PHYSICAL REVIEW A VOLUME 31, NUMBER 3 Some aspects of self-consistent propagator theories MARCH 1985 M. Durga Prasad, Sourav Pal, * and Debashis Mukherjee Theory Group, Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700032, India (Received 17 February 1984) A general method for constructing bases for operator manifolds for any propagator, which satisfy "vacuum annihilation conditions" (VAC's) is developed. This approach is based on the observation that if the transformation of the unperturbed ground state to the correlated ground state is represented as a rotation in the Pock space, the corresponding rotation induced in the basis of the concerned operator space would generate a basis which satisfies VAC's on the correlated ground state. The associated requirements for the Hermiticity of superoperator Hamiltonian would also be met in this new basis. The proposed method is noniterative in that, once the form of the ground- state function is specified, the expansion of the operator manifold satisfying VAC s on the ground state does not require any iterative readjustment. The resultant propagators in this approach are fully linked. It is shown that this theory is equivalent to a propagator formalism in terms of hole- and particle-creation and/or annihilation operators with a modified effective Harniltonian. The self-consistent electron and polarization propagators are considered as examples, and their underly- ing perturbative structures are analyzed. The role of density shift operators and higher-rank opera- tors are discussed. I. INTRODUCTION Propagator methods have become increasingly popular over the last few years for the direct calculation of various difference energies of spectroscopic interest. Various for- mal approaches such as those based on the Feynman- Dyson equation, decoupling procedures of propagator equations of motion, and the superoperator resolvent- based methods have been widely explored for their com- putational feasibilities, and in particular, the electron and polarization propagators have been studied extensively at different levels of approximations. ' Notwithstanding the successes of these methods in simulating the experimental observations, one persistent criticism against these methods has been that the wave functions implicit in the resultant approximate propaga- tors do not always satisfy the Pauli principle (the so-called "lV-representability problem" '). Within the framework of Feynman-Dyson perturbation theory (FDPT, which is the basis of the X-perturbation theory and functional dif- ferentiation based methods), uneven expansion of the numerator and the denominator to different orders as a consequence of the linked-cluster theorem is the cause of this error. Similarly, when the approximations made in the decoupling of higher-order propagators are incon- sistent with the approximate ground state, the resultant propagator is not X-representable. In the superoperator approach, the resultant propagator is not N-representable whenever the concerned operator space does not yield a proper resolution of identity in the presence of the super- operator Hamiltonian 11, 24 — 26 Most of the attempts to derive iV-representable propa- gator approximations were based on the superoperator for- malism. " In this context Manne showed that the operator manifold OM —— [Iat, j U [az j U [a&at, ah, , k ,&l j U ] (1. 1) is complete and yields a resolution of identity with respect to any approximate wave function not orthogonal to the particle-hole vacuum state. Similarly Dalgaard showed that the operator manifold OD — — [IB; j U IBg j U IB; BJ j U IBBJ j U ~ ] is complete for the calculations of polarization propaga- tor, where B; is the set of hole-particIe excitation opera- tors I a& ah, j. While these studies are significant, they have not been of much practical use in answering the ex- isting criticism, since, in any practical computation only a subset of either Ost or OD is used in conjunction with some approximate ground state (g.s. ), and quite often, the resolution of identity does not hold in such a truncated operator manifold when that particular approximation is used to represent the ground state. For example, the trun- cated operator manifold OD — — [IB; j U IB; j ] does not per- mit a proper resolution of identity when the Hartree-Fock ground state is used as vacuum. In a series of papers starting from 1980, Goscinski and %einer analyzed the questions regarding the completeness of operator manifolds and the related question of the reso- lution of identity with particular reference to incomplete operator manifolds. " Their conclusions, as applic- able to the polarization propagator, can be summarized as follows: Given an approximate g.s. %„a truncated opera- tor manifold 0, = [IX; j U IX~ j ] is compete if and only if it is possible to define a new basis [ I Q; j U I Q; j ] for 0, such that the set I O'„Q 4, j is a linearly independent sub- set of the lV-particle Hilbert space and the following equa- tions and their Hermitian adjoints are satisfied: 31 1287