger self-interaction correction (SIC), tested with both full ($\alpha=1$) and globally scaled ($\alpha=1/2$) corrections.
• Spin Treatment: Open-shell singlet states are treated using unrestricted single-determinant calculations, yielding mixed-spin solutions. Spin purification formulas are applied to extract pure singlet energies and dipole moments from the mixed-spin and corresponding triplet states.
• Dipole Calculation: Dipole moments are computed from the state-specific electron density within the projector augmented wave (PAW) formalism.

Key Contributions and Results

1. Basis Set Sensitivity: The study demonstrates that excited-state dipole moments are significantly more sensitive to basis set choice than excitation energies. Even when excitation energies are insensitive to the basis set, dipole moments can vary substantially.

   • Single-augmented LCAO basis sets (aug) often overconfine the electron density, leading to large errors in both the magnitude and orientation of the dipole moment for diffuse Rydberg states.
   • Adding extra diffuse functions (d-aug) improves the description of the spatial extent (variance) but does not guarantee convergence of the dipole moment. The atom-centered nature of LCAO functions limits their flexibility in describing the anisotropic redistribution of density, a limitation not present in plane wave representations.
   • For the most diffuse states, discrepancies between d-aug and PW calculations persist, indicating that localized atomic orbitals may be insufficient for accurate dipole predictions even with extensive augmentation.

2. Functional Dependence: Unlike TDDFT, where dipole moments show strong functional dependence, OO calculations exhibit moderate dependence on the exchange-correlation functional.

   • PBE0: This hybrid functional yields the best agreement with high-level reference values (where available), reducing the median absolute relative error to approximately 6%.
   • PBE: The pure GGA functional provides reasonable results with a median absolute relative error of ~20%.
   • SIC: Contrary to expectations based on excitation energy improvements, the inclusion of Perdew-Zunger SIC (both full and half-scaled) does not improve dipole moment accuracy. Instead, SIC tends to overestimate dipole magnitudes and introduces larger errors than PBE or PBE0. This suggests that restoring the correct $-1/r$ asymptotic behavior of the potential is insufficient for improving dipole moments, which depend on the full density redistribution rather than just the asymptotic tail.
   • Spin Purification Issues: SIC calculations frequently lead to symmetry-breaking solutions in triplet states, complicating the spin purification process for certain states in ammonia and methanol.

3. Benchmarking: The results are benchmarked against theoretical best estimates (TBE) derived from high-order coupled-cluster calculations (CCSDT, CCSDTQP) and complete basis set (CBS) extrapolations. The study highlights that reference data for highly diffuse Rydberg states is often scarce or derived from atom-centered basis sets that may not be sufficiently flexible, underscoring a gap in available benchmarks.

Significance and Claims
The paper asserts that excited-state dipole moments serve as a stricter and more demanding test for orbital-optimized methods than excitation energies. The primary significance lies in demonstrating that:

• State-specific orbital relaxation, inherent to OO methods, is crucial for the accurate prediction of Rydberg state dipole moments, outperforming standard TDDFT implementations.
• Plane wave basis sets are essential for reliably describing the anisotropic density redistribution of diffuse Rydberg states, revealing limitations in commonly used atom-centered basis sets that persist even with double-augmentation.
• Self-interaction correction, while beneficial for excitation energies, does not necessarily improve dipole moment predictions and may introduce practical convergence and symmetry-breaking difficulties.

The authors conclude that while OO/PBE0 calculations with plane waves provide a robust framework for predicting dipole moments of Rydberg states, further developments are needed, including high-level reference data for diffuse states using flexible basis representations and improved SIC formulations (e.g., locally scaled SIC) to address overcorrection and symmetry-breaking issues.