1049:
1057:
1041:
38:. Because the fermions change phase with 360 degree rotation, enhanced symmetry groups that describe band degeneracy and topological properties of magnonic systems are needed, which depend not only on geometric rotation, but on the corresponding fermionic phase factor in representations (for the related mathematical concept, see the
1166:
for rotation by an angle α. This cannot be true for an identity in a point group. Consequently, a group must be used in which rotations by α + 2π are classed as symmetry operations distinct from rotations by an angle α. This group is known as the double group,
318:
of orbital and spin angular momentum. This formula applies with most paramagnetic chemical compounds of transition metals and lanthanides. However, in a complex containing an atom with a single electron in the valence shell, the character,
277:
465:
1452:
Chai, Yan; Guo, Ting; Jin, Changming; Haufler, Robert E.; Chibante, L. P. Felipe; Fure, Jan; Wang, Lihong; Alford, J. Michael; Smalley, Richard E. (1991). "Fullerenes with metals inside".
1135:, the so-called spin-only value. However, for a more accurate prediction spin–orbit coupling must be taken into consideration. This means that the relevant quantum number is
373:
344:
97:} that has two symmetry operations, identity and rotation by 360°. The double group has twice the number of symmetry operations compared to the molecular point group.
393:
185:
491:
312:
173:
1359:
1539:
1436:
109:
of complexes of a metal ion in whose electronic structure there is a single electron (or its equivalent, a single vacancy) in a metal ion's
1189:'; deviations from the spin-only value are greater as the magnitude of spin–orbit coupling is greater for silver(II) than for copper(II).
1200:
shell; the magnetic moments of its complexes have been found to lie in the range 1.63 - 1.81 B.M. at room temperature. The double group
1617:
1594:
1342:
1317:
400:
1389:
470:
The change of sign cannot be true for an identity operation in any point group. Therefore, a double group, in which rotation by
1645:
1084:-electron shell, which can contain up to 10 electrons. The ion is a typical example of a compound with this characteristic.
105:
In magnetochemistry, the need for a double group arises in a very particular circumstance, namely, in the treatment of the
1569:
Figgis, Brian N.; Lewis, Jack (1960). "The magnetochemistry of complex compounds". In Lewis, J.; Wilkins, R.G. (eds.).
1650:
1640:
1381:
39:
20:
493:, is classified as being distinct from the identity operation, is used. A character table for the double group
1128:
46:, that have a single unpaired electron in the metal ion's valence electron shell, and complexes of ions like
1586:
153:
1635:
1428:
1215:
shell. The magnetic properties of octahedral complexes of this ion are treated using the double group
1492:
1227:
1609:
375:
about an axis through that atom is equal to minus the character for a rotation through an angle of
349:
1276:
1518:
Bunker, P.R. (1979), "The Spin Double Groups of
Molecular Symmetry Groups", in Hinze, J. (ed.),
272:{\displaystyle \chi ^{J}(\alpha )={\frac {\sin(J+{1 \over 2})\alpha }{\sin {1 \over 2}\alpha }}}
1088:(1) Six-coordinate complexes of the Cu(II) ion, with the generic formula , are subject to the
322:
1613:
1590:
1535:
1432:
1385:
1338:
1313:
53:
In the specific instances of complexes of metal ions that have the electronic configurations 3
378:
1527:
1500:
1461:
1420:
1305:
1286:
1069:
35:
1234:}. The magnetic properties of the compound are treated using the icosahedral double group
473:
285:
158:
1523:
1496:
1561:
1048:
1114:
orbitals are centrosymmetric the related atomic term symbols can be classified in the
1629:
1504:
1252:
1132:
503:
is as follows. The new symmetry operations are shown in the second row of the table.
106:
86:
1056:
1578:
1040:
176:
149:
1531:
1281:
1089:
90:
315:
31:
1076:(II). The electronic configuration of the central Cu ion can be written as 3
923:
but the column header is shown, for convenience, in two rows, rather than
1193:
1115:
1080:. It can be said that there is a single vacancy, or hole, in the copper 3
130:
43:
1465:
1196:
in the +3 oxidation state. Titanium(III) has a single electron in the 3
1479:
Balasubramanian, K. (1996). "Double group of the icosahedral group (I
1208:
1179:
1159:
1131:
can be ignored and the magnetic moment is then predicted to be 1.73
1073:
1068:
The need for a double group occurs, for example, in the treatment of
138:
122:
118:
89:
for electron spin. A double group is formed by combining a molecular
47:
1092:
effect so that the symmetry is reduced from octahedral (point group
1039:
125:
in the +2 oxidation state, where there is a single vacancy in a
1337:. Mineola, New York: Dover Publications Inc. pp. 245–253.
42:). They were introduced for studying complexes of ions like
460:{\displaystyle \chi ^{J}(2\pi +\alpha )=-\chi ^{J}(\alpha )}
69:, rotation by 360° must be treated as a symmetry operation
50:, which have a single "vacancy" in the valence shell.
