1193:)", Bader charges, or "fuzzy atom" charges. Rather surprisingly, despite the wide variation in the (total) partial charges reproduced by the different estimates, analysis of the resulting Pipek-Mezey orbitals has shown that the localized orbitals are rather insensitive to the partial charge estimation scheme used in the localization process. However, due to the ill-defined mathematical nature of Mulliken charges (and Löwdin charges, which have also been used in some works), as better alternatives are nowadays available it is advisable to use them in favor of the original version.
1209:
1238:
with the "bonds" of the molecule, as visualized by the practicing chemist, the most common approach is to instead consider the interaction between filled and unfilled localized molecular orbitals that correspond to σ bonds, π bonds, lone pairs, and their unoccupied counterparts. These orbitals and typically given the notation σ (sigma bonding), π (pi bonding),
1250:* for unoccupied nonbonding orbital is seldom used), π* (pi antibonding), and σ* (sigma antibonding). (Woodward and Hoffmann use ω for nonbonding orbitals in general, occupied or unoccupied.) When comparing localized molecular orbitals derived from the same atomic orbitals, these classes generally follow the order σ < π <
1237:
requires the analysis of interactions between donor and acceptor orbitals between two molecules or different regions within the same molecule, and molecular orbitals must be considered. Because proper (symmetry-adapted) molecular orbitals are fully delocalized and do not admit a ready correspondence
121:
For molecules with an open electron shell, in which some molecular orbitals are singly occupied, the electrons of alpha and beta spin must be localized separately. This applies to radical species such as nitric oxide and dioxygen. Again, in this case the localized and delocalized orbital descriptions
359:
The optimization of the objective function is usually performed using pairwise Jacobi rotations. However, this approach is prone to saddle point convergence (if it even converges), and thus other approaches have also been developed, from simple conjugate gradient methods with exact line searches, to
93:
For molecules with a closed electron shell, in which each molecular orbital is doubly occupied, the localized and delocalized orbital descriptions are in fact equivalent and represent the same physical state. It might seem, again using the example of water, that placing two electrons in the first
911:
The fourth moment method produces more localized virtual orbitals than Foster-Boys method, since it implies a larger penalty on the delocalized tails. For graphene (a delocalized system), the fourth moment method produces more localized occupied orbitals than Foster-Boys and Pipek-Mezey schemes.
58:
In the water molecule for example, ab initio calculations show bonding character primarily in two molecular orbitals, each with electron density equally distributed among the two O-H bonds. The localized orbital corresponding to one O-H bond is the sum of these two delocalized orbitals, and the
906:
743:
1265:
The localized molecular orbitals that organic chemists often depict can be thought of as qualitative renderings of orbitals generated by the computational methods described above. However, they do not map onto any single approach, nor are they used consistently. For instance, the
1196:
The most important quality of the Pipek-Mezey scheme is that it preserves σ-π separation in planar systems, which sets it apart from the Foster-Boys and
Edmiston-Ruedenberg schemes that mix σ and π bonds. This property holds independent of the partial charge estimate used.
1176:
266:
1189:. Recently, Pipek-Mezey style schemes based on a variety of mathematically well-defined partial charge estimates have been discussed. Some notable choices are Voronoi charges, Becke charges, Hirshfeld or Stockholder charges, intrinsic atomic orbital charges (see
1212:
A list of localized molecular orbitals considered in organic chemistry, showing component atomic orbitals and all shapes of the MOs they constitute. In reality, AOs and MOs, as obtained from computations, are much "fatter" than depicted in these
138:
upon a set of canonical molecular orbitals (CMO). The transformation usually involves the optimization (either minimization or maximization) of the expectation value of a specific operator. The generic form of the localization potential is:
23:
which are concentrated in a limited spatial region of a molecule, such as a specific bond or lone pair on a specific atom. They can be used to relate molecular orbital calculations to simple bonding theories, and also to speed up
758:
595:
1053:
495:
1068:
118:(or linear combination of Slater determinants), and it can be shown that if two electrons are exchanged, such a function is unchanged by any unitary transformation of the doubly occupied orbitals.
