Accurate prediction of inverted singlet-triplet excited states using self-consistent spin-opposite perturbation theory

The violation of Hund’s rule, resulting in an inverted singlet-triplet (INVEST) gap, represents a paradigm shift in photophysics with major implications for OLED technology. INVEST molecules facilitate barrierless reverse intersystem crossing, theoretically permitting 100% internal quantum efficiency without thermal activation. However, accurately predicting negative singlet-triplet energy gaps typically demands prohibitive computational costs. In this study, we evaluate the efficacy of our recently developed one-body Møller-Plesset perturbation theory (OBMP2) and its spin-opposite variant (O2BMP2) as efficient alternatives. Benchmarking against 30 INVEST molecules reveals that O2BMP2, with appropriate spin-opposite scaling, achieves the accuracy of ADC(3) and EOM-CCSD. Furthermore, with the possibility of reducing computational complexity to $N^4$, O2BMP2 provides a robust balance of accuracy and efficiency, making it suitable for the high-throughput screening of next-generation INVEST materials.

💡 Research Summary

The paper addresses a central challenge in the design of next‑generation organic light‑emitting diodes (OLEDs): the accurate prediction of inverted singlet‑triplet (INVEST) energy gaps, where the first excited singlet state (S₁) lies below the lowest triplet state (T₁). Such an inversion violates Hund’s rule, enables barrier‑free reverse intersystem crossing (RISC), and theoretically allows 100 % internal quantum efficiency. However, experimental determination of the S₁‑T₁ gap (ΔE_ST) is difficult because the transition is spin‑forbidden, making high‑level quantum‑chemical calculations indispensable. Conventional linear‑response methods such as TD‑DFT, CIS, or even Δ‑SCF often fail for INVEST systems because they neglect essential double‑excitation character and strong electron correlation, leading to qualitatively wrong (positive) ΔE_ST values. High‑accuracy wave‑function methods (CC2, ADC(2), ADC(3), EOM‑CCSD, NEVPT2, CASPT2) can capture the inversion but scale as N⁶–N⁷, rendering them impractical for high‑throughput screening of thousands to millions of candidate molecules.

The authors propose a computationally efficient alternative based on their previously introduced one‑body Møller‑Plesset perturbation theory (OBMP2) and a spin‑opposite‑scaled variant (O2BMP2). The core idea is to perform a canonical transformation of the electronic Hamiltonian, truncate the Baker‑Campbell‑Hausdorff (BCH) expansion at second order, and retain only one‑body operators and scalar constants after a cumulant decomposition. This yields an effective one‑body Hamiltonian consisting of the usual Hartree–Fock (HF) part plus a correlation potential v_{pq} constructed from MP2 double‑excitation amplitudes. The correlated Fock matrix (\bar{f}{pq}=f{pq}+v_{pq}) is then diagonalized, providing self‑consistent molecular orbitals and orbital energies that already incorporate correlation effects. By scaling the MP2 amplitudes with a spin‑opposite factor (c_{os}) (i.e., (T^{ij}{ab}=c{os},g^{ij}_{ab}/(\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b))), the method selectively enhances opposite‑spin correlation, which stabilizes the singlet state more than the triplet and drives ΔE_ST negative.

Two benchmark sets are examined. Set A contains ten small INVEST molecules previously studied by Loos et al. The authors test several spin‑opposite scaling factors (c_os = 1.0, 1.2, 1.5, 1.8). They find that increasing c_os systematically lowers both S₁ and T₁ energies, but the singlet is stabilized to a greater extent, producing increasingly negative ΔE_ST. The optimal scaling is identified around c_os ≈ 1.7, where the mean absolute deviation (MAD) for the singlet and triplet excitation energies is minimized (≈ 0.06 eV for ΔE_ST). Importantly, using only the first‑order BCH term (equivalent to SOS‑MP2) yields predominantly positive gaps; inclusion of the second‑order BCH term and full self‑consistency is essential to obtain the correct sign for all molecules. This demonstrates that both electron correlation and wave‑function relaxation are crucial for the inversion phenomenon.

Set B comprises twenty medium‑size molecules from the Pollice et al. dataset, which includes systems that not only exhibit INVEST gaps but also possess appreciable fluorescence rates. Reference values (theoretical best estimates, TBEs) are taken from high‑level calculations: ADC(2), ADC(3), EOM‑CCSD, FNO‑EOM‑CCSD, and NEVPT2. The authors also compute Δ‑DFT excitation energies with CAM‑B3LYP and PBE0 for comparison. O2BMP2 with c_os = 1.7 delivers a MAD of 0.031 eV for ΔE_ST, essentially matching the performance of Δ‑CCSD(T) (MAD ≈ 0.039 eV) and approaching that of ADC(3) and EOM‑CCSD (MAD ≈ 0.03 eV). By contrast, Δ‑HF severely over‑stabilizes the singlet (unphysically large negative gaps), Δ‑MP2 fails to invert the gap, and Δ‑CCSD, while capturing the inversion, still shows a sizable error (MAD ≈ 0.225 eV). Thus O2BMP2 achieves high‑level accuracy while retaining a formal scaling of O(N⁴) (after exploiting spin‑opposite MP2‑like algorithms), a dramatic reduction from the O(N⁶)–O(N⁷) scaling of the benchmark wave‑function methods.

Implementation details are noteworthy. The self‑consistent OBMP2/O2BMP2 procedures are built into a locally modified version of the PySCF package. Convergence is accelerated using the maximum‑overlap method (MOM) to target a specific excited state and the direct‑inversion‑in‑the‑iterative‑subspace (DIIS) technique. Typical convergence thresholds of 10⁻⁸ Hartree are reached within a few dozen iterations. The authors provide all underlying data in the Supporting Information, facilitating reproducibility.

In summary, the study demonstrates that O2BMP2, with an appropriately tuned spin‑opposite scaling factor (c_os ≈ 1.7) and full self‑consistency, can predict inverted singlet‑triplet gaps with chemical accuracy (≤ 0.05 eV) at a computational cost comparable to spin‑opposite MP2. This balance of accuracy and efficiency makes O2BMP2 an attractive tool for high‑throughput virtual screening of OLED emitters and other photophysical materials where double‑excitation character and strong correlation are essential. The method’s ability to capture the subtle interplay between opposite‑spin correlation and orbital relaxation positions it as a practical bridge between low‑cost DFT‑based approaches and expensive high‑level wave‑function theories, potentially accelerating the discovery of 100 % internal quantum efficiency emitters.