where the Lagrangian multiplier λ is introduced to constrain the net charge for the MM region as Q. 1 refers to a row vector (1, 0, 0, 0, 1, 0, 0, 0, • • • ) with a value of 1 for only the induced charge on each MM atom. When Q is equal to 0 (as for all test cases in this work), the solution to Eq. ( ) could be formally written as,