compu e p og ams
1650 h ps://doi.o g/10.1107/S1600576724008070 J. Appl. C ys . (2024). 57, 1650–1666
ISSN 1600-5767
Recei ed 7 Feb ua y 2024
Accep ed 15 Augus 2024
Edi ed by T. J. Sa o, Tohoku Uni e si y, Japan
Keywo ds: g oup–subg oup ela ions; pseudo-
symme y; dis o ed s uc u es; g oup heo y
applica ions; compu a ional c ys allog aphy.
Published unde a CC BY 4.0 licence
SUBGROUPS: a compu e ool a he Bilbao
C ys allog aphic Se e o he s udy o pseudo-
symme ic o dis o ed s uc u es
Em e S. Tasci,
a
* Luis Elco o,
b
J. Manuel Pe ez-Ma o,
c
Gemma de la Flo
d
and Mois
I. A oyo
b
a
Physics Enginee ing Depa men , Hace epe Uni e si y, Anka a, Tu
¨ kiye,
b
Depa amen o de Fı
´sica, Facul ad de Ciencia y
Tecnologı
´a, Uni e sidad del Pais Vasco UPV/EHU, Apa ado 644, Bilbao, Spain,
c
Facul ad de Ciencia y Tecnologı
´a,
Uni e sidad del Pais Vasco UPV/EHU, Apa ado 644, Bilbao, Spain, and
d
Ins i u e o Applied Geosciences, Ka ls uhe
Ins i u e o Technology, Ka ls uhe, Ge many. *Co espondence e-mail: [email p o ec ed]
SUBGROUPS is a ee online p og am a he Bilbao C ys allog aphic Se e
(h ps://www.c ys .ehu.es/). I pe mi s he explo a ion o all possible symme ies
esul ing om he dis o ion o a highe -symme y pa en s uc u e, p o ided
ha he ela ion be ween he la ices o he dis o ed and pa en s uc u es is
known. The p og am calcula es all he subg oups o he pa en space g oup
which comply wi h his ela ion. The equi ed minimal inpu is he space-g oup
in o ma ion o he pa en s uc u e and he ela ion o he uni cell o he
dis o ed o pseudo-symme ic s uc u e wi h ha o he pa en s uc u e.
Al e na i ely, he wa e ec o (s) obse ed in he di ac ion da a cha ac e izing
he dis o ion can be in oduced. Addi ional condi ions can be added, including
il e s ela ed o space-g oup ep esen a ions. The p og am p o ides e y
de ailed in o ma ion on all he subg oups, including g oup–subg oup hie a chy
g aphs. I a C ys allog aphic In o ma ion F amewo k (CIF) ile o he pa en
high-symme y s uc u e is uploaded, he p og am gene a es CIF iles o he
pa en s uc u e desc ibed unde each o he chosen lowe symme ies. These
CIF iles may hen be used as s a ing poin s o he e inemen o he dis o ed
s uc u e unde hese possible symme ies. They can also be used o densi y
unc ional heo y calcula ions o o any o he ype o analysis. The powe and
e iciency o he p og am a e illus a ed wi h a ew examples.
1. In oduc ion
The de e mina ion o a dis o ed o pseudo-symme ic
commensu a e s uc u e can be qui e challenging. In such
ypes o s uc u e he di ac ion da a can be quan i a i ely
explained o a good app oxima ion by a s uc u al model o
highe symme y, which we shall call he pa en s uc u e. The
small symme y-b eaking de ia ions om his model gene ally
only in oduce e y weak addi ional ea u es ha can mani es
hemsel es in a wide ange o p ope ies which could be used
o dis inguish he ac ual symme y o he c ys al bu , alas,
hese may be qui e di icul o esol e and assess. Simila
p oblems can happen in densi y unc ional heo y (DFT)
calcula ions when ying o de e mine he g ound s a e o a
ma e ial which de ia es sligh ly om a pa en s uc u e o
highe symme y. In ei he o hese wo cases he pa en
s uc u e may al eady be known as a i ual idealized
a angemen o he amily o ma e ials o which he in es i-
ga ed ma e ial belongs, o i may be a eal s uc u e co e-
sponding o a di e en phase o he same ma e ial whe e he
symme y-b eaking dis o ion is no p esen . I a s a ing
model o he pa en s uc u e is a ailable (which is he case
o hese kinds o si ua ion), as he de ia ions om his model
a e small, he s uc u e de e mina ion can be educed o a i
o he di ac ion da a h ough a e inemen p ocess, o in he
case o DFT calcula ions o an ene gy minimiza ion a ound
he pa en s uc u e in con igu a ion space. Fo hese
p ocesses i is, howe e , con enien , o e en in some cases
necessa y, o make a p io assump ion on some speci ic space-
g oup symme y o he in es iga ed s uc u e, which by
de ini ion should be desc ibed by a subg oup o ha asso-
cia ed wi h he pa en s uc u e. As a i s s ep, i is hen
desi able o enume a e sys ema ically all possible subg oups
o he pa en space g oup which may be consis en wi h he
di ac ion da a and should be checked. We p esen he e he
online p og am SUBGROUPS, eely accessible a he Bilbao
C ys allog aphic Se e (h ps://www.c ys .ehu.es/) (A oyo e
al., 2006a,b, 2011), which has been de eloped o pe o m his
p elimina y ask. The p og am enume a es all possible
symme ies as subg oups o he pa en space g oup, wi h
op ional comp ehensi e in o ma ion abou hem and hei
g oup–subg oup ela ionships. F om a p ac ical poin o iew,
once he lis o possible non-equi alen subg oup symme ies
is p o ided, he p og am op ionally gene a es app op ia e
C ys allog aphic In o ma ion F amewo k (CIF) iles o any
inpu pa en s uc u e wi h i s symme y educed o each o
hese subg oups.
Some o he ea u es o he p esen p og am a e simila o
hose in ISODISTORT (S okes e al., 2016b) and/o in
ISOSUBGROUP (Campbell e al., 2006) om he
ISOTROPY So wa e Sui e (S okes e al., 2016a). Howe e ,
he p esen p og am app oaches he p oblem in a di e en
way, as space-g oup ep esen a ions a e only conside ed
op ionally once he se o possible subg oups has been
calcula ed. SUBGROUPS also has op ional ou pu s, including
g aphs, which can be highly use ul as complemen a y in o -
ma ion. In he nex wo sec ions he main ea u es o he
p og am and i s di e en ou pu s a e explained. Some
examples o applica ion a e hen shown.
