Extension of the CIPSI-Driven CC($P$;$Q$) Approach to Excited Electronic States

This paper extends the CIPSI-driven CC($P$;$Q$) methodology to excited electronic states using the equation-of-motion coupled-cluster formalism, demonstrating its ability to efficiently converge high-accuracy energetics for various molecular systems through inexpensive Hamiltonian diagonalizations in compact configuration interaction spaces.

The Gist

Imagine you are trying to predict exactly how a molecule behaves when it gets excited—like when sunlight hits it and it starts to glow or react. In the world of quantum chemistry, molecules are like incredibly complex orchestras. To understand the music they make (their energy levels), you have to listen to every single instrument (electron) playing at once.

The problem is that the "full orchestra" (a method called Full Configuration Interaction) is so massive that even the world's fastest supercomputers can't play the whole song in a reasonable amount of time.

The Old Way: The "Good Enough" Guess

Scientists have developed a method called Coupled-Cluster (CC) theory. Think of this as a conductor who only listens to the first and second rows of the orchestra (the "singles" and "doubles"). This is fast and usually works well for simple songs.

However, when the music gets complicated—like when the molecule is stretched out or when two electrons jump to new seats at the same time (double excitations)—the conductor misses the crucial notes played by the "third row" (triples). To hear those notes, you'd need to listen to the entire orchestra, which takes forever.

There are shortcuts (like CR-EOMCC(2,3)) that try to guess what the third row is doing based on the first two. But sometimes, the guess is wrong, especially when the molecule is in a tricky, stretched-out state. It's like trying to predict a jazz solo by only looking at the sheet music for the first two bars; you might miss the wild improvisation that happens later.

The New Solution: The "Smart Scout" (CIPSI-Driven CC(P;Q))

This paper introduces a brilliant new strategy. Instead of guessing the third row or listening to the whole orchestra, they use a Smart Scout (called CIPSI).

Here is how the new method works, using a simple analogy:

1. The Scout's Job: Imagine you need to find the most important notes in a massive library of sheet music (the "triply excited determinants"). You don't have time to read every book. So, you send out a Smart Scout (CIPSI) who quickly flips through the books and picks out the top 1% of the most critical notes that actually matter for the song.
2. The Main Conductor (CC(P)): The main conductor (the CC(P) method) now listens to the first and second rows plus those top 1% of critical notes the Scout found. Because the conductor is now hearing the most important "triple" notes, the song sounds much better than before.
3. The Final Polish (CC(Q)): There are still a few tiny notes the Scout missed. The method adds a quick, non-iterative "polish" (the CC(Q) correction) to account for those remaining tiny details.

Why is this a Big Deal?

The authors tested this "Smart Scout" method on three different molecular scenarios:

• CH+ (The Stretchy Ion): Like a rubber band being pulled.
• CH (The Radical): A molecule with a tricky, unpaired electron.
• Water (H2O): Specifically, watching the bond break as water splits into hydrogen and hydroxyl.

The Results:

• Speed: The method is incredibly fast. It only needs to look at a tiny fraction of the possible "triple" notes (sometimes less than 5%) to get the answer.
• Accuracy: It produces results that are almost identical to the "Full Orchestra" method (EOMCCSDT), which is the gold standard but too slow to use.
• Reliability: In situations where the old shortcuts (guessing the triples) failed completely—creating "spurious bumps" or wrong shapes in the energy graphs—the Smart Scout method fixed them perfectly. It smoothed out the curves and gave the right answer, even when the molecule was stretched to its breaking point.

The Bottom Line

Think of this paper as inventing a GPS for quantum chemistry. Before, if you wanted to know the exact energy of an excited molecule, you had to drive every single road in the country (Full CI) or take a shortcut that sometimes led you off a cliff (old approximations).

Now, this new method uses a smart algorithm to find the "highways" (the most important electron configurations) that actually matter. It gets you to the destination (accurate energy) with the same precision as driving every road, but in a fraction of the time. This allows scientists to study complex chemical reactions, like how water breaks apart or how molecules react to light, with a level of accuracy that was previously too expensive to compute.

Technical Summary

1. Problem Statement

Accurately determining excited electronic states, particularly those dominated by multi-electron transitions (e.g., double excitations) or located along bond-stretching coordinates (where multi-reference character is strong), remains a significant challenge in quantum chemistry.

• Limitations of EOMCCSD: The standard Equation-of-Motion Coupled-Cluster with Singles and Doubles (EOMCCSD) method is computationally efficient ($O(N^6)$) but fails for states with significant double-excitation character or strong static correlation.
• Limitations of EOMCCSDT: While EOMCCSDT (including full triples) provides high accuracy, its computational cost ($O(N^8)$) is prohibitive for all but the smallest systems.
• Limitations of Existing Approximations: Perturbative corrections (e.g., CR-EOMCC(2,3)) and active-space methods improve upon EOMCCSD but often fail to accurately approximate EOMCCSDT energetics when the coupling between lower-rank ($n \le 2$) and higher-rank ($n=3$) amplitudes becomes large, which is common in stretched geometries and multi-reference (MR) situations.

The authors aim to develop a method that recovers EOMCCSDT accuracy at a fraction of the computational cost, specifically for excited states, by leveraging the CC(P;Q) framework combined with the Configuration Interaction with Perturbative Selection (CIPSI) algorithm.

2. Methodology

The paper extends the ground-state CIPSI-driven CC(P;Q) methodology to excited states using the EOM-CC formalism. The approach employs a two-step workflow:

A. The CC(P;Q) Framework

The method partitions the excitation space into two subspaces:

1. P Space (Iterative): Contains the reference determinant, all singly and doubly excited determinants, and a selected subset of triply excited determinants. The CC(P) and EOMCC(P) equations are solved iteratively within this space.
2. Q Space (Non-iterative Correction): Contains the remaining triply excited determinants not included in the P space. A state-specific non-iterative correction, $\delta_\mu(P;Q)$, is calculated using generalized moments of the CC(P)/EOMCC(P) equations projected onto the Q space.

