Flexible boundary layer using exchange for embedding theories. I. Theory and implementation

Embedding theory is a powerful computational chemistry approach to exploring the electronic structure and dynamics of complex systems, with QM/MM being the prime example. A challenge arises when trying to apply embedding methodology to systems with diffusible particles, e.g. solvents, if some of them must be included in the QM region, for example in the description of solvent-supported electronic states or reactions involving proton transfer or charge-transfer-to-solvent: without a special treatment, inter-diffusion of QM and MM particles will lead eventually to a loss of QM/MM separation. We have developed a new method called Flexible Boundary Layer using Exchange (FlexiBLE) that solves the problem by adding a biasing potential to the system that closely maintains QM/MM separation. The method rigorously preserves ensemble averages by leveraging their invariance to exchange of identical particles. With a careful choice of the biasing potential, and the use of a tree algorithm to include only important QM and MM exchanges, we find the method has an MM-forcefield-like computational cost and thus adds negligible overhead to a QM/MM simulation. Furthermore, we show that molecular dynamics with the FlexiBLE bias conserves total energy and remarkably, dynamical quantities in the QM region are unaffected by the applied bias. FlexiBLE thus widens the range of chemistry that can be studied with embedding theory.