8.8 Symmetry

MOLPRO can use Abelian point group symmetry only. For molecules with degenerate symmetry, an Abelian subgroup must be used -- e.g., $C_{2v}$ or $D_{2h}$ for linear molecules. The symmetry group which is used is defined in the integral input by combinations of the symmetry elements $x$, $y$, and $z$, which specify which coordinate axes change sign under the corresponding generating symmetry operation. It is usually wise to choose $z$ to be the unique axis where appropriate (essential for $C_2$ and $C_{2h}$). The possibilities in this case are shown in Table 1.

Normally, MOLPRO determines the symmetry automatically, and rotates and translates the molecule accordingly. However, explicit symmetry specification is sometimes useful to fix the orientation of the molecule or to use lower symmetries.

Table 1: The symmetry generators for the point groups

Generators	Point group
(null card)	$C_1$ (i.e. no point group symmetry)
X (or Y or Z)	$C_s$
XY	$C_2$
XYZ	$C_i$
X,Y	$C_{2v}$
XY,Z	$C_{2h}$
XZ,YZ	$D_2$
X,Y,Z	$D_{2h}$

The irreducible representations of each group are numbered 1 to 8. Their ordering is important and given in Tables 2 - 4. Also shown in the tables are the transformation properties of products of $x$, $y$, and $z$. $s$ stands for an isotropic function, e.g., $s$ orbital, and for these groups, this gives also the transformation properties of $x^2$, $y^2$, and $z^2$. Orbitals or basis functions are generally referred to in the format number.irrep, i.e. 3.2 means the third orbital in the second irreducible representation of the point group used.

Table 2: Numbering of the irreducible representations in $D_{2h}$

	$D_{2h}$
No.	Name	Function
1	$A_{g}$	$s$
2	$B_{3u}$	$x$
3	$B_{2u}$	$y$
4	$B_{1g}$	$xy$
5	$B_{1u}$	$z$
6	$B_{2g}$	$xz$
7	$B_{3g}$	$yz$
8	$A_{u}$	$xyz$

Table 3: Numbering of the irreducible representations in the four-dimensional groups

	$C_{2v}$		$C_{2h}$		$D_2$
No.	Name	Function	Name	Function	Name	Function
1	$A_1$	$s, z$	$A_g$	$s, xy$	$A$	$s$
2	$B_1$	$x, xz$	$A_u$	$z$	$B_3$	$x, yz$
3	$B_2$	$y, yz$	$B_u$	$x, y$	$B_2$	$y, xz$
4	$A_2$	$xy$	$B_g$	$xz, yz$	$B_1$	$xy$

Table 4: Numbering of the irreducible representations in the two-dimensional groups

	$C_{s}$		$C_{2}$		$C_{i}$
No.	Name	Function	Name	Function	Name	Function
1	$A^{'}$	$s, x, y, xy$	$A$	$s, z, xy$	$A_g$	$s, xy, xz, yz$
2	$A^{''}$	$z, xz, yz$	$B$	$x, y, xz, yz$	$A_u$	$x, y, z$