Résumé
To this end, we introduce a new variational formulation for the grand potential of a mixed quantum-classical system. Within the Born-Oppenheimer approximation, and neglecting electronic entropy, the quantum solute is described by a product of electronic and nuclear density matrices, both depending parametrically on coordinates of the classical solvent. It can then be shown that a functional of the total density matrix satisfies a variational principle for the grand potential. Using a mean-field approximation, we express the grand potential of the mixed quantum-classical system as a variational problem which depends only on the nuclear density matrix. The nuclei experience an external field generated by the electronic and classical one-particle densities.
In practice, the computation of the grand potential is reduced to a sequence of density optimizations. First, the classical solvent density and the solute electronic density are optimized for a fixed solute nuclear geometry, using the previously reported mixed quantum mechanics/classical procedure. Subsequently, the solute geometry is optimized for a fixed solvent configuration. Finally, the redox potentials of a selection of Benzoquinone/Hydroquinone couples are computed after geometry optimizations. The predictions are in good agreement with QM calculation using a continuum solvent model and with experimental data