CONVOL: Real Space Convolution

Authors: Doug Collins, Trina Norden & Jim Stewart

Contact: Jim Stewart, Department of Chemistry,

University of Maryland, College Park, MD 20742. U.S.A.

CONVOL provides a means for the real space convolution of density with another function by multiplication of their respective structure factors in reciprocal space. Convolution is not explicitly carried out by CONVOL. It is used to generate coefficients for a FOURR calculation. Structure factors from an input bdf are extracted, multiplied by the chosen transform and the products written to the output bdf as structure factors. Fourier synthesis of these new structure factors gives a function which may be described as the pointwise replacement of density by a weighted average. For each point, the average is over the entire unit cell, and the relative shape and placement of the weight function is chosen by the user.

Method

The functions which may be applied to the structure factors are a subset of those in the International Tables for X-Ray Crystallography, Vol. II (1967), Table 2.5.3D, page 72. These convolutions smooth or shape the direct space electron density. For example, this program can be used as an efficient way to do "density averaging" of a low resolution electron density map.

Input consists of a command line for the function from Table 2.5.3D desired, containing the necessary parameters of the function shown in the column headed F(u), and an input bdf containing the observed F and phase values to be modified. Output is a bdf with the modified F and phase values suitable for Fourier transformation. Note that in order to produce the modified map, FOURR must be used with fcal as the specified coefficient type.

To see the use of this program, consider function 3.1. The application of this function corresponds to the direct space process of density smoothing by local averaging. In direct space the density would be modified by finding, for each point in the map, the local average over points within the surrounding parallelepiped specified. Function 3.1 produces the equivalent result by modifying the the amplitude and phase of each reflection and then carrying out the Fourier transform. This process requires only a reflection-by-reflection multiplication in reciprocal space instead of the repeated point-by-point averaging required by the direct space method.

<sf>WARNING</sf>: As in International Tables, parallel coordinates referred to the crystal lattice are used without regard to the possible non-orthogonality of the base. The coordinates ARE fractional coordinates.

The Convolution Functions

f1.2 The density which would result from Fourier transformation of the input F and phase values will be translated to a new origin at ,,.

f2.2 The density at points in direct space will be replaced by ellipsoidally averaged density weighted by an inverse exponential function of the square of the distance from the point. For a factor the weight function along axis i is

The complete weight function is . The effect is similar to applying an overall anisotropic thermal parameter to smear the density.

f3.1 Density at points in direct space is replaced by the average of the density, a, inside a parallelepiped centred on the point. The parallelepiped size is set by parameters, , parallel to the unit cell edges.The interior of the parallelepiped is defined by || < .

f3.2 Similar to f3.1, but the volume is a cylinder parallel to a crystal axis specified by the user. The cylinder is bounded by a parallelepiped with |x(i)| < b(i). The formula takes as along the cylinder axis, and reorientation of the cylinder is accompilshed by tying each to a crystal axis.

f3.3 Similar to f3.1, but the function is an inclusive ellipsoid of constant value a in its interior, 0 elsewhere. Along each axis the ellipsoid is bounded by || <. This figure can be a sphere if the base is orthogonal and, and are properly chosen.

f4 Is similar to f3.3 but the weighting inside the ellipsoid varies linearly with distance from the origin. At its origin the ellipsoid has the value a, at its boundary the value 0. At an interior point ,, the value is a(1-|r|); is the sum of (/)2.

f5 Is similar to f4 but the weighting is quadratically ramped, given by a(1-).

File Assignments

Reads structure factor data from the input archive bdf

Writes modified structure factor data to the output archive bdf

Example

CONVOL parent list 10

f1.2 0 .5 0

In this example the phases in the input bdf will be modified such that the subsequent Fourier map produced from the output bdf will be displaced by 1/2 unit cell along the y direction relative to the map that would be produced by use of the input bdf.

References

International Tables for X-Ray Crystallography, Vol. II, (1967), Sections 2.5.3.4 and 6.3. Edited by J. S. Kasper and K. Lonsdale, The Kynoch Press.

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Edited by M. Abramowitz and I. A. Stegun, Dover Publications, Inc.