04 | Symmetry operations as matrices#

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We can use matrix multiplication and vector addition to linearly transform a set of coordinates (atom positions) from some reference state to another. Generally,

\[ \tilde{x} = \mathbf{W}x+\mathbf{w} \]

where in 3D space W is a 3x3 matrix, x is a column vector containing the reference coordinates and w is a column vector the same shape as x that applies a translational element. For every element of symmetry, there is a matrix \(W\) and vector \(w\) from which we can transform any position from its initial location to another symmetry equivalent location. For all operations in a point group

\[\begin{split}w=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{split}\]

and for this reason be be ignored in discussions of molecular symmetry.

Examples#

Following are a few examples of simple tranformations on single vectors.

Identity transform#

The following transform \(\tilde{x}(W,w)\) returns a vector identical to the input vector \(x\)

\[\begin{split} \begin{bmatrix} x\\ y \\ z \end{bmatrix} = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{split}\]

Inversion#

Coordinates are inverted by negating their values.

\[\begin{split} \begin{bmatrix} -x \\ -y \\ -z \end{bmatrix} = \begin{bmatrix} -1&0&0 \\ 0&-1&0 \\ 0&0&-1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{split}\]

Reflection#

Below is a transform function that reflects all input coordinates across the xy plane.

\[\begin{split} \begin{bmatrix} x \\ y \\ -z \end{bmatrix} = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&-1\\ \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} + \begin{bmatrix} 0\\0\\0 \end{bmatrix} \end{split}\]

Rotation#

Here is a 180º rotation about the z-axis.

\[\begin{split} \begin{bmatrix} -x \\ -y \\ z \end{bmatrix} = \begin{bmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} + \begin{bmatrix} 0\\0\\0 \end{bmatrix} \end{split}\]

Exercises#

Define \(W\) and \(w\) for the following transformations#

Reflection across the \(xz\)-plane
180° rotation about the x axis
180° rotation about an axis exactly between \(x\) and \(y\)
90° rotation about the \(z\)-axis

Describe in words the following transformation:#

\[\begin{split} W= \begin{bmatrix} -1&0&0\\ 0&0&1\\ 0&1&0\\ \end{bmatrix};\; w=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{split}\]

04 | Symmetry operations as matrices

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