Nuclear-electronic orbital second-order coupled cluster for excited states

This paper introduces the NEO-CC2 and NEO-SOS'-CC2 methods for calculating excited states within the nuclear-electronic orbital framework, demonstrating that the scaled-opposite-spin variant achieves near-quantitative accuracy in capturing vibrational, electronic, and mixed excitations at a significantly lower computational cost than existing high-accuracy approaches.

The Gist

Imagine you are trying to predict how a molecule vibrates and dances when it gets excited by energy. Usually, scientists treat the heavy atomic nuclei (like protons) as solid, stationary balls and only the tiny, fast-moving electrons as quantum waves. This is like watching a dance where the heavy dancers are glued to the floor, and only the light, airy dancers are allowed to float and spin.

But in reality, especially for light atoms like hydrogen, the "heavy" protons also wiggle, tunnel through barriers, and act like quantum waves. The Nuclear-Electronic Orbital (NEO) framework is a new way of thinking where we treat these protons just like electrons—giving them their own quantum "dance moves."

However, calculating these quantum dances for both electrons and protons at the same time is incredibly expensive, like trying to simulate a whole orchestra playing perfectly in real-time. It requires supercomputers and takes forever.

The Problem: The "Cheap" vs. "Expensive" Dilemma

Scientists have two main tools:

1. The "Cheap" Tool (NEO-TDDFT): It's fast and good for basic moves, but it misses the fancy, complex choreography (like overtones and double jumps). It's like a dance instructor who can teach you the basic steps but can't choreograph a complex routine.
2. The "Expensive" Tool (NEO-EOM-CCSD): It's incredibly accurate and captures every nuance of the dance, but it's so computationally heavy that you can only use it on tiny molecules. It's like hiring a world-class choreographer and a full orchestra, but you can only afford to do it for a single duet.

The Solution: A "Smart" Middle Ground

This paper introduces a new method called NEO-CC2 and its upgraded cousin, NEO-SOS′-CC2. Think of these as a "smart, budget-friendly choreographer" that uses a clever trick to get 90% of the accuracy of the expensive method but runs 1,000 times faster.

Here is how they work, using simple analogies:

1. The "Approximate" Dance (NEO-CC2)

The authors developed a method that approximates the complex interactions between electrons and protons. It's like using a simplified map to navigate a city. It gets you to the right neighborhood, but sometimes you miss the exact street address.

• The Result: It works okay, but it tends to overestimate how much energy is needed for the protons to jump to higher vibrational states. It's like guessing the price of a ticket and always being a little too high.

2. The "Scaling" Trick (NEO-SOS′-CC2)

This is the real magic of the paper. The authors realized that the "cheap" method gets the ground state (the resting dance) and the excited state (the jumping dance) wrong in different ways.

• The Analogy: Imagine you are trying to balance a scale. The "cheap" method puts too much weight on the ground side. To fix this, they introduced a scaling factor—a special knob they can turn.
  • They turn the knob to reduce the weight on the ground state.
  • They turn the knob to increase the weight on the excited state.
• The Result: By tweaking this knob (specifically for how electrons and protons interact), they fixed the balance. Suddenly, the "cheap" method became nearly as accurate as the "expensive" one.

What Did They Test?

They tested this new method on several molecular "dance floors":

• Positronium Hydride: A molecule made of a proton, two electrons, and an anti-matter particle (a positron). Here, they proved their method could match the "gold standard" results almost perfectly.
• Triatomic Molecules (HCN, HNC, FHF⁻, HeHHe⁺): These are molecules with three atoms. They looked at how the hydrogen atom vibrates.
  • The Victory: The old "cheap" method (TDDFT) failed to predict "overtones" (dancing to the beat of the beat) and "combination bands" (mixing two different dance moves).
  • The New Method: Their new NEO-SOS′-CC2 successfully predicted these complex moves. It could see the "double jumps" where an electron and a proton get excited at the same time, something the cheaper methods completely missed.

Why Does This Matter?

This is a breakthrough because it allows scientists to study complex chemical reactions involving quantum protons (like in enzymes or fuel cells) without needing a supercomputer for every single calculation.

In summary:
The authors built a high-speed, high-accuracy simulation tool for molecules where protons act like quantum waves. They did this by taking a fast but slightly inaccurate method and adding a "tuning knob" (scaling factor) that corrects the errors. Now, we can watch the full, complex quantum dance of molecules in a fraction of the time it used to take.

Technical Summary

1. Problem Statement

The Nuclear-Electronic Orbital (NEO) framework treats specific nuclei (typically protons) quantum mechanically alongside electrons, allowing for the simultaneous description of electronic and nuclear quantum effects (e.g., tunneling, zero-point energy, and proton delocalization). While ground-state NEO methods are well-developed, excited-state methods face significant challenges:

• Accuracy vs. Cost Trade-off: Affordable methods like NEO-Time-Dependent Density Functional Theory (NEO-TDDFT) are limited to single-excitation character. They fail to capture higher-order vibrational states (overtones, combination bands) and mixed electron-proton double excitations due to the underlying adiabatic approximation.
• Computational Bottleneck: High-accuracy methods like NEO-EOM-CCSD (Equation-of-Motion Coupled Cluster Singles and Doubles) or NEO-MRCI (Multireference Configuration Interaction) can capture these complex features but scale poorly ($O(N^6)$ or higher), making them prohibitively expensive for larger systems.
• Correlation Deficiencies: Existing lower-cost NEO methods often struggle with electron-proton correlation, leading to systematic errors in excitation energies.

