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Tables of subgroups, site groups and interchange groups of symmetry point groups and the construction of SALCs
Chemical Physics 12 (1976) 355-365 0 North-Holland Publishing Company
TABLES OF SUBGROUPS, SITE GROUPS AND INTERCHANGE GROUPS OF SYMMETRY POINT GROUPS AND THE CONSTRUCTION OF SALCs* E. RYTTER* Chemistry Civision, Argonne.Natioual
Loboratory.
Argonne, Illinois 60439,
USA
Received 24 July 1915 Revised manuscript received 24 October 1975
A table listing subgroups and supergroups of 43 chemicaliy important symmetry point groups is presented. Other tables give the site groups and the corresponding interchange groups for those point groups which are of finite order. It is shown how the tables may be used to facilitate the structure determination of met&li_eand and related compounds from spectroscopic data and how the construction of symmetry adapted linear combinations, SALCs, may be simplified. The tables aiso may be used in factor group analysis.
1. Introduction In a metal-ligand or similar compound, the opera tions of the symmetry elements passing through the mass center of a ligand constitute the site symmetry point group (here abbreviated to site group). The site group must be a subgroup of the molecular point group, but not ail these subgroups can be site groups. Furthermore, the site group is a subgroup of the point group of the free ligand. Rules like these are helpful in the construction of molecular models. The necessary information of subgroups and site groups of a point group is presented in tabular form (tables 1 and 2.~. Another important application of site groups is in the construction of symmetry adapted linear combinations, SALCs [ I]. The interchange group, G,, which is the simplest group interchanging symmetrically equivalent ligands of site symmetry G,, is needed iu Ahe calculations. The molecular point group then may be written as G=Gf-GS,
(1)
* Work performed under the auspices of the-U..% Energy Research and Development Administration. * Postdoctoral appointment at Argonne National Laboratory 1974-1975.0n!eavefromInstitute ofInorganicCbemistry, University of Trondheim, N-7034 Trondheim-NTH, Norway;
where the dot represents either a direct product, X, a semidirect product,~, or a weak direct product 12-61. The projection operator $$) for construction of SALCs in the group G is I’d”
=
c
pr)~,
R
R
where xg) is the character corresponding to the operaiionR in the representation f’. Flurry [6] has shown that introduction of the site and interchange groups yields (3) The representations I” and F” must correlate to F 171. This equation greatly simplifies the construction of SALCs. To apply the formula it is necessary to know corresponding site and interchange groups in a point group. These data are provided in tables 2 and 3. Recently, Dounay and Turrell [8] presented tables of oriented site symmetries in space groups and showed how the tables can be used to obtain structural information from vibration spectra. In this procedure, the subgroups of the point grouPs of molecules or ions in the crystal are-needed. Donnay and Turrell, however, list only_ the subgroups, acceptable as site groups in the space.group, of the 32 crystallographic point
E. .?Qtter/Tables of mb-,
356 Table
sire and interclrange groups
1
Subgroups and supcrgroups of 43 point groups. AH subgroups of the finitc groups except the groups themsciws, 512 of&d &o Of&d ate listed, The number of symmctric?llp nonequivalent positions is givcrl for each subgroup
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I
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l
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a
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-
f
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.. I
.-
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and
groups. The symmetries of the molecules or ions naturally are not limited to these crystallographic point groups. It therefore is useful to have a table of subgroups of other chemically imporrant point groups also from a crystallographic point of view.
2. Subgroups and supergroups The subgroups of 43 point groups, including the infinite groups Dmh and C&, are summarized in table 1. The subgroups arc found by following the row down~vards to the right from the point group and from each entry in this row going do~vliwards to the left to a corresponding subgroup. The entries give the number of symmetrically nonequivalent positions of the subgroup in the supergroup, This number does not include different orientations of axes and planes of the subgrcmp. For instance, the three positions of C,, in Dsh, one for each of the axes CzP C$, and c’;, account for the figure three in the table. However, if the axis of C& is coincident with C$, the planes crv(xz) and uv@z) may be correlated to oh and a, in two ways. These possibilities are not included in table I. When it is taken into account that each group is a trivial subgroup of itself, all subgroups of the 41 finite groups are given, except Sr2 of Dti and SIO of DTd. Table 1 also may be used to find several supergroups of a given point group.
3. Site groups and interchange groups Far from ali the subgroups of the 41 finite groups of table 1 also are site groups. The site groups are listed in table 2. The table includes every possible site and may be used in a similar way as table 1, although the point groups now appear along a diagonal. Furthermore, there are two entries for each site group, one on each side of the diagonal. The entries on the right hand side show the multiplicities of the site groups, i.e., the number of times a given site is repeated by the point group symmetry. This number is the order of the interchange group, G,: One set of interchange groups is given to the left of the diagonal. Excluded are G; for the sites in a general position (Gr = G) and for sites with the full
symmetry of the point group (Gr= Cl). Infour cases there is no interchange group fulfilling the condition C = G, -G,.The corresponding entries in table 2 are given the sign 4. In several cases there may be more than one interchange group corresponding to a given site. Moreover, the positions of Gt and/or of GS in G may not be unique. Specification of all the interchange groups as well as of positions is given in table 3 when ambiguities exist. The interchange groups may be found algebraicafIy. As an example, two interchange groups for the C& site in Dsh are calculated. One abelian factoring ofQh is f&4] Dstl = D6 X Ci
=C~AC;XCi=C3XC~"C;XCi.
(41
The orientationa conventions are consistent with Altmann [4] . From the multiplication tables of C,, and C”,, , the following factoring may be found c~v=c~xc,=c~xc,“=c,‘xc~‘,
(51
C2h=C,XCs=C2Xci=CsXCi.
(6)
Combination of eqs. (4) and (6) and using the commutative property of the direct produtit, yield
=csxc;,=s,xc;,.
(8)
Eq. (8) is derived from (7) by using (5) and standard expressions [4] for C, and Ss. Note that the orientations in eqs. (5) and (6) are relative to the two-fold axis and therefore must be changed when applied to Q.h* The two interchange groups Cs and S, given in eq. (8) are consistent with the groups listed in table 3. Even though the algebraic procedure works, it gener-_ aily was found more convenient to find the interchange groups by systematic, geometric inspection.
