162:
239:
1416:
1283:
254:(the element), we imagine taking each point in a molecule and then moving it out the same distance on the other side. In summary, the inversion operation projects each atom through the centre of inversion and out to the same distance on the opposite side. The inversion center is a point in space that lies in the geometric center of the molecule. As a result, all the cartesian coordinates of the atoms are inverted (i.e.
144:
The identity operation corresponds to doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity operation. The identity operation is denoted by
153:. In the identity operation, no change can be observed for the molecule. Even the most asymmetric molecule possesses the identity operation. The need for such an identity operation arises from the mathematical requirements of group theory.
774:
of an axis can be regarded as a number of times that, for the least rotation which gives an equivalent configuration, that rotation must be repeated to give a configuration identical to the original structure (i.e. a 360° or
722:
658:
1048:
965:
524:
213:
If the plane of symmetry bisects the angle between two 2-fold axes perpendicular to the principal axis, it is designated as a dihedral mirror plane, which is indicated by a subscript
883:
825:
565:
173:(sigma). Its orientation relative to the principal axis of the molecule is indicated by a subscript. The plane must pass through the molecule and cannot be completely outside it.
1487:
1120:
involves two operation steps: a proper rotation followed by reflection through a plane perpendicular to the rotation axis. The improper rotation is represented by the symbol
1504:
are additionally possible. These are rotations or reflections together with partial translation. These operations may change based on the dimensions of the crystal lattice.
1262:
1214:
477:
1083:
1001:
918:
345:
305:
131:
need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.
169:
The reflection operation is carried out with respect to symmetry elements known as planes of symmetry or mirror planes. Each such plane is denoted as
104:
is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasizes its usefulness.
1270:, not symmetry operations. The rotation axis of the highest order is known as the principal rotation axis. It is conventional to set the Cartesian
197:
If the plane of symmetry is perpendicular to the principal axis, it is designated as a horizontal mirror plane, which is indicated by a subscript
1412:. Note that if any two operations are carried out in succession the result is the same as if a single operation of the group had been performed.
1312:
rotation axis which passes through the carbon atom and the midpoints between the two hydrogen atoms and the two chlorine atoms. Define the
675:
611:
1541:
758:
molecule is rotated by 180° about an axis passing through the oxygen atom, no detectable difference before and after the
77:
of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some
1577:
1398:
366:
1010:
927:
493:
230:
Through the reflection of each mirror plane, the molecule must be able to produce an identical image of itself.
412:
1516:
355:
246:
molecule. All of the fluorine atoms change their position to opposite side with respect to the sulfur center
439:
851:
793:
533:
1639:
729:
Here the molecule can be rotated into equivalent positions around an axis. An example of a molecule with
1444:
1131:
is the order. Since the improper rotation is the combination of a proper rotation and a reflection,
1223:
54:
31:
of an object that leaves the object looking the same after it has been carried out. For example, a
28:
1360:
rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the
1178:
447:
1593:
1056:
974:
891:
66:
1432:. In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular
314:
277:
1087:
is the identical configuration because it gives the original structure, and it is called an
177:
If the plane of symmetry contains the principal axis of the molecule (i.e., the molecular
161:
8:
243:
109:
1620:
1531:
1317:
181:-axis), it is designated as a vertical mirror plane, which is indicated by a subscript
120:
85:
1644:
1573:
1536:
1117:
392:
50:
1551:
1508:
1501:
1267:
354:
Examples of molecules that have an inversion center include certain molecules with
78:
74:
46:
1289:
70:
1572:. Great Britain Oxford University Press: W.H. Freeman and Company. p. 404.
1441:
axes which pass half-way between the C-H bonds and six mirror planes. Note that
1633:
128:
1520:
116:
93:
20:
1519:
with the addition symmetry operations produce the 230 crystallographic
1497:
238:
1282:
1415:
377:
97:
62:
42:
115:
Symmetry operations can be collected together in groups which are
1422:
101:
1368:
plane permutes the chlorine atoms. The four symmetry operations
425:
1313:
58:
1274:-axis of the molecule to contain the principal rotation axis.
