Crystallographic and Spectroscopic Symmetry Notations B. D. Sharma California State University, Los Angeles, CA 90032 and Los Angeles Pierce College, Woodland Hills, CA 91371 Applications of symmetry are of fundamental importance t o chemists (1,2,3). The development of the ideas of symmetry has rested in the hands of mathematicians and crystallographers. The notation used by spectroscopists is that of Schijenflies (4). This notation is most common. Crystallographers use the Hermann-Maupuin notations (5). Even though hoth notations describe the same symmetry. there are distinct differences in the manner of dexrilnion. This difference can lead to confusion in correlating the two notations. This is particularly true for the correlation of the spectroscopist's alternating axes (rotatory-reflection axes) and crystallographer's inversion axes. An article in THIS JOURNAL (6) while pointing out common errors in the description of these axes, points to the demonstration of Sr, by the combination of Ca and ah as a ooint of confusion. This is orimarilv the r< oithe la~k~fcieardistinction between themeanin& of the Schoenflies notation and Hermann-Maueuin rlotatlm. It so happens that the symbols Sa, C4, a i d r b in the Schoenflies notation have two different meanings, for each of these symbols. The notation of Hermann-Mauguin has just one meaning for each of the symbols it employs. Hermann-Mauguin Notation In this notation no attention is paid torepresent the mirror o~eration.the rotation ooeration and the inversion ooeration t;y any diitinctive symh;,ls. This point has been addiessed to hv Donnav. and Donnav 17). Each svmbol in the Hermanni\;lauguin notation refeistd the symmetry element. For example, the symbol m means the mirror plane (not mirror operation) perpendicular to onefold axis. I t is understood that if a mirror plane exists, one is carrying out the mirror operation. The symbol 2 refers to a twofold rotation axis. The existence of a twofold rotation axis and a mirror plane perpendicular to this twofold rotation axis is implied by
-.m2
The
symbol Zis the fourfold inversion axis. Finally, the symbol i(i) r&mesents the center of inversion (center uf symmetry,. The thirty-two possible crystallograuhic point groups are represented in a manner as to show the combinationof essential symmetry elements inherent in each point group. There are no special symbols for each of the point groups or for the operations. Sch6enflies Notation In the Schoenflies notation, a t the very outset, one needs t o reckon with two different ways of writing the symbols, namely the boldface symbols (or italic symbols) and the ordinary symbols. These symbols employing the same letter and number mean two entirely different ideas. The symbol Cp stands for the point group, and C4 stands for both the fourfold rotation axis and the operation involving the rotation of 2d4. Thus the symbol C, implies the n-fold symmetry axis and the n-fold rotation operation. An operation is distinct from the symmetry element. The operation C, is further differentiated by the symbolism C,k to mean rotation of 2 n k h . The symbol a stands for both the mirror plane and mirror operation. These dual meanings, if not kept in mind, can lead to confusion as the carrying out of a particular operation does not mean the existence of the corresponding symmetry element. 554
Journal of Chemical Education
The same holds true for the inversion operation and center of inversion (center of svmmetrvj symbolized a i i. Finally, the symbol S, describes the poinfgr&p with just n-fold alternating (rotatory-reflection) axis. The symbol S, means both the n-fold alternating (rotatory-reflection) axis and the operation of 2aln rotation combined with mirror operation in a horizontal plane perpendicular to the line a l m i which the rotation operation was envisared. Note that we did not use the words "mirror plane." e his-distinction of an operation as opposed to symmetry element is of paramount importance. The symbol S,, specifically refers to the operation in which one combines a rotation operation of 2nkln with the mirror operation in a horizontal plane in the manner described above. Combination of Operations There are only three operations, namely the mirror operation, the rotation operation, and the inversion operation. Combination of a mirror operation in a horizontal plane with inversion operation through a point in this plane is instructive of the differences in the meaning of operation and symmetry element that we have referred to above. In an &thogonal coordinate system with xy plane as the horizontal plane, the combination of oh operation and inversion operation i implies the following. i
'h
XYZ-(X,Y, operatlo"
-Z)-(-X, operation
-y,z)
The resulting coordinates are xyz and ( - x , - y , 2). This set corresponds to the existence of a twofold rotation axis coinciding with the z-axis of the orthogonal coordinate system. In other words, we are looking at a finite object belonging to the point group Cz (Schoenflies notation) or 2 (Hermann-Mauguin notation). This point group does not contain a mirror plane and a center of inversion (center of symmetry). However, a finite object belonging to just the point group Cz (or 2) can be brought into self-confidence by the combination of a mirror operation and an inversion operation, in either
.,.--.. n.rlnr
In matrix notation the transformation i i given by the product of the transformation matrices corresponding to the mirror ooeration and the inversion ooeration. in either order. This is shown below:
0 -1
i operation
0 0 -1 ah operation
Cp operation
or
rh
operation
i operation
C2 operation
On the other band, the combmation of horizontal mirror plane
oh and center of inversion (center of symmetry) i implies the following.
