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The use of accelerated sampling methods such as metadynamics has shownsignificant advantage in calculations that involve infrequent events, whichwould otherwise require sampling a prohibitive number of configurations todetermine, e.g., difference in free energies between two or more chemicallydistinct states such as in the calculation of acid dissociation constants$K_a$. In this case, the most common method is to bias the system via a singlecollective variable (CV) representing the $coordination~number$ of the protondonor group, which yields results in reasonable agreement with experiments.Here we study the deprotonation of acetic acid using the reactive force fieldReaxFF and observe a significant dependence of $K_a$ on the simulation box sizewhen biasing only the coordination number CV, which is due to incompletesampling of the deprotonated state for small simulation systems, andinefficient sampling for larger ones. Incorporating a second CV representingthe distance between the H$_3$O$^+$ cation and the acetate anion results in asubstantially more efficient sampling both accelerating the dynamics andvirtually eliminating the computational box size dependence.

Advantages of multi-dimensional biasing in accelerated dynamics: application to the calculation of the acid $pK_a$ for acetic acid