Constraints & Restraints II. Molecular Geometry

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Molecular Geometry Restraints

Concepts

The earliest versions of the Rietveld program had little scope for control of molecular geometry and did not contain restraint functions. Bond lengths could only be constrained using rather complex quadratic expressions whose coefficients were pre-calculated by the user. This simply reflected the preponderance of inorganic structure refinement in the early use of the program.

With the move to structure solution and refinement of organic and other molecular systems, the need for geometrical restraint functions became obvious. Several types of geometrical restraint can be envisaged:

• A bond length can be restrained to a known distance, e.g. aliphatic C-C distance as 1.54 Å
• A bond angle can be restrained to a typical angle, e.g. C-C-C angle for sp³ carbon atom can be restrained to the tetrahedral value of 109.4°;
• Two (or more) bond lengths can be restrained to be equal, e.g. the 3 C-H distances of the methyl group -CH₃ can be assumed to be equal;
• Four (or more) atoms can be be restrained to have a particular torsion angle;
• Four (or more) atoms can be be restrained to lie in a plane (which is a special case of the torsion angle restraint), e.g. the phenyl group -C₆H₅ (also abbreviated to -Ph or -φ) can be assumed to be flat.

Geometrical restraint functions of the types listed above are very useful in the early stages of the refinement of a crystal structure. Care should be taken over the deviation allowed for each restraint. Analysis of the Cambridge Crystallographic Database (which will be discussed in a later section) shows that the distribution of bond lengths about a mean value is usually much smaller than a related distribution of bond angles about the mean, e.g. it might be reasonable to restrain a C-C distance to, say, 1.54 ± 0.02 Å while the tetrahedrally-coordinated C atom in the same molecule may have its internal bond angles restrained to 109 ± 2 °. In other words, distance restraints can often be applied more strictly (or rigidly) than, say, angle ones.

Many modern Rietveld programs are now equipped with geometric restraint functions. When planar and torsional restraints are unavailable, an equivalent restraint can usually be effected using angle or distance restraints (or a combination of both), though usually with less convenience. Thus, by a phenyl ring can be restrained to be planar by setting all of the internal C-C-C angles to 120° (and all of the external C-C-H angles to 120° also). This is less satisfactory than a true planarity restraint, since the internal phenyl C-C-C angles normally deviate from 120° but the ring still remains flat.

Case Study

The refinement of the structure of phase II of sulphur hexafluoride gives us one of the earliest instances of the use of restraints. Given the correct space-group symmetry, the refinement of the low-temperature crystal structure presents few problems. However, the structure was initially solved and refined in space group P-1 with 6 molecules per unit cell! This was only possible because the molecular geometry of each molecule can be restrained as follows:

• All S-F bond lengths are restrained to be equal. For a single molecule, the use of 5 equality restraints are required;
• In addition, all S-F bond lengths can be restrained to the room-temperature value (obtained by Rietveld refinement of phase I). This requires only 1 additional distance restraint given the equality restraints above;
• All F-S-F bond angles restrained to 90° for cis F-S-F atoms and to 180° for trans F-S-F atoms.

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