4. Construction
of SALCs
The application of eq. (3) in t& formation of symmetry adapted linear combinations Is not atways straightforward. The question is which representations are to be correlated. The derivation of (3) is
-358
z
_
Table 2 Site gr&ps of 41 finite point groups. All site groups except the groups themsc&cs arc listed. The multiplicity of each site is given to the tight of the diagonal and one interchange group for each site _eroup is given to the left
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* indicates that mutual exctusion of Roman and infrared Octivities f indicates that no interchange group fulfills G = G, G,. ’ GI = G for GS = c,
otcws
for these
groups.
l
basedon
[5] G=GI.GSandon
(0 = xg’, . $“I
XR
1
where the operation k in G is the product of b and ?i? GE-and Cs, ~e~ectiy~~y. Irr she use bf eq. (3}, P&r 5 is opkted on the basis-functions of a given Ii&d a+ ploper linear combinations are formed _
(9)
belonging to the irreducible representations of GS. It follows that the representations I?’ in eq. (3) which are at’ interest, are irreducible. Furthermore, each b&s function (a pair for doubte degenerate species, etc.) in GS may be treated separately by k$& j_ _ Consider now the correlations (see applicatirlns)
E. Rytter/TabIes
of Sub-. site and interchange groUpS
359
Table 3 Site groups and interchange groups of symmetry point groups. Only those site groups are liste G which have several interchange groups or where posirionl ambiguity may occur for either the site group or the interchange group. The positions in G are given in parentheses Symmetry group, G
Site group, GS
Interchange group, GI
Oh
c4v
D3.
c3v GVG)
,
s6
~,D2d(Ci,4),Clh.Dzh(3i72)
C,(Q)
0. Td 0. Th
0
czcc;)
T
Td
c3v
D2.
CSbh)
s4
c2 cc;,
D6d
C6VA2 D6.
&h
St2
Gv C2vG) Gvc;) C&ah) c&J c&j,,
06
Gv c6h &I
C&)
c6,
‘%vbd)
C,bd)
c6,
c3vbv)
c6
Cp ci
CS
c6.
s6
Gv c2 CS
Dsh
D4d
CS G,
Ds,
csh
c4v
c2m
c2cc;>
c4-G &l &r SE
CS
D4h
c2, &iv
c4v C2vG) C,,(G) c&,) C&> c&d)
Symmetry proup,
d
Site group, GS
Interchange group, GI
I
D2h
There are four equivalent basis functions in A’, one on each site, which are to be gombined by F&i’) into basis functions of AI, B2 and .E of G. For the symmetry species E, eq. (3) becomes
which may be found ~IIthe book by Fateley et al. I?] *. -t
core the~peciaf application of 2 ijctor 2 vvhieh far so& groupsmust bcused in the correlationsbetwccn GSand G [71.
.. .
From the definition of the projection operator in eq. (2), it is obvious that
E. RyrterfTobles of sub-, sire and intercl~onge groups
this function belonging to A’ and A” of C, are projected out first. It then may be argued that the operation equally well could have been performed on, e.g., the part belonging
to A’ thus giving no contribu-
tion to A”. This is equivalent to say that each part of eq. (11) may be used separately on any function. It follows that
(‘2) Furthermore, @(Bz+B3) D2
= j%%’
the following decomposition
is valid:
+ $3’ 2
D2
The operator Pp interchanges symmetrically equivalent functions, &-re on each site. Consequently, these functions generate a regular representation with one function for each of the irreducible representations in the interchange group [9]. It is possible therefore to use each term of eq. (14) separately to form basis functions in E of D2d. This procedure takes care of orthogonalization of degenerate basis functions, but not of orientation and normalization. If the combined formula of eq. (12) is used instead, only half of the four C, basis functions
are combined
into a linear combination. To obtain a linear combination of the other half, the formula must be applied to a representative of the second ser. The two SALCs then formed constitute the appropriate degenerate pair in D,,. The correlations (see applications) W,)
W4v) /
E
Fb?
Gr(D3) -
A1+E
F 1U -
A2+E
-
@>@generates a degenerate pair and i#rj may be op?vrated on each of the degenerate fun&ions. The result will be two functions, each being a linear combination of one basis of Fzr and one of Fzu since A1(D3) correlates to these representations of G(Q Appropriate linear combinations therefore must be taken to obtain tile pure Fzg and Fzu SALCs. The easiest way to find the remaining Fzg and F2u SALCs is to apply Pzl) 1x1 . the same way, but for two other equivalent D3 subgroups of 0,. In summary, the projection operator $9 = P(r-‘)-$%(I*“) may he used to find basis function belongGI GS ing to r of G by using the operators corresponding to any of the irreducible representations of Gj and Gs that correlates to f’. However, when more than one irreducible representation in G correlate to the same representations T” and T“‘, appropriate linear combinations must be taken to discriminate between these representations in G. After the basis functions for a given site is formed by ?$“‘I, it is feasible to orient the functions on the oth& sites so that equivalent functions are transformed into each other by G,. To obtain full factoring of secular determinants, degenerate pairs must be oriented in the same way. This may be achieved by proper orientation of functions on the sites. The present method is especially powerful for the construction of vibrational symmetry coordinates for complexes with polyatomic ligands and a detailed analysis of vibrations of metal-ligand compounds will be published [ !O] . The paper will contain a systematic approach to group theoretical treatment and structure determination from vibrational spectra.
