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Crystallographic Restriction Theorem and Molecular Symmetry

By Julie Geng, V Form

Author’s Note: 

As many of my peers and teachers know, I am a big chemistry “nerd”. While I am engrossed in organic synthesis, I aim to broaden my perspective by making the connections between chemistry and other STEM subjects. Interdisciplinary studies are indispensable nowadays, as many scientific breakthroughs are results of the collective efforts of specialists in various interrelated fields.


As exemplified by the copious examples in “Linear Algebra: A Modern Introduction” by David Poole, linear algebra is widely applied to many fields of science. Concepts in linear algebra are widely applied to the molecular and crystalline symmetry in organic chemistry. The transformation matrices are used to describe the internal symmetry of molecules, while the concept of trace as a similar invariant is used to substantiate the crystallographic restriction theorem.

Molecular Symmetry

A symmetrical water molecule

Do you remember the molecular model you played with in your sophomore chemistry class? Did you ever notice that some molecules are more symmetrical than others?

Imagine carrying out some operation on a molecule. If the final configuration is indistinguishable from the initial one, then the operation is a symmetry operation for that object. By definition, a symmetry operation is an action that leaves an object looking the same after it has been carried out. Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is performed.

There are five symmetry elements that a molecule may possess. The first one is identity, E. The identity operation consists of rotating by 360◦, and the corresponding symmetry element is the entire molecule. Every molecule has at least this element.

The second type is an n-fold axis of rotation. If rotating by 360◦/n leaves the molecule unchanged, the molecule is said to contain a Cn axis.

Screen Shot 2014-01-11 at 1.53.10 PM


If a reflection in the plane leaves a molecule looking the same, then it contains a symmetry element, plane of symmetry.

Screen Shot 2014-01-11 at 1.54.33 PM

Xenon tetrafluoride

The fourth symmetry element is a centre of symmetry, meaning an inversion through the centre of symmetry leaves the molecule unchanged.

Screen Shot 2014-01-11 at 1.54.22 PM

Dinitrogen difluoride

The last symmetry element is Sn, an n-fold improper rotation axis. The improper inversion operation consists of rotating through an angle 360◦/n about the axis, followed by reflecting a plane perpendicular to the axis.

Screen Shot 2014-01-11 at 1.55.04 PM


Linear Transformation and Molecular Symmetry

Matrices can be used to map one set of coordinates or functions onto another set. Matrices used for this purpose are called transformation matrices. We can multiply transformation matrices by the basis that describes the atomic orbitals of a molecule to accomplish the various symmetry operations.

Consider a two-dimensional molecule:
1. The identity operation (E):
The identity operation rotate the molecule by 360◦, leaving the molecule unchanged. The transformation matrix corresponding to this operation is thus the identity matrix:

Screen Shot 2014-01-25 at 6.40.36 PM

2. Reflection in a plane (σ):

Reflection in x and y-axis can be accomplished by the following two matrices respectively:

 Screen Shot 2014-01-14 at 9.52.49 PM


Screen Shot 2014-01-14 at 9.52.55 PM

3. Inversion (I)

An inversion is a reflection in the 45◦ axis, and the transformation matrix is:

Screen Shot 2014-01-15 at 7.03.25 PM

4. Rotation about an axis (Cn)

In two dimensions, the transformation matrix for a rotation by an angle θ about the origin is:

Screen Shot 2014-01-14 at 9.45.56 PM

5. Improper Inversion (Sn)

To carry out two transformation, we just multiply the two transformation matrices together to obtain an overall transformation matrix.

Symmetry in Crystalline Structures and Crystallographic Restriction Theorem 

Symmetry does not only exist in molecular structures but also in crystalline structures. Crystals have an ordered internal arrangement of atoms. This ordered arrangement displays internal symmetry. For example, the atoms are arranged in a symmetrical fashion on a three dimension network called lattice. Besides internal symmetry, there are three types of external symmetry a crystal can possess: rotation, reflection, and inversion. The rotational symmetry of crystals is further discussed. Rotational symmetry also exists in many wallpaper design as illustrated in the following images.

