Dimensionality dependence of energy of heavy positive ions and the 1/Z expansion

With interactions chosen to satisfy Poisson’s equation in d dimensions, the form of the total energy E(Z,N,d) of an atomic ion having nuclear charge Ze and N electrons is considered for large Z within the context of the 1/Z expansion. In particular, the form of the coefficients of this expansion is displayed at large N, the nth order coefficient being shown to be proportional to $N^{n+\alpha }$ where α is given explicitly and depends solely on dimensionality. The singular point is at d=4, in contrast to the situation when the 1/r Coulomb interaction is directly taken over into d dimensions.