A2”E
___ F2s
’ 5”
361
A1+E
constitute a slightly more complicated example. A degenerate pair of functions on six equivalent sites are going to be combined into four triplets of degenerate basis functions. However, in this case a species in Gt correlates to more than one species in G for a given species in GS. According to the example above, each factor of the projection operator, e.g., Pr)@(-“, 03 C4v’ may be applied separately. The G, operator
5. Applications 5.1. Co(NO&This complex is assumed to have six symmetrically equivalent NO2 ligands, octahedrally coordinated to cobalt. The site group of NO2 must be a subgroup of C,, and from table 1 it is seen that the possible site symmetries are c2v, C,. C, and Ct. To find corresponding total symmetries each of these four groups is localized in table 2 and the column above each group is searched for entries with multiplicity six. The possible symmetry groups of Co(N02)~- are
362
E. RytterfTables of sub-, site and interchange gmups
found to the left of each entry and the combinations of site arid totai groups when are: C2,,(Td, Th,L&), C2(~D@D3d),
C.&v9 c&pD3&D3h)y
and cl(c6,
D,* % C3”, C3h).
The skeletal symmetry is octahedral and the total symmetry group therefore must be a subgroup of 0,. This leaves eight of the fifteen combinations above after inspection of the subgroups listed in table 1. Two more possibilities, c2(&) and C,(C,,), can be eliminated because the site groups are not consistent with both the skeletal and the total symmetries. Six combinations are left, namely C2v(Td, T,,), C,(7),
Td !--I- 1
T,, (4,)
I+. l_PossibIc Co(NO&structures. Rotation of NO7 in the symmetry groups shown ”
-the.direcrion of the arr~ws+es .. in parentheses.
Table 4 The number of NO2 scissoring and Co-N stretching fundamentals for different Co(NO& models Co(NO&symmetry
NO* site symmetry
NOa scissoring
Td
C2v
1
1
1
1
1
I
Th
c2v c2
:I
Co-N stretching
cs
2
2
L$
Cs
2
2
&I
Cl
3
3
S6
Cl
2
2
c&d) and Cl (D3, S6). All may be visualized in more than one way by orientation of the NO2 ligands
on the sites. Some simplifications are achieved if it is assumed that the nitrogen atoms and the two-fold axes of the ligands point towards cobalt. Consequently, the only freedom of the ligands is rotation about the Co-N axes. This rotation gives rise to seven different structures (fig. 1). The appearance of two D3ri isomers is reflected by the number 2 in the entry in table 1 for the C, subgroup of C2,. Three of the models (Z’, D3 and S6) contain a general angle of rotation of NO, and fig. 1 shows how they may be obtained from the other models. The structure with symmetry T may be derived from either Td or Th, the &?$6 structure from Th or Dsd and the D3 structure from either of the D3d structures. Nakagawa and Shimanouchi [ 1 I- 151 have discussed Co(NO&structures with the same assumptions as above and they erroneously stated that Th, T and S, are the only possible symmetries. They recorded the infrared spectra of solid Na3Co(N02)6 and K~CO(NO~)~ and found two NO2 scissoring and two Co-N stretching frequencies for the sodium compound compared to one of each for the potassium compound. The symmetry of the cobalt complex in K,Co(NO.& is known [ 16) to be Th and Eiakagawa and Shimanouchi attributed the splittings in Na3Co(ru’02)6 to lowering of the symmetry from Th to s6. The number of NO2 scissoring and Co-N stretching fundamentals is compared in table 4 for all seven Co(N02)i- models. It is seen that D,, sym-
E. RyttedTabks
af sub-, site
and interchatige
groups
363
C,). The intercixangegroupsD,
and 5, are given in table 3 and the correlations j7j between Gg, G and G, are shown in table 5. aeration of$?) on the basis functions ?r and n* Ieaves each of &em invariant and therefore they belong to the spoke A’ of C,_ They correlate to AL. B2, and E ofDZd which then will conrain the final SALCs. From fig. 2 it may be noticed that all operations of D2 or $4 transform a part of a basis into another part without change of sign.This isvalid for the rr as well as for the x’ basis and they therefore form equivalent SAL& Proceeding with T, operation with PEj yields @iI
=q,*
z nf f $1 + X5’*f #**,
,fs2’
qp
se r’
_
*”
+ n”’
-
n”f’9
gr
= j$y”
z r’
_
n”
-
R”’
+
n”“,
$j
zz @g&f
z It*
*
x9( _
r’,,
-
n”“_
Fig. 2. The CrC$ (&d) Structure.
metry also is consistent with the observations and at the same time possesses higher symmetry than 8,.
(19
5.2. cxgfinear comb~a~~ons a~te~a~~ve~~ are formed using $fl instead of pgj if the degenerate pair is orientedlin the same way. In Crow- there are also 7~~and rr,” orbitals orthogonal to those already treated and all these orbitals may Weract with the metal orbit&. The ?I~ and XT sets belong to the site symmetry species A” and the proper SALCs easily may be eondiucted foliowing the standard procedure given for the A orbit&.
These
Recently, Rkh and Hoffma~n [I71 perfo~~n~~ a charge-Mative extended Hkkekatcutation for this ion. The idealized equilibrium structure has ,W2e symmetry and is shown kt fig. 2 together with the numeration of the four Oi- ligands and the oxygen atoms. Each @and has two orbitals, zr and n*, directed rowxds the central atom. The orientation of the a* orbital, which may be taken as pI - p2 is dependent on the numeration of atoms I and 2 of a @and. In order to obtain symmetry adapted linear cornbinations of the orbitals, it first is observed that the site symmetry of Oi- IS * Cs (table 2 shows that the only sites with rnu~t~~~ic~ty4 have symmetry C, or
Tabie S
Correlations of the site and interchange group symmetry sputies with the irreducible representations of Dzd for C&” B &d(G)
A
A
Bt
A
A
B
5t
B
Ba+B3
E
5.3. XY6 As a more complicated example, p orbitals on the ligands of an octahedral complex are consideredlf. The orientation of the basis functions is defined in Eg. 3 and the correlations behveen the site group c&,, the total group 0, and the interchange groups D, or S, (tabbfe 3) are given in tabfe 6. Operation with the projection operators of the site group, Pdl;:, l” = AL I ___, on pX, p,,. and pz shows that pz belongs to AL and &, p,,) constitutes a degenerate pair of species E. The pz orbit& of site symmetry speciesAt corre-
* RIirry f6] uad tile saxa 5xanpte in the original paper on d~om~s~ion of prajeetion operators. Howover, he complicates the eaktiation by ~fr~u~n~ the site symmetry & in oh and the corresponding “interchange group” IJj& Furthermore, hc only finds one component of each dcgencrate basis, Nis second example, the external symmetry coordinates of the tetrahcdnl XY4 model, conkains several errors.