2. The p og am SUBGROUPS
The p og am SUBGROUPS is a ailable in he G oup–
Subg oup Rela ions o Space G oups sec ion o he Bilbao
C ys allog aphic Se e (BCS).
The ma hema ical p ocedu es and algo i hms used by he
p og am a e explained in de ail in Appendix A, whe eas in he
ex we mainly ocus on he usage o he p og am and ypes o
possible applica ions.
2.1. Basic inpu and ou pu
The inpu o he p og am SUBGROUPS may seem some-
wha complex a i s glance, bu i can be di ided in o wo
pa s: (i) a minimal inpu equi ed o un he p og am, and (ii)
a se o op ional il e s (see Sec ion 3), based on di e en
c i e ia, which can e ine and na ow down he esul s
ob ained in (i). As minimal inpu , SUBGROUPS only equi es
he pa en space g oup and he ela ion o he la ice o he
in es iga ed s uc u e wi h ha o he pa en s uc u e o be
speci ied. The de e mina ion o his la ice ela ion om he
di ac ion da a o he s uc u e is usually a he s aigh -
o wa d. The symme y o he dis o ed s uc u e, as i is
necessa ily desc ibed by a subg oup o he pa en space g oup,
mus ha e a la ice which, apa om a possible s ain, mus be
ei he he same la ice o a subla ice. I s uni cell is hen
de ined wi h espec o he pa en uni cell as a supe cell,
which gene a es a subse o he pa en la ice ansla ions. In
he simples case whe e he pa en la ice is man ained in he
dis o ed s uc u e, his supe cell coincides wi h he pa en
uni cell.
The pa en space g oup is in oduced using i s se ial numbe
acco ding o In e na ional Tables o C ys allog aphy, Vol. A
(hence o h e e ed o as ITA) (A oyo, 2016), and he
p og am assumes ha he space g oup and i s uni cell a e o
be conside ed in he s anda d/de aul se ing used in he BCS.
This se ing is he one ound in ITA, bu in he case o space
g oups which ha e mo e han one desc ip ion in ITA, a ixed
choice among hem is made.
1
The basis ec o s which de ine
he supe cell o he dis o ed s uc u e a e in oduced as linea
combina ions o he basis ec o s o he con en ional uni cell
o he pa en space g oup, wi h he possibili y o adding some
cen ing.
Al e na i ely, he la ice ela ion be ween he wo s uc-
u es can be in oduced in ecip ocal space, indica ing he
p ima y modula ion wa e ec o o ec o s p esen in he
dis o ion. In he simples case, i he la ice is main ained, his
modula ion ec o should be null. In many cases i can be a
single non-ze o wa e ec o o a se o hem (symme y ela ed
o no by he ope a ions o he pa en space g oup). As he
p og am is only in ended o commensu a e s uc u es, he
inpu wa e ec o s should be commensu a e and desc ibed
wi h espec o he con en ional ecip ocal uni cell o he
pa en space g oup. The subg oups sough mus be compa ible
wi h hese modula ion wa e ec o s such ha he p oduc o
each modula ion ec o wi h he ansla ions o ming he
subg oup mus be an in ege . As we will see in Sec ion 3.4, his
al e na i e o m o in oducing he la ice o he dis o ed
s uc u e is especially con enien i one wan s o es ic he
possible symme ies, ollowing Landau heo y, o hose ha
can esul om a dis o ion ans o ming acco ding o one o
mo e speci ic i educible ep esen a ions o he pa en space
g oup.
The p og am calcula es and lis s all he possible subg oups
o he pa en space g oup which ha e he ansla ion subg oup
de ined by he inpu supe cell. The subg oups a e classi ied
acco ding o conjugacy classes wi h espec o he pa en space
g oup, and he i s ou pu only lis s one subg oup (chosen
a bi a ily) as he ep esen a i e o each conjugacy class ( he
p ocess is explained in de ail in Sec ion 2.2). This i s lis can
he e o e be conside ed as an enume a ion o he conjugacy
classes desc ibing all possible dis inc compa ible symme ies.
compu e p og ams
J. Appl. C ys . (2024). 57, 1650–1666 Em e S. Tasci e al. �SUBGROUPS ool a he BCS 1651
1
Fo space g oups wi h mo e han one desc ip ion in ITA, he ollowing
se ings a e chosen as s anda d in he BCS: unique axis b se ing, cell choice 1
o monoclinic space g oups, hexagonal axes se ing o hombohed al space
g oups, and o igin choice 2 (o igin a 1) o cen osymme ic space g oups
wi h mo e han one con en ional choice o he o igin.
All subg oups wi hin a conjugacy class a e physically
equi alen , i.e. hey can be associa ed wi h physically
equi alen domain-like dis o ed s uc u es which a e ela ed
by he los ope a ions. The e o e, o he pu pose o
enume a ing dis inc possible symme ies o he in es iga ed
s uc u e, a single subg oup o each conjugacy class, as shown
in his i s lis , is su icien . Howe e , i he p og am is used o
iden i y he ela ion be ween a pa en s uc u e and a speci ic
known dis o ed s uc u e, hen he space g oup used o he
desc ip ion o he dis o ed s uc u e may no be he one
chosen by he p og am as ep esen a i e o he conjugacy
class. In his case, he app op ia e subg oup mus be ound
wi hin he lis o conjuga e subg oups, which can also be
p o ided by he p og am (see Sec ion 2.2).