1178:
With species such as the square-planar complex of the
1407:
476:
403:
381:
352:
325:
288:
188:
161:
117:- shell. This occurs, for example, with the elements
1044:
Sub-structure at the center of an octahedral complex
1192:A double group is also used for some compounds of
485:
459:
387:
367:
338:
306:
271:
167:
1052:Structure of a square-planar complex ion such as
179:for angular momentum by an angle α is given by
1604:Vulfson, Sergey G.; Arshinova, Rose P. (1998).
1267:which both contain a single unpaired electron.
1520:The Permutation Group in Physics and Chemistry
1255:. This has been illustrated for the species CH
16:Use of mathematical groups in magnetochemistry
1405:Foëx, D.; Gorter, C. J.; Smits, L.J. (1957).
1378:Point group character tables and related data
1251:Double groups may be used in connection with
1222:When a cerium(III) ion is encapsulated in a C
1211:(III) ion, Ce, has a single electron in the 4
1204:is used to classify their electronic states.
8:
1522:, Lecture Notes in Chemistry, vol. 12,
1182:(II) ion the relevant double group is also
1573:. New York: Interscience. pp. 400–451.
1335:Group Theory in Chemistry and Spectroscopy
1312:. New York: Wiley. pp. 289–294, 376.
141:(III) which has a single electron in the 4
133:(III) which has a single electron in the 3
475:
442:
408:
402:
380:
351:
330:
324:
287:
253:
229:
211:
193:
187:
160:
1055:
1047:
943:Character tables for the double groups T
508:
1360:"Spin-orbit coupling and double groups"
1297:
1483:) and its application to fullerenes".
1310:Chemical Applications of Group Theory
1158:for a rotation by an angle of α + 2π
346:, for a rotation through an angle of
85:. This arises from the nature of the
7:
1376:Salthouse, J.A.; Ware, M.J. (1972).
34:for the quantitative treatment of
14:
1454:The Journal of Physical Chemistry
1032:are given in Salthouse and Ware.
1558:Introduction to Magnetochemistry
1060:An atom or ion (red) held in a C
902:The symmetry operations such as
510:Character table: double group
454:
448:
429:
414:
239:
220:
205:
199:
175:, for rotation of a molecular
19:For mathematical details, see
1:
1571:Modern Coordination Chemistry
1333:Tsukerblat, Boris S. (2006).
1127:(2) To a first approximation
1072:of 6-coordinate complexes of
368:{\displaystyle 2\pi +\alpha }
1585:. Oxford Chemistry Primers.
1505:10.1016/0009-2614(96)00849-4
1532:10.1007/978-3-642-93124-6_4
1100:to tetragonal (point group
1667:
1606:Molecular Magnetochemistry
1382:Cambridge University Press
314:; angular momentum is the
18:
1425:Chemistry of the Elements
1423:; Earnshaw, Alan (1997).
1226:cage, the formula of the
339:{\displaystyle \chi ^{J}}
21:Finite subgroups of SU(2)
1485:Chemical Physics Letters
1587:Oxford University Press
1556:Earnshaw, Alan (1968).
1409:. Paris: Masson et Cie.
388:{\displaystyle \alpha }
1162:is equal to minus the
1065:
1053:
1045:
487:
461:
389:
369:
340:
308:
273:
169:
129:-electron shell, with
1646:Theoretical chemistry
1429:Butterworth-Heinemann
1421:Greenwood, Norman N.
1154:is half-integer, the
1059:
1051:
1043:
488:
486:{\displaystyle 2\pi }
462:
390:
370:
341:
309:
307:{\displaystyle J=L+S}
274:
170:
168:{\displaystyle \chi }
1610:Taylor & Francis
1228:endohedral fullerene
474:
401:
379:
350:
323:
286:
186:
159:
1579:Orchard, Anthony F.
1497:1996CPL...260..476B
1466:10.1021/j100173a002
1230:is written as {Ce@C
1129:spin–orbit coupling
1070:magnetic properties
940:in a single row .
919:belong to the same
520:
1526:, pp. 38–56,
1384:. pp. 55–57.
1277:Molecular symmetry
1066:
1054:
1046:
509:
483:
457:
385:
365:
336:
304:
269:
165:
30:was introduced by
1651:Materials science
1641:Molecular physics
1541:978-3-540-09707-5
1460:(20): 7564–7568.
1438:978-0-08-037941-8
1306:Cotton, F. Albert
897:
896:
267:
261:
237:
40:formal definition
26:The concept of a
1658:
1623:
1600:
1583:Magnetochemistry
1574:
1565:
1545:
1544:
1515:
1509:
1508:
1476:
1470:
1469:
1449:
1443:
1442:
1427:(2nd ed.).