orientation of the original motif. Renetition of the onerations d t w r i l ~ t dahwe brings ahout the ~~Wcuincidence. 'l'ltt. onlrr uithe combinatio~lor the two unerationi 1s immaterial: thus. one may first invert the motif through a point to give imagi: nary motif and then rotate n-fold counterclockwise this imaginary motif, to give a real motif which is indistinguishable from the original orientation of the motif. The transformation matrix is
This gives the following coordinates which correspond to the point group C2h (Schoenflies notation) or 2 / m (Hermann-Mauguin). We shall not attempt to arrive at symmetry equivalence for all combinations since definitive works exist in this area (4, 5). However, we shall now present the correlation of the crystallographic inversion axes with spectroscopist's alternating axes (rotatory-reflection axes). In so doing we shall show that the Sq operation is indeed a comhination of a,, (mirror operation) and Cq (4-fold rotation operation), which is different from the combination of ah (mirror plane) and Cq (4-fold rotation axes). Spectroscopist's Alternating (Rotatory-Reflection) Axes Point groups possessing just alternating axes, symbolized as S,, involve an n-fold, counterclockwise rotation of a motif to give an imaeinarv motif and immediate reflection of this im&inar) motif it; the plane perpendirulnr to the line id rotarion tori\,e re01 motif \i,hich is indistineuishnhlr from the originnl orienlntiun ot'the motif. Oprr,it~unsare conrinued in this nlanner to arrive at self-coincidrntw. Thc order ui two operations is immaterial. One may reflect the motif in a horizontal plane to give imaginary reflected motif and rotate the imaginary reflected motif by n-fold counterclockwise rotation to give real motif. The transformation matrix is
[:a i]
cos a -sin a
[.si; a
ah
operation
-,.
H]
cos a -sin a
=
C, operation
: a
I]
-[.:
-em a
cos a -sin a [si;a
cO
[
i
a
C, operation
][
1 0
ah
;] =
operation
iSi;.
cos a -sin a
0
S, operation
where a is 2 r l n and rotation is counterclockwise, (facing the positive sense of z-axis of the orthogonal coordinate system). I t is, therefore, clear that S, operation is a comhination of ah operation and C, operation. The repetition of S, operation generates the S, point group. However, it must be kept in mind that the combination of ah symmetry element (mirror plane) and C, symmetry element (n-fold rotation axis) leads to the point group CA, which contains the symmetry element
S" .
-
0
Differences between Crystallographic lnversion Axes and the Spectroscopist's Alternating (Rotatory-Reflection) Axes The operations found in ii and S , point groups are clearly different from each other. All the S , and ii point groups are not unique. In Tahle 1, we relate S , to Ti and equivalent symmetry. Tahle 2 gives the relation of Ti to S , and equivalent symmetry. I t is to be noted that S , point groups with n odd are, in fact, C,h point groups. The Ti point groups with n odd Table 1. Relatlon of S. to ii and Equivalent Symmetrya
5-2
s3 S4
ii
Equivalent Symmetry
2
CS
1
i(i)Center of
e
CS.
-
);(
-
-
10
Se
14
3
S7
8
So
SIO SII
SIP
-
):(
permitted in crystals)
Cs1(3Xl)
(i)
):( y
(E)
Sf4
S-s
Sm
Unique
Cn, (not permined in crystals) Unique (not permitted in crystals) 18 Csn permitted in crystair) 5 C5)(5X 1)(not permitted in crystals) 11 22 Clrh (;;;)(not permitted in crystals.) 12 Unique (not permined in crystals) 26 Cia. ( E i ( n d pemined in cryrtais) 7 C,, (7 X 1)(not permitted in crystals) 30 C,,, (not permined in crystals) 16 Unique (not permitted in crystais)
-
Sa
inversion or Center of Symmetq
(i)
C5,
Ss
513
Crystallographic lnversion Axes . . I'oinr groupi possessing just inversion axes, symboii7ed as ii involve an 11 -fold counterclockwisc rotation of a motif to ti\,e an imaginary motif and immediate inversion of this imaginary motif through a point about which the point group is generated to give a real motif which is indistinguishable from the
sin a
where u is (2aln) and rotation is counterclockwise, (facing the positive sense of the z-axis of the orthogonal coordinate system).
S,
81
!IX[-j -: :] -cos a
Sn
cos a -sin a 0
81
0 -1
S, operation
or
sin a
-
1
'The reader's mention io drawn to J. D.H. Donnay. Jwmslof Washinmon Academy 01SCMCBS, 25, 476 (19351 and referencestherein.