Fig. 1 shows as an example he lis o conjugacy classes
p o ided by he p og am o he pa en space g oup P421m
(No. 113) and a supe cell de ined by 2a, 2b,cwi h espec o
he pa en uni cell. Each lis ed subg oup is unambiguously
de ined by i s space-g oup ype in he second column and he
choice o uni cell and o igin (shown in he hi d column)
which would ans o m he ope a ions o his subg oup o he
pa en space g oup o he s anda d se ing o he indica ed
space-g oup ype. This ans o ma ion is desc ibed by a
ma ix–column pai (P,p) and consis s o wo pa s: a linea
pa Pgi en by a (3 �3) ma ix, and an o igin shi p= (p
1
,
p
2
,p
3
) gi en by a (3 �1) column ec o . This ans o ma ion
(P,p) is de ined wi h espec o he uni cell (a
p
,b
p
,c
p
) and
o igin O
p
o he pa en space g oup in he ollowing o m:
ðas;bs;csÞ ¼ ðap;bp;cpÞP;
Os¼Opþp1apþp2bpþp3cp;
whe e (a
s
,b
s
,c
s
) and O
s
a e he uni -cell basis ec o s and
o igin, espec i ely, o which he subg oup ope a ions ake
he s anda d/de aul o m a ailable in he BCS and in ITA.
This means ha he ans o med basis ec o s a e de e mined
by he column coe icien s o he ma ix P(no by i s ow
coe icien s). This ans o ma ion o he s anda d se ing is in
gene al no unique and he p og am jus makes a a he
a bi a y choice among he possible ones. He eina e , speci ic
ans o ma ions (P,p) a e o en w i en in he o m (P
11
a+
P
21
b+P
31
c,P
12
a+P
22
b+P
32
c,P
13
a+P
23
b+P
33
c;p
1
,p
2
,p
3
).
We s ess ha he space-g oup symbol in he second column
in Fig. 1 is in gene al insu icien o de ine he subg oup. In
mos cases he ans o ma ion (P,p) is also necessa y o
elimina e any ambigui y. I is he applica ion o he in e se o
his ans o ma ion o he ope a ions o he space-g oup ype,
exp essed in i s s anda d se ing, which yields he se o
ope a ions o he pa en space g oup (in i s s anda d se ing)
cons i u ing he de ined subg oup. In ac , as shown in Fig. 1,
he e can be di e en subg oups belonging o di e en
conjugacy classes which ha e he same space-g oup ype, and
i is only he ans o ma ion lis ed in he hi d column ha
dis inguishes hem.
The p og am can depic he g oup–subg oup hie a chy
among he lis ed symme ies in he o m o a g aph, as shown
in Fig. 2 o he lis o subg oups in Fig. 1. We s ess ha his
g oup–subg oup g aph ep esen s g oup–subg oup ela ions
be ween unspeci ied subg oups belonging o he co esponding
compu e p og ams
1652 Em e S. Tasci e al. �SUBGROUPS ool a he BCS J. Appl. C ys . (2024). 57, 1650–1666
Figu e 1
Possible symme ies, as ob ained wi h SUBGROUPS, o a dis o ed s uc u e ha ing P421m(No. 113) as i s pa en space g oup and wi h a p imi i e uni
cell app oxima ely gi en by 2a, 2b,cwi h espec o he pa en uni cell. Only a ep esen a i e subg oup o each conjugacy class o subg oups is lis ed.
See he ex o u he explana ions abou each column in his able.
conjugacy classes. The e o e, in gene al hey do no imply a
g oup–subg oup ela ion be ween he speci ic subg oups ha
ha e been chosen as ep esen a i es in he accompanying lis .
I will be shown in Sec ion 2.2 ha de ailed g aphs o g oup–
subg oup ela ions be ween speci ic subg oups can be
ob ained i he subg oups wi hin each conjugacy class a e
lis ed. SUBGROUPS gene a es all hese g aphs using he
open g aph isualiza ion sys em G aph iz (Gansne & No h,
2000).
The ou h column in Fig. 1 indica es he subg oup index
wi h espec o he pa en g oup, i.e. he a io be ween he
numbe o ope a ions in he pa en space g oup and hose in
he subg oup (e en hough he numbe o ope a ions in he
g oups can be in ini e, hei a io is s ill well de ined as he
numbe o cose s in he decomposi ion o he pa en space
g oup wi h espec o he subg oup). This subg oup index
gi es he numbe o domain s a es ha can be expec ed in a
dis o ed phase wi h his symme y. I is shown decomposed as
he p oduc o wo ac o s, he i s one ela ing he wo la ices
(klassengleich index) and he second one he wo poin g oups
( ansla ionengleich index). The klassengleich index is de e -
mined by he la ice ela ion in oduced by he use and
he e o e is he same o all lis ed subg oups. The klassen-
gleich index de e mines he numbe o dis inc domain s a es
ela ed by los la ice ansla ions (no dis inguishable in
di ac ion expe imen s), while he ansla ionengleich index
gi es he numbe o dis inc o ien a ion domain s a es asso-
cia ed wi h los o a ion, e lec ion, o o-in e sion o in e sion
symme y ope a ions.
2.2. De ailed desc ip ion o he subg oups o ming each
conjugacy class
The i s ou pu o he p og am includes a link o de ailed
in o ma ion on each conjugacy class ( i h column in Fig. 1).
The subg oups belonging o he conjugacy class o which he
chosen subg oup is a ep esen a i e a e enume a ed. As an
example, Fig. 3 shows he lis ob ained wi h his op ion o he
conjugacy class wi h space-g oup ype Cm, which is lis ed in
Fig. 1. I is e y impo an o s ess ha , in gene al, his lis
may no include he whole se o subg oups wi hin he
conjugacy class, as i is es ic ed o he conjuga e subg oups
ha a e compa ible wi h he la ice ela ion in oduced in he
inpu . Conjuga e subg oups belonging o he same conjugacy
class, bu wi h di e en o ien a ions such ha hei supe cell
does no coincide wi h ha o he inpu , o wi h symme y-
ela ed o a ed modula ion wa e ec o (s) di e en om hose
in oduced, a e no lis ed by he p og am.
Inspec ing he ans o ma ion o he s anda d se ing o
each subg oup in Fig. 3, one can see he o ien a ion o i s
monoclinic axis wi h espec o he pa en uni cell and he
posi ion o i s s anda d o igin, i.e. he posi ion o he mi o
plane wi hin he pa en s uc u e. Thus he subg oups Cm a e
dis inguished by ei he he loca ion o he p ese ed mi o
compu e p og ams
J. Appl. C ys . (2024). 57, 1650–1666 Em e S. Tasci e al. �SUBGROUPS ool a he BCS 1653
Figu e 2
A g oup–subg oup g aph o he symme ies shown in Fig. 1. The g aph
e e s o conjugacy classes o subg oups a he han speci ic subg oups.
