Rotational Splittings in Diatomic Molecules of Interest to Searches for New Physics

This paper presents a theoretical model that integrates relativistic four-component wavefunctions with Hund's case (a) rotational Hamiltonians to calculate $\Lambda$-splittings in diatomic molecules like PtH, ThF$^+$, and TaO$^+$, providing qualitative agreement with experiments and predicting a 9 kHz splitting for TaO$^+$ that offers benefits for reducing systematic uncertainty in searches for new physics while posing potential depolarization challenges.

The Gist

The Big Picture: Hunting for Ghosts in the Machine

Imagine the Standard Model of physics as a massive, incredibly detailed instruction manual for how the universe works. It explains almost everything we see: atoms, light, gravity. But scientists suspect there are "ghosts" in the machine—mysterious forces or particles that the manual doesn't mention. These ghosts might explain why the universe is made of matter instead of antimatter, or why time seems to flow only one way.

To find these ghosts, scientists use diatomic molecules (molecules made of just two atoms) as ultra-sensitive probes. Think of these molecules as tiny, spinning tops. If a "ghost" force touches them, the way they spin changes in a very specific, tiny way.

The Problem: The "Spinning Top" is Too Stiff

In these experiments, scientists need to line up (polarize) these spinning molecules using an electric field, like using a magnet to line up iron filings.

However, these molecules have a natural "stiffness" or internal wobble called $\Lambda$-splitting (Lambda-splitting).

• The Analogy: Imagine trying to balance a spinning top on a table. If the top has a heavy, uneven weight inside, it wobbles on its own even before you touch it. This wobble makes it hard to control.
• The Science: This internal wobble creates a tiny energy gap between two states of the molecule. If this gap is too big, you need a huge electric field to force the molecule to line up. But huge electric fields create "noise" (systematic errors) that can hide the ghost signals scientists are looking for.

The Goal: Scientists want molecules where this internal wobble is almost non-existent (tiny $\Lambda$-splitting). This way, a gentle electric field can line them up perfectly without creating noise.

The Solution: A New Mathematical Map

The authors of this paper (Sunaga and Fleig) created a new theoretical map (a computer model) to predict exactly how big this "wobble" is for different molecules.

They didn't just guess; they built a sophisticated model that combines:

1. Relativistic Quantum Mechanics: Because these molecules contain heavy atoms (like Platinum, Thorium, and Tantalum), the electrons move so fast they need Einstein's relativity to be described correctly.
2. Rotational Physics: How the whole molecule spins.

They treated the molecule's wavefunction (its quantum "fingerprint") as a complex mixture of different possibilities, rather than a single simple state. This is crucial because heavy atoms are messy; their electrons don't stick to simple rules.

The Test Drive: Three Molecules

To check if their map was accurate, they tested it on three specific molecules:

1. Platinum Hydride (PtH): The "Control Group"

• Why: We already know the answer for this one from past experiments.
• Result: The authors' model matched the real-world data almost perfectly. This proved their map works. It's like driving a new GPS through a city you know well to make sure the directions are right.

2. Thorium Fluoride Cation (ThF+): The "Heavy Hitter"

• Why: This is a leading candidate for future experiments.
• Result: The model predicted the wobble size correctly, confirming that ThF+ is a good candidate, though its wobble is still a bit "loud" (around 5–10 MHz).

3. Tantalum Oxide Cation (TaO+): The "Star Discovery"

• Why: This is the new kid on the block.
• Result: The model predicted something amazing: The wobble is incredibly tiny—only about 9 kHz.
• The Analogy: If the wobble in ThF+ is like a car engine idling, the wobble in TaO+ is like a whisper.
• Why this matters: Because the wobble is so small, scientists can line up TaO+ molecules with a very gentle electric field. This reduces "noise" and makes the experiment much cleaner.

The Catch: The "Ramp-Up" Problem

There is a small downside to having such a tiny wobble.

