The formal infinite series for the probability amplitudes are transformed by (i) a regrouping procedure which separates out "repetitive" terms from all orders beyond the second and combines them to produce a common factor multiplying all orders, (ii) a procedure of summation to a closed form (essentially an analytical continuation) replacing the above common factor by a generalized energy denominator, and (iii) unlimited repetition of (i) and (ii). Procedure (ii) is based on the generalized energy quantity ${\mathcal{E}}_{\mathrm{gh}\cdots \mathrm{np}}=E_p+\Sigma \stackrel{}{q\ne \mathrm{gh}\cdots \mathrm{np}}\frac{V_{\mathrm{pq}}{V}_{\mathrm{qp}}}{\mathrm{}\left(q\right)\mathrm{}}+\Sigma \stackrel{}{\mathrm{qr}\ne \mathrm{gh}\cdots \mathrm{np}}\frac{V_{\mathrm{pq}}{V}_{\mathrm{qr}}{V}_{\mathrm{rp}}}{\mathrm{}\left(q\right)\mathrm{}\left(r\right)\mathrm{}}+\cdots$

and the formal identity ${[E-{\mathcal{E}}_{\mathrm{gh}\cdots \mathrm{np}}]}^{-1\mathrm{}}=\frac{1}{\mathrm{}\left(p\right)\mathrm{}}\left[1+\Sigma \stackrel{}{q\ne \mathrm{gh}\cdots n}\frac{V_{\mathrm{pq}}{V}_{\mathrm{qp}}}{\mathrm{}\left(p\right)\mathrm{}\left(q\right)\mathrm{}}+\Sigma \stackrel{}{\mathrm{qr}\ne \mathrm{gh}\cdots n}\frac{V_{\mathrm{pq}}{V}_{\mathrm{qr}}{V}_{\mathrm{rp}}}{\mathrm{}\left(p\right)\mathrm{}\left(q\right)\mathrm{}\left(r\right)\mathrm{}}+\cdots \right]$

employing the notation $\mathrm{}\left(x\right)=E-E_x$.

• Received 8 June 1948

DOI:https://doi.org/10.1103/PhysRev.74.664