. _
-
364<
E. Rytterj’Tables of sub-, site and interchbngegroups
-_
Fig. 3. The defined orientations of the px andp, ligand basis functions of the xY6 octahedral model viewed along three of the
three-fold axes. The positive parts of the pz orbit& point towards the central atom.
Table 6 Correlations of the site and interchange group symmetry spe-
interchange group is used (table 6). However, it is much easier to fiid the right linear combinations for, e.g., Fzg and Fzu ( correlates to At of D3) than for Fls and I$ (correlates to A, of s6). This is because every basis function in Fzg must be transferred into itself by inversion since the character is 3 while every basis function in Ftu must be transferred into itself with opposite sign by inversion(character -3). Therefore the interchange group LI3 is employed in spite of the fact that the E part of the projection operator does not distinguish between any of the four triply degenerated species in oh. The Fls and Fzu functions are calculated as an example, but the treatment is equivalent for the Fig and F,, functions. The %I) operator is used on pX. +pY1 and perI pul instead of on Pxl and F,,~. This is just a matter of orientation of the degenerate pair and has the advantage that all 12 symmetrically equivalent functions are generated. The result with x1 representing&l, etc., is
cies with the irreducible model
PD~‘)(XICY1) = x1+X2’X3+x4+x5+Xs
late with Al,, E g, and -Flu of oh and these species therefore will contain the SAL& The linear combinations easiest are foilnd by application of the s,5 projection operators as described by Flurry [6]. However, note that each of the rivee parts of i’yu+Eu) used separately yields all the triple degenerate functions in Flu and in orthogonal form. Similar arguments are valid for the Eg functions. Alternatively, the intcrchanga group D, may be used, but since E of this group correlates to both Ep and Flu of oh it is necessary to operate with kh: on four of the symmetrically equivalent & orbital% e.g.,pzl, pz2, pz4, and pt5 _From the result the appropriate linear combinations belor,ging to Eg and F~, can be found. A similar pzocedure is necessary for the degenerate (p,,p,,) pair. At first, it may seem unimportant which
Basis
C4vGs)
representations
of 01, for the XYe
+YttY2+Y3+Y4+Y5+Y(j*
Oh(G).
Ag -
AS.
-E
Ag
A2+E
-
AI+E
As+Etz AsiEg A, Au
-
and ~~~i)(~,l-Y1):x1+*2+x3-x4-xg-xg
EP
-
A+E.
Eu A,,+E; AI;+EU
(16)
-,$-Y2-~3*~4+r,+~,.
(17)
The first of these linear combinations belongs to Fzg and the second to Fzu since the contribution to the character by kversion is +I and - 1, respectively. If the starting functions had been x1 andYI it had been necessary to take linear combinations of the two expressitins formed iri order to get the-character contrihutions +I- and -l_ &I operator, To avoid the complications with tile PD .~ 3 : --.
365
E. Rytter!Tablesofsub-,sire and inrerchangegrorrps
the P(A1) operator corr~spond~g to two other threefold %es in 0, (see fig. 3) is applied to (xl+y1) and (x1--yt). The result for atl the degenerate functions in Fzgand Fzu is
The author gratefully ac~owledges the finantiial support from GeneraIdirektdr Roif Qstbyes fund and Royal Norwegian Council for Scientific and Industria1 Research.
References [iI F.A. Cotton, Chemical Applications of Group Theory (Wiley-Interscience,
-&+&+~~-~4+~$*~fj’ $$p&) =xl+x2+x3-x4-xs-x6 - Y1-Y2-Y3+yaiy5 +y69 WLt)
=XI-XZ+X3-X4+X~-X6
+Yl -v, -u3 -y4 +$ +y6, (p,(F,J
=XI-yX3-X4+XSfX6
+Y1+~~ -u, --u,-v,
+vs-
These SALCs are consistent with those published by Gray and Beach [I 81. Neither of the functions are normalized and orthogonahzation is only achieved between the different irreducible representations since only the PErI operator was used.
New York, 1971).
121L. Jansen and M. Boon, Theory of
Fini!eGroups. Ap-
plications in Physics (North-Holland.
Amsterdam,
1967). S.L. Aitmann, Phil. Trans. Roy. Sot. A 225 (1963) 216. S.L. Altmann, Rev. Mod. Phys. 35 (1963) 641. R. Kopelman, J. Chem. Phys. 47 (1967) 2631. R-L. ~lurry,Theoret. Chim.Acta 31(1973) 221. W.G. Fat&y, F.R. Dolkh, N.T. McDevitt and F.F_ Bentley, Infrared nnd Raman Selection Rules for Molccutar and Lattice Vibrations: The Correlation hicthod OViley-Interscience, New York, 1972). [81 J.D.H. Donnay and G. Turrell, Chem. Phys. 6 (1974) 1. 191 E.B. Wilson, J.C. De&r and P.C. Cross, Molecuiar Vibrations (hfffiraw-HitI, New York, 1955). IlO1 E. Rytter, to be published. fill f. Naka~awa, T. Shim~~ouc~i and K. Yrunanki, Inorg. Chem. 3 (1964) 772. and T. Shimanouchi, 22 (1966) 1707. I131 I. Nakagawa and T. Shimanouchi,
11215. Nakagawa
Spcctrochim.
Acta
Spectrochim.
Acta
A23 (1967) 2099. (141 I. Nakagnwa, T. Shimanouchi and K. Yamasaki. Inorg.
Acknowledgement fhe author is indebted to Dr. D. Gruen for stimulating the author’s interest in coordination chemistry and for interesting discussions in connection with the present work.
Chem. 7 (1968) 1337. 1. Nakagawa,Can. J. Spectrosc. I8 (1973) I. PI 1161 hf. Dricl and H.J. Vcrwcci, Z. Krist. A95 (1936) 308. [I71 N. Riisch and R. Hoffmann, Inorg. Chem. 13 (1974) 2656. [181 H.B. Gray and N.A. Beach.J. Am. Chem. Sot. 85 (1963) 2922.
