Relativistic KRCI calculations of symmetry violating interaction constants for YbX (X: Cu, Ag and Au) molecules

This paper reports relativistic KRCI calculations of parity-odd and time-reversal-odd interaction constants, along with first-time determinations of hyperfine structure constants, for the ground electronic states of YbCu, YbAg, and YbAu molecules.

The Gist

Imagine the universe as a giant, perfectly balanced scale. For a long time, physicists believed this scale was perfectly symmetrical: if you flipped the universe like a mirror image (Parity) or played the movie of the universe backward (Time-Reversal), the laws of physics would look exactly the same.

However, modern physics suspects there's a tiny, hidden weight on one side of the scale that we haven't found yet. This "weight" is a violation of those symmetries, and finding it could explain why our universe is made of matter instead of being an empty void of antimatter.

This paper is like a team of master architects (the researchers) building a highly detailed, microscopic model of a specific type of "suspicious" molecule to see if they can spot that hidden weight.

Here is the breakdown of their work in everyday terms:

1. The Detective Work: Hunting for "Ghost" Particles

The researchers are looking for two specific things:

• The Electron's "Lopsided" Shape (eEDM): Imagine an electron as a tiny, perfect sphere. If it has an "electric dipole moment," it's like the sphere is slightly squashed or has a positive side and a negative side, making it lopsided. This lopsidedness is a smoking gun for new physics.
• The "Whisper" Between Nucleus and Electron (Scalar-Pseudoscalar Interaction): This is a subtle, invisible handshake between the center of the atom (the nucleus) and the electron. It's a specific type of interaction that shouldn't happen if the universe is perfectly symmetrical.

2. The Magnifying Glass: Why YbX Molecules?

To find these tiny effects, you need a magnifying glass. In the atomic world, heavy, polar molecules are the best magnifying glasses.

• The team chose YbX molecules, where "Yb" is Ytterbium (a heavy rare-earth metal) and "X" is a metal partner: Copper (Cu), Silver (Ag), or Gold (Au).
• Think of these molecules as heavy, spinning tops. Because they are heavy and have an uneven charge distribution, they create a massive internal electric field. It's like a storm inside the molecule. When the electron spins in this storm, any tiny "lopsidedness" gets amplified, making it easier to detect.

3. The Simulation: Building a Digital Twin

Since we can't easily measure these tiny effects in a lab right now, the researchers built a super-accurate digital twin of these molecules using a supercomputer.

• The Method (KRCI): They used a method called "Kramers-restricted Configuration Interaction." Imagine trying to predict the weather. You could just guess, or you could simulate every single air molecule. This method simulates the behavior of every single electron in the molecule, accounting for the fact that they are moving near the speed of light (relativity).
• The Blueprint (Basis Sets): They used different levels of detail for their blueprint, ranging from a "double-zeta" (a rough sketch) to a "quadruple-zeta" (a hyper-realistic 3D render). They found that for the Gold molecule (YbAu), they needed the most detailed blueprint to get the right answer, whereas the Copper and Silver molecules were a bit more forgiving.

4. The Findings: What Did They Discover?

• The "Gold" Surprise: They found that the YbGold molecule behaves very differently from the others. The effects they were looking for were much smaller in YbGold. Why? It's like two people pulling a rope in opposite directions with equal strength; the rope doesn't move. In YbGold, the Ytterbium and Gold atoms cancel each other out, making the signal very weak.
• The "Silver" and "Copper" Twins: The YbSilver and YbCopper molecules showed strong, similar signals, making them excellent candidates for future experiments.
• New Maps (Hyperfine Structure): The paper also provided the first-ever "maps" of the magnetic fingerprints (hyperfine structure) for the atoms inside these molecules. This is crucial because if scientists want to catch these molecules in a lab, they need to know exactly how to tune their lasers to grab them. It's like knowing the exact radio frequency to tune into a specific station.

5. Why Does This Matter?

This paper is a roadmap for future experiments.

• Experimentalists are currently building machines to catch these molecules and measure them.
• The theorists (the authors) have provided the "theoretical target." They are saying, "If you build your machine to look for these specific values, you might find the new physics we are all hunting for."
• If they find these symmetry violations, it could rewrite the textbooks on how the universe began and why we exist.

In a nutshell: The authors used a supercomputer to simulate heavy metal molecules with extreme precision. They figured out exactly how these molecules should behave if the laws of physics are slightly broken, and they provided the first-ever detailed magnetic maps for these molecules, guiding experimentalists on where to look for the next big discovery in physics.

Technical Summary

1. Problem Statement and Motivation

The search for the intrinsic electric dipole moment of the electron ($eEDM$, denoted as $d_e$) is a primary avenue for probing physics beyond the Standard Model (BSM), specifically testing violations of Parity (P) and Time-reversal (T) symmetries.

