13 Symmetry

The Mrcc program can handle Abelian point group symmetry. The handling of symmetry can be controlled by keywords cmpgrp (see page cmpgrp) and symm (see page symm). 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 B2g{}_{2g}. For the A{}^{{}^{\prime}} and A′′{}^{{}^{\prime\prime}} irreps of CsC_{s} group use A' and A", respectively (apostrophe and quotation mark).

Character table for the C1C_{1} point group

E\;\;\;E
A 11 xx, yy, zz, RxR_{x}, RyR_{y}, RzR_{z},
x2x^{2}, y2y^{2}, z2z^{2}, xyxy, xzxz, yzyz

Character table for the CiC_{i} point group

E\;\;\;E i\;\;\;i
A1g{}_{1g} 11 11 RxR_{x}, RyR_{y}, RzR_{z}, x2x^{2}, y2y^{2}, z2z^{2}, xyxy, xzxz, yzyz
A1u{}_{1u} 11 -1-1 xx, yy, zz

Character table for the CsC_{s} point group

E\;\;\;E σh\;\;\;\sigma_{h}
A{}^{{}^{\prime}} 11 11 xx, yy, RzR_{z}, x2x^{2}, y2y^{2}, z2z^{2}, xyxy
A′′{}^{{}^{\prime\prime}} 11 -1-1 zz, RxR_{x}, RyR_{y}, yzyz, xzxz

Character table for the C2C_{2} point group

E\;\;\;E C2\;\;C_{2}
A 11 11 zz, RzR_{z}, x2x^{2}, y2y^{2}, z2z^{2}, xyxy
B 11 -1-1 xx, yy, RxR_{x}, RyR_{y}, yzyz, xzxz

Character table for the C2vC_{2v} point group

E\;\;\;E C2\;\;C_{2} σh\;\;\;\sigma_{h} σv\;\;\;\sigma_{v}
A1{}_{1} 11 11 11 11 zz, x2x^{2}, y2y^{2}, z2z^{2}
B1{}_{1} 11 -1-1 11 -1-1 yy, RxR_{x}, yzyz
B2{}_{2} 11 -1-1 -1-1 11 xx, RyR_{y}, xzxz
A2{}_{2} 11 11 -1-1 -1-1 RzR_{z}, xyxy

Character table for the C2hC_{2h} point group

E\;\;\;E C2(z)\;C_{2}(z) i\;\;\;i σh\;\;\;\sigma_{h}
Ag{}_{g} 11 11 11 11 RzR_{z}, x2x^{2}, y2y^{2}, z2z^{2}, xyxy
Bg{}_{g} 11 -1-1 11 -1-1 RxR_{x}, RyR_{y}, xzxz, yzyz
Au{}_{u} 11 11 -1-1 -1-1 zz
Bu{}_{u} 11 -1-1 -1-1 11 xx, yy

Character table for the D2D_{2} point group

E\;\;\;E C2(z)\;C_{2}(z) C2(y)\;C_{2}(y) C2(x)\;C_{2}(x)
A 11 11 11 11 x2x^{2}, y2y^{2}, z2z^{2}
B1{}_{1} 11 11 -1-1 -1-1 zz, RzR_{z}, xyxy
B2{}_{2} 11 -1-1 11 -1-1 yy, RyR_{y}, xzxz
B3{}_{3} 11 -1-1 -1-1 11 xx, RxR_{x}, yzyz

Character table for the D2hD_{2h} point group

E\;\;\;E C2(z)\;C_{2}(z) C2(y)\;C_{2}(y) C2(x)\;C_{2}(x) i\;\;\;i σxy\;\;\;\sigma_{xy} σxz\;\;\;\sigma_{xz} σyz\;\;\;\sigma_{yz}
Ag{}_{g} 11 11 11 11 11 11 11 11 x2x^{2}, y2y^{2}, z2z^{2}
B1g{}_{1g} 11 11 -1-1 -1-1 11 11 -1-1 -1-1 RzR_{z}, xyxy
B2g{}_{2g} 11 -1-1 11 -1-1 11 -1-1 11 -1-1 RyR_{y}, xzxz
B3g{}_{3g} 11 -1-1 -1-1 11 11 -1-1 -1-1 11 RxR_{x}, yzyz
Au{}_{u} 11 11 11 11 -1-1 -1-1 -1-1 -1-1
B1u{}_{1u} 11 11 -1-1 -1-1 -1-1 -1-1 11 11 zz
B2u{}_{2u} 11 -1-1 11 -1-1 -1-1 11 -1-1 11 yy
B3u{}_{3u} 11 -1-1 -1-1 11 -1-1 11 11 -1-1 xx