TABLES OF SUBGROUPS, SITE GROUPS AND INTERCHANGE GROUPS OF SYMMETRY POINT GROUPS AND THE CONSTRUCTION OF SALCs* E. RYTTER* Chemistry Civision, Argonne.Natioual
Loboratory.
Argonne, Illinois 60439,
USA
Received 24 July 1915 Revised manuscript received 24 October 1975
A table listing subgroups and supergroups of 43 chemicaliy important symmetry point groups is presented. Other tables give the site groups and the corresponding interchange groups for those point groups which are of finite order. It is shown how the tables may be used to facilitate the structure determination of met&li_eand and related compounds from spectroscopic data and how the construction of symmetry adapted linear combinations, SALCs, may be simplified. The tables aiso may be used in factor group analysis.
1. Introduction In a metal-ligand or similar compound, the opera tions of the symmetry elements passing through the mass center of a ligand constitute the site symmetry point group (here abbreviated to site group). The site group must be a subgroup of the molecular point group, but not ail these subgroups can be site groups. Furthermore, the site group is a subgroup of the point group of the free ligand. Rules like these are helpful in the construction of molecular models. The necessary information of subgroups and site groups of a point group is presented in tabular form (tables 1 and 2.~. Another important application of site groups is in the construction of symmetry adapted linear combinations, SALCs [ I]. The interchange group, G,, which is the simplest group interchanging symmetrically equivalent ligands of site symmetry G,, is needed iu Ahe calculations. The molecular point group then may be written as G=Gf-GS,
(1)
* Work performed under the auspices of the-U..% Energy Research and Development Administration. * Postdoctoral appointment at Argonne National Laboratory 1974-1975.0n!eavefromInstitute ofInorganicCbemistry, University of Trondheim, N-7034 Trondheim-NTH, Norway;
where the dot represents either a direct product, X, a semidirect product,~, or a weak direct product 12-61. The projection operator $$) for construction of SALCs in the group G is I’d”
=
c
pr)~,
R
R
where xg) is the character corresponding to the operaiionR in the representation f’. Flurry [6] has shown that introduction of the site and interchange groups yields (3) The representations I” and F” must correlate to F 171. This equation greatly simplifies the construction of SALCs. To apply the formula it is necessary to know corresponding site and interchange groups in a point group. These data are provided in tables 2 and 3. Recently, Dounay and Turrell [8] presented tables of oriented site symmetries in space groups and showed how the tables can be used to obtain structural information from vibration spectra. In this procedure, the subgroups of the point grouPs of molecules or ions in the crystal are-needed. Donnay and Turrell, however, list only_ the subgroups, acceptable as site groups in the space.group, of the 32 crystallographic point
E. .?Qtter/Tables of mb-,
356 Table
sire and interclrange groups
1
Subgroups and supcrgroups of 43 point groups. AH subgroups of the finitc groups except the groups themsciws, 512 of&d &o Of&d ate listed, The number of symmctric?llp nonequivalent positions is givcrl for each subgroup
.
.
.
_
.
.
.
-
.
*
.
.
.
I
.
.
I
.
*
.
l
.
.
a
*
.
-
f
:
.. I
.-
.
.
and
groups. The symmetries of the molecules or ions naturally are not limited to these crystallographic point groups. It therefore is useful to have a table of subgroups of other chemically imporrant point groups also from a crystallographic point of view.
2. Subgroups and supergroups The subgroups of 43 point groups, including the infinite groups Dmh and C&, are summarized in table 1. The subgroups arc found by following the row down~vards to the right from the point group and from each entry in this row going do~vliwards to the left to a corresponding subgroup. The entries give the number of symmetrically nonequivalent positions of the subgroup in the supergroup, This number does not include different orientations of axes and planes of the subgrcmp. For instance, the three positions of C,, in Dsh, one for each of the axes CzP C$, and c’;, account for the figure three in the table. However, if the axis of C& is coincident with C$, the planes crv(xz) and uv@z) may be correlated to oh and a, in two ways. These possibilities are not included in table I. When it is taken into account that each group is a trivial subgroup of itself, all subgroups of the 41 finite groups are given, except Sr2 of Dti and SIO of DTd. Table 1 also may be used to find several supergroups of a given point group.
3. Site groups and interchange groups Far from ali the subgroups of the 41 finite groups of table 1 also are site groups. The site groups are listed in table 2. The table includes every possible site and may be used in a similar way as table 1, although the point groups now appear along a diagonal. Furthermore, there are two entries for each site group, one on each side of the diagonal. The entries on the right hand side show the multiplicities of the site groups, i.e., the number of times a given site is repeated by the point group symmetry. This number is the order of the interchange group, G,: One set of interchange groups is given to the left of the diagonal. Excluded are G; for the sites in a general position (Gr = G) and for sites with the full
symmetry of the point group (Gr= Cl). Infour cases there is no interchange group fulfilling the condition C = G, -G,.The corresponding entries in table 2 are given the sign 4. In several cases there may be more than one interchange group corresponding to a given site. Moreover, the positions of Gt and/or of GS in G may not be unique. Specification of all the interchange groups as well as of positions is given in table 3 when ambiguities exist. The interchange groups may be found algebraicafIy. As an example, two interchange groups for the C& site in Dsh are calculated. One abelian factoring ofQh is f&4] Dstl = D6 X Ci
=C~AC;XCi=C3XC~"C;XCi.
(41
The orientationa conventions are consistent with Altmann [4] . From the multiplication tables of C,, and C”,, , the following factoring may be found c~v=c~xc,=c~xc,“=c,‘xc~‘,
(51
C2h=C,XCs=C2Xci=CsXCi.
(6)
Combination of eqs. (4) and (6) and using the commutative property of the direct produtit, yield
=csxc;,=s,xc;,.
(8)
Eq. (8) is derived from (7) by using (5) and standard expressions [4] for C, and Ss. Note that the orientations in eqs. (5) and (6) are relative to the two-fold axis and therefore must be changed when applied to Q.h* The two interchange groups Cs and S, given in eq. (8) are consistent with the groups listed in table 3. Even though the algebraic procedure works, it gener-_ aily was found more convenient to find the interchange groups by systematic, geometric inspection.