• Experimental Context: Heavy open-shell polar molecules are preferred over atoms for $eEDM$ searches due to their large internal effective electric fields ($\mathcal{E}_{eff}$). Molecules like YbX (where X = Cu, Ag, Au) have recently been identified as promising candidates for such experiments.
• Theoretical Gap: Interpreting experimental results requires precise theoretical values for P-odd and T-odd interaction constants:
  • $W_d$: Related to the effective electric field ($\mathcal{E}_{eff}$) and $d_e$.
  • $W_s$: Characterizes the scalar-pseudoscalar (S-PS) nucleon-electron interaction.
  • Hyperfine Structure (HFS) Constants: Essential for laser cooling and trapping experiments (optical selection rules), yet previously unreported for YbX molecules.
• Limitations of Previous Work: While a recent study by Polet et al. (2024) reported $W_d$ and $W_s$ using the Fock-space coupled-cluster (FSCCSD) method, there was a need for independent verification using a different high-level relativistic method (Configuration Interaction) and a comprehensive report on HFS constants.

2. Methodology

The authors performed relativistic calculations using the DIRAC software suite.

• Electronic Structure Method:
  • Kramers-Restricted Configuration Interaction (KRCI): Specifically limited to single and double excitations (KRCISD). This method accounts for electron correlation effects within a four-component relativistic framework.
  • Wavefunction: The CI wavefunction is a linear combination of determinantal functions built from a reference Dirac-Hartree-Fock (DHF) state.
• Basis Sets:
  • Utilized uncontracted Dyall's core-valence basis sets: cv2z, cv3z, and cv4z (double, triple, and quadruple-zeta quality).
  • Frozen Core: Orbitals with energies lower than $-1$ Hartree ($E_h$) were frozen, resulting in 27 active electrons. For the cv2z basis set, a cutoff of $-2$ $E_h$ was used to match previous literature (increasing active electrons to 33).
• Generalized Active Space (GAS) Technique:
  • To manage computational cost while capturing correlation, two distinct GAS configurations were tested:
    • G-C1: Active space partitioned into paired, unpaired, and virtual orbitals.
    • G-C2: A redistribution where the $6s^2$ electrons of Yb are included in the active space alongside the unpaired electron of the X atom.
• Physical Quantities Calculated:
  • $W_d$ and $W_s$: Calculated via expectation values of the eEDM and S-PS interaction Hamiltonians.
  • HFS Constants: Calculated the parallel ($A_\parallel$) and perpendicular ($A_\perp$) components of the magnetic dipole hyperfine structure constants for various isotopes ($^{171,173}$Yb, $^{63,65}$Cu, $^{107,109}$Ag, $^{197}$Au).

3. Key Contributions

1. First-Principles Calculation of HFS Constants: This work provides the first reported values for the parallel and perpendicular components of the magnetic dipole HFS constants for YbCu, YbAg, and YbAu molecules. These are critical for future laser-cooling experiments.
2. Independent Verification of Interaction Constants: The study provides an independent calculation of $W_d$ and $W_s$ using the KRCISD method, offering a crucial cross-check against the existing FSCCSD results.
3. Systematic Basis Set and Configuration Analysis: The authors systematically analyzed the sensitivity of results to basis set size (cv2z to cv4z) and the treatment of active orbitals (G-C1 vs. G-C2), identifying the most reliable computational protocols for these heavy systems.

4. Key Results

• Interaction Constants ($W_d$ and $W_s$):
  • Agreement with Literature: For YbCu and YbAg, the G-C1 configuration yielded results within ~3% of the FSCCSD literature values. However, for YbAu, G-C1 showed significant deviation, while G-C2 provided results much closer to the literature (absolute differences of ~3.5% for $W_d$ and ~0.7% for $W_s$).
  • Magnitude Trends: $W_d$ and $W_s$ values for YbCu and YbAg are of similar magnitude. YbAu exhibits significantly smaller values due to the counter-balancing of large, opposite-signed contributions from the constituent atoms.
  • Basis Set Sensitivity: YbAu showed higher sensitivity to the size of the virtual orbital space and basis set size compared to YbCu and YbAg.
  • Recommendation: The authors recommend the G-C2 configuration using the cv3z basis set as the most reliable set of values (highlighted in bold in their tables).
• Hyperfine Structure Constants:
  • Calculated $A_\parallel$ and $A_\perp$ for all relevant isotopes.
  • Isotopic Differences: The absolute differences in HFS components between $^{171}$Yb and $^{173}$Yb are substantial (ranging from ~2200 to ~3300 MHz depending on the molecule), indicating significant variations in spin density near the nucleus.
  • Configuration Sensitivity: The HFS constants for Yb isotopes showed a maximum variation of ~5% between G-C1 and G-C2. For the X atoms (Cu, Ag, Au), the deviation was higher (up to 22% for Cu), highlighting the importance of active orbital redistribution in HFS calculations.

5. Significance and Impact

• Support for $eEDM$ Searches: By providing accurate $W_d$ and $W_s$ values, this work enables the conversion of future experimental energy shift measurements into precise constraints on the electron's electric dipole moment ($d_e$).
• Enabling Laser Cooling: The newly reported HFS constants are a prerequisite for designing laser cooling and trapping schemes for YbX molecules. Without these values, optical selection rules and momentum transfer strategies cannot be effectively implemented.
• Methodological Validation: The study demonstrates that the KRCISD method, particularly with the G-C2 active space treatment, is a robust alternative to coupled-cluster methods for calculating symmetry-violating properties in heavy diatomic molecules.
• Future Directions: The results serve as a benchmark for future theoretical improvements (e.g., including triple excitations) and are essential for the experimental community planning to use YbX molecules in next-generation fundamental physics tests.