Character Tables

According to the Scrödinger equation

$$\hat{H}_{vib}\Phi_k = E_{k}\Phi_k,$$ (17)

each eigenfunction $\Phi$ is associated with a certain energy level $E_k$.

Therefore, molecular eigenfunctions and energy levels can be labelled with a symmetry index $k$ which indicates the point symmetry group of the molecule.

The quantitative characteristic of the labelling is a character table which shows the behavior of the molecular wavefunctions under the symmetry operations of the molecular symmetry point group. Since only certain combinations of symmetry elements occur in the various point groups and since some of their symmetry elements are consequence of others, only certain combinations of symmetry properties of the vibrational (and electronic) wavefunctions are possible. Following Mulliken, in the molecular spectroscopy these combinations of symmetry properties are called symmetry types, or species. In the formal group theory the same combinations are called irreducible representations of the group.

As an example we first consider the character table of the $C_s$ symmetry group which is shown in Table 3

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Table 3: Character Table for the $C_{s}$ Group

$C_s$	$E$	$\sigma(xy)$	$h=2$
$A'$	+1	+1	$x$, $y$
$A''$	+1	$-1$	$z$

Here the first line shows the symmetry operations of the group, $E$ and $\sigma(xy)$, where $(xy)$ indicates the reflection mirror plain. The first column indicates the irreducible representations of the group $A'$ and $A''$, while +1 and $-1$ is used for indication the symmetric and antisymmetric behavior of the wavefunctions with respect to the corresponding symmetry operation. Note, that in every normal vibration and eigenfunction there are species (irreducible representations) which are symmetric under all symmetry operations permitted within a group. These species are called totally symmetric and usually indicated by $A$, or $A_1$, or $A'$. Particularly for the $C_s$ group the totally symmetric species is indicated by $A'$ and presented in the second line in Table 3. It is seen, that the group $C_s$ has two species, $A'$ and $A''$.

The last column in the table indicate the group order, $h=2$ and the simple functions of the coordinates $x, y, z$ which belongs to a certain irreducible representation. These functions are very important, because they represent the symmetry of $p_x$, $p_y$, and $p_x$ atomic orbitals which as we know are used for building the molecular orbitals. Therefore, these coordinates provide a simple way of understanding which species a normal mode, or wavefunction belongs to.

For instance, consider the plane, but non-linear molecule of hydrazoic acid, $N_3H$ which belongs to the $C_s$ group. It has, according to Table 3 normal vibrations which are symmetric, or antisymmetric with respect to the molecular plane. During the former, all atoms remain in the plane, during the latter, they move in lines perpendicular to the plane.

As another example consider the character table of the $C_{2v}$ symmetry group which is shown in Table 4

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Table 4: Character Table for the $C_{2v}$ Group

$C_{2v}$	$E$	$C_2$	$\sigma_v$	$\sigma_v'$	$h=4$
$A_1$	+1	+1	+1	+1	$z$	$x^2$, $y^2$, $z^2$
$A_2$	+1	+1	$-1$	$-1$		$xy$
$B_1$	+1	$-1$	+1	$-1$	$x$	$xz$
$B_2$	+1	$-1$	$-1$	+1	$y$	$yz$

As seen from Table 4, the $C_{2v}$ group has four species (irreducible representations). The totally symmetric species is called in this case $A_1$. Each of the other $A_2$, $B_1$ and $B_2$ species are used to denote one-dimensional (non-degenerate) representations. $A$ is used if the character under the principal rotation is +1, while $B$ is used if the character is $-1$. If other higher dimensional representations are permitted, letter E denotes a two-dimensional irreducible representation and $T$ denotes a three-dimensional representation. The symmetry species $A_1$, $A_2$, $B_1$, and $B_2$ summarize the symmetry properties of the vibrational, or electronic molecular wavefunctions of a for polyatomic molecule. They are analogue to the symmetry labels $\Sigma$, $\Pi$, $\Delta$ which are used for diatomic molecules.

