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11: Molecular Symmetry

In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical transformation which turns a molecule into an indistinguishable copy of itself is called a symmetry operation. A symmetry operation can consist of a rotation about an axis, a reflection in a plane, an inversion through a point, or some combination of these.

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 Cn 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 C3v, 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

Figure \(\PageIndex{1}\): Two views of the ammonia molecule.

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 C3v 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.

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