02 | Atomic coordinates

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Molecules and materials are three-dimensional. The chemistry of materials is reliant upon our knowledge of the arrangement of their constituent atoms in space. This axiom that the properties of a material are defined by the atomic structure is the central tenet of much of modern materials chemistry. The vast majority of our knowledge of atomic coordinates in both molecules and materials originates from a single experimental technique: X-ray scattering. Thus a basic understanding of how atomic coordinates are used and derived from experiment is a necessary for our common pursuit of the ubiquitous structure-property relationships that engross our work as chemists.

This extends to liquids and amorphous materials as well. The structures in solution are very often inferred from the structures can can be isolated in the solid state. Likewise much or our direct measurements of polymeric structures and glass comes from either crystalline analogues of the sames solids and materials or total scattering techniques and the derivation of structural models that best fit their intensity distributions functions.

Data Structures

There are many formats for recording and manipulating atomic structure data but we will focus on two of the most common. Crystallographic information files (cif) are used to record the coordinates of a periodic structures along with all the symmetry rules required for the structure to tile in 3D space. For molecules which are not periodic and fundamental zero-dimensional a simpler format is available where the element x, y, and z coordinates are simply listed these files have the .xyz format.

XYZ files

Below is an example of the contents of a .xyz file for methane (CH₄). Reliable structural data for many simple molecules can be found in the NIST Web Book. These structures are computed and geometry optimized. The Computational Chemistry Comparison and Benchmark Database also provides experimental and computed structures for many simple molecules.

These xyz files are structures as follow:

1. The first line holds a single integer that is equal to the number of atoms in the file

2. The seconds line is a comment line. In this case, here is show the CAS number for methane.

3. Each subsequent line holds the element and its xyz coordinates separated by a space. The coordinates are reported in Ångstroms.

Here is an example of the contents of an xyz file. This can be opened and read by any plain text editor (notepad, textedit, vim, nano, atom, sublime, vscode…)

5         
74-82-8-3d
C 1.058200 0.935200 0.810300
H 1.416500 1.570000 -0.004800
H 1.207300 1.448200 1.764400
H -0.006200 0.728200 0.670200
H 1.615400 -0.005600 0.811400

These coordinates are in Cartesian Space; all three axes are mutually orthogonal and the distances between increments are all equivalent. We’ll see when describing periodic structures, the Cartesian coordinate system is not always best suited for describing atom positions.

Atom positions are vectors

While we often think of atomic coordinates as describing a point in space, it is often useful to also consider them as position vectors. Vector calculus provides many convenient methods for describing the relationships between atomic positions and their transformations. Atoms coordinates can be described as a position vector this is a set of scalers from which the coordinates system’s basis vectors can be multiplied to describe a translation from the coordinate system’s origin to the specified location. That is generally[1],

\[ \vec{X} = x \bf{\hat{x}} + y \bf{\hat{y}} +z \bf{\hat{z}} \]

where the location of a point can be found by multiplying the three basis vectors of the coordinate system (\(\bf{\hat{x}}\), \(\bf{\hat{y}}\), and \(\bf{\hat{z}}\)) by \(x\), \(y\), and \(z\) respectively then adding these three component vectors together. For a cartesian coordinate system all three basis vectors are of unit length and mutually orthogonal such that locating a position in space from \(x\), \(y\), and \(z\) alone is trivial.

\[\begin{split} \bf{ \hat{x}} = \bf{i} = \begin{bmatrix} 1\\0\\0 \end{bmatrix};\; \bf{\hat{y}} = \bf{j} = \begin{bmatrix} 0\\1\\0 \end{bmatrix};\; and \; \bf{\hat{z}} = \bf{k} = \begin{bmatrix} 0\\0\\1 \end{bmatrix} \end{split}\]

For the 2D molecule benzene, two position vectors for nearest neighbor carbons atoms are shown in Fig. 1 and labeled as C1 and C2.

Bond Distances

The difference vector between any two sets of atomic coordinates points in the direction from the first atom to the second and its magnitude is equivalent to the interatomic distance, Fig. 1. In certain instances this also corresponds to a bond length.

Fig. 1 Bond distance calculation form coordinates in the 2D molecule benzene. The C-C bond distance can be calculate from position vectors as the magnitude of the difference vector \(\vec{C_1}-\vec{C_2}\).

For the two carbon atoms specified in Fig. 1, \(\vec{C_1}\) and \(\vec{C_2}\) their difference vector is

\[ \vec{C_1}-\vec{C_2} = [x_1-x_2, y_1-y_2, z_1-z_2]. \]

Note that this difference vector originates at \(\vec{C_2}\) points to the position \(\vec{C_1}\) Since we are in a Cartesian coordinate system the carbon-carbon bond length is

\[ |\vec{C_1}-\vec{C_2}| = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2} \]

Bond Angle

We can also easily obtain bond angles from atomic position vectors. While the bond length in independent of the direction of the difference vector from which is derived is pointing, bond angles are not. Thus extra care must be taken with regards to the definition of our difference vectors when calculating bond angles.

Fig. 2 The bond angle between H1, C2, and C1 can be calculated from the dot product of the vectors \(\vec{H_1}-\vec{C_2}\) and \(\vec{C_1}-\vec{C_2}\)

For example, we can calculate the bond angle defined by the three bonded atoms H2, C2, and C1 with positions \(\vec{H_2}\), \(\vec{C_2}\), and \(\vec{C_1}\) using the dot product. These difference vectors are shown Fig. 2. Ultimately, \(\theta\) in radians can be calculates as

\[\begin{split} \vec{X_1} = \vec{H_1}-\vec{C_2}\\ \vec{X_2} = \vec{C_1}-\vec{C_2}\\ \end{split}\]

\[ \vec{X_1} \cdot \vec{X_2} = |\vec{X_1}||\vec{X_2}|\cos{\theta}; \]

thus,

\[ \theta = \cos{\left( \frac{\vec{X_1} \cdot \vec{X_2}} {|\vec{X_1}||\vec{X_2}|} \right)}. \]

Exercises

1. For the xyz file contents shown for methane in the above text, show that the C-H bond length is equal to 1.09 Å.

2. Show that the H-C-H bond angle is equal to 109.5°.

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[1]

Much of the mathematics and syntax used here is a condensed version of what can by found in the International Tables of Crystallography. Notation is as much as possible kept consistent with these texts. For more comprehensive documentation of visit https://it.iucr.org and in particular Volume A.