289:
99:
265:, there must be a point in the center of an object that is the inversion center. Inversion consists of passing each point through the center of inversion and out to the same distance on the other side of the molecule. In the inversion operation for 3D coordinates, the inversion center is the origin (0,0,0). When an object is inverted, the position vector of a point in an object, ⟨x,y,z⟩, is inverted to ⟨-x,-y,-z⟩.
231:
inversion about a point on the axis. These definitions are equivalent because inversion about a point is equivalent to rotation by 180° about any axis, followed by mirroring about a plane perpendicular to that axis. The symmetry elements for improper rotation are the rotation axis, and either the mirror plane, the inversion point, or both. The improper rotation group of order 2
277:
152:
93:. It corresponds to an operation of 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 element. An object having no symmetry elements other than E is called asymmetric. Such an object is necessarily chiral.
230:
is the composition of a rotation about an axis, and reflection in a plane perpendicular to that axis. The order in which the rotation and reflection are performed does not matter (that is, these operations commute). Improper rotation is also defined as the composition of a rotation about an axis, and
288:
173:
Rotational symmetry, also known as radial symmetry, is represented by an axis about which the object rotates in its corresponding symmetry operation. A group of proper rotations is denoted as
79:
of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.
415:"Definition of symmetry elements in space groups and point groups. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry"
75:
of the object. The elements of this symmetry group should not be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of
480:
366:
127:. In a molecule that also has an axis of symmetry, a mirror plane that includes the axis is called a vertical mirror plane and is labeled
532:
505:
304:
symmetry. Its symmetry elements are: a vertical rotation axis, a horizontal plane, and an inversion point at the center.
103:
71:
employing the symmetry element that leave the object unchanged. The set containing these operations form one of the
76:
330:
118:
276:
141:. A vertical mirror plane that bisects the angle between two C2 axes is called a dihedral mirror plane,
98:
335:
68:
168:
44:
500:. Published in Great Britain by Oxford University Press: W.H. Freeman and Company. p. 405.
562:
538:
528:
501:
476:
436:
362:
356:
227:
468:
426:
256:
56:
52:
32:
414:
325:
134:, while one perpendicular to the axis is called a horizontal mirror plane and is labeled
40:
36:
24:
72:
556:
320:
220:
48:
67:, a symmetry element corresponds to a set of symmetry operations, which are the
385:
431:
440:
542:
472:
294:
20:
151:
522:
315:
60:
64:
87:
The identity symmetry element is found in all objects and is denoted
16:
Point, line, or plane about which a molecule or crystal is symmetric
182:, where the degrees of rotation that restore the object is 360/n (
150:
97:
106:
molecule is asymmetric: it has no symmetries except the identity.
47:
can take place. In particular, a symmetry element can be a
527:. Elaine Moore (3rd ed.). Boca Raton: CRC Press.
380:
378:
219:
notation is also used for the related, more abstract,
408:
406:
8:
524:Solid state chemistry : an introduction
458:
456:
430:
465:Space Groups for Solid State Scientists
347:
272:
7:
463:Burns, Gerald; Glazer, A.M. (2013).
413:Wolff, P.M. de; et al. (1989).
390:Online Dictionary of Crystallography
355:Robert G. Mortimer (10 June 2005).
55:(either proper and improper), or a
358:Mathematics for Physical Chemistry
14:
361:. Academic Press. pp. 276–.
282:Example of vertical mirror plane.
419:Acta Crystallographica Section A
287:
275:
1:
123:Mirror planes are denoted by
104:bromochlorofluoroiodomethane
297:molecule, an object having
59:. For an object such as a
579:
498:ATKINS' PHYSICAL CHEMISTRY
254:
166:
116:
432:10.1107/S0108767389002230
331:Hermann-Mauguin notation
119:Reflection (mathematics)
473:10.1016/c2011-0-05712-5
261:For inversion, denoted
521:Smart, Lesley (2005).
496:Atkins, Peter (2006).
