Solution of the phase problem at non-atomic resolution by the phantom derivative method

For a given unknown crystal structure (the target), n random structures, arbitrarily designed without any care for their chemical consistency and usually uncorrelated with the target, are sheltered in the same unit cell as the target structure and submitted to the same space-group symmetry. (These are called ancil structures.) The composite structures, whose electron densities are the sum of the target and of the ancil electron densities, are denoted derivatives. No observed diffraction amplitudes are available for them: in order to emphasize their unreal nature, the term phantom is added. The paper describes the theoretical basis by which the phantom derivative method may be used to phase the target structure. It may be guessed that 100-300 ancil structures may be sufficient for phasing a target structure, so that the phasing technique may be denoted as the multiple phantom derivative method. Ancil phases and amplitudes may be initially combined with observed target magnitudes to estimate amplitudes and phases of the corresponding phantom derivative. From them suitable algorithms allow one to obtain poor target phase estimates, which are often improved by combining the indications arising from each derivative. Probabilistic criteria are described to recognize the most reliable target phase estimates. The method is cyclic: the target phase estimates just obtained are used to improve amplitudes and phases of each derivative, which, in their turn, are employed to provide better target phase estimates. The method is a fully ab initio method, because it needs only the experimental data of the target structure. The term derivative is maintained with reference to SIR-MIR (single isomorphous replacement-multiple isomorphous replacement) techniques, even if its meaning is different: therefore the reader should think of the phantom derivative method more as a new method than as a variant of SIR-MIR techniques. The differences are much greater than the analogies. The paper also describes how phantom derivatives may be used for improving structure models obtained via other ab initio or non-ab initio techniques. The method is expected to be insensitive to the structural complexity of the target and to the target experimental data resolution, provided it is better than 4-6 Å.