On the Role of Electronic Correlation and State‐Specific Environment Polarization in Singlet–Triplet Gap Inversion

Molecules characterized by an inverted singlet-triplet gap ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Δ</mml:mi> <mml:msub><mml:mi>E</mml:mi> <mml:mi>ST</mml:mi></mml:msub> <mml:mo><</mml:mo> <mml:mn>0</mml:mn></mml:mrow> <mml:annotation>$$ \Delta {E}_{\mathrm{ST}}<0 $$</mml:annotation></mml:semantics> </mml:math> ) hold potential for optoelectronic applications. Electronic correlation and environmental polarization are key factors influencing negative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Δ</mml:mi> <mml:msub><mml:mi>E</mml:mi> <mml:mi>ST</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ \Delta {E}_{\mathrm{ST}} $$</mml:annotation></mml:semantics> </mml:math> , and the latter is gaining attention for its possible role in "mimicking" correlation contributions to yield negative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Δ</mml:mi> <mml:msub><mml:mi>E</mml:mi> <mml:mi>ST</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ \Delta {E}_{\mathrm{ST}} $$</mml:annotation></mml:semantics> </mml:math> . However, a comprehensive study of solvation effects on both structures and energy gaps is still lacking. In this work, we evaluate computational strategies for calculating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Δ</mml:mi> <mml:msub><mml:mi>E</mml:mi> <mml:mi>ST</mml:mi></mml:msub> <mml:mo><</mml:mo> <mml:mn>0</mml:mn></mml:mrow> <mml:annotation>$$ \Delta {E}_{\mathrm{ST}}<0 $$</mml:annotation></mml:semantics> </mml:math> gaps, incorporating electronic correlation and solvent polarization in molecules exhibiting singlet-triplet inversion. Using RMS-CASPT2 as a benchmark, we demonstrate that double-hybrid density functionals and mixed-reference spin-flip TD-DFT (MRSF-TD-DFT) can partially recover electronic correlation. Furthermore, we investigate solvation effects on both singlet and triplet excited states, highlighting the limitations of linear-response schemes in continuum solvation models. We finally develop a protocol combining electronic correlation and state-specific solvent polarization using double-hybrid functionals and the Vertical Excitation Model (VEM), leveraging its Lagrangian implementation to compute structures and adiabatic energies. Applying our B2PLYP/VEM(UD) protocol to larger systems with experimentally observed negative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Δ</mml:mi> <mml:msub><mml:mi>E</mml:mi> <mml:mi>ST</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ \Delta {E}_{\mathrm{ST}} $$</mml:annotation></mml:semantics> </mml:math> gaps, we quantitatively reproduce experimental emissive and non-radiative transition rates.