The g aph can be gene a ed wi h o wi hou he nume ical labels gi en o
he subg oups in he lis shown in Fig. 1 ( i s column). The de aul op ion
wi h hese nume ical labels is shown he e.
Figu e 3
A lis o subg oups o space g oup P421m(No. 113) belonging o he conjugacy class o subg oups o ype Cm, lis ed in he hi d ow o Fig. 1. The i s
subg oup is he one used as a ep esen a i e in he lis o conjugacy classes in Fig. 1. See he ex o an explana ion o each column.
plane and/o i s o ien a ion. Two subg oups keep he mi o
plane pe pendicula o he di ec ion [110], while he o he wo
subg oups keep he mi o plane pe pendicula o he di ec-
ion ½110�, bo h di ec ions being symme y equi alen in he
pa en space g oup. A column wi h in o ma ion on he index
o he subg oup is also p esen in his able, bu in his case i is
edundan , as he index is necessa ily he same o all conju-
ga e subg oups.
The ou pu shown in Fig. 3, simila o ha in Fig. 1, includes
some op ions. In he i h column (‘Symme y ope a ions’),
he e a e wo op ions ha allow iewing he se o symme y
ope a ions o ming each o he lis ed subg oups, ei he in plain
ex o ma o in ma ix o m. Fig. 4 shows as an example he
lis o ope a ions ob ained o he i s subg oup in Fig. 3 when
he ‘Ma ix o m’ is selec ed.
Once a conjugacy class is lis ed, he g oup–subg oup ela-
ionship o each speci ic subg oup wi hin he class, including
bo h subg oups and supe g oups, can be ob ained using he
op ional bu ons in he column en i led ‘Se o subg oups’
(Fig. 3). Figs. 5 and 6 show how his in o ma ion is gi en o
compu e p og ams
1654 Em e S. Tasci e al. �SUBGROUPS ool a he BCS J. Appl. C ys . (2024). 57, 1650–1666
Figu e 4
Symme y ope a ions (o gene al posi ions) o he subg oup wi h space-g oup ype Cm and lis ed as 3.1 in he i s ow o Fig. 3, as ob ained when
clicking in he column ‘Symme y ope a ions’ (‘Ma ix o m’). Only one ope a ion is lis ed, as a ep esen a i e, o each se o ope a ions di e ing by
la ice ansla ions o he subg oup. In his simple case he lis is educed o wo ope a ions. They a e desc ibed in di e en o ma s, including he Sei z
no a ion, and a e gi en bo h in he basis gene a ing he la ice o he subg oup (le -hand columns) and in he pa en uni -cell basis ( igh -hand columns).
In his la e case hey can be iden i ied as ope a ions o he pa en space g oup in i s s anda d se ing.
Figu e 5
A g oup–subg oup g aph showing he supe g oups and subg oups o one
o he Cm subg oups o P421m(No. 113), namely he one lis ed as 3.1 in
Fig. 3. The nume ical labels a e hose iden i ying he subg oups in he
lis ings p o ided o each conjugacy class.
Figu e 6
A lis o he subg oups appea ing in he g aph o Fig. 5, as ob ained when clicking on he bu on ‘Lis o subg oups’ o he subg oup indexed 3.1 in he
ou pu shown in Fig. 3.
he case o he subg oup numbe ed 3.1 in Fig. 3. The g oup–
subg oup hie a chy is depic ed g aphically by clicking on
‘G aph o subg oups’ (Fig. 5). This subg oup has wo dis inc
supe g oups o ype Cmm2. Full in o ma ion on he subg oups
p esen in he g aph (including hei unambiguous de ini ion)
is p o ided when clicking on ‘Lis o subg oups’ (Fig. 6). As in
he p e ious lis ing o Fig. 3, he ou pu includes addi ional
op ions o u he in o ma ion.
2.3. I educible ep esen a ions o he pa en space g oup
compa ible wi h each subg oup
The bu on ‘Ge i eps’, which is gene ally a ailable o
each subg oup in he lis ings p o ided by he p og am (see
Figs. 1, 3 and 6), is a di ec link o he p og am Ge _i eps, also
a ailable as a s andalone online p og am in he BCS. By
calling his p og am o one pa icula subg oup one ge s all
he i educible ep esen a ions (i eps) o he pa en g oup
which a e compa ible wi h he chosen subg oup (de ails o he
calcula ions and ma hema ical de ini ions o he e ms used
he ein can be ound in Appendix A). These a e he i eps
which can cha ac e ize he deg ees o eedom ha , in
acco dance wi h he Von Neumann p inciple, a e symme y
allowed, and he e o e hey a e necessa ily se ee in a
dis o ed s uc u e wi h i s symme y desc ibed by his
subg oup. The p og am p o ides no only he i eps bu also
he subspace o di ec ion equi ed wi hin he i ep space and
he iso opy subg oup (o epike nel) associa ed wi h his i ep
and di ec ion. In he case o complex i eps, he p og am
conside s hei combina ion in o physically ( eal) i educible
ep esen a ions.
The i ep labels used by he p og am a e hose o he i ep
abula ions a ailable in he BCS and in he ISOTROPY
So wa e Sui e. This is he CDML no a ion [C acknell, Da ies,
Mille and Lo e; C acknell (1979)] which is used by all
p og ams in hese wo web acili ies. In he case o physically
i educible ep esen a ions, he i ep labels o he wo i eps
being combined a e pu oge he o o m a single label.
Fig. 7 shows he ou pu ob ained by calling Ge _i eps in he
case o he subg oup o ype Cm numbe ed 3.1 in Fig. 3. The
ac ual i ep ma ices which a e conside ed o he desc ip ion
o he ele an i eps can be consul ed by clicking on he
co esponding bu on ‘ma ices o he i eps’ in he las
column o he able. This is a di ec link o he da abase
REPRESENTATIONS SG, also a ailable in he BCS, whe e
he ma ix o m ha is being used o he i eps can be
e ie ed. Al hough he CDML i ep labels a e he same as
hose in he ISOTROPY So wa e Sui e, he speci ic ma ix
o m o he i eps conside ed in REPRESENTATIONS SG
may be di e en . The e o e i is impo an o s ess ha he
o de pa ame e di ec ion in he second column o he able in
Fig. 7, which depends on he ma ix choice o he i ep, is no
necessa ily he same as he one ha may be ob ained using he
p og ams o he ISOTROPY So wa e Sui e.