• The Analogy: Imagine a spinning top that is too perfectly balanced. If you try to speed it up too quickly, it might get confused and fall over (depolarize).
• The Science: In a real experiment, scientists have to ramp up the rotation of the molecules. If the internal wobble is too small, the molecule might lose its alignment during this process. The authors note that while TaO+ is theoretically perfect for reducing noise, scientists will need to be very careful with how they spin it up to avoid losing the signal.

The Conclusion

This paper is a blueprint for the next generation of physics experiments.

By creating a reliable way to calculate these tiny internal wobbles, the authors have identified TaO+ as a potentially superior candidate for hunting new physics. It offers a "quiet" environment where the faint whispers of new laws of physics might finally be heard, provided scientists can handle the delicate task of spinning it up without dropping the ball.

In short: They built a better calculator, found a molecule (TaO+) that is almost perfectly quiet, and told the world, "Hey, if you want to find new physics, try spinning this one!"

Technical Summary

1. Problem Statement

Diatomic molecules with low-lying $^3\Delta_1$ electronic states (such as HfF$^+$, ThF$^+$, and TaO$^+$) are premier platforms for searching for physics beyond the Standard Model, specifically for measuring the electron's electric dipole moment (EDM) and nuclear magnetic quadrupole moments (indicating CP-violation).

A critical requirement for these experiments is the ability to polarize molecules using small external electric fields. This polarization relies on the mixing of opposite parity states ($e$ and $f$ components) within a rotational level. The efficiency of this mixing is inversely proportional to the energy separation between these parity doublets, known as $\Lambda$-splitting (or $\Lambda$-doubling).

• The Challenge: While small $\Lambda$-splitting is desirable for easy polarization, it must be accurately predicted to avoid systematic errors (e.g., depolarization during rotation ramp-up) and to design appropriate experimental traps.
• The Gap: Existing theoretical methods often rely on scalar relativistic approximations or perturbative treatments of spin-orbit coupling, which may lack the precision required for heavy, multi-reference molecular systems where relativistic effects are dominant.

2. Methodology

The authors present a theoretical model that integrates relativistic four-component wavefunctions with the traditional rotational Hamiltonian based on Hund's case (a).

• Hamiltonian Formulation:
  The total Hamiltonian is $\hat{H} = \hat{H}_{ELE} + \hat{H}_{ROT}$.
  • $\hat{H}_{ELE}$: Derived from a four-component Dirac-theory framework, accurately including spin-orbit (SO) coupling in the zeroth-order wavefunctions.
  • $\hat{H}_{ROT}$: The rotational Hamiltonian expressed in Hund's case (a) basis, utilizing the operator $\hat{N} = \hat{J} - \hat{L} - \hat{S}$.
• Basis Set and Multireference Character:
  • The model uses signed basis states $|e/f J \Omega \Lambda \Sigma\rangle$.
  • Crucially, it accounts for the multireference character of the wavefunctions. The physical states are treated as linear combinations of basis functions: $|\Psi\rangle = \sum C_i |e/f J \Omega \Lambda_i \Sigma_i\rangle$.
  • For molecules with complex electronic structures (ThF$^+$, TaO$^+$), the model incorporates configuration weights ($d$). The matrix elements are weighted by the overlap of the target electronic configurations (e.g., $7s^2$ vs. $7s7p$ in ThF$^+$, or $5d^2$ in TaO$^+$).
• Mechanism of $\Lambda$-Doubling:
  The paper clarifies that for $^3\Delta_1$ states, direct rotational coupling does not cause splitting. Instead, the splitting arises indirectly:
  1. Spin-orbit coupling mixes the target $^3\Delta_1$ state with a nearby $^{2S+1}\Pi_1$ state.
  2. Rotational operators ($\hat{J}_\pm \hat{L}_\mp$) couple this mixed $\Pi_1$ component to a nearby $^{2S+1}\Sigma_0$ state.
  3. This second-order interaction induces the $\Lambda$-splitting.
• Approximations:
  • Vibrational overlap integrals are approximated as unity ($\langle v|v'\rangle \approx 1$) near equilibrium, justified by the small changes in spectroscopic constants between coupled states.
  • Hyperfine interactions are neglected as they are minute compared to rotational couplings.

