Functional analysis of the energy

MP perturbation theory is generally used only for the ground state energy. The zeroth-order approximation for the wavefunction is taken to be the lowest-energy HF determinant. Nevertheless, the function E z) obtained from this perturbation series, when considered as a function over the complex z-plane, contains within it the full energy spectrum of eigenstates with the same symmetry as the ground state. This is the consequence of a theorem presented by Katz [14]  

Now consider the ground state function E (z). Starting at the origin, trace a path in the z-plane that circles about the point zoi This point is a branch point singularity, which means that a 360° circuit will lead to a new Riemann sheet corresponding to the fimction E (z). Similarly, following E ) (z) on a path that circles zq2 leads to E (z), and so on. 

It is interesting that the position of Zc is strongly dependent on the basis set. In the case of F if the cc-pVDZ basis, which lacks the dilfiise functions of the augmented basis sets, is used instead, then the singularity on the negative real axis seems to disappear [5,7,20]. Apparently, difliise functions are needed to describe the autoionization. 

It is important to note that a path along the negative real axis could also show an avoided crossing between the ground state and the first excited state. In Section 4.3, we present empirical evidence indicating that 

---