1417:
1411:
1410:
1402:
1396:
1395:
1373:
1367:
1366:
1364:
1355:
1349:
1348:
1330:
1324:
1323:
1302:
1287:Magnetochemistry
1031:
1027:
1020:
1013:
1006:
999:
992:
985:
978:
971:
964:
957:
950:
946:
868:
834:
800:
766:
732:
698:
664:
635:
528:
521:
515:
498:
492:
490:
489:
484:
466:
464:
463:
458:
447:
446:
413:
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386:
374:
372:
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345:
343:
342:
337:
335:
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313:
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305:
278:
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268:
266:
262:
254:
245:
238:
230:
212:
198:
197:
174:
172:
171:
166:
93:with the group {
73:, in a separate
36:magnetochemistry
1666:
1665:
1661:
1660:
1659:
1657:
1656:
1655:
1626:
1625:
1620:
1603:
1597:
1577:
1568:
1555:
1553:
1551:Further reading
1548:
1542:
1517:
1516:
1512:
1482:
1478:
1477:
1473:
1451:
1450:
1446:
1439:
1431:. p. 971.
1419:
1418:
1414:
1404:
1403:
1399:
1392:
1375:
1374:
1370:
1362:
1357:
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1109:
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989:
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348:
347:
326:
321:
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283:
246:
213:
189:
184:
183:
157:
156:
137:shell and with
103:
24:
17:
12:
11:
5:
1664:
1662:
1654:
1653:
1648:
1643:
1638:
1628:
1627:
1618:
1595:
1562:Academic Press
1552:
1549:
1547:
1546:
1540:
1510:
1491:(3): 476–484.
1480:
1471:
1444:
1437:
1412:
1397:
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1368:
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1256:
1248:
1245:
1238:
1231:
1223:
1186:
1176:
1175:
1171:
1148:
1133:Bohr magnetons
1125:
1121:
1104:
1095:
1064:fullerene cage
1061:
1037:
1034:
1022:
1015:
1008:
1001:
994:
987:
980:
973:
966:
959:
952:
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1631:
1624:
1621:
1619:90-5699-535-9
1615:
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1598:
1596:0-19-879278-6
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1408:
1401:
1398:
1393:
1387:
1383:
1380:. Cambridge:
1379:
1372:
1369:
1361:
1358:Lipson, R.H.
1354:
1351:
1346:
1344:0-486-45035-X
1340:
1336:
1329:
1326:
1321:
1319:0-471-17570-6
1315:
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1298:
1292:
1288:
1285:
1283:
1280:
1278:
1275:
1274:
1270:
1268:
1254:
1253:free radicals
1247:Free radicals
1246:
1244:
1241:
1237:
1229:
1220:
1218:
1214:
1210:
1205:
1203:
1199:
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136:
132:
128:
124:
120:
116:
112:
108:
107:paramagnetism
100:
98:
96:
92:
88:
87:wave function
84:
80:
76:
72:
68:
64:
60:
56:
51:
49:
45:
41:
37:
33:
29:
22:
1636:Group theory
1605:
1602:
1582:
1576:
1570:
1567:
1557:
1554:
1519:
1513:
1488:
1484:
1474:
1457:
1453:
1447:
1424:
1415:
1406:
1400:
1391:0-521-081394
1377:
1371:
1353:
1334:
1328:
1309:
1300:
1250:
1239:
1235:
1221:
1216:
1212:
1206:
1201:
1197:
1191:
1183:
1177:
1168:
1163:
1155:
1151:
1144:
1140:
1136:
1118:
1111:
1106:
1101:
1093:
1081:
1077:
1067:
1036:Applications
942:
937:
931:
924:
920:
916:
910:
903:
901:
864:
830:
796:
762:
728:
694:
660:
653:
646:
640:
631:
625:
619:
614:
608:
603:
597:
592:
577:
567:
558:
550:
542:
535:
524:
511:
494:
469:
281:
177:wavefunction
150:group theory
147:
142:
134:
126:
114:
110:
104:
94:
82:
78:
74:
70:
66:
62:
58:
54:
52:
28:double group
27:
25:
1282:Point group
1090:Jahn-Teller
91:point group
1630:Categories
1293:References
316:vector sum
101:Background
81:operation
32:Hans Bethe
1164:character
1156:character
1150:(3) When
1110:). Since
481:π
452:α
440:χ
436:−
427:α
421:π
406:χ
383:α
363:α
357:π
328:χ
264:α
251:
243:α
218:
203:α
191:χ
163:χ
154:character
77:from the
1581:(2003).
1524:Springer
1308:(1971).
1271:See also
1259:F and CH
1194:titanium
1139:, where
1116:subgroup
1028:and R(3)
131:titanium
79:identity
1493:Bibcode
1160:radians
145:shell.
1616:
1593:
1538:
1435:
1388:
1341:
1316:
1209:cerium
1180:silver
1074:copper
282:where
152:, the
139:cerium
123:silver
119:copper
1363:(PDF)
1145:L + S
921:class
113:- or
75:class
65:and 4
1614:ISBN
1591:ISBN
1536:ISBN
1433:ISBN
1386:ISBN
1339:ISBN
1314:ISBN
1207:The
1021:, D
947:, O
909:and
121:and
95:E, R
1528:doi
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