739:
1364:
plane permutes the hydrogen atoms while reflection in the
16:
Geometric transformation which produces an identical image
1171:, an inversion operation about an inversion center. When
1266:
Rotation axes, mirror planes and inversion centres are
717:{\displaystyle {\tfrac {360^{\circ }}{3}}=120^{\circ },}
653:{\displaystyle {\tfrac {360^{\circ }}{2}}=180^{\circ },}
391:). Examples of molecules without inversion centers are
1515:
symmetry operations. Combinations of operations of the
1021:
938:
856:
798:
680:
616:
538:
498:
1447:
1226:
1181:
1059:
1013:
977:
930:
894:
854:
796:
678:
614:
536:
496:
450:
317:
280:
1104:
is an order of three, and is often referred to as a
262:). The symbol used to represent inversion center is
156:
1481:
1256:
1208:
1077:
1042:
995:
959:
912:
877:
819:
716:
652:
559:
518:
471:
339:
299:
1631:
1158:, a reflection operation about a mirror plane.
1111:
266:. When the inversion operation is carried out
250:In an inversion through a centre of symmetry,
127:In the context of molecular symmetry, quantum
1145:and a perpendicular plane exist separately.
1043:{\displaystyle 3\times {\tfrac {2\pi }{3}}.}
960:{\displaystyle 2\times {\tfrac {2\pi }{3}},}
519:{\displaystyle {\tfrac {360^{\circ }}{n}}}
837:represents the first rotation around the
242:Inversion operation is shown here with a
237:
160:
1632:
1614:
1567:
233:
1617:Chemical Applications of Group Theory
1555:Chemical applications of group theory
139:
1610:
1608:
1606:
1542:Crystallographic restriction theorem
878:{\displaystyle {\tfrac {2\pi }{3}},}
820:{\displaystyle {\tfrac {2\pi }{3}}.}
593:. It is equivalent to the Identity (
560:{\displaystyle {\tfrac {2\pi }{n}},}
112:with respect to symmetry operations.
13:
1594:"Symmetry elements and operations"
1511:may be considered as representing
1414:
1281:
788:proper rotation, which rotates by
586:is a rotation through 360°, where
577:is omitted if it is equal to one.
14:
1656:
1603:
779:rotation). An example of this is
442:. Such operations are denoted by
1482:{\displaystyle S_{4}^{2}=C_{2}.}
157:Reflection through mirror planes
1586:
1561:
426:Proper rotation operations or
1:
1546:
1517:crystallographic point groups
1257:{\displaystyle S_{n}^{2n}=E.}
1209:{\displaystyle S_{n}^{n}=E,}
1112:Improper rotation operations
134:
108:Physical properties must be
7:
1526:
1491:
1277:
1138:will always exist whenever
413:trigonal pyramidal geometry
10:
1661:
1570:ATKINS' PHYSICAL CHEMISTRY
670:is a rotation of 120°, as
606:is a rotation of 180°, as
472:{\displaystyle C_{n}^{m},}
81:(a point, line or plane).
1078:{\displaystyle C_{3}^{3}}
996:{\displaystyle C_{3}^{3}}
913:{\displaystyle C_{3}^{2}}
767:operation is observed.
340:{\displaystyle i^{n}=-E}
270:times, it is denoted by
29:geometric transformation
1615:Cotton, Albert (1990).
573:times. The superscript
300:{\displaystyle i^{n}=E}
96:of atoms such that the
1568:Atkins, Peter (2006).