Screen Shot 2014-01-25 at 7.01.18 PM

Rotational Symmetry in Wallpaper Design

The crystallographic restriction theorem was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. One way to prove this theorem in 2D and 3D is by using matrix. A more complicated proof is needed for higher dimensions when the consideration of quasicrystals becomes necessary.

The matrix proof of crystallographic restriction theorem uses the concept of trace, a matrix property.

Definition: The trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right of A.

Screen Shot 2014-01-25 at 7.01.36 PM

Rotational Symmetry in Wallpaper Design

Trace is a similarity invariant, meaning the value of the trace does not change under a similarity transformation of the form: A^'=BAB^(-1),  A similarity transformation is a conformal mapping whose transformation matrix  A^'=BAB^(-1), where A and A′ are similar matrices.In this case, A corresponds to the lattice basis, and A’ corresponds to the rotation matrix. Similarity transformations transform objects in space to similar objects such as moving from one lattice point to another lattice point in a crystalline structure.

Because the lattice basis and the rotation matrix are similar, they share the same trace. Because the rotation matrix describes the transformation of the lattice points, it has to contain all integers that are multiples of the set distance between two neighboring lattice points. As a result, the trace is an integer. For a 2D rotation, the trace is 2cos θ. Therefore, cos θ has to equal an integer. The rotation angles at which cosθ equals integers are 60◦, 90◦, 120◦, and 180◦. These rotation angles corresponds to 6-, 4-, 3-, and 2-fold rotation.

The following two examples demonstrate the validity of this proof.

Consider a 60◦ (6-fold) rotation matrix with respect to an orthonormal basis in 2D: Screen Shot 2014-01-15 at 2.17.46 PM

The trace is 1, an integer. As a result, 6-fold rotation is possible.

However, consider a 45◦(8-fold) rotation matrix still with respect to an orthonormal basis:

Screen Shot 2014-01-15 at 2.17.53 PMThe trace is 1/ 2, not an integer. As a result, 8-fold rotation is not possible.

The proof for 3D rotations can be conducted in a similar fashion.

A Final Chemistry Remark 

Many proofs done by mathematicians have confirmed the soundness of crystallographic restriction theorem. Nevertheless, quasicrystals can occur with other symmetries, such as 5-fold; these were not discovered until 1982, when a diffraction pattern out of a quasicrystal was first seen by the Israeli scientist Dan Shechtman, who won the 2011 Nobel Prize in Chemistry for his discovery. For further information, please click here.

Julie Geng is a V Former from from Shanghai, China; she lives in Gaccon. She is obsessed with chemistry and enjoys the show Breaking Bad.


1. Vallance, Claire. MOLECULAR SYMMETRY, GROUP THEORY, & APPLICATIONS. N.p.: n.p., n.d. PDF.

2. “Matrix Trace.” — from Wolfram MathWorld. N.p., n.d. Web. 25 Jan. 2014.

3. “Similarity Transformation.” — from Wolfram MathWorld. N.p., n.d. Web. 25 Jan. 2014.

4. “Similar Matrices.” — from Wolfram MathWorld. N.p., n.d. Web. 25 Jan. 2014.

5. “The Symmetry of Crystals. The Restriction Theorem.” The Symmetry of Crystals. The Restriction Theorem. N.p., n.d. Web. 25 Jan. 2014.

6. “Introduction & Symmetry Operations.” Introduction & Symmetry Operations. N.p., n.d. Web. 25 Jan. 2014.

Image Credits:

1. http://bytesizebio.net/wp-content/uploads/2010/01/eureka-lab-cartoon.gif

2. http://www.bridgesmathart.org/art-exhibits/bridges2008/strauss4.jpg

3. http://www.math.washington.edu/~julia/mathday09/e53.jpg

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