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Number 7 July 1982
555
Relation of iito
Table 2.
-1n
S, S2
2
s,
-
-3 -45 = -6 7 8 -
1
9
-
lo
11 12 13 14 15 16
1
-
S . and Equivalent Symmetry a
Equivalent Symmetry i(center of symmetry or center ot inversion) 1
(Cs) m
s,
3 x i(c3d Unique s,o 5 x i(C,) (not permined in crystals) 3 Sp -(Can) m S,, 7 X i (C,d (not permitted in crystals) S. Unique (not permitted in crystals) SIB 9 X 1 (Cgd(not permitted in crystals) 5 Ss - (CSh)(not permined in crystals) m SZ2 11 X i (C,,,) (not permined in crystals) ST2 Unique (not permined in crystals) see 13 x i(cqad(not permitted in crystals)
S4
7 (C7d (not permitted in crystals) m
S7 SSa
S,.
15 X i (Clsi) (not permined in oystals) Unique (not permined in crystals)
* T k rea4w'b attention is drawn to J. D. H. Wnnay, OISC~~~B 25.476 S,
Thecoordinates (x,y,z); (-y,x, -2); (-x,-y,z);and(y,;x, -2) constitute a collection of points in a motif correspondmg to the existence of the symmetry axis S4 in the point group Sa. S42 operation is CZoperation and S44 operation is the ever present identity operation E.
Jwrmlof Wsh;ngton Acadsmy
(1935) and references therein.
are C,; point groups. S, point groups with n even can be subdivided into two classes, depending upon whether n is a multiple of 4 or not. S, point groups with n even but not multiple of 4 are C., point groups. ii point groups with n even 11
but not multiple of 4 are C,?h point groups. The most important result is that both ?i and S. point groups with n multiple of 4 are identical and are unique. The only inversion axes or alternating axes (rotatory-reflection axes) that are of con_sequen_ceare those with&multipkof 4. Thus, we have Sq = 4; S g 18; SI2= 12; S m 16; S 2 0 1 20 and so on. In these point groups there is no n-fold rotation axis, no mirror plane, and no center of inversion (center of symmetry), even though these can be generated by either the combination of oh operation with C, operation or the combination of the inversion operation with n-fold rotation operation and repetition of these combinations. These point groups contain, coincident with the alternating axes or inversion axes, two-fold (Cz), n12-fold Cn, and nl4-fold C,ro-4 tation symmetry axes. 2 The transformations of a general point, other than thepoint about which the point group is generated, in Sq 1 4 point groups by S, operation and the operation due to 4 axis are as follows.
-
Thecoordinates (x, y,z); (y, -x, -2); (-x, -y,z);and (-y, x, -2)-as a set represent the symmetry axis 4 in the point group 4. These coordinates are identical to the ones derived for the Sqpoint group. However, the operation S4' is not the same as 41. In fact, it is theoperation 43 that correspo~dsto S41and Sd3c_orrespondsto 4l. The operation Sd2= CZ= 42 and S44 = E = 44. Using this method one can show that point groups 4 S4;$ Sat;1 2 L S12and SO on. An actual molecule that has been studied extensively and is of practical use, that possesses the symmetry axis 4 = Sq is S4N4 (8).
- -
556
Journal of Chemical Education
Acknowledgment
Literature Cited
The author is grateful to California Institute of Technology for continued access to the library facilities. The author would be remiss if special acknowledgment is not made to Professor Galen D. Stucky for sending reprints of papers of Professor J. D. H. Donnay along with reviews of the original version of this paper, at which time the author was not aware of these papers.
111 Railhausen. C.J. and Gray. H. B.,"Mc,ledar OrbitalTheory."The Benjamin Cummins, 1964. 121 Cottmi. P.A., "Chemical Applications of Group Theory." lnterseienee Puhliahera, N e w Ysrk, 1971. 131 Jaffe. H.H. and Orchin. M.. '"Symmetry in Chemistry: John Wiley & Sons, N.Y., 1965. 141 Wilmn, R., D e c k J. C.. and Cnas. P. C.. "Molecular Vibrations: Mffirsv-Hill Bmk Curnpany, 1961.Chap5. (5) Henry,N. F.M. and I.unsdale, K.. IEdiloral."lnfernatiunalTsbles EorX~RayCrystalhgaphy." Vol 1.2nd Ed.,Kynoch Prerr, Birminpham, England, 1965. 161 Ladd. M. P.C.. J. CHEMRDUC.,57.686 119801. 171 Dennay, J. D. H.and Dunnsy.G.,Aclo Cryslollo~.,A28,SllO119721. 181 Sharma. R. D.and 0snuhue.J.Aclo Cryrlollngr., 16,891 11963).
.'
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Number 7
July 1982
557