960:
753:
The fourth moment (FM) procedure is analogous to Foster-Boys scheme, however the orbital fourth moment is used instead of the orbital second moment. The objective function to be minimized is
144:
1217:
Organic chemistry is often discussed in terms of localized molecular orbitals in a qualitative and informal sense. Historically, much of classical organic chemistry was built on the older
98:
electrons in the second bond is not the same as having four electrons free to move over both bonds. However, in quantum mechanics all electrons are identical and cannot be distinguished as
1585:
Kleier, Daniel; J. Chem. Phys. 61, 3905 (1974) (1974). "Localized molecular orbitals for polyatomic molecules. I. A comparison of the
Edmiston-Ruedenberg and Boys localization methods".
405:
1274:
are generally treated as a filled σ(out) orbital and an unfilled pure p orbital, even though the lone pairs of water could be described analogously by filled σ(out) and p orbitals (
588:
1703:
Pipek, János; Mezey, Paul G. (1989). "A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions".
354:
298:
1200:
While the usual formulation of the Pipek-Mezey method invokes an iterative procedure to localize the orbitals, a non-iterative method has also been recently suggested.
325:
1620:
1889:
Alcoba, Diego R.; Lain, Luis; Torre, Alicia; Bochicchio, Roberto C. (15 April 2006). "An orbital localization criterion based on the theory of "fuzzy" atoms".
47:
lead to delocalized orbitals that, in general, extend over an entire molecule and have the symmetry of the molecule. Localized orbitals may then be found as
901:{\displaystyle \langle {\hat {L}}_{\text{FM}}\rangle =\sum _{i}\langle \phi _{i}|({\hat {x}}-\langle i|{\hat {x}}|\phi _{i}\rangle )^{4}|i\rangle }
738:{\displaystyle \langle {\hat {L}}_{\text{FB}}\rangle =\sum _{i}\langle \phi _{i}|({\hat {x}}-\langle i|{\hat {x}}|i\rangle )^{2}|\phi _{i}\rangle }
1284:
67:
1391:
44:
2029:
1230:
1063:
Pipek-Mezey localization takes a slightly different approach, maximizing the sum of orbital-dependent partial charges on the nuclei:
965:
410:
1942:
Høyvik, Ida-Marie; Jansik, Branislav; Jørgensen, Poul (3 April 2013). "Pipek–Mezey localization of occupied and virtual orbitals".
1171:{\displaystyle \langle {\hat {L}}\rangle _{\textrm {PM}}=\sum _{A}^{\textrm {atoms}}\sum _{i}^{\textrm {orbitals}}|q_{i}^{A}|^{2}}
2055:
1515:
Høyvik, Ida-Marie; Jansik, Branislav; Jørgensen, Poul (2012). "Trust Region
Minimization of Orbital Localization Functions".
1749:
Lehtola, Susi; Jónsson, Hannes (8 January 2014). "Pipek–Mezey orbital localization using various partial charge estimates".
1287:). In other words, the type of localized orbital invoked depends on context and considerations of convenience and utility.
923:
1550:
Boys, S. F. (1960). "Construction of
Molecular orbitals to be minimally variant for changes from one molecule to another".
261:{\displaystyle \langle {\hat {L}}\rangle =\sum _{i=1}^{n}\langle \phi _{i}\phi _{i}|{\hat {L}}|\phi _{i}\phi _{i}\rangle }
327:
is a molecular spatial orbital. Many methodologies have been developed during the past decades, differing in the form of
1186:
1182:
33:
1480:
Leonard, Joseph M.; Luken, William L. (1982). "Quadratically
Convergent Calculation of Localized Molecular Orbitals".
1270:
of water are usually treated as two equivalent sp hybrid orbitals, while the corresponding "nonbonding" orbitals of
2060:
2050:
1787:
Knizia, G. (2013). "Intrinsic Atomic
Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts".
111:
375:
1234:
1190:
135:
52:
29:
1312:
Hirst, D. M.; Linington, Mary E. (1970). "Localized orbitals for the oxygen and nitric oxide molecules".
25:
1614:
1222:
1985:
Heßelmann, Andreas (10 May 2016). "Local
Molecular Orbitals from a Projection onto Localized Centers".
500:
1806:
1712:
1666:
1594:
1559:
1419:
1851:
Cioslowski, J. (1991). "Partitioning of the orbital overlap matrix and the localization criteria".
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1967:
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1497:
1329:
131:
115:
48:
330:
274:
1445:
Lehtola, Susi; Jónsson, Hannes (2013). "Unitary
Optimization of Localized Molecular Orbitals".
