12: Molecular Symmetry

The Ammonia Molecule

We shall introduce the concepts of symmetry and group theory by considering a concrete example–the ammonia molecule NH₃. In any symmetry operation on NH₃, the nitrogen atom remains fixed but the hydrogen atoms can be permuted in 3!=6 different ways. The axis of the molecule is called a C₃ axis, since the molecule can be rotated about it into 3 equivalent orientations, $120^\circ$ apart. More generally, a C_n axis has n equivalent orientations, separated by $2\pi/n$ radians. The axis of highest symmetry in a molecule is called the principal axis. Three mirror planes, designated $\sigma_1,\sigma_2,\sigma_3$, run through the principal axis in ammonia. These are designated as $\sigma_v$ or vertical planes of symmetry. Ammonia belongs to the symmetry group designated C_3v, characterized by a three-fold axis with three vertical planes of symmetry.

Let us designate the orientation of the three hydrogen atoms in Figure $\PageIndex{1}$ as {1, 2, 3}, reading in clockwise order from the bottom. A counterclockwise rotation by 120$^\circ$, designated

by the operator C₃, produces the orientation {2, 3, 1}. A second counterclockwise rotation, designated $C_3^2$, produces {3, 1, 2}. Note that two successive counterclockwise rotations by 120$^\circ$ is equivalent to one clockwise rotation by 120$^\circ$, so the last operation could also be designated $C_3^{-1}$. The three reflection operations $\sigma_1,\sigma_2,\sigma_3$, applied to the original configuration {1, 2, 3} produces {1, 3, 2}, {3, 2, 1} and {2, 1, 3}, respectively. Finally, we must include the identity operation, designated E, which leaves an orientation unchanged. The effects of the six possible operations of the symmetry group C_3v can be summarized as follows:

$$E\{1,2,3\}=\{1,2,3\} C_3\{1,2,3\}=\{2,3,1\}$$

$$C_3^2\{1,2,3\}=\{3,1,2\} \sigma_1\{1,2,3\}=\{1,3,2\}$$

$$\sigma_2\{1,2,3\}=\{3,2,1\} \sigma_3\{1,2,3\}=\{2,1,3\}$$

We have thus accounted for all 6 possible permutations of the three hydrogen atoms.

The successive application of two symmetry operations is equivalent to some single symmetry operation. For example, applying C₃, then $\sigma_1$ to our starting orientation, we have

$$\sigma_1 C_3\{1,2,3\}=\sigma_1\{2,3,1\}=\{2,1,3\}$$

But this is equivalent to the single operation $\sigma_3$. This can be represented as an algebraic relation among symmetry operators

$$\sigma_1 C_3=\sigma_3$$

Note that successive operations are applied in the order right to left when represented algebraically. For the same two operations in reversed order, we find

$$C_3 \sigma_1 \{1,2,3\} = C_3 \{1,3,2\} = \{3,2,1\} = \sigma_2 \{1,2,3\}$$

Thus symmetry operations do not, in general commute

$$A B \not\equiv B A \label{1}$$

although they may commute, for example, $C_3$ and $C_3^2$.

The algebra of the group $C_{3v}$ can be summarized by the following multiplication table.

$$\begin{matrix} & 1^{st} & E & C_3 & C_3^2 & \sigma_1 &\sigma_2 &\sigma_3 \\ 2^{nd} & & & & & & & \\ E & &E &C_3 &C_3^2 &\sigma_1 &\sigma_2 &\sigma_3 \\ C_3& &C_3 &C_3^2 &E &\sigma_3 &\sigma_1 &\sigma_2 \\ C_3^2& & C_3^2 &E &C_3 &\sigma_2 &\sigma_3 &\sigma_1 \\ \sigma_1& &\sigma_1 &\sigma_2 &\sigma_3 &E &C_3 &C_3^2 \\ \sigma_2 & &\sigma_2 &\sigma_3 &\sigma_1 &C_3^2 &E &C_3 \\ \sigma_3 & &\sigma_3 &\sigma_1 &\sigma_2 &C_3 &C_3^2 &E \end{matrix}$$

Notice that each operation occurs once and only once in each row and each column.

Group Theory

In mathematics, a group is defined as a set of g elements $\mathcal{G} \equiv \{G_1,G_2...G_h\}$ together with a rule for combination of elements, which we usually refer to as a product. The elements must fulfill the following four conditions.

1. The product of any two elements of the group is another element of the group. That is $G_iG_j=G_k$ with $G_k\in\mathcal{G}$
2. Group multiplication obeys an associative law, $G_i(G_jG_k)=(G_iG_j)G_k\equiv G_iG_jG_k$
3. There exists an identity element E such that $EG_i=G_iE=G_i$ for all i.
4. Every element $G_i$ has a unique inverse $G_i^{-1}$, such that $G_iG_i^{-1}=G_i^{-1}G_i=E$ with $G_i^{-1}\in\mathcal{G}$.

The number of elements h is called the order of the group. Thus $C_{3v}$ is a group of order 6.

