A ug 1 99 9 Approximate Solution of the Representability Problem

Approximate solution of the ensemble representability problem for density operators of arbitrary order is obtained. This solution is closely related to the " Q condition " of A. J. Coleman. The representability conditions are formulated in orbital representation and are easy for computer implementation. They are tested numerically on the base of CI calculation of simple atomic and molecular systems. In the last decade a number of Post HF methods of high precision have been developed [1]. It is unlikely, however, that in the nearest future these methods in their present form have any chance to be applied for electronic structure calculations of extensive molecular and crystalline systems. Instead much more simple DFT methods [2] based on the work of Kohn [3] are becoming the main tool for calculations of such systems. There exists an alternative approach in density functional theory which progress had been restricted by serious mathematical problems connected with the so-called representability property of 2-electron density. After the famous theorem of Coleman [4] who solved the representability problem for 1-density operators, there appeared a number of papers with attempts to generalize Coleman's result to treat q-density operators (for q ≥ 2) [5, 6, 7, 8, 9, 10]. These attempts, however, did not end in results of practical importance. It even became accepted that the problem is far too complicated to find any applications for electronic structure calculations. In present work the necessary conditions for Fermion representability closely related to the Q condition of Coleman [11, 12] are obtained. Some relevant calculations of small atomic and molecular systems are discussed. An alternative approach to approximate treament of the representability problem is developed in [13] (and references therein).