PHYSICAL REVIEW A VOLUME 39, NUMBER 7 Generalized saddle-point method for Feshbach resonances APRIL 1, 1989 Mirosfaw Bylicki Instytut Fizyki, Unimersytet Mikofaja Kopernika, ul. Grudziqdzka 5, PL-87-100 Torun, Poland (Received 27 July 1988) The mini-max principle is extended to work for the approximations to resonances in the square- integrable function space. The hole-projection (or saddle-point) technique for Feshbach resonances, introduced previously by Chung [Phys. Rev. A 20, 1743 11979)], is derived from the mini-max prin- ciple. Limits of applicability of the method are discussed and its generalization based on the Feshbach-type projector technique is given. The generalized method is applied to the case of the 1s2s2p 'P' resonance of He I. INTRODUCTION The main obstacle in the application of variational methods to resonances in many-electron systems is the existence of an infinite space of states having the same symmetry as a resonance and lying below it. In general, the variational manifold should not contain these states. In order to remove them, the variational space is usually made orthogonal to some approximations of these states. Projection is the best method of orthogonalization. It is used in Feshbach-type projection-operator methods. ' There exist also less accurate techniques such as the quasi — projection-operator method and the one-particle projection-operator technique. In the latter one, intro- duced by Nicolaides, the orthogonality of a trial func- tion to the one-configurational approximations of lower- lying states is obtained by an appropriate orthogonaliza- tion of orbitals. One-particle projectors of the same type have been used also by Chung within his saddle-point, or hole- projection, technique. Chung's method originates from the physical intuition that vacancies should exist in the wave functions describing resonances. His papers are concerned with how to build these holes into a trial func- tion and how to optimize them. The hole-projection is treated by him as a way of building-in the vacancy and not a method of orthogonalization. The presence of the vacancy is believed to stop a variational breakdown. That is why that method is reputed to be of a general na- ture. However, there are many instances in which it does not work. Some of them are discussed in Sec. III of this paper. A novelty introduced by Chung is that the energy of a resonance is maximized with respect to variations of the function describing a hole. An appropriate theorem was precisely proven for a one-particle system only. One should realize that in this case the function describing a vacancy and total wave function of the system belong to the same one-electron space. Therefore the one-particle projector is a good projector operating in the Hilbert space of states of the entire system. Moreover, there are no autoionizing states in such a system. Hence the prescription for the energy of an excited state given by Chung's theorem follows directly from the well-known mini-max principle. The proof of Chung's theorem for a many-particle system is unsatisfactory. The main aim of this work is to properly interpret the saddle-point technique. Thus the prescription presented in Sec. III is strictly the same as that given by Chung. However, it is derived in a quite different way, starting from the mini-max principle. This new derivation is simpler and more rigorous than that of Chung. More- over, limitations of the method are clearly visible. It is indicated that the saddle-point technique should be gen- eralized to avoid these limitations. Such a generalization consisting in the application of the Feshbach-type projec- tors within the mini-max principle scheme is presented in Sec. IV. Some numerical results obtained using this gen- eralized method are given in Sec. V. The mini-max principle is valid for a discrete spectrum only. It is not applicable to resonance states because of continua of states which are not square integrable and lie below resonance levels. In Sec. II the mini-max principle is extended to work for the square-integrable approxima- tions to resonances. II. MINI-MAX PRINCIPLE FOR RESONANCES We are interested in resonant states of an ¹ lectron system autoionizing via Coulomb interaction between electrons. The Hamiltonian & is taken to be nonrela- tivistic and Hermitian. The domain of A can be divided into subspaces corresponding to distinct eigenvalues of operators commuting with . &, as, e.g. , parity, total spin, and total orbital angular momentum. In this work we consider one such subspace. Furthermore, we want to describe the resonance by a square-integrable function and therefore instead of & we consider an operator H which is the representation of & in a space 2) of square- integrable ¹ lectron functions of a given symmetry. The best square-integrable approximations E, and g, to the energy and to the wave function of the resonance are a solution of the equation HP=Eg . The spectrum of H consists of eigenvalues corresponding to truly bound states and resonances and also to some scattering states, which are simulated in A. 39 3316 1989 The American Physical Society