A set of quantities which obeys the group multiplication table is called a representation of the group. Because of the possible noncommutativity of group elements [cf. Eq (1)], simple numbers are not always adequate to represent groups; we must often use matrices. The group $C_{3v}$ has three irreducible representations, or IR’s, which cannot be broken down into simpler representations. A trivial, but nonetheless important, representation of any group is the totally symmetric representation, in which each group element is represented by 1. The multiplication table then simply reiterates that $1\times 1=1$. For $C_{3v}$ this is called the $A_1$ representation:

$$A_1: E=1,C_3=1,C_3^2=1,\sigma_1=1,\sigma_2=1,\sigma_3=1 \label{2}$$

A slightly less trivial representation is $A_2$:

$$A_2: E=1,C_3=1,C_3^2=1,\sigma_1=-1,\sigma_2=-1,\sigma_3=-1 \label{3}$$

Much more exciting is the E representation, which requires $2\times 2$ matrices:

$$E= \begin{pmatrix} 1 &0 \\0 &1 \end{pmatrix} \qquad C_3=\begin{pmatrix} -1/2 &-\sqrt{3}/2 \\ \sqrt{3}/2 &-1/2 \end{pmatrix} \\ C_3^2=\begin{pmatrix} -1/2 &\sqrt{3}/2 \\ -\sqrt{3}/2 &-1/2 \end{pmatrix} \qquad \sigma_1=\begin{pmatrix} -1 &0 \\0 &1 \end{pmatrix} \\ \sigma_2=\begin{pmatrix} 1/2 &-\sqrt{3}/2 \\ -\sqrt{3}/2 &-1/2 \end{pmatrix} \qquad \sigma_3=\begin{pmatrix} 1/2 &\sqrt{3}/2 \\ \sqrt{3}/2 &-1/2 \end{pmatrix} \label{4}$$

The operations $C_3$ and $C_3^2$ are said to belong to the same class since they perform the same geometric function, but for different orientations in space. Analogously, $\sigma_1, \sigma_2$ and $\sigma_3$ are obviously in the same class. E is in a class by itself. The class structure of the group is designated by $\{E,2C_3,3\sigma_v\}$. We state without proof that the number of irreducible representations of a group is equal to the number of classes. Another important theorem states that the sum of the squares of the dimensionalities of the irreducible representations of a group adds up to the order of the group. Thus, for $C_{3v}$, we find $1^2+1^2+2^2=6$.

The trace or character of a matrix is defined as the sum of the elements along the main diagonal:

$$\chi(M)\equiv\sum_kM_{kk} \label{5}$$

For many purposes, it suffices to know just the characters of a matrix representation of a group, rather than the complete matrices. For example, the characters for the E representation of $C_{3v}$ in Eq (4) are given by

$$\chi(E)=2,\quad \chi(C_3)=-1, \quad \chi(C_3^2)=-1, \\ \chi(\sigma_1)=0, \quad \chi(\sigma_2)=0, \quad \chi(\sigma_3)=0 \label{6}$$

It is true in general that the characters for all operations in the same class are equal. Thus Eq (6) can be abbreviated to

$$\chi(E)=2,\quad \chi(C_3)=-1, \quad \chi(\sigma_v)=0 \label{7}$$

For one-dimensional representations, such as $A_1$ and $A_2$, the characters are equal to the matrices themselves, so Equations $\ref{2}$ and $\ref{3}$ can be read as a table of characters.

The essential information about a symmetry group is summarized in its character table. We display here the character table for $C_{3v}$

$$\begin{matrix} C_{3v} &E &2C_3 &3\sigma_v & & \\\hline A_1 &1 &1 &1 &z &z^2,x^2+y^2 \\A_2 &1 &1 &-1 & & \\E &2 &-1 &0 &(x,y) &(xy,x^2-y^2),(xz,yz) \end{matrix}$$

The last two columns show how the cartesian coordinates x, y, z and their products transform under the operations of the group.

Group Theory and Quantum Mechanics

When a molecule has the symmetry of a group $\mathcal{G}$, this means that each member of the group commutes with the molecular Hamiltonian

$$[\hat G_i,\hat H]=0 \quad i=1...h \label{8}$$

where we now explicitly designate the group elements $G_i$ as operators on wavefunctions. As was shown in Chap. 4, commuting operators can have simultaneous eigenfunctions. A representation of the group of dimension d means that there must exist a set of d degenerate eigenfunctions of $\hat H$ that transform among themselves in accord with the corresponding matrix representation. For example, if the eigenvalue $E_n$ is d-fold degenerate, the commutation conditions (Equation $\ref{2}$) imply that, for $i=1...h$,

$$\hat G_i \hat H \psi_{nk} = \hat H \hat G_i \psi_{nk}=E_n \hat G_i \psi_{nk} \; \text{for} \;k=1...d \label{9}$$

Thus each $\hat G_i \psi_{nk}$ is also an eigenfunction of $\hat H$ with the same eigenvalue $E_n$, and must therefore be represented as a linear combination of the eigenfunctions $\psi_{nk}$. More precisely, the eigenfunctions transform among themselves according to

$$\hat G_i \psi_{nk}=\sum_{m=1}^d D(G_i)_{km}\psi_{nm} \label{10}$$

where $D(G_i)_{km}$ means the $\{k,m\}$ element of the matrix representing the operator $\hat G_i$.

