D2h E C2 C2' C2" i sh sv sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1 1 1 1 1 1 1 1 ... ... T...T ....... T...T...T ........... T...T...T...T B1g 1 1 -1 -1 1 1 -1 -1 ..T ... .T... ....... .T...T... ........... .T...T...T... B2g 1 -1 -1 1 1 -1 1 -1 .T. ... ..T.. ....... ..T...T.. ........... ..T...T...T.. B3g 1 -1 1 -1 1 -1 -1 1 T.. ... ...T. ....... ...T...T. ........... ...T...T...T. Au 1 1 1 1 -1 -1 -1 -1 ... ... ..... ...T... ......... ...T...T... ............. B1u 1 1 -1 -1 -1 -1 1 1 ... ..T ..... ..T...T ......... ..T...T...T ............. B2u 1 -1 -1 1 -1 1 -1 1 ... .T. ..... .T...T. ......... .T...T...T. ............. B3u 1 -1 1 -1 -1 1 1 -1 ... T.. ..... T...T.. ......... T...T...T.. ............. Symmetry of Rotations and Cartesian products Ag 2d+3g+4i+5k+6m x2−y2, z2, (x2−y2)2−4x2y2, z2(x2−y2), z4, x2(x2−3y2)2−y2(3x2−y2)2, z2((x2−y2)2−4x2y2), z4(x2−y2), z6 B1g R+d+2g+3i+4k+5m Rz, xy, xy(x2−y2), xyz2, xy(x2−3y2)(3x2−y2), xyz2(x2−y2), xyz4 B2g R+d+2g+3i+4k+5m Ry, xz, xz(x2−3y2), xz3, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz3(x2−3y2), xz5 B3g R+d+2g+3i+4k+5m Rx, yz, yz(3x2−y2), yz3, yz((5+2√5)x2−y2)((5−2√5)x2−y2), yz3(3x2−y2), yz5 Au f+2h+3j+4l xyz, xyz(x2−y2), xyz3 B1u p+2f+3h+4j+5l z, z(x2−y2), z3, z((x2−y2)2−4x2y2), z3(x2−y2), z5 B2u p+2f+3h+4j+5l y, y(3x2−y2), yz2, y((5+2√5)x2−y2)((5−2√5)x2−y2), yz2(3x2−y2), yz4 B3u p+2f+3h+4j+5l x, x(x2−3y2), xz2, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), xz2(x2−3y2), xz4 Notes: α The order of the D2h point group is 8, and the order of the principal axis (C2) is 2. The group has 8 irreducible representations. β The D2h point group is also known as Vh. The letter V derives from German ‘Vierergruppe’ (group of four) for the Klein four-group, to which D2 is isomorphic. γ The D2h point group is canonically generated by C2, C2′ and i. Another common choice picks the three mirror planes. Generally, any triple out of the 7 nontrivial elements will generate the group provided at least one has negative parity. δ The D2h group has three nonequivalent C2 axes. The one labelled “C2” is the principal axis (z in canonical orientation). The other two are perpendicular to the principal axis. By convention, the C2′ is the x axis and C2″ the y axis. ε The D2h group has three nonequivalent mirror planes. The one labelled σh is orthogonal to the principal C2 axis and thus corresponds to the xy plane. The other two contain the principal axis. By convention, the σv is the xz plane and the σd the yz plane. ζ The lowest nonvanishing multipole moment in D2h is 4 (quadrupole moment). η This is an Abelian point group (the commutative law holds between all symmetry operations). The D2h group is Abelian because it satisfies the sufficient condition to contain no axes of order higher than two. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. θ The naming of irreducible representations in this group as B1,B2,B3 is purely conventional. ι There are no symmetry elements of an order higher than 2 in this group. The symmetry-adapted Cartesian products in the table above are needlessly complicated; rather, any simple product will do. κ All characters are integers because the order of the principal axis is 1,2,3,4 or 6. Such point groups are also referred to as “crystallographic point groups”, as they are compatible with periodic lattice symmetry. There are exactly 32 such groups: C1,Cs,Ci,C2,C2h,C2v,C3,C3h,C3v,C4,C4h,C4v,C6,C6h,C6v,D2,D2d,D2h,D3,D3d,D3h,D4,D4h,D6,D6h,S4,S6,T,Td,Th,O,Oh.
| D1h | ||
| C2 C2v C2h D2 | D2h | D2d Ci |
| D3h | ||
| D4h |
This Character Table for the D2h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.