In he hi d column o Fig. 7 one can consul he iso opy
subg oup associa ed wi h each o he lis ed compa ible i eps.
These i ep iso opy subg oups o he pa en space g oup, also
speci ied by a space-g oup ype and a ans o ma ion (P,p) o
i s s anda d se ing (gi en in a sho -hand no a ion), a e he
symme ies ha would only esul om he p esence o a
dis o ion in he s uc u e, ans o ming acco ding o he
co esponding i ep ( es ic ed o he indica ed subspace). By
de ini ion, all hese subg oups mus be supe g oups o he
ac ual subg oup being analysed, o coincide wi h i . In he
la e case his symme y b eak can be he esul o a phase
ansi ion ul illing he Landau heo y condi ion (Landau &
Li shi z, 2013; Cowley, 1980) o a single i ep desc ibing he
ans o ma ion p ope ies o i s o de pa ame e . Dis o ed
s uc u es e y o en comply wi h his Landau assump ion,
and he e o e subg oups ha can be eached by he onse o a
single i ep and appea in hei Ge _i eps ou pu as he
iso opy subg oup o one o he i eps a e mo e p obable. We
shall see in Sec ion 3 ha his condi ion can be applied as a
il e . The lis o compa ible i eps and espec i e iso opy
subg oups o he subg oup Cm (2a2b, 2a+ 2b,c; 1/4, 1/4, 0)
shown in Fig. 7 includes his subg oup i sel o he ou -
dimensional i ep labelled X
1
(o X1 – due o o ma limi a-
ions some ou pu pages do no show he numbe s in he i ep
labels as subsc ip s bu as o dina y on s). Hence, his
subg oup sa is ies he Landau condi ion, as i can be eached
by he p esence o a dis o ion acco ding o his single i ep
X
1
, es ic ed wi hin a wo-dimensional subspace. The bold
cha ac e s o he wo wa e ec o s (0, 1/2, 0) and (1/2, 0, 0) o
he i ep s a in Fig. 7 indica e ha bo h o hem a e in ol ed
in he dis o ion.
The link o he p og am Ge _i eps also pe mi s he use o
ob ain a g aphic ep esen a ion o all he in e media e
subg oups o he chosen subg oup, showing hei g oup–
subg oup hie a chy. In he case o he subg oups which a e
lis ed as iso opy subg oups, he g aph also indica es he
associa ed i ep. As an example, Fig. 8 depic s he g aph ha
can be ob ained as a complemen o he ou pu shown in Fig. 7.
One can see ha he end symme y Cm can be eached wi h
he single i ep X
1
.
The wo subg oups o ype Cmm2 in Fig. 8, numbe ed 6 and
7 (also lis ed in Fig. 6), a e iso opy subg oups o he i ep X
1
,
i.e. he same i ep o which he Cm subg oup is also an
compu e p og ams
J. Appl. C ys . (2024). 57, 1650–1666 Em e S. Tasci e al. �SUBGROUPS ool a he BCS 1655
Figu e 7
I educible ep esen a ions (i eps) o he space g oup P421m(No. 113)
which a e compa ible wi h i s subg oup o ype Cm (2a2b, 2a+ 2b,c;
1/4, 1/4, 0), lis ed as 3.1 in Fig. 3. This is he ou pu ob ained by clicking on
he bu on ‘Ge i eps’. The ou pu lis s he wa e ec o s in ol ed o each
i ep (in bold), he i ep label and he equi ed di ec ion wi hin he i ep
space. Fo each i ep, he iso opy subg oup o epike nel is also indica ed.
See he ex o mo e de ails.
iso opy subg oup. This can be easily checked by calling
Ge _i eps o hese wo subg oups in he ou pu shown in
Fig. 6. Bu , in he case o hese wo highe subg oups, he
di ec ion wi hin he i ep space is u he es ic ed o a single
ee pa ame e . In such cases, he p og am only indica es o
he i ep he iso opy subg oup co esponding o he mos
gene al dis o ion/di ec ion allowed. In con as , he in e -
media e subg oup o ype Pm shown in Fig. 8 does no include
any i ep label because i is no an iso opy subg oup o any
i ep. The g aph shows ha his symme y can only be a ained
h ough he combina ion o dis o ions acco ding o a leas
wo o he h ee i eps associa ed wi h i s h ee immedia e
supe g oups, namely M
1
M
3
, M
5
and GM
5
.
2.4. Gene a ion o CIF iles o he pa en s uc u e unde he
selec ed subg oups
The a he comp ehensi e symme y in o ma ion p o ided
by he p og am as explained abo e can be e y use ul when
in es iga ing a dis o ed o pseudo-symme ic s uc u e.
Howe e , in many cases he i s and mos impo an p oblem
is he ac ual de e mina ion o he dis o ed o pseudo-
symme ic s uc u e, ei he using di ac ion da a o h ough
ene gy minimiza ion in DFT calcula ions. To acili a e a
s aigh o wa d use o he p og am when dealing wi h his
ype o p oblem, he lis o symme ies, as in he example in
Fig. 1, includes in he las column an op ion o each lis ed
subg oup which in oduces an au oma ic link o ano he ool
o he BCS, namely TRANSTRU. I a CIF ile o he pa en
s uc u e is hen uploaded, his op ion pe mi s he au oma ic
gene a ion o a se o CIF iles, one o each o he selec ed
subg oups, whe e he pa en s uc u e is desc ibed unde he
subg oup symme y in he s anda d se ing o i s space g oup
ype. The CIF iles can hen be used in e inemen s using
di ac ion da a o in DFT ene gy minimiza ions, cons ained
o hese al e na i e symme ies. No e ha o monoclinic and
iclinic symme ies he s anda d uni cell ha SUBGROUPS
may ha e chosen can be qui e inapp op ia e, depending on he
me ics o he pa en la ice. I is hen con enien o ans o m
he CIF ile o a desc ip ion wi h a mo e adequa e uni cell.