4. Construction
of SALCs
The application of eq. (3) in t& formation of symmetry adapted linear combinations Is not atways straightforward. The question is which representations are to be correlated. The derivation of (3) is
-358
z
_
Table 2 Site gr&ps of 41 finite point groups. All site groups except the groups themsc&cs arc listed. The multiplicity of each site is given to the tight of the diagonal and one interchange group for each site _eroup is given to the left
. . ..e.. . . . ...8 .
...4..
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. . . . . * . . . . . . . . ;t:::::: ,m.. . . .1.
.
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L.“....
6
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b......
6
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16
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I l-.2-t
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GI
.*...**.* .
D:,: O& ..*.. .. . . * . . .. .
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02
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24
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1
24
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-1
.**...
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. 12
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. . . . * . . ,..... . . . .
c6.
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* indicates that mutual exctusion of Roman and infrared Octivities f indicates that no interchange group fulfills G = G, G,. ’ GI = G for GS = c,
otcws
for these
groups.
l
basedon
[5] G=GI.GSandon
(0 = xg’, . $“I
XR
1
where the operation k in G is the product of b and ?i? GE-and Cs, ~e~ectiy~~y. Irr she use bf eq. (3}, P&r 5 is opkted on the basis-functions of a given Ii&d a+ ploper linear combinations are formed _
(9)
belonging to the irreducible representations of GS. It follows that the representations I?’ in eq. (3) which are at’ interest, are irreducible. Furthermore, each b&s function (a pair for doubte degenerate species, etc.) in GS may be treated separately by k$& j_ _ Consider now the correlations (see applicatirlns)
E. Rytter/TabIes
of Sub-. site and interchange groUpS
359
Table 3 Site groups and interchange groups of symmetry point groups. Only those site groups are liste G which have several interchange groups or where posirionl ambiguity may occur for either the site group or the interchange group. The positions in G are given in parentheses Symmetry group, G
Site group, GS
Interchange group, GI
Oh
c4v
D3.
c3v GVG)
,
s6
~,D2d(Ci,4),Clh.Dzh(3i72)
C,(Q)
0. Td 0. Th
0
czcc;)
T
Td
c3v
D2.
CSbh)
s4
c2 cc;,
D6d
C6VA2 D6.
&h
St2
Gv C2vG) Gvc;) C&ah) c&J c&j,,
06
Gv c6h &I
C&)
c6,
‘%vbd)
C,bd)
c6,
c3vbv)
c6
Cp ci
CS
c6.
s6
Gv c2 CS
Dsh
D4d
CS G,
Ds,
csh
c4v
c2m
c2cc;>
c4-G &l &r SE
CS
D4h
c2, &iv
c4v C2vG) C,,(G) c&,) C&> c&d)
Symmetry proup,
d
Site group, GS
Interchange group, GI
I
D2h
There are four equivalent basis functions in A’, one on each site, which are to be gombined by F&i’) into basis functions of AI, B2 and .E of G. For the symmetry species E, eq. (3) becomes
which may be found ~IIthe book by Fateley et al. I?] *. -t
core the~peciaf application of 2 ijctor 2 vvhieh far so& groupsmust bcused in the correlationsbetwccn GSand G [71.
.. .
From the definition of the projection operator in eq. (2), it is obvious that
E. RyrterfTobles of sub-, sire and intercl~onge groups
this function belonging to A’ and A” of C, are projected out first. It then may be argued that the operation equally well could have been performed on, e.g., the part belonging
to A’ thus giving no contribu-
tion to A”. This is equivalent to say that each part of eq. (11) may be used separately on any function. It follows that
(‘2) Furthermore, @(Bz+B3) D2
= j%%’
the following decomposition
is valid:
+ $3’ 2
D2
The operator Pp interchanges symmetrically equivalent functions, &-re on each site. Consequently, these functions generate a regular representation with one function for each of the irreducible representations in the interchange group [9]. It is possible therefore to use each term of eq. (14) separately to form basis functions in E of D2d. This procedure takes care of orthogonalization of degenerate basis functions, but not of orientation and normalization. If the combined formula of eq. (12) is used instead, only half of the four C, basis functions
are combined
into a linear combination. To obtain a linear combination of the other half, the formula must be applied to a representative of the second ser. The two SALCs then formed constitute the appropriate degenerate pair in D,,. The correlations (see applications) W,)
W4v) /
E
Fb?
Gr(D3) -
A1+E
F 1U -
A2+E
-
@>@generates a degenerate pair and i#rj may be op?vrated on each of the degenerate fun&ions. The result will be two functions, each being a linear combination of one basis of Fzr and one of Fzu since A1(D3) correlates to these representations of G(Q Appropriate linear combinations therefore must be taken to obtain tile pure Fzg and Fzu SALCs. The easiest way to find the remaining Fzg and F2u SALCs is to apply Pzl) 1x1 . the same way, but for two other equivalent D3 subgroups of 0,. In summary, the projection operator $9 = P(r-‘)-$%(I*“) may he used to find basis function belongGI GS ing to r of G by using the operators corresponding to any of the irreducible representations of Gj and Gs that correlates to f’. However, when more than one irreducible representation in G correlate to the same representations T” and T“‘, appropriate linear combinations must be taken to discriminate between these representations in G. After the basis functions for a given site is formed by ?$“‘I, it is feasible to orient the functions on the oth& sites so that equivalent functions are transformed into each other by G,. To obtain full factoring of secular determinants, degenerate pairs must be oriented in the same way. This may be achieved by proper orientation of functions on the sites. The present method is especially powerful for the construction of vibrational symmetry coordinates for complexes with polyatomic ligands and a detailed analysis of vibrations of metal-ligand compounds will be published [ !O] . The paper will contain a systematic approach to group theoretical treatment and structure determination from vibrational spectra.
