Conical intersections in solution with polarizable embedding: Integral-exact direct reaction field

A common strategy to exploring the properties and reactivity of complex systems is to use quantum mechanics/molecular mechanics (QM/MM) embedding, wherein a QM region is defined and treated with electronic structure theory and the remainder of the system is treated with a forcefield. Of equal importance to the level of quantum chemical theory used for the QM region and the quality of the forcefield is the treatment of the coupling between the QM and MM subsystems. The state-of-the-art is to use a polarizable forcefield for the MM region and mutually couple the QM wavefunction and MM induced dipoles, in addition to the usual electrostatic embedding, yielding a polarizable embedding (QM/MM-Pol) approach. However, extensions to electronic excited states are less well developed, and we showed previously that current popular QM/MM-Pol approaches exhibit issues of root flipping and/or incorrect descriptions of electronic crossings in multistate calculations.[J. Chem. Theory Comput. 14, 2137 (2018)] Here we demonstrate a solution to these problems with an integral-exact reformulation of the Direct Reaction Field approach of Thole and Van Duijnen (QM/MM-IEDRF). The resulting embedding potential includes one- and two-electron operators, and many-body dipole-induced dipole interactions, and thus includes a natural description of the screening of electron-electron interactions by the MM induced dipoles. Pauli repulsion from the environment is mimicked by effective core potentials on the MM atoms. Inherent to the DRF approach is the assumption that MM dipoles respond instantaneously to the positions of the QM electrons, therefore dispersion interactions are captured approximately. All electronic states are eigenfunctions of the same Hamiltonian while the polarization induced in the environment and the associated energetic stabilization are unique to each state. This allows for a consistent definition of transition properties and state crossings. We demonstrate QM/MM-IEDRF by exploring the influence of a (polarizable) inert xenon matrix environment on the conical intersection underlying cis-trans isomerization of ethylene.