TRANSTRU as a s andalone p og am can also be used o his
pu pose. One jus needs o in oduce he same g oup o he
g oup–subg oup pai , and he desi ed change in uni cell.
F om a p ac ical poin o iew, once he lis o possible
subg oups/symme ies is ob ained in he i s s ep explained in
Sec ion 2.1, and a e applying, i necessa y, some o he
a ailable il e s (see Sec ion 3), he use can skip all he
op ional de ailed in o ma ion abou he subg oups and go
di ec ly o his las op ion o gene a e app op ia e CIF iles o
he desi ed symme ies.
3. Fil e s
In o de o educe he numbe o po en ial symme ies
gene a ed by he p og am, di e en il e s can be applied
based on di e en c i e ia. These il e s can be in oduced on
he i s inpu page and se e o na ow down he enume a ion
p ocess.
We s ess ha hese il e s a e applied a e he p og am
ob ains he ull lis o subg oups, which is done ma hema ically
wi hou any il e . This means ha he applica ion o any o he
il e s does no educe he compu ing ime. The p og am in
ac may ail in cases whe e he numbe o possible un il e ed
subg oups is ex emely la ge, equi ing a e y long compu ing
ime, e en i he il e ed se we e small. As he complexi y o
he b anches belonging o he g oup–candida e-subg oup
ees inc eases exponen ially, and since each o hese chains is
handled sepa a ely p oceeding h ough maximal subg oups all
he way down o P1, a highe subg oup index migh esul in a
long wai ing ime.
The il e s ha can be applied can be di ided in o di e en
ca ego ies as ollows.
3.1. Maximal subg oups
The simples il e s a e hose ha limi he lowes symme y
o be conside ed. Wi hou hem he p og am lis s all
subg oups up o he lowes possible one. Al e na i ely, he lis
can be limi ed o he maximal subg oups, i.e. hose subg oups
o which no in e media e supe g oup exis s among hose
subg oups calcula ed by he p og am. Subg oups can also be
compu e p og ams
1656 Em e S. Tasci e al. �SUBGROUPS ool a he BCS J. Appl. C ys . (2024). 57, 1650–1666
Figu e 8
A g oup–subg oup g aph showing all he in e media e subg oups
be ween he speci ied subg oup o ype Cm and he pa en space g oup
P421m(No. 113), as ob ained calling he p og am Ge _i eps. The g aph
shows all in e media e subg oups, including hose wi h la ices di e en
om he one ha is equi ed o he subg oup Cm. In he case o hose
subg oups which a e iso opy subg oups lis ed in Fig. 7, he co e-
sponding i ep is also indica ed. The numbe s o each subg oup a e hose
in he lis p o ided by he p og am (no shown he e), whe e he
subg oups a e ully de ined.
limi ed o hose being pola , non-pola , cen osymme ic o
non-cen osymme ic, e c.
3.2. Displaci e dis o ions
Ano he impo an il e exis s o s uc u es ha ing e y
ew independen a oms on special posi ions. In hese cases,
some o he subg oups ma hema ically calcula ed by he
p og am canno be a ained by displaci e dis o ions, i.e. by
any kind o co ela ed a omic displacemen s. The eason is
ha , i all he a oms occupy special posi ions, a easible
subg oup mus necessa ily inc ease he numbe o ee pa a-
me e s necessa y o de ine hei posi ions wi h espec o all i s
immedia e supe g oups, o he wise his symme y can ne e be
a ained by he displacemen s o hese a oms because one o
he supe g oups wi h he same numbe o ee pa ame e s
would be ealized (see Appendix A2). The occupied Wycko
posi ions can be speci ied and he p og am hen d ops om
he lis all hese ‘impossible’ symme ies, while he subg oups
only a ainable by some la ice s ain, i exis ing, can be
included o excluded.
As an example le us conside a pa en s uc u e wi h space
g oup Pm3m(No. 221) and h ee symme y-independen
a oms on he Wycko posi ions 1a, 1band 3d,i.e. he ideal
p o o ype s uc u e o a pe o ski e. I we a e in e es ed in
possible dis o ed pe o ski es which keep he pa en la ice,
and he e o e we in oduce as ‘supe cell’ he same pa en uni
cell, he numbe o possible dis inc symme ies (conjugacy
classes o subg oups) p o ided by SUBGROUPS wi hou
applying any il e is 33 (including Pm3mi sel as a i ial case
ma hema ically ul illing he subg oup condi ion). I , howe e ,
he h ee men ioned Wycko posi ions a e in oduced as he
only occupied ones, he lis is hen educed o 19 subg oups.
Op ionally, he subg oups which a e only a ainable h ough
la ice s ains can also be excluded and he lis is hen educed
o 12 classes (always including he pa en space g oup). The e
a e he e o e 11 space g oups which can desc ibe he
symme y o a dis o ed pe o ski e esul ing om a displaci e
dis o ion ha (app oxima ely) keeps he pa en la ice. Thei
g oup–subg oup hie a chy is shown in Fig. 9 ( he o de ing
does no indica e he index le els bu has been a anged wi h
espec o maximal subg oup chains). We s ess ha his
op ional il e is in ended o limi he possible symme ies o
hose caused by a omic displacemen s. The e o e i should no
be applied i he dis o ion may include some ype o o de –
diso de phenomenon, wi h he occupancy o some a omic
si es a ying be ween he pa en and dis o ed s uc u es.
