Harker Kasper Inequalities Research Articles

On the Harker–Kasper inequalities

It is shown that two unitary structure factors generate a two-parameter family of Harker–Kasper inequalities and that the strongest of these coincides with the third-order determinant Karle–Hauptman inequality.

Some thoughts on Harker–Kasper inequalities

The approach by Harker & Kasper [Acta Cryst. (1948), 1, 70-75] which led to the first inequality relationships between structure factors has not previously been applied to the space group P1 and there seems to have been a view that it could not give useful results for that space group. The idea has also been advanced that Harker-Kasper inequalities are contained within the complete set of determinantal inequalities derived by Karle & Hauptman [Acta Cryst. (1950), 3, 181-187]. It is shown that the Harker-Kasper approach can be applied in space group P1 and gives inequality relationships which are distinct, at least in form, from those derived from determinants. Indeed, in some cases, a simple Harker-Kasper inequality can be more effective than an equally simple determinantal inequality in restricting the allowed values of three-phase invariants.

Direct methods – from birth to maturity

The award of the Nobel Prize for Chemistry to two pioneers of direct methods, H. Hauptman and J. Karle, has been a recognition of the importance that these methods have attained, in crystallography in particular and in science generally. The development of direct methods is traced from its first beginnings with Harker-Kasper inequalities and the Karle & Hauptman determinantal inequalities. The range of application of these methods to centrosymmetric structures was much increased by the introduction of the sign relationship by Sayre, Cochran and Zachariasen and an ACA Monograph by Hauptman & Karle to which the origins of the application of direct methods may be traced. Sayre's paper, in 1952, developed an exact equation which applied to both centrosymmetric and non-centrosymmetric structures as did the Karle & Hauptman determinantal inequalities. However, it was the derivation of the probability distribution for an individual three-phase relationship by Cochran in 1955 and the tangent formula by Karle & Hauptman in 1956 which provided the weaponry to tackle non-centrosymmetric structures, but nothing decisive was done until, in 1964, I. L. Karle & J. Karle solved the first non-centrosymmetric structure with direct methods using their symbolic addition procedure. The advent of the computer allowed automatic multisolution procedures, such as MULTAN, SHELX and SIMPEL, to take over as the main tools for the solution of small structures. Various developments since about 1970 have slowly introduced direct-methods concepts into the field of solving macromolecular structures, starting with the maximum-determinant method of Tsoucaris and progressing to the still experimental maximum-entropy method. Various predictions are made about the progress of direct methods in the next few years. A new tangent formula, the Sayre tangent formula, which gives most of the benefit of the maximum-entropy method but is much easier to apply, is suggested as a possible new source of progress. Other progress may be made in the association of direct methods with physical methods, such as isomorphous replacement and anomalous scattering, following work which has already been done over the past few years.

The joint probability distribution applied to a weak sign relationship

The probability of the sign of the product Eh - kEh + k is derived by the mathematical device of the joint probability distribution. Some previous formulae, in contrast with Harker-Kasper inequalities, are criticized. Some experimental tests fit quite well the theory developed here.

A method for direct structure determinations inP1 and related groups

In centrosymmetric symmorphic space groups the correct Σ2 solution need not be one with the best consistency. In these space groups the correct Σ2 solution can be found using a new criterion related to the Harker–Kasper inequalities: {\rm HKC} = \sum_{\bf h} \sum_{\bf k}{{(|U_{{\bf h}}| - |U_{{\bf k}}|)^{2}}\over{(1 - |U_{\bf{h + k}}|) (1 - |U_{\bf{h - k}}}|)}\ {\rm sign}\,({\bf{h + k}})\ {\rm sign}\,({\bf{h - k}}) in which the summations over h and k are restricted to those terms for which |Uh|, |Uh + k|, |Uh − k| are large and |Uk| is small. The weights in HKC are consistent with the Harker–Kasper inequalities for T.

The relation between space group symmetry and the properties of direct solutions based on the Σ2formula. A method for using centrosymmetric Σ2solutions for solving non-centrosymmetric structures

The refinement criteria (consistency criteria) of the Σ2 and tangent formulae are discussed. From these criteria the properties of Σ2 solutions in various space groups are evaluated. Structure determinations can then be divided into five categories according to the properties of the most consistent Σ2 solution. In three of these, trivial phases are obtained. For solving non-centrosymmetric structures in which the Σ2 formula produces centrosymmetric phases (such as structures in P21), a new method (Σ2P method) is proposed which makes use of both Σ2 and Patterson techniques. In structures for which the most consistent Σ2 solution corresponds to the trivial set of phases ψhkl = 0, Harker–Kasper inequalities or other methods have to be used in order to select the most reliable solution from the different sets of phases built up by means of the Σ2 relation. In an appendix, new weighting schemes for the tangent refinement are proposed.

The crystal structure of spermine tetrahydrochloride

+ + + + Crystals of spermine tetrahydrochloride, NH3[CH2]3NHE[CHz]4NH2[CHz]3NH3.4C1-, are monoclinic, space group P21/c, with 2 formula units in a cell with dimensions: a = 8.60, b = 7.88, c = 15.26 .~, fl= 121 ° 30'. The structure has been determined by the interpretation of Patterson projections and application of Harker-Kasper inequalities and then refined by three-dimensional differential syntheses. The charged molecule of spermine is not zigzag planar, two of the four bonds between carbon atoms and imino nitrogen atoms being in gauche conformations. The average value of the C-C single bonds is 1522+0009 ~. The C-N bonds are equal within experimental error with an average value of 1496 + 0.009/~. The two bond angles within the chain adjacent to the bonds in gauche conformation appear significantly larger than the tetrahedral values. + + The strongest interactions in the crystals occur between the NH2 and NH3 groups and the chloride ions. The distances N . . . CI as well as their directions indicate the formation of N-H • • • CI hydrogen bonds.

Corrections of Grison's paper on the Harker–Kasper inequalities and to Zachariasen's paper on the `Statistical method'

Corrections of Grison's paper on the Harker–Kasper inequalities and to Zachariasen's paper on the `Statistical method'

Application of our linear inequalities and some remarks on B. S. Magdoff's paper on `Forbidden reflections in the Harker–Kasper inequalities'

Application of our linear inequalities and some remarks on B. S. Magdoff's paper on `Forbidden reflections in the Harker–Kasper inequalities'

A graphical method for applying Harker–Kasper inequalities to structure determination

A graphical method for applying Harker–Kasper inequalities to structure determination

Forbidden reflections in the Harker–Kasper inequalities

Forbidden reflections in the Harker–Kasper inequalities

On the derivation of Harker–Kasper inequalities

On the derivation of Harker–Kasper inequalities

The application of the Harker–Kasper method of phase determination

The Harker-Kasper inequalities and related ones are applied to the data on oxalic acid dihydrate given by Robertson and Woodward. The technique of obtaining some forty signs of the structure amplitudes is developed in detail. Some general remarks on the method conclude the paper.