3. Key Contributions

1. Unified Theoretical Framework: The authors developed a code that combines high-precision four-component relativistic electronic structure data with a rigorous rotational Hamiltonian treatment, specifically tailored for the multireference nature of heavy molecular ions.
2. Quantification of Configuration Weights: The study explicitly demonstrates how the "weight" of specific electronic configurations ($d$) reduces the effective rotational coupling matrix elements, a factor often overlooked in simpler models.
3. Validation and Prediction: The method was validated against existing experimental data for PtH and ThF$^+$ and used to make the first robust theoretical prediction for TaO$^+$.

4. Results

A. Platinum Hydride (PtH)

• Purpose: Verification of the method against known experimental data.
• Findings: The model successfully reproduced the order of magnitude of experimental $\Lambda$-splittings for various electronic states ($\Omega = 1/2, 3/2, 5/2$).
• Sensitivity: The results showed significant dependence on the input energy gaps between coupled states. When experimental energy gaps were used, the agreement improved, though some overestimation occurred, highlighting the sensitivity of the perturbation to energy denominators.

B. Thorium Fluoride Cation (ThF$^+$)

• Context: A leading candidate for electron EDM searches.
• Findings:
  • The calculated $\Lambda$-splitting for the $^3\Delta_1$ state (rotational ground state $J=1$) ranges from 2.2 MHz to 4.5 MHz (depending on the choice of angular momentum quantum number $L$ and inclusion of configuration weights).
  • Including the electronic configuration weights ($d$) reduced the splitting by approximately half compared to models ignoring them.
  • The results qualitatively agree with experimental values ($5.29(5)$ MHz and $10.0$ MHz from different sources), confirming the model's reliability for order-of-magnitude predictions in complex systems.

C. Tantalum Oxide Cation (TaO$^+$)

• Context: A promising candidate for nuclear CP-violation searches via the nuclear magnetic quadrupole moment.
• Findings:
  • The predicted $\Lambda$-splitting for the $^3\Delta_1$ ground state at $J=1$ is approximately 9 kHz (specifically 7.9–8.7 kHz depending on input parameters).
  • Comparison: This is significantly smaller than ThF$^+$ (MHz scale).
  • Reasons for small splitting:
    1. Larger energy gap between the $^3\Delta_1$ and the coupling $^3\Sigma_0$ state.
    2. Weaker contribution of the $\Pi_1$ state to the $^3\Delta_1$ state (weaker SO mixing).
    3. Smaller overlap of the $5d^2$ electronic configurations between the coupled states.

5. Significance and Implications

• Experimental Design: The predicted ~9 kHz splitting for TaO$^+$ is a double-edged sword:
  • Advantage: It allows for full molecular polarization with very small external electric fields, reducing systematic uncertainties related to field inhomogeneity.
  • Challenge: The splitting is so small that it may lead to depolarization during the "rotation ramp-up" phase of an experiment if not carefully managed.
• Trap Requirements: The small splitting implies that TaO$^+$ experiments can potentially avoid the need for large ring traps required for molecules with larger splittings, simplifying ion dynamics studies.
• General Applicability: The developed methodology is applicable to other target molecules for CP-violation searches, such as TaN, WC, and PbO. It provides a robust tool for estimating $\Omega$-doublings in systems where Hund's case (a) is a good approximation but relativistic and multireference effects are non-negligible.

In conclusion, this paper provides a critical theoretical bridge between high-level relativistic electronic structure calculations and the practical rotational dynamics required for next-generation precision tests of fundamental physics.