1483:
1419:
1286:
1258:
1210:
1167:is usually denoted as
1154:is usually denoted as
1079:
1044:
997:
961:
914:
879:
821:
718:
654:
561:
520:
473:
440:rotation about an axis
367:square planar geometry
341:
301:
247:
166:
1484:
1418:
1333:plane as containing
1285:
1259:
1211:
1080:
1045:
998:
962:
915:
880:
822:
719:
655:
562:
521:
474:
411:) and molecules with
342:
302:
241:
164:
1557:, Wiley, 1962, 1971
1445:
1344:plane as containing
1224:
1179:
1057:
1011:
975:
928:
892:
852:
794:
676:
612:
534:
494:
448:
315:
278:
165:Reflection operation
1462:
1244:
1196:
1074:
1005:is the rotation by
992:
922:is the rotation by
909:
750:) molecule. If the
465:
356:octahedral geometry
244:sulfur hexafluoride
234:Inversion operation
1640:Physical chemistry
1621:Wiley-Interscience
1532:Molecular symmetry
1479:
1448:
1420:
1287:
1254:
1227:
1206:
1182:
1175:is an even number
1075:
1060:
1040:
1035:
993:
978:
957:
952:
910:
895:
875:
870:
817:
812:
714:
696:
650:
632:
557:
552:
516:
514:
469:
451:
337:
297:
248:
167:
140:Identity Operation
121:permutation groups
90:symmetry operation
86:molecular symmetry
84:In the context of
25:symmetry operation
1619:. United States:
1537:Crystal structure
1502:glide reflections
1268:symmetry elements
1118:improper rotation
1034:
951:
869:
811:
695:
631:
551:
513:
488:is a rotation of
438:refers to simple
415:(general formula
393:cyclopentadienide
369:(general formula
358:(general formula
1652:
1625:
1624:
1612:
1601:
1600:
1598:
1590:
1584:
1583:
1565:
1509:Bravais lattices
1488:
1486:
1485:
1480:
1475:
1474:
1461:
1456:
1440:
1431:
1411:
1396:
1388:
1380:
1371:
1367:
1363:
1359:
1350:
1343:
1339:
1332:
1328:
1311:
1302:
1273:
1263:
1261:
1260:
1255:
1243:
1235:
1219:
1215:
1213:
1212:
1207:
1195:
1190:
1174:
1170:
1166:
1157:
1153:
1144:
1137:
1130:
1126:
1103:
1094:
1089:identity element
1086:
1084:
1082:
1081:
1076:
1073:
1068:
1051:
1049:
1047:
1046:
1041:
1036:
1030:
1022:
1004:
1002:
1000:
999:
994:
991:
986:
968:
966:
964:
963:
958:
953:
947:
939:
921:
919:
917:
916:
911:
908:
903:
886:
884:
882:
881:
876:
871:
865:
857:
845:
836:
828:
826:
824:
823:
818:
813:
807:
799:
787:
778:
773:
766:
757:
749:
738:symmetry is the
737:
725:
723:
721:
720:
715:
710:
709:
697:
691:
690:
681:
669:
661:
659:
657:
656:
651:
646:
645:
633:
627:
626:
617:
605:
596:
592:
585:
576:
572:
568:
566:
564:
563:
558:
553:
547:
539:
527:
525:
523:
522:
517:
515:
509:
508:
499:
487:
480:
478:
476:
475:
470:
464:
459:
421:
410:
409:
408:
405:
390:
375:
364:
350:
346:
344:
343:
338:
327:
326:
310:
306:
304:
303:
298:
290:
289:
273:
269:
265:
261:
257:
253:
225:
216:
209:
200:
193:
184:
180:
172:
152:
148:
79:symmetry element
75:point reflection
47:regular triangle
40:
39:
35:
1660:
1659:
1655:
1654:
1653:
1651:
1650:
1649:
1630:
1629:
1628:
1613:
1604:
1596:
1592:
1591:
1587:
1580:
1566:
1562:
1549:
1529:
1498:screw rotations
1494:
1470:
1466:
1457:
1452:
1446:
1443:
1442:
1439:
1433:
1430:
1426:
1410:
1401:
1390:
1382:
1379:
1373:
1369:
1365:
1361:
1358:
1352:
1349:
1345:
1341:
1338:
1334:
1330:
1327:
1321:
1310:
1304:
1301:
1297:
1293:
1290:Dichloromethane
1280:
1271:
1236:
1231:
1225:
1222:
1221:
1217:
1191:
1186:
1180:
1177:
1176:
1172:
1168:
1165:
1159:
1155:
1152:
1146:
1143:
1139:
1136:
1132:
1128:
1125:
1121:
1114:
1102:
1096:
1092:
