Leveraging the Bias‐Variance Tradeoff in Quantum Chemistry for Accurate Negative Singlet‐Triplet Gap Predictions: A Case for Double‐Hybrid DFT

Molecules that have been suggested to violate the Hund's rule, having a first excited singlet state ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{S}}_1 $$</mml:annotation></mml:semantics> </mml:math> ) energetically below the triplet state ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mrow><mml:mtext>T</mml:mtext></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{T}}_1 $$</mml:annotation></mml:semantics> </mml:math> ), 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msub> <mml:mo>-</mml:mo> <mml:msub><mml:mrow><mml:mtext>T</mml:mtext></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{S}}_1-{\mathrm{T}}_1 $$</mml:annotation></mml:semantics> </mml:math> 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mo>></mml:mo> <mml:mn>100</mml:mn> <mml:mspace></mml:mspace> <mml:mtext>meV</mml:mtext></mml:mrow> <mml:annotation>$$ >100\kern0.3em \mathrm{meV} $$</mml:annotation></mml:semantics> </mml:math> ) despite very low standard deviations ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mo>≈</mml:mo> <mml:mn>10</mml:mn> <mml:mspace></mml:mspace> <mml:mtext>meV</mml:mtext></mml:mrow> <mml:annotation>$$ \approx 10\kern0.3em \mathrm{meV} $$</mml:annotation></mml:semantics> </mml:math> ). 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.