关注：

1) 结构优化过程中子群到高对称性超群的转变

2) crystal structure determination

If G is a group and H is a subgroup of G, then G is a supergroup of H.

If H is a maximal subgroup of G, then G is a minimal supergroup of H.

题记：

Since occupation of some special positions in one space group makes it equivalent to one of its supergroups,

a full relaxation of ions in alower symmetry phase would degenerate itself into (or equivalently relax to) a higher symmetry phase (Figs. 1).

参考网址：

程序

http://www.cryst.ehu.es/cryst/supergroups.html  

程序说明

http://www.cryst.ehu.es/cryst/super_theory.html

The crystallographic problem

  The problem of the determination of the supergroups of a given space group is of rather general interest. It is useful for search of overlooked symmetries in crystal structure determination, or in the analysis of successive phase transitions, where the introduction of a hypothetical (supergroup) parent phase can be important.

Another important application is related with the detection of pseudosymmetries in crystal structures as a method of predicting structural phase transitions at higher temperatures (Igartua, Aroyo, Perez-Mato, 1996).

There are few papers treating the supergroups of space groups in some detail (Koch, 1984). The existing listings of minimal supergroups of space groups are not complete as they provide only a list of those space groups G which contain a space group H as a maximal non-isomorphic subgroup

(International Tables for Crystallography, vol. A, 1992 (ITA)).

This information is in general not sufficient because it does not include all the possible supergroups of H isomorphic to G. The presented program determines all possible supergroups G_i of H which belong to the space group type of G.

SUPERGROUPS determines all possible supergroups G_i systematically inverting the data on maximal subgroups of space groups (International Tables for Crystallography, vol. A1 (ITA1)). This data has been prepared in CIF-like format in computer-readable form.

Following the procedure based in normalizers of space groups (Koch 1984, Wondratschek 1996) the program calculates all supergroups G_i > H, G_i ~ G.

Supergroup: S; Subgroup: G: G <S

The decomposition of S with respect to G:

S = G + q₂G + q₃G + ... + q_nG

Supergroups S⁽ⁱ⁾ of G, isomorphic to S:

S⁽ⁱ⁾ = a_i^-1 S a_i, S⁽ⁱ⁾ ~ S  

Decomposition of S⁽ⁱ⁾ with respect to G:

S⁽ⁱ⁾ = a_i^-1 G a_i +                   a_i^-1 q₂ G a_i +                  a_i^-1 q₃ G a_i + ... +                  a_i^-1 q_n G a_i =
= Gⁱ + q₂ⁱ Gⁱ +                  q₃ⁱ Gⁱ + ... +                  q_nⁱ Gⁱ,                   q_k^j =  a_j^-1 q_k a_j

S⁽ⁱ⁾ decomposition with respect to G:

S⁽ⁱ⁾ = G + q_i,2 G + q_i,3 G + ... +  q_i,n G

==> TWO sets of supergroups S⁽ⁱ⁾ of G:

• Supergroups conjugated under N(G) (the normalizer of G)
  S⁽ⁱ⁾ = a_i^-1 S a_i, a_i in N(G)

  
  

• Supergroups not conjugated under N(G)
  S⁽ⁱ⁾ = a_i^-1 S a_i, a_i not in N(G)

http://www.ncbi.nlm.nih.gov/pubmed/11326116