1069:
1064:
1058:
1055:
1054:
1052:
1023:
1020:
1012:
1009:
1008:
1006:
987:
982:
976:
973:
972:
970:
940:
937:
929:
926:
925:
923:
904:
899:
893:
890:
889:
887:
858:
855:
853:
850:
849:
847:
844:
838:
835:
829:
800:
797:
795:
792:
791:
789:
786:
780:
776:
771:
765:
759:
755:
751:
747:
743:
736:
730:
705:
701:
686:
682:
679:
677:
674:
673:
671:
668:
662:
641:
637:
622:
618:
615:
613:
610:
609:
607:
604:
598:
594:
587:
584:
578:
574:
570:
540:
537:
535:
532:
531:
529:
504:
500:
497:
495:
492:
491:
489:
486:
482:
460:
455:
449:
446:
445:
443:
436:proper rotation
432:
420:
416:
406:
403:
402:
400:
396:
389:
385:
381:
374:
370:
363:
359:
348:
322:
318:
316:
313:
312:
308:
285:
281:
279:
276:
275:
271:
267:
263:
259:
255:
251:
236:
224:
218:
214:
208:
202:
198:
192:
186:
182:
178:
170:
159:
150:
146:
142:
137:
71:Euclidean plane
37:
33:
32:
17:
12:
11:
5:
1658:
1648:
1647:
1642:
1627:
1626:
1602:
1585:
1578:
1559:
1548:
1545:
1528:
1525:
1493:
1490:
1478:
1473:
1469:
1465:
1460:
1455:
1451:
1437:
1428:
1405:
1377:
1356:
1347:
1336:
1325:
1308:
1299:
1295:
1279:
1276:
1253:
1250:
1247:
1242:
1239:
1234:
1230:
1205:
1202:
1199:
1194:
1189:
1185:
1163:
1150:
1141:
1134:
1123:
1113:
1110:
1100:
1095:). Therefore,
1072:
1067:
1063:
1039:
1033:
1029:
1026:
1019:
1016:
990:
985:
981:
956:
950:
946:
943:
936:
933:
907:
902:
898:
874:
868:
864:
861:
842:
833:
816:
810:
806:
803:
784:
763:
753:
745:
734:
713:
708:
704:
700:
694:
689:
685:
666:
649:
644:
640:
636:
630:
625:
621:
602:
582:
556:
550:
546:
543:
512:
507:
503:
484:
468:
463:
458:
454:
431:
430:-fold rotation
424:
418:
398:
387:
383:
372:
361:
336:
333:
330:
325:
321:
296:
293:
288:
284:
235:
232:
228:
227:
220:
211:
204:
195:
188:
158:
155:
141:
138:
136:
133:
125:
124:
113:
15:
9:
6:
4:
3:
2:
1657:
1646:
1643:
1641:
1638:
1637:
1635:
1623:. p. 23.
1622:
1618:
1611:
1609:
1607:
1595:
1589:
1581:
1579:0-7167-8759-8
1575:
1571:
1564:
1560:
1558:
1556:
1553:
1544:
1543:
1539:
1538:
1534:
1533:
1524:
1522:
1518:
1514:
1513:translational
1510:
1505:
1503:
1499:
1496:In crystals,
1489:
1476:
1471:
1467:
1463:
1458:
1453:
1449:
1436:
1424:
1417:
1413:
1409:
1404:
1400:
1394:
1386:
1376:
1355:
1324:
1319:
1315:
1307:
1303:. There is a
1291:
1284:
1275:
1269:
1264:
1251:
1248:
1245:
1240:
1237:
1232:
1228:
1203:
1200:
1197:
1192:
1187:
1183:
1162:
1149:
1119:
1109:
1107:
1099:
1090:
1070:
1065:
1061:
1037:
1031:
1027:
1024:
1017:
1014:
988:
983:
979:
954:
948:
944:
941:
934:
931:
905:
900:
896:
872:
866:
862:
859:
841:
832:
814:
808:
804:
801:
783:
768:
762:
741:
733:
727:
711:
706:
702:
698:
692:
687:
683:
665:
647:
642:
638:
634:
628:
623:
619:
601:
597:) operation.
590:
581:
554:
548:
544:
541:
510:
505:
501:
466:
461:
456:
452:
441:
437:
429:
423:
414:
394:
379:
368:
357:
352:
334:
331:
328:
323:
319:
294:
291:
286:
282:
245:
240:
231:
223:
212:
207:
196:
191:
176:
175:
174:
163:
154:
132:
130:
129:wavefunctions
122:
118:
114:
111:
107:
106:
105:
103:
99:
95:
91:
87:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
30:
26:
22:
1616:
1588:
1569:
1563:
1554:
1552:F. A. Cotton
1550:
1540:
1535:
1530:
1521:space groups
1512:
1506:
1495:
1434:
1421:
1407:
1402:
1392:
1384:
1374:
1353:
1322:
1305:
1288:
1265:
1160:
1147:
1115:
1105:
1097:
1088:
839:
830:
781:
769:
760:
731:
728:
726:and so on.
