Linear scaling quantum theories are reviewed especially focusing on the method adopted in GAMESS. The three key translation equations of the fast multipole method (FMM) are deduced from the general polypolar expansions given earlier by Steinborn and Ruedenberg. Simplifications are introduced for the rotation-based FMM that lead to a very compact FMM formalism. The OPS (optimum parameter searching) procedure, a stable and efficient way of obtaining the optimum set of FMM parameters, is established with complete control over the tolerable error ε. In addition, a new parallel FMM algorithm, requiring virtually no inter-node communication, is suggested which is suitable for the parallel construction of Fock matrices in electronic structure calculations.