As an example we consider normal vibrations of the formaldehyde molecule $H_2CO$ which belongs to the group $C_{2v}$. It is seen that the three normal vibrations $\nu_1$, $\nu_2$, and $\nu_3$ are totally symmetric and thus belong to species $A_1$. The vibrations $\nu_4$ and $\nu_5$ belong to species $B_1$ (if we call the plane of the molecule the $xz$ plane), and $\nu_6$ belongs to species $B_2$. There is no normal vibration of species $A_2$ in this case. However, in more complicated molecules belonging to the same group there also can be normal vibrations belonging to species $A_2$.

Let us now consider the symmetry of electronic orbitals. As we know, lowercase Greek letters $\sigma$, $\pi$, etc are used for denoting the symmetries of orbitals in diatomic molecules. Similarly, the lowercase Latin letters $a_1$, $a_2$, $b_1$, and $b_2$ are used for denote the symmetry of orbitals in polyatomic molecules which belong to the $A_1$, $A_2$, $B_1$, and $B_2$ irreducible representations, respectively. Alternatively, one says that the wavefunctions $a_1$, $a_2$, $b_1$, and $b_2$ span the irreducible representations $A_1$, $A_2$, $B_1$, and $B_2$. The functions in the 5-th and 6-th columns in Table 4 represent the symmetry of different $p$ and $d$ atomic orbitals which span a certain irreducible representation.

For instance, the symmetry of electronic wavefunctions in the $H_2O$ molecule are as follows. The atomic orbitals of the $O$ atom are: $O2p_x$, $O2p_y$, and $O2p_z$. Assuming that the molecular plane is $YZ$ we can see that the orbital $O2p_x$ change sign under a 180$^0$ rotation, $C_2$ and under the reflection $\sigma_v'$, but remains the same under the reflection $\sigma_v$. Therefore, this orbital belongs to the $B_1$ irreducible representation. As we shall see, any molecular orbital built from this atomic orbital will be a $b_1$ orbital. It can also be seen in the similar way that $O2p_y$ orbital changes sign under $C_2$, but remain the same after $\sigma_v'$, thus it belongs to $B_2$ and can contribute to $b_2$ molecular orbital. Similarly, it can be shown that $O2p_z$ belongs to the $A_1$ irreducible representation.

Finally, consider the character table of the $C_{3v}$ symmetry group which is shown in Table 5

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Table 5: Character Table for the $C_{3v}$ Group

$C_{3v}$	$E$	$2C_3$	$3\sigma_v$	$h=6$
$A_1$	+1	+1	+1	$z$	$z^2$, $x^2+y^2$
$A_2$	+1	+1	$-1$
E	+2	$-1$	0	($x,y$)	($xz$, $x^2-y^2$), ($xz, yz$)

There are several new features of the Character Table 5 compared with the Character Tables 3 and 4.

First of all, the number of symmetry operations $h=6$ is now not equal to the number of possible irreducible representations (3). That is because, some of the symmetry operations in Table 5 can be combined into classes, which means that they are of the same type (for example, rotations) and can be transferred into one another by a symmetry operation of the same group. For instance, the 3-fold rotations $C_3^+$ and $C_3^-$ belong to the same class because the can be transformed to each other by reflection in the bisecting plane. Therefore, these two rotations are put to the same cell in Table 5. Also three vertical planes of mirror reflection $\sigma_v$, $\sigma_v'$, and $\sigma_v''$ belong to the same class because they can be transformed to each other by 3-fold rotation. All these mirror planes are put to another cell in Table 5.

There is an important theorem of group theory states that:

Number of symmetry species is equal to the number of classes.

There are three classes of symmetry operations in $C_{3v}$ group shown in the first line in Table 5 and, therefore, there are three symmetry species which are shown in the first column. It is seen that all elements of each symmetry class have the same symmetry characters.