A2”E
___ F2s
’ 5”
361
A1+E
constitute a slightly more complicated example. A degenerate pair of functions on six equivalent sites are going to be combined into four triplets of degenerate basis functions. However, in this case a species in Gt correlates to more than one species in G for a given species in GS. According to the example above, each factor of the projection operator, e.g., Pr)@(-“, 03 C4v’ may be applied separately. The G, operator
5. Applications 5.1. Co(NO&This complex is assumed to have six symmetrically equivalent NO2 ligands, octahedrally coordinated to cobalt. The site group of NO2 must be a subgroup of C,, and from table 1 it is seen that the possible site symmetries are c2v, C,. C, and Ct. To find corresponding total symmetries each of these four groups is localized in table 2 and the column above each group is searched for entries with multiplicity six. The possible symmetry groups of Co(N02)~- are
362
E. RytterfTables of sub-, site and interchange gmups
found to the left of each entry and the combinations of site arid totai groups when are: C2,,(Td, Th,L&), C2(~D@D3d),
C.&v9 c&pD3&D3h)y
and cl(c6,
D,* % C3”, C3h).
The skeletal symmetry is octahedral and the total symmetry group therefore must be a subgroup of 0,. This leaves eight of the fifteen combinations above after inspection of the subgroups listed in table 1. Two more possibilities, c2(&) and C,(C,,), can be eliminated because the site groups are not consistent with both the skeletal and the total symmetries. Six combinations are left, namely C2v(Td, T,,), C,(7),
Td !--I- 1
T,, (4,)
I+. l_PossibIc Co(NO&structures. Rotation of NO7 in the symmetry groups shown ”
-the.direcrion of the arr~ws+es .. in parentheses.
Table 4 The number of NO2 scissoring and Co-N stretching fundamentals for different Co(NO& models Co(NO&symmetry
NO* site symmetry
NOa scissoring
Td
C2v
1
1
1
1
1
I
Th
c2v c2
:I
Co-N stretching
cs
2
2
L$
Cs
2
2
&I
Cl
3
3
S6
Cl
2
2
c&d) and Cl (D3, S6). All may be visualized in more than one way by orientation of the NO2 ligands
on the sites. Some simplifications are achieved if it is assumed that the nitrogen atoms and the two-fold axes of the ligands point towards cobalt. Consequently, the only freedom of the ligands is rotation about the Co-N axes. This rotation gives rise to seven different structures (fig. 1). The appearance of two D3ri isomers is reflected by the number 2 in the entry in table 1 for the C, subgroup of C2,. Three of the models (Z’, D3 and S6) contain a general angle of rotation of NO, and fig. 1 shows how they may be obtained from the other models. The structure with symmetry T may be derived from either Td or Th, the &?$6 structure from Th or Dsd and the D3 structure from either of the D3d structures. Nakagawa and Shimanouchi [ 1 I- 151 have discussed Co(NO&structures with the same assumptions as above and they erroneously stated that Th, T and S, are the only possible symmetries. They recorded the infrared spectra of solid Na3Co(N02)6 and K~CO(NO~)~ and found two NO2 scissoring and two Co-N stretching frequencies for the sodium compound compared to one of each for the potassium compound. The symmetry of the cobalt complex in K,Co(NO.& is known [ 16) to be Th and Eiakagawa and Shimanouchi attributed the splittings in Na3Co(ru’02)6 to lowering of the symmetry from Th to s6. The number of NO2 scissoring and Co-N stretching fundamentals is compared in table 4 for all seven Co(N02)i- models. It is seen that D,, sym-
E. RyttedTabks
af sub-, site
and interchatige
groups
363
C,). The intercixangegroupsD,
and 5, are given in table 3 and the correlations j7j between Gg, G and G, are shown in table 5. aeration of$?) on the basis functions ?r and n* Ieaves each of &em invariant and therefore they belong to the spoke A’ of C,_ They correlate to AL. B2, and E ofDZd which then will conrain the final SALCs. From fig. 2 it may be noticed that all operations of D2 or $4 transform a part of a basis into another part without change of sign.This isvalid for the rr as well as for the x’ basis and they therefore form equivalent SAL& Proceeding with T, operation with PEj yields @iI
=q,*
z nf f $1 + X5’*f #**,
,fs2’
qp
se r’
_
*”
+ n”’
-
n”f’9
gr
= j$y”
z r’
_
n”
-
R”’
+
n”“,
$j
zz @g&f
z It*
*
x9( _
r’,,
-
n”“_
Fig. 2. The CrC$ (&d) Structure.
metry also is consistent with the observations and at the same time possesses higher symmetry than 8,.
(19
5.2. cxgfinear comb~a~~ons a~te~a~~ve~~ are formed using $fl instead of pgj if the degenerate pair is orientedlin the same way. In Crow- there are also 7~~and rr,” orbitals orthogonal to those already treated and all these orbitals may Weract with the metal orbit&. The ?I~ and XT sets belong to the site symmetry species A” and the proper SALCs easily may be eondiucted foliowing the standard procedure given for the A orbit&.
These
Recently, Rkh and Hoffma~n [I71 perfo~~n~~ a charge-Mative extended Hkkekatcutation for this ion. The idealized equilibrium structure has ,W2e symmetry and is shown kt fig. 2 together with the numeration of the four Oi- ligands and the oxygen atoms. Each @and has two orbitals, zr and n*, directed rowxds the central atom. The orientation of the a* orbital, which may be taken as pI - p2 is dependent on the numeration of atoms I and 2 of a @and. In order to obtain symmetry adapted linear cornbinations of the orbitals, it first is observed that the site symmetry of Oi- IS * Cs (table 2 shows that the only sites with rnu~t~~~ic~ty4 have symmetry C, or
Tabie S
Correlations of the site and interchange group symmetry sputies with the irreducible representations of Dzd for C&” B &d(G)
A
A
Bt
A
A
B
5t
B
Ba+B3
E
5.3. XY6 As a more complicated example, p orbitals on the ligands of an octahedral complex are consideredlf. The orientation of the basis functions is defined in Eg. 3 and the correlations behveen the site group c&,, the total group 0, and the interchange groups D, or S, (tabbfe 3) are given in tabfe 6. Operation with the projection operators of the site group, Pdl;:, l” = AL I ___, on pX, p,,. and pz shows that pz belongs to AL and &, p,,) constitutes a degenerate pair of species E. The pz orbit& of site symmetry speciesAt corre-
* RIirry f6] uad tile saxa 5xanpte in the original paper on d~om~s~ion of prajeetion operators. Howover, he complicates the eaktiation by ~fr~u~n~ the site symmetry & in oh and the corresponding “interchange group” IJj& Furthermore, hc only finds one component of each dcgencrate basis, Nis second example, the external symmetry coordinates of the tetrahcdnl XY4 model, conkains several errors.