3.3. Landau condi ion
The mos impo an il e ha he p og am p o ides is
p obably he one ha es ic s he enume a ion o subg oups
o hose ha can be a ained wi h a Landau- ype phase an-
si ion, i.e. o hose subg oups co esponding o symme y
b eaks which can be explained by he p esence o a dis o ion
ans o ming acco ding o a single i educible ep esen a ion
o he pa en space g oup. This means ha , ollowing Landau
heo y (Landau & Li shi z, 2013; Cowley, 1980), a single o de
pa ame e acco ding o a single i ep can be in oduced o
desc ibe a phase ansi ion be ween he wo symme ies. This
il e can be undamen al o es ic ing a huge numbe o
ma hema ically possible symme ies o jus a ew which can be
conside ed mos p obable om a physical iewpoin . In he
case o he example shown in Sec ion 2 o a pa en space g oup
P421m, his il e is ine ec i e since all he subg oups lis ed in
Fig. 1 ul il he Landau condi ion, bu in he second example
conside ed in Sec ion 3.2 o a pa en space g oup Pm3mand
compu e p og ams
J. Appl. C ys . (2024). 57, 1650–1666 Em e S. Tasci e al. �SUBGROUPS ool a he BCS 1657
Figu e 9
A g oup–subg oup g aph showing all he possible subg oup symme ies
ha can ha e a dis o ed pe o ski e s uc u e, i.e. a s uc u e wi h pa en
space g oup Pm3m(No. 221) and occupied Wycko posi ions 1a, 1band
3d(o 3c), i he dis o ion is o displaci e ype and he la ice is main-
ained. Symme ies only a ainable by a la ice s ain a e no included.
Figu e 10
A g oup–subg oup g aph showing all he possible subg oup symme ies
ha can ha e a displaci e dis o ed pe o ski e s uc u e as he esul o a
Landau- ype phase ansi ion wi h a single o de pa ame e , wi h he
la ice main ained.
wi h he la ice main ained, he numbe o symme ies educes
om 32 o 23. In he case o a pe o ski e-like s uc u e and a
displaci e dis o ion, he 11 possible symme ies men ioned in
Sec ion 3.2 educe o nine. Thei g oup–subg oup hie a chy
(see Fig. 10) shows ha he e a e i e which a e maximal
symme ies and hese would be he i s ones o explo e.
3.4. Dis o ions acco ding o one o se e al speci ic i eps
I he wa e ec o (s) op ion is used o in oduce he la ice
ela ion be ween pa en and dis o ed s uc u es, he
enume a ion o possible symme ies can be limi ed o hose
esul ing om a dis o ion ans o ming acco ding o one o
mo e speci ic i eps associa ed wi h he inpu wa e ec o (s).
The il e ing o symme ies esul ing om he simul aneous
p esence o mo e han one i ep is limi ed o a single wa e-
ec o o an i ep s a o wa e ec o s.
Fig. 11 shows he g aph ob ained o he subg oups o he
space g oup Pm3m, which he p og am enume a es i his il e
is applied o he i ep GM4wi h k= (0, 0, 0). This i ep is he
one associa ed wi h any kind o pola displaci e dis o ion,
and one ecognizes in he igu e all he space g oups ha ha e
been obse ed in pe o ski e-like compounds exhibi ing some
p ope e oelec ic phases due o a pola dis o ion.
4. Examples o applica ion
These examples a e all explained in mo e de ail in he u o ial
o he p og am, which is a ailable on i s webpage (h ps://
jou nals.iuc .o g/b/se ices/abou .h ml).
4.1. Symme y o he low- empe a u e phase o ulle ene–
cubane
C ys als ha include molecules o bo h ulle ene and
cubane a e known o c ys allize a high empe a u es
acco ding o he Fm3m(No. 225) space g oup, wi h he
diso de ed ulle enes cen ed on he si e 4a(000) and he
diso de ed cubane molecules on 4b(1
2
1
2
1
2). A low empe a u e,
as hese molecules become o de ed, he sys em exhibi s a
phase ansi ion in o an o ho hombic phase. F om powde
di ac ion expe imen s, he inal low- empe a u e phase was
epo ed o be a non-cen ed o ho hombic s uc u e, wi h he
pa ame e s o i s p imi i e uni cell sa is ying he app oxima e
ela ions a’b’a
c
/2
1/2
, while c’2a
c
, whe e a
c
is he cell
pa ame e o he cubic phase (Pekke e al., 2005). Howe e ,
he space g oup o his phase could no be de e mined and he
s uc u e emained unknown o se e al yea s (Bo el e al.,
2006). I ook i e yea s inally o iden i y he space g oup and
de e mine he co esponding s uc u e (Bo el e al., 2011).
Ob iously, i he possible symme y o his phase could ha e
been es ic ed o a minimal se o space g oups, he e would
ha e been a be e chance o succeeding in he in e p e a ion
and analysis o i s di ac ion diag am when his s uc u e was
ini ially in es iga ed. I is shown below ha , using
SUBGROUPS, he mos p obable space g oups consis en
wi h he obse ed cell pa ame e s can be educed o wo. One
o hem is indeed he symme y g oup o he s uc u e ha was
inally de e mined in 2011.
The me ics o he epo ed p imi i e o ho hombic uni cell
clea ly indica e ha i s ela ion wi h he cubic cell o he non-
dis o ed pa en s uc u e mus be o he o m a
s
=a/2 b/2,
b
s
=a/2 + b/2, c
s
= 2c, whe e a,band cde ine he con en ional
cen ed cubic uni cell o he pa en s uc u e. In oducing jus
he pa en space g oup Fm3mand his supe cell (as p imi i e)
in he i s inpu page o he p og am, SUBGROUPS p o ides
qui e a long lis o 99 possible subg oups ( igo ously, a lis o
conjugacy classes, as explained abo e), which is consis en
wi h he inpu supe cell. No e ha he p og am pe mi s he
use o de ine he supe cell wi h some cen ing. The e o e,
ins ead o he p imi i e supe cell indica ed abo e, one can
in oduce a C-cen ed supe cell a
s
=a,b
s
=b,c
s
= 2cand he
esul is jus he same, as bo h supe cells de ine he same
la ice. The cubic symme y o he pa en space g oup also
means ha he in e change o cell pa ame e s in he supe cell
de ini ion will esul in he same lis o conjugacy classes o
subg oups.
