O$_4^+$ Has $D_{2h}$ Symmetry

Our example will the O$_4^+$ cation, which has $D_{2h}$ point group symmetry, as shown in Figure 1. The Cartesian coordinates of this molecule are given in Table 1, and the character table for $D_{2h}$ is given in Table 2. From the table, we can see that there are eight distinct symmetry operations for this point group: the identity ($E$), three different $C_2$ rotation axes, a center of inversion ($i$), and three mirror planes ($\sigma$). You can easily verify that O$_4^+$ possesses all of these symmetry properties. These symmetry operations form the columns of the table. There are also eight rows, or irreducible representations, labeled $A_g$, $B_{1g}$, $\ldots$, $B_{3u}$. The 1's and -1's in the table indicate whether the irreducible representation (or irrep, for short) is symmetric or antisymmetric for that symmetry operation.

Figure 1: O$_4^+$ Cation

Table 1: O$_4^+$ Cartesian Coordinates

Table 2: Character Table for Point Group $D_{2h}$

$D_{2h}$	$E$	$C_2(z)$	$C_2(y)$	$C_2(x)$	$i$	$\sigma(xy)$	$\sigma(xz)$	$\sigma(yz)$
$A_g$	1	1	1	1	1	1	1	1		$x^2,y^2,z^2$
$B_{1g}$	1	1	-1	-1	1	1	-1	-1	$R_z$	$xy$
$B_{2g}$	1	-1	1	-1	1	-1	1	-1	$R_y$	$xz$
$B_{3g}$	1	-1	-1	1	1	-1	-1	1	$R_x$	$yz$
$A_{u}$	1	1	1	1	-1	-1	-1	-1
$B_{1u}$	1	1	-1	-1	-1	-1	1	1	$z$
$B_{2u}$	1	-1	1	-1	-1	1	-1	1	$y$
$B_{3u}$	1	-1	-1	1	-1	1	1	-1	$x$