. _
-
364<
E. Rytterj’Tables of sub-, site and interchbngegroups
-_
Fig. 3. The defined orientations of the px andp, ligand basis functions of the xY6 octahedral model viewed along three of the
three-fold axes. The positive parts of the pz orbit& point towards the central atom.
Table 6 Correlations of the site and interchange group symmetry spe-
interchange group is used (table 6). However, it is much easier to fiid the right linear combinations for, e.g., Fzg and Fzu ( correlates to At of D3) than for Fls and I$ (correlates to A, of s6). This is because every basis function in Fzg must be transferred into itself by inversion since the character is 3 while every basis function in Ftu must be transferred into itself with opposite sign by inversion(character -3). Therefore the interchange group LI3 is employed in spite of the fact that the E part of the projection operator does not distinguish between any of the four triply degenerated species in oh. The Fls and Fzu functions are calculated as an example, but the treatment is equivalent for the Fig and F,, functions. The %I) operator is used on pX. +pY1 and perI pul instead of on Pxl and F,,~. This is just a matter of orientation of the degenerate pair and has the advantage that all 12 symmetrically equivalent functions are generated. The result with x1 representing&l, etc., is
cies with the irreducible model
PD~‘)(XICY1) = x1+X2’X3+x4+x5+Xs
late with Al,, E g, and -Flu of oh and these species therefore will contain the SAL& The linear combinations easiest are foilnd by application of the s,5 projection operators as described by Flurry [6]. However, note that each of the rivee parts of i’yu+Eu) used separately yields all the triple degenerate functions in Flu and in orthogonal form. Similar arguments are valid for the Eg functions. Alternatively, the intcrchanga group D, may be used, but since E of this group correlates to both Ep and Flu of oh it is necessary to operate with kh: on four of the symmetrically equivalent & orbital% e.g.,pzl, pz2, pz4, and pt5 _From the result the appropriate linear combinations belor,ging to Eg and F~, can be found. A similar pzocedure is necessary for the degenerate (p,,p,,) pair. At first, it may seem unimportant which
Basis
C4vGs)
representations
of 01, for the XYe
+YttY2+Y3+Y4+Y5+Y(j*
Oh(G).
Ag -
AS.
-E
Ag
A2+E
-
AI+E
As+Etz AsiEg A, Au
-
and ~~~i)(~,l-Y1):x1+*2+x3-x4-xg-xg
EP
-
A+E.
Eu A,,+E; AI;+EU
(16)
-,$-Y2-~3*~4+r,+~,.
(17)
The first of these linear combinations belongs to Fzg and the second to Fzu since the contribution to the character by kversion is +I and - 1, respectively. If the starting functions had been x1 andYI it had been necessary to take linear combinations of the two expressitins formed iri order to get the-character contrihutions +I- and -l_ &I operator, To avoid the complications with tile PD .~ 3 : --.
365
E. Rytter!Tablesofsub-,sire and inrerchangegrorrps
the P(A1) operator corr~spond~g to two other threefold %es in 0, (see fig. 3) is applied to (xl+y1) and (x1--yt). The result for atl the degenerate functions in Fzgand Fzu is
The author gratefully ac~owledges the finantiial support from GeneraIdirektdr Roif Qstbyes fund and Royal Norwegian Council for Scientific and Industria1 Research.
References [iI F.A. Cotton, Chemical Applications of Group Theory (Wiley-Interscience,
-&+&+~~-~4+~$*~fj’ $$p&) =xl+x2+x3-x4-xs-x6 - Y1-Y2-Y3+yaiy5 +y69 WLt)
=XI-XZ+X3-X4+X~-X6
+Yl -v, -u3 -y4 +$ +y6, (p,(F,J
=XI-yX3-X4+XSfX6
+Y1+~~ -u, --u,-v,
+vs-
These SALCs are consistent with those published by Gray and Beach [I 81. Neither of the functions are normalized and orthogonahzation is only achieved between the different irreducible representations since only the PErI operator was used.
New York, 1971).
121L. Jansen and M. Boon, Theory of
Fini!eGroups. Ap-
plications in Physics (North-Holland.
Amsterdam,
1967). S.L. Aitmann, Phil. Trans. Roy. Sot. A 225 (1963) 216. S.L. Altmann, Rev. Mod. Phys. 35 (1963) 641. R. Kopelman, J. Chem. Phys. 47 (1967) 2631. R-L. ~lurry,Theoret. Chim.Acta 31(1973) 221. W.G. Fat&y, F.R. Dolkh, N.T. McDevitt and F.F_ Bentley, Infrared nnd Raman Selection Rules for Molccutar and Lattice Vibrations: The Correlation hicthod OViley-Interscience, New York, 1972). [81 J.D.H. Donnay and G. Turrell, Chem. Phys. 6 (1974) 1. 191 E.B. Wilson, J.C. De&r and P.C. Cross, Molecuiar Vibrations (hfffiraw-HitI, New York, 1955). IlO1 E. Rytter, to be published. fill f. Naka~awa, T. Shim~~ouc~i and K. Yrunanki, Inorg. Chem. 3 (1964) 772. and T. Shimanouchi, 22 (1966) 1707. I131 I. Nakagawa and T. Shimanouchi,
11215. Nakagawa
Spcctrochim.
Acta
Spectrochim.
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Acknowledgement fhe author is indebted to Dr. D. Gruen for stimulating the author’s interest in coordination chemistry and for interesting discussions in connection with the present work.
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