Graphical method for deriving an effective interaction with a new vertex function

Introducing a new vertex function, (Z) over cap (E), of an energy variable E, we derive a new equation for the effective interaction. The equation is obtained by replacing the (Q) over cap box in the Krenciglowa-Kuo (KK) method with (Z) over cap (E). This new approach can be viewed as an extension of the KK method. We show that this equation can be solved both in iterative and noniterative ways. We observe that the iteration procedure with (Z) over cap (E) brings about fast convergence compared to the usual KK method. It is shown that, as in the KK approach, the procedure of calculating the effective interaction can be reduced to determining the true eigenvalues of the original Hamiltonian H and they can be obtained as the positions of intersections of graphs generated from (Z) over cap (E). We find that this graphical method yields always precise results and reproduces any of the true eigenvalues of H. The calculation in the present approach can be made regardless of overlaps with the model space and energy differences between unperturbed energies and the eigenvalues of H. We find also that (Z) over cap (E) is a well-behaved function of E and has no singularity. These characteristics of the present approach ensure stability in actual calculations and would be helpful to resolve some difficulties due to the presence of poles in the (Q) over cap box. Performing test calculations, we verify numerically theoretical predictions made in the present approach.