In the following, we will use a multi-index notation which facilitates compact and open-ended expressions of the QM/MM terms. A multi-index denoted by α, β, γ, etc., is a 3-tuple consisting of three nonnegative integers (e.g. α = (α x , α y , α z )) that are associated with the x, y, and z Cartesian components, respectively. For instance, α = (0, 0, 0) denotes zeroth order with respect to all Cartesian components, α = (1, 0, 0) is first order with respect to the x component, α = (0, 0, 2) is second order with respect to the z component, etc. The short-range electrostatic QM-MM interaction energy can then be written as