The goal of this dissertation has been to develop a method that enables one to calculate accurate, rigorous lower bounds to the eigenvalues of the standard nonrelativistic spin-free Hamiltonian for an atom with N electrons. Lower bounds are necessary in order to complement upper bounds obtained from the Hartree-Fock and Rayleigh-Ritz techniques. Without accurate lower bounds, it is impossible to estimate the error of the approximate values of the energies. By combining two heretofore distinct methods and using the symmetry properties of the Hamiltonian, this goal has been achieved. The first of the two methods is the method of intermediate problems. By beginning with an appropriately chosen "base operator" H⁰, one forms a sequence of intermediate Hamiltonians Hᵏ, k = 1,2,..., whose corresponding eigenvalues form a sequence of lower bounds to the eigenvalues of the original Hamiltonian H. Complications which occurred in this method due to the stability of essential spectra under compact perturbations were later surmounted with the use of abstract separation of variables by D. W. Fox. The second technique, the effective field method, provides a lower bound operator to the interelectron repulsion term in H that is of the form of a sum of separable potentials. This latter technique reduces the eigenvalue problem for H⁰ to a sum of single particle operators. With the use of a special potential, the Hulthen potential, one may construct an explicitly solvable base problem from the effective field method, if one uses the method of intermediate problems to calculate lower bounds to non-S states. This base problem is then suitable as a starting point for the method of intermediate problems with the Fox modifications. The eigenvalues of the new base problem are already comparable to the celebrated Thomas-Fermi energies. The final part of the dissertation provides a practical procedure for determining the physically realizable spectra of the intermediate operators. This is accomplished by restricting the Hamiltonian to subspaces of proper physical symmetry so that the resulting lower bounds will be to eigenvalues of physical significance.