How to calculate molecular geometry

The relevant atomic coordinates for the calculation of molecular geometry consist of the values of x, y, and z, obtained as precisely as possible, for all atoms. From values of x, y, and z we can derive geometrical information such as bond distances, bond angles, torsion angles, and the mean planes through groups of atoms. In this way the geometry of the molecule or ion is found. 

The reader should appreciate that because atoms are always vibrating, the atomic positions found in an X-ray diffraction experiment represent the average positions of the atoms during vibration. The atomic parameters include atomic displacement parameters (described in Chapter 13) which give some measure of the amplitude of this 

Because atomic arrangements in crystal structures are periodic from unit cell to unit cell, one part of a molecule may lie in one unit cell, and another part may lie in an adjacent unit cell. If there is an atom at x. because of this translational symmetry there is another at 1 + x, and another ai n + x, where n is any integer. Values of x, y, and that are reported usually correspond to a complete and distinct molecule so that, for convenience, some atomic coordinates may have negative values, and other atomic coordinates may have values greater than 1.0000. The symmetry of the space group and the identical contents of adjacent unit cells can lead to a diagram of the complete crystal structure and an analysis not only of the molecular structure, but also of its surroundings. 

Typical sets of atomic coordinates are given in Tables 11.1 and 11.2 for the alkali halides and copper sulfate pentahydrate, respectively. Fractional atomic coordinates are listed separately from unit cell dimensions 

FIGURE 11.2. A convenient way to draw the contents of a unit cell, given the atonii coordinates, is shown here, (a) The outline of the unit cell is drawn to scale in twc dimensions as shown. It is then divided into one tenths in each dimension, by meaIl of a ruler (any scale, inches, centimeters), inclined as shown, so that each side can b--divided into ten parts, (b) The result is a grid on which the positions of atoms can hr plotted, as shown. In the third dimension, if the third unit-cell axis is perpendicular to the plane of the paper, Pythagoras theorem can be used to measure interatomi-distances if it is not perpendicular, only an approximate estimate can be made. 

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