Leveraging the Bias-Variance Tradeoff in Quantum Chemistry for Accurate Negative Singlet-Triplet Gap Predictions: A...

Molecules that have been suggested to violate the Hund's rule, having a first excited singlet state ( ${\text{S}}_1$ ) energetically below the triplet state ( ${\text{T}}_1$ ), are rare. Yet, they hold the promise to be efficient light emitters. Their high-throughput identification demands exceptionally accurate excited-state modeling to minimize qualitatively wrong predictions. We benchmark twelve ${\text{S}}_1-{\text{T}}_1$ energy gaps to find that the local-correlated versions of ADC(2) and CC2 excited state methods deliver excellent accuracy and speed for screening medium-sized molecules. Notably, we find that double-hybrid DFT approximations (e.g., B2GP-PLYP and PBE-QIDH) exhibit high mean absolute errors ( $>100\text{meV}$ ) despite very low standard deviations ( $\approx 10\text{meV}$ ). Exploring their parameter space reveals that a configuration with 75% exchange and 55% correlation, which reduces the mean absolute error to below 5 meV, but with an increased variance. Using this low-bias parameterization as an internal reference, we correct the systematic error while maintaining low variance, effectively combining the strengths of both low-bias and low-variance DFT parameterizations to enhance overall accuracy. Our findings suggest that low-variance DFT methods, often overlooked due to their high bias, can serve as reliable tools for predictive modeling in first-principles molecular design. The bias-correction data-fitting procedure can be applied to any general problem where two flavors of a method, one with low bias and another with low variance, have been identified a priori.