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.
| Point Group | Schönfliess Symbol in adf | Irreducible representations in single-group symmetry | Irreducible representations in double-group symmetry |
| C1 | NOSYM | A | A1/2 |
| 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 A1.u A2.u E.u T1.u T2.u | E1/2.g U3/2.g E5/2.g 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 Delta.g Delta.u Phi.g Phi.u | J1/2.g J1/2.u J3/2.g J3/2.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 E1 E2 ... odd n: without B | A1/2 A1/2* |
| Cnh | C(NH), n must be 2 | even n: A.g B.g A.u B.u E1.g E1.u E2.g E2.u ... odd n: AA AAA EE1 EE2 ... EEE1 EEE2 ... | A1/2.g A1/2.g* A1/2.u A1/2.u* |
| Cnv | C(NV), n<9 | A1 A2 B1 B2 E1 E2 E3 ... odd n: without B1 and B2 | E1/2 E3/2 E5/2 ... for even n also: An/2 An/2* |
| Dn | D(N), n<9 | n=2: A B1 B2 B3 other: A1 A2 B1 B2 E1 E2 E3 ... odd n: without B1 B2 | E1/2 E3/2 ... for 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 (≠2): A1.g A2.g B1.g B2.g E1.g E2.g E3.g ... A1.u A2.u B1.u ... odd n: AA1 AA2 EE1 EE2 ... AAA1 AAA2 EEE1 EEE2 .... | even n: E1/2.g E1/2.u E3/2.g E3/2.u ... odd n: E1/2 E3/2 E5/2 ... |
| Dnd | D(ND), n<9 | n=2: A1 A2 B1 B2 E1 other: A1.g A2.g E1.g E2.g ... E(n-1)/2.g A1.u A2.u E1.u E2.u ... E(n-1)/2.u | even n: E1/2 E3/2 ... odd n: E1/2.g E1/2.u E3/2.g E3/2.u .... An/2.g An/2.u An/2.g* An/2.u* |
| Schönfliess symbols and the labels of the irreducible representations. | |||
Most labels are easily associated with the notation usually encountered in literature. Exceptions are AA, AAA, EE1, EEE1, EE2, EEE2, etc etera. 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 Care 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).
