ON THE EXPANSION OF KARLE-HAUPTMAN DETERMINANTS INTO MULTIPLETS AND THE MAXIMUM ENTROPY RULE. Oscar E. Piro,¹ C. de Rango,² G. Tsoucaris² and G. Navaza², ¹Dept. Fisica, Ftad. C. Exactas, Univ. Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina, ²Laboratoire de Physique, Faculte de Pharmacie, Universite de Paris-Sud 92290 Chatenay-Malabry, France

Improved Direct Methods formulations employing besides triplets, quartet and quintet products of structure factors (phase invariants) clearly suggest the useful phase information conveyed by higher-order multiplets in structure determination procedures. These formulations can be obtained from the first few terms in the expansion of large joint probability distribution (j.p.d.) of structure factors constructed on the basis of K-H determinants, combined with the reciprocal space Maximum Entropy rule. In this connection, it has been pointed out that the power of current Direct Methods to solve large crystal structures is limited by the usual practice of approximating large j.p.d.'s by piecing together marginal distributions of single invariants. It is therefore interesting to describe how the full phasing power of large j.p.d.'s is reached by the progressive incorporation of phase information due to higher-order multiplets. As a contribution to this goal, we disclose herewith the group theoretical structure of the contributing phase invariants in the expansion of an arbitrary m-order K-H determinant into multiplets up to the order m. Implications and possible applications of the results will be discussed.