This work presents a general mathmatical framework for the well-known nonlocal Quasicontinuum method, namely multiresolution molecular mechanics (MMM). The generalization is with respect to propose a novel summation rule SR^MMM that is derived mathematically and consistently for general finite element type. The procedure for obtaining SR^MMM is developed based on deriving and then fitting the atomic energy distribution within an finite element under the constraint of a given shape function. To validate the generalized framework, test problems including non-local harmonic and anharmonic models undergoing tensile, shear and bending deformations will be solved using linear, bilinear and quadratic elements, respectively. Results obtained using the proposed SR^MMM is compared with Gauss-quadrature-like summation rule SR^GQ for accuracy. Through error structure analyses, it is found that the proposed SR^MMM outperforms SR^GQ even when the latter employs more quadrature points than the former. In addition, SR^MMM is consistently decomposed into bulk and surface summation rules SR^B and SR^S, with SR^B for the bulk region and SR^S to capture surface effects. Furthermore, the dynamics performance and adaptivity of MMM will also be discussed.