663:
599:
588:
579:
435:
433:
427:
353:
311:is even and
249:
229:
221:
205:
189:
168:
143:
126:
89:
83:
24:
18:
1399:point group
94:permutation
67:translation
61:across its
21:mathematics
1634:Categories
1547:References
1329:axis, the
569:performed
117:isomorphic
55:reflection
49:about its
1397:form the
1340:and the
1320:with the
1318:co-linear
1216:but when
1106:threefold
1028:π
1018:×
945:π
935:×
863:π
805:π
707:∘
688:∘
643:∘
624:∘
545:π
506:∘
351:is odd.
332:−
135:Molecules
110:invariant
1645:Symmetry
1527:See also
1492:Crystals
1278:Examples
846:axis by
378:ethylene
274:, where
260:–x,–y,–z
98:molecule
63:diagonal
43:rotation
1500:and/or
1423:Methane
1220:is odd
1085:
1053:
1050:
1007:
1003:
971:
967:
924:
920:
888:
885:
848:
827:
790:
724:
672:
660:
608:
567:
530:
526:
490:
479:
444:
376:), and
102:crystal
73:, or a
69:of the
36:⁄
1576:
1314:z axis
1127:where
1108:axis.
969:while
770:Order
481:where
59:square
51:center
1597:(PDF)
740:water
347:when
307:when
256:x,y,z
92:is a
57:of a
45:of a
41:turn
27:is a
1574:ISBN
1507:The
1389:and
1351:. A
386:C=CH
88:, a
65:, a
53:, a
23:, a
1346:CCl
1316:as
1116:An
703:120
684:360
639:180
620:360
591:= 1
528:or
502:360
422:).
365:),
258:to
149:or
119:to
100:or
19:In
1636::
1605:^
1523:.
1427:CH
1425:,
1393:yz
1391:σ(
1385:xz
1383:σ(
1381:,
1372:,
1366:xz
1362:yz
1342:yz
1335:CH
1331:xz
1298:Cl
1294:CH
1292:,
434:A
417:AB
371:AB
360:AB
226:).
210:).
194:).
1599:.
1582:.
1477:.
1472:2
1468:C
1464:=
1459:2
1454:4
1450:S
1438:4
1435:S
1429:4
1408:v
1406:2
1403:C
1395:)
1387:)
1378:2
1375:C
1370:E
1357:2
1354:C
1348:2
1337:2
1326:2
1323:C
1309:2
1306:C
1300:2
1296:2
1272:z
1252:.
1249:E
1246:=
1241:n
1238:2
1233:n
1229:S
1218:n
1204:,
1201:E
1198:=
1193:n
1188:n
1184:S
1173:n
1169:i
1164:2
1161:S
1156:σ
1151:1
1148:S
1142:n
1140:C
1135:n
1133:S
1129:n
1124:n
1122:S
1101:3
1098:C
1093:E
1091:(
1071:3
1066:3
1062:C
1038:.
1032:3
1025:2
1015:3
989:3
984:3
980:C
955:,
949:3
942:2
932:2
906:2
901:3
897:C
873:,
867:3
860:2
843:3
840:C
834:3
831:C
815:.
809:3
802:2
785:3
782:C
777:π
775:2
772:n
764:2
761:C
756:O
754:2
752:H
748:O
746:2
744:H
742:(
735:2
732:C
712:,
699:=
693:3
667:3
664:C
648:,
635:=
629:2
603:2
600:C
595:E
589:n
583:1
580:C
575:m
571:m
555:,
549:n
542:2
511:n
485:n
483:C
467:,
462:m
457:n
453:C
428:n
419:3
407:5
404:−
401:H
399:5
397:C
395:(
388:2
384:2
382:H
380:(
373:4
362:6
349:n
335:E
329:=
324:n
320:i
309:n
295:E
292:=
287:n
283:i
272:i
268:n
264:i
252:i
222:d
219:σ
217:(
215:d
206:h
203:σ
201:(
199:h
190:v
187:σ
185:(
183:v
179:z
171:σ
151:I
147:E
123:.
38:3
34:1
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