Section 13.1.5.2. Information gain from ideal noncrystallographic symmetry

Rossmann & Blow (1962) wrote, “The effect of noncrystallographic symmetry … results in decreasing the size of the structure to be determined, while the number of observable intensities remains the same. This `redundancy' in information might be used to help solve a structure.” This idea is developed below.

First, it will be shown that (in the absence of noncrystallographic symmetry) there is a constant ratio between the number of independent measurements required to specify the scattering density at a chosen resolution and the volume of the asymmetric unit. Then the effect of noncrystallographic symmetry on this ratio is discussed. The importance of the ratio (volume of symmetry-constrained unit/volume of asymmetric unit) is stressed. Another ratio is developed – available no. of measurements/ideally required no. of measurements – and this is referred to as the overdetermination ratio.

Consider a noncentrosymmetric crystal whose asymmetric unit volume is and whose diffraction data have been measured to a resolution d. If the multiplicity of the space group (number of asymmetric units in the primitive unit cell) is Z, the volume of reciprocal space, , per point of the primitive reciprocal lattice is given by where V is the volume of the primitive unit cell.

The number of independent orders of diffraction (the number of independent intensities) within the resolution sphere of radius 1/d is given by In this formula, Friedel's law is supposed to apply. A set of 2Z reflections have identical intensity due to the combined effects of Friedel's law and crystal symmetry. When is expressed in terms of , the multiplicity factor disappears.

To calculate the scattering density over the volume at resolution d, independent quantities need to be specified (say, the real and imaginary parts of each of the independent structure factors). The required number of measurements is If only the diffracted intensities can be measured, they provide exactly half the measurements required to calculate the density at resolution d.

In what follows, it is assumed that the required number of measurements, R, to specify the scattering density at the chosen resolution is proportional to the volume over which the density must be specified. This is true when the volume is a crystallographic asymmetric unit [equation (13.1.5.2)], and it agrees with another analysis discussed below. Following this argument, the overdetermination ratio where X is the volume whose density is unknown.

Next, consider that ideal noncrystallographic symmetry applies. The crystal asymmetric unit contains N identical subunits and no other scattering matter. Since the symmetry is noncrystallographic, it is never possible to fit the subunit volumes together so as to fill the unit cell exactly. The volume assigned to each subunit, U, has to be less than , leaving some parts of the unit cell not assigned to any subunit. In the case of ideal noncrystallographic symmetry, these regions are necessarily empty. In this case , which is less than , so from equation (13.1.5.3) Even where N is only 2, more intensity data are available than the number of measurements ideally required to specify the electron density at resolution d.

A more sophisticated analysis of the number of variables required to define a structure with noncrystallographic symmetry has been made in terms of sets of orthogonal `eigendensity functions', which satisfy the noncrystallographic symmetry (Crowther, 1967). Any structure satisfying the symmetry requirements can be constructed from the appropriate set of eigendensities. Crowther (1969) demonstrated that the number of eigendensities m is approximately

The structure is specified by m weights, which are applied to the m allowed eigendensities (which depend only on the symmetry constraints), so the overdetermination ratio the same result as before, showing that the two methods of analysis approximately agree.