303:
2025:
2002:
1959:
1916:
1822:
1766:
1682:
1532:
1462:
1387:
920:
Edmiston-Ruedenberg localization maximizes the electronic self-repulsion energy by maximizing
82:
70:. The Boys and Edmiston-Ruedenberg localization methods mix these orbitals to give equivalent
37:
20:
1994:
1951:
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Edmiston, Clyde; Ruedenberg, Klaus (1963). "Localized Atomic and
Molecular Orbitals".
2044:
1360:
1872:
1834:
1633:
Introduction to
Computational Chemistry by Frank Jensen 1999, page 228 equation 9.27
1501:
1333:
590:. In one dimension, the Foster-Boys (FB) objective function can also be written as
1971:
1928:
107:
1651:
1347:
Duke, Brian J. (1987). "Linnett's double quartet theory and localised orbitals".
1229:, this simple model of bonding is supplemented by semi-quantitative results from
372:) localization method minimizes the spatial extent of the orbitals by minimizing
1226:
1571:
1431:
1998:
28:
electronic structure calculations by taking advantage of the local nature of
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2006:
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1911:
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1302:
Levine I.N., “Quantum Chemistry” (4th ed., Prentice-Hall 1991) sec.15.8
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1955:
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1818:
1762:
1678:
1606:
1528:
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1724:
66:
For multiple bonds and lone pairs, different localization procedures
59:
localized orbital for the other O-H bond is their difference; as per
497:. This turns out to be equivalent to the easier task of maximizing
1801:
81:
in water, while the Pipek-Mezey method preserves their respective
1652:"Orbital localization using fourth central moment minimization"
1048:{\displaystyle {\hat {L}}=|{\vec {r}}_{1}-{\vec {r}}_{2}|^{-1}}
1650:
Høyvik, Ida-Marie; Jansik, Branislav; Jørgensen, Poul (2012).
1246:(unoccupied nonbonding orbital, "empty p orbital"; the symbol
490:{\displaystyle {\hat {L}}=|{\vec {r}}_{1}-{\vec {r}}_{2}|^{2}}
89:
Equivalence of localized and delocalized orbital descriptions
1262:*) < π* < σ* when ranked by increasing energy.
955:{\displaystyle \langle {\hat {L}}_{\text{ER}}\rangle }
122:
are equivalent and represent the same physical state.
1382:. Chichester, England: John Wiley and Sons. pp.
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503:
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of the delocalized orbitals, given by an appropriate
1225:models of bonding. To account for phenomena like
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1593:(10). Journal of Chemical Physics: 3905–3919.
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1242:(occupied nonbonding orbital, "lone pair"),
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1619:: CS1 maint: numeric names: authors list (
1987:Journal of Chemical Theory and Computation
1789:Journal of Chemical Theory and Computation
1751:Journal of Chemical Theory and Computation
1517:Journal of Chemical Theory and Computation
1447:Journal of Chemical Theory and Computation
1276:for further discussion, see the article on
400:{\displaystyle \langle {\hat {L}}\rangle }
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1349:Journal of Molecular Structure: THEOCHEM
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2024:. Oxford, UK: Oxford University Press.
1378:Introduction to Computational Chemistry
1295:
1612:
1285:sigma-pi and equivalent-orbital models
110:must have a form which satisfies the
32:. Localized orbitals in systems with
7:
45:ab initio quantum chemistry methods
1944:Journal of Computational Chemistry
1891:Journal of Computational Chemistry
14:
1853:Journal of Mathematical Chemistry
1233:. However, the understanding of
583:{\displaystyle \sum _{i}^{n}^{2}}
300:is the localization operator and
1181:Pipek and Mezey originally used
1705:The Journal of Chemical Physics
1587:The Journal of Chemical Physics
1231:Hückel molecular orbital theory
368:The Foster-Boys (also known as
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1361:10.1016/0166-1280(87)80072-6
34:periodic boundary conditions
17:Localized molecular orbitals
1659:Journal of Chemical Physics
1282:and the discussion above on
2077:
1187:mathematically ill defined
349:{\displaystyle {\hat {L}}}
293:{\displaystyle {\hat {L}}}
1572:10.1103/RevModPhys.32.300
1552:Reviews of Modern Physics
1432:10.1103/RevModPhys.35.457
1412:Reviews of Modern Physics
320:{\displaystyle \phi _{i}}
112:Pauli exclusion principle
2022:Stereoelectronic Effects
1999:10.1021/acs.jctc.6b00321
1235:stereoelectronic effects
2056:Computational chemistry
1482:Theoretica Chimica Acta
1314:Theoretica Chimica Acta
1191:intrinsic bond orbitals
68:give different orbitals
1374:Jensen, Frank (2007).
