Distribution Fitting Methods

Abstract

The measured intensities alone are not sufficient to determine the structure. The solubility of the phase problem of a structure determination is dependent on the Amount of a Priori Structure Information (APSI) - i.e. atomicity, the known number of atoms, known molecular fragment, etc. Therefore the optimal procedure for the structure determination can be formulated as an optimal transmission of APSI among all phases. The mathematical condition for “best” utilization op APSI can be found in [1] as:

$$ s = {V^{ - 1}}\int\limits_V {{{\left( {{\rho _{calc}} - {\rho _{exaxt}}} \right)}^{ \to 2}}dV = {V^{ - 3}}\sum {\left( {{{\left| {{E_H}} \right|}^2}\operatorname{var} {\varphi _H}} \right) = \min imum}} $$

where variances of phases can be derived from the recurent formula:

$$ \operatorname{var} {\varphi _H} = {\left( {\sum {{{\left( {{b_j}\quad {V_j} + \sum {{a_{ij}}\quad \operatorname{var} {\varphi _{ij}}} } \right)}^{ - 1}}} } \right)^{ - 1}} $$

following from the Graph of Phase Relations (GPR). The V_i is the variance of the i-th seminvariant used for calculation of the phase φ_H in the (n+1)-th level, var_φH is a variance of the phase φ_H and var_φij are variances of known phases in preceding levels. The Theory of Seminvariant Graphs described in [2] gives a procedure how to find the optimal choice of seminvariants, their optimal utilizations, and how to find the best starting set supposing given set of seminvariants [3,4,5].