The authors aim to develop a method that bridges this gap: offering the accuracy of high-level wavefunction methods for excited states with the computational efficiency of second-order approximations.

2. Methodology

The authors introduce and implement two new methods within the Q-Chem 6.3 software package:

A. NEO-CC2 (Nuclear-Electronic Orbital Second-Order Coupled Cluster)

• Theory: This is an excited-state variant of NEO-CC2. It retains the full single-excitation ($T_1$) amplitude equations from NEO-CCSD but truncates the double-excitation ($T_2$) amplitude equations at the second order.
• Implementation: Excited states are obtained via coupled cluster response theory by diagonalizing the Jacobian matrix. To avoid the cost of full diagonalization, the Löwdin partitioning technique is used to reduce the problem to an effective single-excitation space, incorporating the effects of double excitations perturbatively.
• Limitation: Standard NEO-CC2 tends to underestimate electron-proton correlation energy, leading to poor accuracy for certain systems.

B. NEO-SOS′-CC2 (Scaled-Opposite-Spin NEO-CC2)

• Theory: To correct the correlation deficiencies of standard CC2, the authors implement a Scaled-Opposite-Spin (SOS) approach adapted for multicomponent systems.
• Scaling Factors: The method introduces distinct scaling factors for ground-state ($c_{gs}$) and excited-state ($c_{ex}$) correlations. Crucially, it separates scaling for:
  • Same-spin electronic components (set to zero, following standard SOS).
  • Opposite-spin electronic components.
  • Electron-proton components (denoted as $c_{ep}$).
• Key Innovation: The authors found that for systems with quantum protons, increasing ground-state correlation (as done in standard SOS) degrades excited-state accuracy. Therefore, they optimized the scaling to either decrease ground-state electron-proton correlation or increase excited-state electron-proton correlation relative to the ground state.

3. Key Contributions

1. Development of NEO-CC2/NEO-SOS′-CC2: The first implementation of second-order coupled cluster methods for excited states within the multicomponent NEO framework.
2. Handling Mixed Excitations: The methods successfully capture mixed electron-proton double excitations (vibronic excitations), a feature missing in NEO-TDDFT and NEO-TDHF.
3. Optimized Scaling Strategy: The identification that standard SOS scaling (increasing correlation) is detrimental for protonic excited states. The proposed NEO-SOS′-CC2 uses specific scaling factors ($c_{gs}^{ep} = 0.91, c_{ex}^{ep} = 1.0$) to balance ground and excited state descriptions, significantly improving accuracy.
4. Computational Efficiency: The methods retain the favorable scaling of CC2 (potentially $O(N^4)$ with density fitting), offering a robust alternative to the $O(N^6)$ NEO-EOM-CCSD.

4. Results and Benchmarking

The methods were benchmarked on four systems: Positronium Hydride (PsH), HeHHe+, HCN, HNC, and FHF−.

A. Positronium Hydride (PsH)

• Context: Electrons and positrons treated quantum mechanically; proton is classical.
• Findings:
  • Standard NEO-CC2 significantly underestimated correlation energy (MUE of 11.3 mHa vs. NEO-FCI).
  • NEO-SOS′-CC2 (with $c_{ep}=1.3$) achieved near-quantitative accuracy (MUE of 3.0 mHa), comparable to the much more expensive NEO-EOM-CCSD.

B. Quantum Proton Systems (HeHHe+, HCN, HNC, FHF−)

• Context: One quantum proton and electrons treated quantum mechanically.
• Vibrational Features:
  • NEO-CC2: Captured overtones and combination bands qualitatively but overestimated fundamental frequencies (e.g., bending modes in HeHHe+ were ~200 cm⁻¹ too high).
  • NEO-SOS′-CC2: With optimized scaling ($c_{gs}^{ep}=0.91, c_{ex}^{ep}=1.0$), the method achieved near-quantitative agreement with reference FGH (Fourier Grid Hamiltonian) and NEO-MRCI data.
  • Comparison: NEO-SOS′-CC2 outperformed NEO-TDDFT, which failed to predict overtones correctly (predicting them at ~10x the fundamental frequency) and often got the state ordering wrong.
• Mixed Electron-Proton Excitations:
  • In HCN, the method successfully predicted mixed electron-proton double excitations (e.g., simultaneous excitation of an electron and a protonic bending mode).
  • The energy gaps between pure electronic excitations and mixed vibronic excitations matched those of NEO-EOM-CCSD within ~0.1 eV.

5. Significance and Conclusion

This work represents a major step forward in multicomponent quantum chemistry.

• Accuracy: NEO-SOS′-CC2 provides accuracy approaching that of high-level methods (NEO-EOM-CCSD, NEO-MRCI) for excited states, including complex features like overtones, combination bands, and mixed vibronic excitations.
• Efficiency: By utilizing the CC2 approximation and SOS scaling, the method offers a computationally affordable route ($O(N^5)$ or $O(N^4)$) to these high-accuracy results, making it applicable to larger molecular systems where NEO-EOM-CCSD is intractable.
• Physical Insight: The study highlights the critical importance of treating ground-state and excited-state electron-proton correlations differently. The finding that reducing ground-state correlation scaling improves excited-state accuracy is a novel and crucial insight for future multicomponent method development.

In summary, NEO-SOS′-CC2 is a promising ab initio alternative to NEO-TDDFT, capable of delivering quantitative accuracy for excited-state spectra in systems with significant nuclear quantum effects.