# 13 Symmetry

The Mrcc program can handle Abelian point group symmetry. The handling of symmetry can be controlled by keywords cmpgrp (see page 12) and symm (see page 12). In the following we give the character tables used by the program. The symmetry of electronic states can be specified by keyword symm using either the serial number of the irrep or its symbol. The serial number of an irrep is given by its position in the below tables as appropriate. To specify the state symmetry by the symbol of the irrep replace the superscripts in the irrep symbol by lowercase letters, e.g., give B2g for B${}_{2g}$. For the A${}^{{}^{\prime}}$ and A${}^{{}^{\prime\prime}}$ irreps of $C_{s}$ group use A’ and A", respectively (apostrophe and quotation mark).

Character table for the $C_{1}$ point group

$\;\;\;E$
A $1$ $x$, $y$, $z$, $R_{x}$, $R_{y}$, $R_{z}$,
$x^{2}$, $y^{2}$, $z^{2}$, $xy$, $xz$, $yz$

Character table for the $C_{i}$ point group

$\;\;\;E$ $\;\;\;i$
A${}_{1g}$ $1$ $1$ $R_{x}$, $R_{y}$, $R_{z}$, $x^{2}$, $y^{2}$, $z^{2}$, $xy$, $xz$, $yz$
A${}_{1u}$ $1$ $-1$ $x$, $y$, $z$

Character table for the $C_{s}$ point group

$\;\;\;E$ $\;\;\;\sigma_{h}$
A${}^{{}^{\prime}}$ $1$ $1$ $x$, $y$, $R_{z}$, $x^{2}$, $y^{2}$, $z^{2}$, $xy$
A${}^{{}^{\prime\prime}}$ $1$ $-1$ $z$, $R_{x}$, $R_{y}$, $yz$, $xz$

Character table for the $C_{2}$ point group

$\;\;\;E$ $\;\;C_{2}$
A $1$ $1$ $z$, $R_{z}$, $x^{2}$, $y^{2}$, $z^{2}$, $xy$
B $1$ $-1$ $x$, $y$, $R_{x}$, $R_{y}$, $yz$, $xz$

Character table for the $C_{2v}$ point group

$\;\;\;E$ $\;\;C_{2}$ $\;\;\;\sigma_{h}$ $\;\;\;\sigma_{v}$
A${}_{1}$ $1$ $1$ $1$ $1$ $z$, $x^{2}$, $y^{2}$, $z^{2}$
B${}_{1}$ $1$ $-1$ $1$ $-1$ $y$, $R_{x}$, $yz$
B${}_{2}$ $1$ $-1$ $-1$ $1$ $x$, $R_{y}$, $xz$
A${}_{2}$ $1$ $1$ $-1$ $-1$ $R_{z}$, $xy$

Character table for the $C_{2h}$ point group

$\;\;\;E$ $\;C_{2}(z)$ $\;\;\;i$ $\;\;\;\sigma_{h}$
A${}_{g}$ $1$ $1$ $1$ $1$ $R_{z}$, $x^{2}$, $y^{2}$, $z^{2}$, $xy$
B${}_{g}$ $1$ $-1$ $1$ $-1$ $R_{x}$, $R_{y}$, $xz$, $yz$
A${}_{u}$ $1$ $1$ $-1$ $-1$ $z$
B${}_{u}$ $1$ $-1$ $-1$ $1$ $x$, $y$

Character table for the $D_{2}$ point group

$\;\;\;E$ $\;C_{2}(z)$ $\;C_{2}(y)$ $\;C_{2}(x)$
A $1$ $1$ $1$ $1$ $x^{2}$, $y^{2}$, $z^{2}$
B${}_{1}$ $1$ $1$ $-1$ $-1$ $z$, $R_{z}$, $xy$
B${}_{2}$ $1$ $-1$ $1$ $-1$ $y$, $R_{y}$, $xz$
B${}_{3}$ $1$ $-1$ $-1$ $1$ $x$, $R_{x}$, $yz$

Character table for the $D_{2h}$ point group

$\;\;\;E$ $\;C_{2}(z)$ $\;C_{2}(y)$ $\;C_{2}(x)$ $\;\;\;i$ $\;\;\;\sigma_{xy}$ $\;\;\;\sigma_{xz}$ $\;\;\;\sigma_{yz}$
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${}_{3u}$ $1$ $-1$ $-1$ $1$ $-1$ $1$ $1$ $-1$ $x$