210:= 72º rotation). The
159:
107:
154:
101:
69:rigid transformations
336:Schoenflies notation
155:Mirror planes of XeF
169:Rotational symmetry
163:Rotational symmetry
57:center of inversion
45:symmetry operations
386:"Symmetry element"
160:
108:
482:978-0-12-394400-9
368:978-0-08-049288-9
228:improper rotation
196:= 120º rotation,
189:= 180º rotation,
570:
547:
546:
518:
512:
511:
493:
487:
486:
460:
451:
450:
448:
447:
434:
410:
401:
400:
398:
397:
382:
373:
372:
352:
291:
279:
257:Point reflection
203:= 90º rotation,
53:axis of rotation
29:symmetry element
578:
577:
573:
572:
571:
569:
568:
567:
553:
552:
551:
550:
535:
520:
519:
515:
508:
495:
494:
490:
483:
462:
461:
454:
445:
443:
412:
411:
404:
395:
393:
384:
383:
376:
369:
354:
353:
349:
344:
326:Crystallography
312:
305:
303:
292:
283:
280:
271:
259:
253:
245:
216:
208:
201:
194:
187:
179:
171:
165:
158:
146:
139:
132:
121:
115:
85:
73:symmetry groups
25:crystallography
17:
12:
11:
5:
576:
574:
566:
565:
555:
554:
549:
548:
533:
513:
506:
488:
481:
452:
425:(7): 494–499.
402:
374:
367:
346:
345:
343:
340:
339:
338:
333:
328:
323:
318:
311:
308:
307:
306:
301:
293:
286:
284:
281:
274:
270:
267:
255:Main article:
252:
249:
240:
214:
206:
199:
192:
185:
177:
167:Main article:
164:
161:
156:
144:
137:
130:
117:Main article:
114:
111:
110:
109:
84:
81:
15:
13:
10:
9:
6:
4:
3:
2:
575:
564:
561:
560:
558:
544:
540:
536:
534:0-7487-7516-1
530:
526:
525:
517:
514:
509:
507:0-7167-8759-8
503:
499:
492:
489:
484:
478:
474:
470:
466:
459:
457:
453:
442:
438:
433:
428:
424:
420:
416:
409:
407:
403:
391:
387:
381:
379:
375:
370:
364:
360:
359:
351:
348:
341:
337:
334:
332:
329:
327:
324:
322:
319:
317:
314:
313:
309:
300:
296:
290:
285:
278:
273:
268:
266:
264:
258:
250:
248:
246:
244:
239:
234:
229:
224:
222:
218:
217:
209:
202:
195:
188:
181:
180:
170:
162:
153:
149:
147:
140:
133:
126:
120:
113:Mirror planes
112:
105:
100:
96:
95:
94:
92:
91:
82:
80:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
523:
516:
497:
491:
467:. Elsevier.
464:
444:. Retrieved
422:
418:
394:. Retrieved
392:. 2021-09-25
389:
357:
350:
321:Group theory
298:
262:
260:
242:
237:
236:
232:
225:
221:cyclic group
212:
211:
204:
197:
190:
183:
175:
174:
172:
142:
135:
128:
124:
122:
89:
88:
86:
77:fixed points
49:mirror plane
43:about which
28:
18:
235:is denoted
446:2021-09-29
396:2021-09-25
342:References
441:0108-7673
295:Ferrocene
251:Inversion
21:chemistry
563:Symmetry
557:Category
543:56661923
316:Symmetry
310:See also
83:Identity
61:molecule
269:Gallery
65:crystal
541:
531:
504:
479:
439:
365:
63:or a
51:, an
41:plane
39:, or
33:point
31:is a
539:OCLC
529:ISBN
502:ISBN
477:ISBN
437:ISSN
363:ISBN
37:line
27:, a
23:and
469:doi
427:doi
226:An
19:In
559::
537:.
475:.
455:^
435:.
423:45
421:.
417:.
405:^
388:.
377:^
302:10
247:.
223:.
148:.
102:A
35:,
545:.
510:.
485:.
471::
449:.
429::
399:.
371:.
299:S
263:i
243:n
241:2
238:S
233:n
215:n
213:C
207:5
205:C
200:4
198:C
193:3
191:C
186:2
184:C
178:n
176:C
157:4
145:d
143:σ
138:h
136:σ
131:v
129:σ
125:σ
90:E
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
