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The frequency check, utilizing three-dimensional superpositions of the Patterson and vector verification maps, assigns relative values to positions in electron-density space which are consistent with both the Patterson function and the space group symmetry of the crystal system. These relative values-which represent the extent of peak coincidence between a Patterson and electron-density function-facilitate the choice of the first tentative atom position, a decisive factor in the success of the vector verification procedure. Employing a bivalued Patterson map, each three-dimensional superposition resulting in a relative value requires one to five seconds on a large solid state computer. (Author)

A three-dimensional crystal structure analysis of 3,3,4,4-tetrahydrofurantetrol determined that this molecule exists in a twisted configuration (44 ring twist), that the space group is P(4)2 sub 1c with eight molecules per unit cell, and that the molecules lie in layers which are held together and interconnected by hydrogen bonds. The trial model was obtained by a new method called vector verification. The method systematically checks, by applying space group symmetry and other conditions, each point in a three-dimensional grid in electron density space. At each grid point, a series of vectors is computed and compared with the observed Patterson function, and only if the vectors exist in the Patterson map can the grid point be considered as a possibility for an atomic position. The trial model was refined using full matrix least squares to a final model having an agreement index of 0.11. (Author)

The details of the γ class of syntheses are given in this paper. The theory of the standard syntheses namely, |F|, exp[iα], exp[2iα], 1/|F| and 1/F is refined to the next higher order of approximation. The results are then applied to work out the ratio ρ of the strength of the peak at the unknown to that at the known atomic positions in the α, γ, α', β' and γ' syntheses. Finally a general comparison of the various syntheses is made using the above ratio as the criterion for judging their relative merits.

If a minimum function is computed from overlap on a single-weight peak in the Patterson function, the resultant map will, ideally, show a single image of the structure. However it is difficult to locate single-weight peaks in practice and overlap on a multiple peak produces several heavily-overlapped parallel images of the structure. The procedure described in this paper is as follows: (i) A minimum function is constructed from a multiple peak. (ii) The Fourier coefficients of the minimum function are found and, from these and the observed intensities, a map is computed which shows by what vectors the several images in the maximum function are displaced. (iii) The minimum function is displaced by the vectors found in (ii) and the minimum function of the result gives an approximation to the structure. An application to a model structure is described and possible developments of the method are indicated.

This paper gives an account of the results of testing out the various types of syntheses (the α, β and γ classes in their general and modified forms) which were proposed earlier for the deconvolution of the Patterson function when a part of the structure is known. The tests have been carried out on the centrosymmetric projections of the structures L-tyrosine hydrochloride, L-tyrosine hydrobromide and L-cystine dihydrobromide. As expected from theory, all the syntheses reveal the unknown part of the structure. However, it is rather difficult to decide from the present tests as to which of these is the best. A modified form of the phase synthesis has also been tested and, as expected from theory, it resembles closely the Fourier synthesis.

It has been shown by Jacobson, Wunderlich & Lipscomb (1961) that the modification function M2(s)= s 2 exp (--n2s2/p) applied to structure amplitudes results in a vector distribution corresponding to the Patterson function of the gradient of the electron density. The negative excursions resulting from sharpening by M2(s) were significantly attenuated and the sharpness of main peaks enhanced by using the function (k+s 2) exp (-z~2s2/p) which was equivalent to adding to the 'gradient' Patterson function a contribution from that sharpened in the usual way [M (s)=exp (-n2sE/p)]. Waser & Schomaker (1953) refer to the use of the function Ma(s)=s 4 exp (-n2s2/p), 'which has proved most useful in affording a Patterson function with near maximal resolving power' and whose transform is virtually unaffected by finite cut-off. This function was used in the determination of the crystal structure of hydroxy-L-proline by Donohue & Trueblood (1952) and more recently by Hodgkin & Maslen (1961), who