Schrödinger equation in terms of linearly averaged position moments

Linearly averaged position moment (LAPM) [{R_1}^{k_1} {R_2}^{k_2} ... {R_3N}^{k_3N}] is defined as the linear (not quadratic) average of the position moment operator {R_1}^{k_1} {R_2}^{k_2} ... {R_3N}^{k_3N} over the N-electron wave function Ψ({Rj}), where {Rj} are 3N Cartesian coordinates of electrons and {kj} are non‐negative integers. When all the LAPM’s are well defined, it is shown that the Schrödinger equation is equivalent to a set of an infinite number of equations between LAPM’s involving the potential‐energy operator. The kinetic energy operator does not appear. The spherical polar representation of the LAPM equation is also presented. Illustrations are given for simple one‐ and two‐electron atoms, where the LAPM equation is applied to the determination of approximate wave functions and associated energie