Schönfliess symbols and symmetry labels

A survey of all point groups that are recognized by ADF is given below. The table contains the Schönfliess symbols together with the names of the subspecies of the irreducible representations as they are used internally by ADF. The subspecies names depend on whether single-group or double-group symmetry is used. Double-group symmetry is used only in relativistic spin-orbit calculations.

Note that for some input of TDDFT (Response) calculations, other conventions apply for the subspecies. This is explicitly mentioned in the discussion of that application.

Table 16 Schönfliess symbols and the labels of the irreducible representations.
Point Schönfliess Irreducible representations Irreducible representations
Group Symbol in ADF in single-group symmetry in double-group symmetry
R3 ATOM s p d f s1/2 p1/2 p3/2 d3/2 d5/2 f5/2 f7/2
Td T(D) A1 A2 E T1 T2 E1/2 U3/2 E5/2
Oh O(H) A1.g A2.g E.g T1.g T2.g E1/2.g U3/2.g E5/2.g
    A1.u A2.u E.u T1.u T2.u E1/2.u U3/2.u E5/2.u
C∞v C(LIN) Sigma Pi Delta Phi J1/2 J3/2 J5/2 J7/2
D∞h D(LIN) Sigma.g Sigma.u Pi.g Pi.u J1/2.g J1/2.u J3/2.g J3/2.u
    Delta.g Delta.u Phi.g Phi.u J5/2.g J5/2.u J7/2.g J7/2.u
Ci C(I) A.g A.u A1/2.g A1/2.u
Cs C(S) AA AAA A1/2 A1/2*
Cn C(N), n must be 2 A B A1/2 A1/2*
Cnh C(NH), n must be 2 A.g B.g A.u B.u A1/2.g A1/2.g* A1/2.u A1/2.u*
Cnv C(NV), n<9 A1 A2 B1 B2 E1 E2 E3 … E1/2 E3/2 E5/2 …
    odd n: without B1 and B2 odd n also: An/2 An/2*
Dn D(N), n<9 n=2: A B1 B2 B3 E1/2 E3/2 …
    other: A1 A2 B1 B2 E1 E2 E3 …  
    odd n: without B1 B2 odd n also: An/2 An/2*
Dnh D(NH), n<9 n=2: A.g B1.g B2.g B3.g A.u B1.u B2.u B3.u even n: E1/2.g E1/2.u
    other: A1.g A2.g B1.g B2.g E1.g E3/2.g E3/2.u …
    E2.g E3.g … A1.u A2.u B1.u …  
    odd n: AA1 AA2 EE1 EE2 … odd n: E1/2 E3/2 E5/2 …
    AAA1 AAA2 EEE1 EEE2 ….  
Dnd D(ND), n<9 even n: A1 A2 B1 B2 E1 … even n: E1/2 E3/2 …
    odd n: A1.g A2.g E1.g E2.g … E(n-1)/2.g odd n: E1/2.g E1/2.u E3/2.g E3/2.u …
    A1.u A2.u E1.u E2.u … E(n-1)/2.u An/2.g An/2.u An/2.g* An/2.u*

Most labels are easily associated with the notation usually encountered in literature. Exceptions are AA, AAA, EE1, EEE1, EE2, EEE2, etcetera. They stand for A’, A’‘, E1’, E1’‘, and so on. The AA, etc. notation is used in ADF to avoid using quotes in input files in case the subspecies names must be referred to.

The symmetry labeling of orbitals may depend on the choice of coordinate system. For instance, B1 and B2 representations in Cnv are interchanged when you rotate the system by 90 degrees around the z-axis so that x-axis becomes y-axis and vice-versa (apart from sign).

Labels of the symmetry subspecies are easily derived from those for the irreps. For one-dimensional representations they are identical, for more-dimensional representations a suffix is added, separated by a colon: For the two- and three-dimensional E and T representations the subspecies labels are obtained by adding simply a counting index (1, 2, 3) to the name, with a colon in between; for instance, the EE1 irrep in the Dnh pointgroup has EE1:1 and EE1:2 subspecies. The same holds for the two-dimensional representations of C∞v and D∞h . For the R3 (atom) point group symmetry the subspecies are p:x, p:y, p:z, d:z2, d:x2-y2, etc.

All subspecies labels are listed in the Symmetry section, very early in the ADF output. To get this, perform a quick run of the molecule using the STOPAFTER key (for instance: stopafter config).

Molecular orientation requirements

ADF requires that the molecule has a specific orientation in space, as follows:

  • The origin is a fixed point of the symmetry group.
  • The z-axis is the main rotation axis, xy is the \(\sigma\)h -plane (axial groups, C(s)).
  • The x-axis is a C2 axis (D symmetries).
  • The xz-plane is a \(\sigma\)v -plane (Cnv symmetries).
  • In Td and Oh the z-axis is a fourfold axis (S4 and C4 , respectively) and the (111)-direction is a threefold axis.

If the user-specified symmetry equals the true symmetry of the nuclear frame (including electric field and point charges) the program will adapt the input coordinates to the above requirements, if necessary. If no symmetry has been specified at all ADF assumes you have specified the symmetry of the nuclear frame, accounting for any fields. If a subgroup has been specified for the molecular symmetry the input coordinates will be used as specified by the user. If a Z-matrix input is given this implies for the Cartesian coordinates: first atom in the origin, second atom on the positive x-axis, third atom in the xy-plane with positive y value.