Secondly, the symmetry species E in Table 5 is a double degenerate one. These species cannot be characterized simply by +1, or $-1$, as for non-degenerate case. As we know, the wavefunctions which belong to a degenerate vibration are neither symmetric, nor antisymmetric with respect to the symmetry operation of the group, but in general can be transformed as a linear combination of each other as

$$\Phi_{v1}' = d_{11}\Phi_{v1} + d_{12}\Phi_{v2} + d_{13}\Phi_{v3} + \cdots,$$

$$\Phi_{v2}' = d_{21}\Phi_{v1} + d_{22}\Phi_{v2} + d_{23}\Phi_{v3} + \cdots,$$ (18)

$$\Phi_{v3}' = d_{31}\Phi_{v1} + d_{32}\Phi_{v2} + d_{33}\Phi_{v3} + \cdots,$$

$$\cdots = \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots,$$

where the primed wavefunctions in the lhs are ones after the symmetry operation while the non-primed wavefunctions in the rhs are the initial ones. In case of a double-degenerate state the number of the wavefunctions and the number of equations in eq. (18) is of cause equal to two.

It can be shown, that for characterization of the behavior of the degenerate eigenfunctions under symmetry operations it is sufficient to label every symmetry operation with the value

$$\chi = d_{11} + d_{22} + d_{33} + \cdots$$ (19)

which is the sum of the diagonal expansion coefficients in the set of equations in eq. (18).

The values $\chi$ in eq. (19) (as well as $\lambda =\pm 1$ symmetric indices for non-degenerate species) are called characters of the irreducible representation. These characters are given in the third line in Table 5. As you can see the characters of the degenerate eigenfunctions are not limited by the values $\pm 1$, but can take other integer numbers including zero.

Note, that the character of identity operator $E$ is always equal to the degeneracy of the state. Therefore, for a $C_{3v}$ molecule any orbitals with a symmetry label $a_1$ and $a_2$ is non-degenerate, while a doubly degenerate pair of orbitals belong to $e$ representation. Because there is not characters greater than 2 in Table 5 we can assume that no triply degenerate orbitals can occur in any $C_{3v}$ molecule.

So far, we dealt with the symmetry classification of individual atomic orbitals. It is important to note that the same technique may be applied to the linear combinations of atomic orbitals which are used for building the molecular orbitals. This allows to classify the molecular energy states and molecular orbitals with respect to the symmetry transformations of the molecule.

As an example, we consider the linear combinations of electronic wavefunctions which belong to different representations in Table 5.

Particularly, for $NH_3$ case the combination

$$s_1 = s_a + s_b + s_c,$$ (20)

where $s_a$, $s_b$, and $s_c$ are $s$-orbitals of three hydrogen atoms, belongs to the species $a_1$.

The combinations

$$s_2 = -s_a + \frac{1}{2}(s_b + s_c)$$ (21)

$$s_3 = s_b-s_c$$

belongs to the doubly degenerate species $e$.

For proving this statement let us consider the transformation of the combinations in eq. (21) under $C_3^+$ and $\sigma_v$ symmetry operations of the group

Rotation $C_3^+$:

$$s_2' = -s_b + \frac{1}{2}(s_c+ s_a)$$ (22)

$$s_3' = s_c-s_a$$

This can be easily proved from eqs. (21) and (22) that

$$s_2' = -\frac{1}{2}s_2 - \frac{3}{4}s_3$$ (23)

$$s_3' = s_2 - \frac{1}{2}s_3$$

Reflection $\sigma_v$: (over the plane containing $N-H_a$ bond)

$$s_2' = -s_a + \frac{1}{2}(s_b+ s_c) = s_2$$ (24)

$$s_3' = s_c-s_b = -s_3$$

Similar expressions can be obtained for the symmetry operations $C_3^-$, $\sigma_v'$, and $\sigma_{v}''$. It is seen that the wavefunctions $s_2$ and $s_3$ are transformed as a linear combination of each other and thus span the species E.

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