The character of the identity operation E immediately shows the degeneracy of the eigenvalues of that symmetry. The $C_{3v}$ character table reveals that $NH_3$, and other molecules of the same symmetry, can have only nondegenerate and two-fold degenerate energy levels. The following notation for symmetry species was introduced by Mulliken:

1. One dimensional representations are designated either A or B. Those symmetric wrt rotation by $2\pi/n$ about the $C_n$ principal axis are labeled A, while those antisymmetric are labeled B.
2. Two dimensional representations are designated E; 3, 4 and 5 dimensional representations are designated T, F and G, respectively. These latter cases occur only in groups of high symmetry: cubic, octahedral and icosohedral.
3. In groups with a center of inversion, the subscripts g and u indicate even and odd parity, respectively.
4. Subscripts 1 and 2 indicate symmetry and antisymmetry, respectively, wrt a $C_2$ axis perpendicular to $C_n$, or to a $\sigma_v$ plane.
5. Primes and double primes indicate symmetry and antisymmetry to a $\sigma_h$ plane.

For individual orbitals, the lower case analogs of the symmetry designations are used. For example, MO’s in ammonia are classified $a_1,a_2$ or e.

For ammonia and other $C_{3v}$ molecules, there exist three species of eigenfunctions. Those belonging to the classification $A_1$ are transformed into themselves by all symmetry operations of the group. The 1s, 2s and $2p_z$ AO’s on nitrogen are in this category. The z-axis is taken as the 3-fold axis. There are no low-lying orbitals belonging to $A_2$. The nitrogen $2p_x$ and $2p_y$ AO’s form a two-dimensional representation of the group $C_{3v}$. That is to say, any of the six operations of the group transforms either one of these AO’s into a linear combination of the two, with coefficients given by the matrices (4). The three hydrogen 1s orbitals transform like a $3\times 3$ representation of the group. If we represent the hydrogens by a column vector {H1,H2,H3}, then the six group operations generate the following algebra

$$\begin{matrix} E=\begin{pmatrix} 1 &0 &0 \\0 &1 &0 \\0 &0 &1 \end{pmatrix} & C_3=\begin{pmatrix} 0 &1 &0 \\0 &0 &1 \\1 &0 &0 \end{pmatrix} \\ C_3^2=\begin{pmatrix} 0 &0 &1 \\1 &0 &0 \\0 &1 &0 \end{pmatrix} & \sigma_1=\begin{pmatrix} 1 &0 &0 \\0 &0 &1 \\0 &1 &0 \end{pmatrix} \\ \sigma_2=\begin{pmatrix} 0&0 &1 \\0 &1 &0 \\1 &0 &0 \end{pmatrix} & \sigma_3=\begin{pmatrix} 0&1 &0 \\1 &0 &0 \\0 &0 &1 \end{pmatrix} \end{matrix} \label{11}$$

Let us denote this representation by $\Gamma$. It can be shown that $\Gamma$ is a reducible representation, meaning that by some unitary transformation the $3 \times 3$ matrices can be factorized into blockdiagonal form with $2 \times 2$ plus $1 \times 1$ submatrices. The reducibility of $\Gamma$ can be deduced from the character table. The characters of the matrices (Equation $\ref{11}$) are

$$\Gamma: \qquad \chi(E)=3, \quad \chi(C_3)=0, \quad \chi_(\sigma_v)=1 \label{12}$$

The character of each of these permutation operations is equal to the number of H atoms left untouched: 3 for the identity, 1 for a reflection and 0 for a rotation. The characters of $\Gamma$ are seen to equal the sum of the characters of $A_1$ plus E. This reducibility relation is expressed by writing

$$\Gamma=A_1\oplus E \label{13}$$

The three H atom 1s functions can be combined into LCAO functions which transform according to the IR’s of the group. Clearly the sum

$$\psi=\psi_{1s}(1)+\psi_{1s}(2)+\psi_{1s}(3) \label{14}$$

transforms like $A_1$. The two remaining linear combinations which transform like E must be orthogonal to (Equation $\ref{14}$) and to one another. One possible choice is

$$\psi'=\psi_{1s}(2)-\psi_{1s}(3), \quad \psi''=2\psi_{1s}(1)-\psi_{1s}(2)-\psi_{1s}(3) \label{15}$$

Now, Equation $\ref{14}$ can be combined with the N 1s, 2s and $2p_z$ to form MO’s of $A_1$ symmetry, while Equation $\ref{15}$ can be combined with the N $2p_x$ and $2p_y$ to form MO’s of E symmetry. Note that no hybridization of AO’s is predetermined, it emerges automatically in the results of computation.

Contributors and Attributions

• Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)