The final energy is given by: $E^{(P+Q)}_\mu = E^{(P)}_\mu + \delta_\mu(P;Q)$.

B. The CIPSI-Driven Selection

To define the P space efficiently without including all possible triples (which would be too expensive):

1. CIPSI Run: A preliminary CIPSI calculation is performed to generate a compact wave function $|\Psi^{(CIPSI)}\rangle$. This involves iterative Hamiltonian diagonalizations in increasingly large subspaces until a user-defined determinant count ($N_{det}(in)$) is reached.
2. Extraction of Triples: The triply excited determinants present in the CIPSI wave function are extracted.
3. Construction of P Space:
   • For the ground state and excited states of the same symmetry, the P space includes all singles/doubles plus the triples extracted from the ground-state CIPSI run.
   • For excited states of different symmetries, the P space includes the same singles/doubles, but the specific triples are extracted from CIPSI runs targeting the lowest-energy states of those specific symmetries.
4. Q Space: Defined as all triply excited determinants not captured by the CIPSI selection.

C. Algorithmic Workflow

1. Run CIPSI to identify dominant triply excited determinants based on $N_{det}(in)$.
2. Construct the P space (Singles + Doubles + Selected Triples).
3. Solve CC(P) and EOMCC(P) equations iteratively in the P space.
4. Calculate the non-iterative correction $\delta_\mu(P;Q)$ using the Q space.
5. Sum to obtain the final CC(P;Q) energy.

3. Key Contributions

• First Excited-State Extension: This work presents the first implementation of the CIPSI-driven CC(P;Q) approach specifically for excited states (EOMCC).
• Efficiency via Selection: It demonstrates that accurate EOMCCSDT energetics can be recovered by including only a small fraction (often <20%) of the total triply excited determinants in the iterative steps, provided they are selected intelligently via CIPSI.
• Handling Multi-Reference Character: The method successfully addresses the failure modes of perturbative triples corrections (like CR-EOMCC(2,3)) in regions of strong static correlation (bond stretching) and for states with significant double-excitation character.
• State-Specific Adaptation: The methodology allows for the selection of triples based on the specific symmetry and character of the target state, improving convergence for difficult excited states.

4. Results

The method was tested on three distinct molecular systems:

A. CH+ Ion (Vertical Excitations)

• Test Cases: Equilibrium and stretched geometries ($R = 2R_e$).
• Findings:
  • Standard EOMCCSD (equivalent to CC(P;Q) with $N_{det}(in)=1$) failed significantly for states dominated by double excitations (errors up to 144 millihartree).
  • CR-EOMCC(2,3) corrections improved results but still showed large errors (up to 63 millihartree) for specific states at stretched geometries.
  • CIPSI-Driven CC(P;Q): With $N_{det}(in) = 1,000$ (capturing ~0.7–3.7% of total triples), errors dropped to <1 millihartree. Increasing to $N_{det}(in) = 5,000$ reduced errors to <0.2 millihartree, effectively matching EOMCCSDT.

B. CH Radical (Adiabatic Excitations)

• Test Cases: Ground state and three lowest doublet excited states ($A^2\Delta$, $B^2\Sigma^-$, $C^2\Sigma^+$).
• Findings:
  • EOMCCSD errors were large (up to 44 millihartree) for states dominated by two-electron transitions.
  • CR-EOMCC(2,3) reduced errors but left residual errors of ~5–8 millihartree for $A^2\Delta$ and $B^2\Sigma^-$.
  • CIPSI-Driven CC(P;Q): Using $N_{det}(in) = 5,000$ (capturing ~13–19% of triples), the method recovered EOMCCSDT energetics within 0.05–0.37 millihartree for all states.

C. Water (H2O) Potential Energy Surfaces

• Test Cases: Ground and 11 excited-state PES cuts along the O–H bond dissociation coordinate (1.3 to 4.4 bohr).
• Findings:
  • In the dissociation region ($R_{OH} > 2.0$ bohr), nearly all states exhibited strong MR character.
  • CR-EOMCC(2,3) failed for specific triplet states (e.g., $3^3A''$), producing spurious bumps in the potential energy surface and massive non-parallelity errors (NPE) of ~42 millihartree.
  • CIPSI-Driven CC(P;Q): With $N_{det}(in) = 5,000$, the method eliminated the spurious bumps and reduced NPEs to <1.5 millihartree for the problematic states, accurately reproducing the EOMCCSDT surfaces.
  • Limitation: Convergence was slower for high-lying excited states ($3^1A'$ and $3^3A'$) because the current implementation uses P spaces tailored to the lowest state of a given symmetry.

5. Significance

This paper establishes the CIPSI-driven CC(P;Q) approach as a robust, black-box alternative to full EOMCCSDT for excited states.

• Computational Efficiency: It achieves EOMCCSDT-level accuracy with computational costs significantly lower than full EOMCCSDT, as the iterative steps operate in spaces much smaller than the full triples manifold.
• Reliability: It overcomes the limitations of perturbative triples corrections in multi-reference regimes, making it suitable for studying bond breaking and complex excited states where traditional methods fail.
• Future Outlook: The authors suggest that further efficiency gains can be achieved by developing state-specific P spaces (tailoring CIPSI runs to specific excited states rather than just the ground state) and by using truncated CIPSI variants.

In summary, the work successfully bridges the gap between the high accuracy of EOMCCSDT and the computational feasibility of lower-order methods, providing a powerful tool for spectroscopy and photochemistry research.