The lis o possible symme ies is ex emely la ge because
by de aul he p og am lis s up o he lowes possible
symme y wi h space g oup P1. Bu as he la ice is epo ed o
be o ho hombic, we can include he il e ha only highe -
symme y subg oups up o he o ho hombic c ys al amily
should be lis ed. The lis is hen educed o 62 subg oups, bu
mos o hem can s ill be disca ded as he lis includes all
subg oups belonging o c ys al sys ems highe han he
o ho hombic. Fu he mo e, among he o ho hombic space
g oups a e also lis ed hose no belonging o he holohed y,
wi h poin g oups 222 and mm2. As maximal symme ies a e
usually ealized, we a e going o assume ha he ele an
poin -g oup symme y is he maximal one wi hin he o ho-
hombic class, i.e. he holohed y mmm (i his assump ion we e
o u n ou o be unsuccess ul, one could hen always p oceed
simila ly wi h he o he wo possible o ho hombic poin
compu e p og ams
1658 Em e S. Tasci e al. �SUBGROUPS ool a he BCS J. Appl. C ys . (2024). 57, 1650–1666
Figu e 11
A g oup–subg oup g aph showing all he possible subg oup symme ies
which can ha e a dis o ed s uc u e wi h pa en space g oup Pm3m(No.
221) as he esul o a dis o ion acco ding o he pola i ep GM4wi h
k= (0, 0, 0). The di e en symme ies co espond o di e en o de
pa ame e di ec ions wi hin he h ee-dimensional i ep space.
MP¼X
j
Rj:ð10Þ
No e ha , in gene al, he numbe n
w
o occupied inde-
penden Wycko posi ions is di e en o di e en subg oups.
Fo each subg oup, he p og am spli s each Wycko posi ion
in he o iginal subse (in oduced by he use ) in o o bi s
which co espond o di e en Wycko posi ions in he
subg oup.
Op ionally, i is also possible o keep he subg oups ha a e
a ainable by la ice s ains. The p ocedu e used is exac ly he
same, bu in he calcula ion o he deg ees o eedom in
equa ion (9) we mus add he s ain deg ees o eedom o he
space g oup, i.e. he numbe o ee cell pa ame e s no ixed
by symme y, which depends on he c ys al sys em: one o
cubic g oups, wo o e agonal, igonal and hexagonal
g oups, h ee o o ho hombic g oups, ou o monoclinic
g oups, and six o iclinic g oups.
A3. Fil e ing wi h espec o he i educible ep esen a ions
This il e can be used when he ela ion be ween he pa en
and he dis o ed s uc u e has been in oduced ia he
wa e ec o (s) op ion ins ead o he supe cell op ion. The
p og am shows he se o i eps o he wa e ec o s and he
use can choose one o mo e i eps. SUBGROUPS will es ic
he ou pu o hose subg oups calcula ed ollowing he algo-
i hm gi en in Sec ion A1 which desc ibe a dis o ion ha
ans o ms acco ding o he chosen i eps. This il e can be
used in combina ion wi h he il e ing wi h espec o he
Wycko posi ions. In his combina ion, he use can choose
among he i eps which ha e a non-ze o mul iplici y in he
decomposi ion o he mechanical ep esen a ion o a leas
one gi en Wycko posi ion.
Fo e e y chosen �i ep and e e y subg oup ob ained
wi hou il e s, he algo i hm used by SUBGROUPS o check
whe he he subg oup can be ealized by he ac ion o an o de
pa ame e ha ans o ms acco ding o he i ep is ou lined
below.
Fi s we iden i y he symme y ope a ions o he subg oup
(one ope a ion o each Relemen o he poin g oup)
exp essed in he basis o he pa en space g oup. Nex we ake
he ma ices o he i ep o he pa en g oup D
�
(R), abula ed
in he da abase REPRESENTATIONS SG o he BCS
(Elco o e al., 2017). These ma ices o m a ep esen a ion o
he subg oup ( he subduced ep esen a ion in o he subg oup)
ha , in gene al, is educible in o i eps o he subg oup. The
chosen subg oup can be ob ained hough a dis o ion ha
ans o ms unde he chosen i ep i he mul iplici y o he
i ial i ep is di e en on ze o in he decomposi ion o he
subduced ep esen a ion. In a e y simila way o he
app oach used in he applica ion o he il e by Wycko
posi ions, we can equi e ha a non-ze o o de pa ame e is
kep in a ian by all he ma ices o he (subduced) ep e-
sen a ion. I is hen possible o w i e an equa ion iden ical o
equa ion (8) which, ins ead o he ma ices Rj
wi, includes he
ma ices D
�
(R) o he i ep and whe e he ec o now
ep esen s he componen s o he o de pa ame e in a space
whose dimension is he dimension o he i ep. As in he
p eceding sec ion [see equa ion (10)], we de ine o each
subg oup he ollowing ma ix,
M�¼X
R
D�ðRÞ:ð11Þ
I he ank o he ma ix o a gi en subg oup is highe han
he co esponding ank o he ma ix o any o i s supe g oups
in he lis , he subg oup is included in he inal lis because his
means ha he mul iplici y o he iden i y i ep in he
decomposi ion o he subduced i ep is highe in he subg oup
han he co esponding mul iplici y in all i s supe g oups.
The e a e mo e deg ees o eedom in he subg oup han in all
i s supe g oups. To calcula e he gene al o de pa ame e ha
co esponds o he dis o ion desc ibed by he subg oup, he
p og am akes as many linea ly independen ows o M
�
as i s
ank. The o de pa ame e belongs o he subspace spanned
by hese ec o s and can be pa ame ized as a linea combi-
na ion o hese ec o s using ee (un es ic ed) magni udes.
The subg oup a he bo om o he g oup–subg oup ee
co esponds o he ke nel o he i ep. The ank o M
�
in his
case is he dimension o M
�
( he dimension o he i ep) and
he o de pa ame e is an a bi a y poin in space wi h he
dimension o he i ep.
A4. Landau condi ion
When he use chooses he ‘Landau condi ion’ in he main
inpu , SUBGROUPS lis s all he subg oups ha can be
ob ained as a esul o a dis o ion ha ans o ms acco ding
o a single i ep o he pa en space g oup. I can be applied
only when he use has in oduced a single modula ion
wa e ec o o a se o ec o s in he same s a o , when he
supe cell op ion is used, i he supe cell can be de i ed om a
se o wa e ec o s ha belong o a single s a . The p og am
iden i ies he se o subg oups o e e y i ep ollowing he
p ocedu e explained in Sec ion A3 and me ges hese lis s in o
a single inal lis o subg oups.
Funding in o ma ion
The ollowing unding is acknowledged: Go e nmen o he
Basque Coun y (g an No. IT1458-22).
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