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136:unitary transformation
134:(LMO) are obtained by
53:unitary transformation
2020:Kirby, A. J. (2002).
1223:orbital hybridization
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1204:In organic chemistry
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30:electron correlation
1811:2013arXiv1306.6884K
1717:1989JChPh..90.4916P
1671:2012JChPh.137v4114H
1599:1974JChPh..61.3905K
1564:1960RvMP...32..296B
1424:1963RvMP...35..457E
1155:
916:Edmiston-Ruedenberg
126:Computation methods
61:Valence bond theory
49:linear combinations
1865:10.1007/BF01166933
1494:10.1007/BF00581477
1326:10.1007/BF01045967
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132:molecular orbitals
116:Slater determinant
21:molecular orbitals
2061:Molecular physics
2051:Quantum chemistry
1956:10.1002/jcc.23281
1950:(17): 1456–1462.
1903:10.1002/jcc.20373
1819:10.1021/ct400687b
1795:(11): 4834–4843.
1763:10.1021/ct401016x
1679:10.1063/1.4769866
1607:10.1063/1.1681683
1529:10.1021/ct300473g
1459:10.1021/ct400793q
1453:(12): 5365–5372.
1393:978-0-470-01187-4
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38:Wannier functions
26:post-Hartree–Fock
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1993:(6): 2720–2741.
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1355:(3–4): 319–330.
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83:σ and π symmetry
74:in ethylene and
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2031:978-0198558934
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2012:
1977:
1934:
1897:(5): 596–608.
1878:
1859:(1): 169–178.
1840:
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1757:(2): 642–649.
1730:
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1665:(22): 244114.
1635:
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1558:(2): 296–299.
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1488:(2): 107–132.
1472:
1437:
1418:(3): 457–465.
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838:
835:
829:
826:
820:
816:
810:
806:
802:
797:
793:
789:
786:
774:
771:
764:
750:
747:
734:
729:
725:
720:
714:
710:
706:
703:
699:
692:
689:
682:
678:
675:
672:
666:
663:
657:
653:
647:
643:
639:
634:
630:
626:
623:
611:
608:
601:
577:
573:
569:
564:
560:
555:
548:
545:
538:
532:
528:
524:
521:
516:
511:
507:
484:
479:
472:
465:
462:
455:
450:
443:
440:
432:
428:
422:
419:
396:
390:
387:
381:
365:
362:
342:
339:
314:
310:
286:
283:
257:
252:
248:
242:
238:
233:
226:
223:
216:
210:
206:
200:
196:
192:
187:
182:
179:
176:
172:
168:
165:
159:
156:
150:
127:
124:
90:
87:
13:
10:
9:
6:
4:
3:
2:
2073:
2062:
2059:
2057:
2054:
2052:
2049:
2048:
2046:
2033:
2027:
2023:
2016:
2013:
2008:
2004:
2000:
1996:
1992:
1988:
1981:
1978:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1938:
1935:
1930:
1926:
1922:
1918:
1913:
1908:
1904:
1900:
1896:
1892:
1885:
1883:
1879:
1874:
1870:
1866:
1862:
1858:
1854:
1847:
1845:
1841:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1803:
1798:
1794:
1790:
1783:
1781:
1777:
1772:
1768:
1764:
1760:
1756:
1752:
1745:
1743:
1741:
1739:
1737:
1735:
1731:
1726:
1722:
1718:
1714:
1710:
1706:
1699:
1697:
1693:
1688:
1684:
1680:
1676:
1672:
1668:
1664:
1660:
1653:
1646:
1644:
1642:
1640:
1636:
1630:
1627:
1622:
1616:
1608:
1604:
1600:
1596:
1592:
1588:
1581:
1578:
1573:
1569:
1565:
1561:
1557:
1553:
1546:
1543:
1538:
1534:
1530:
1526:
1522:
1518:
1511:
1508:
1503:
1499:
1495:
1491:
1487:
1483:
1476:
1473:
1468:
1464:
1460:
1456:
1452:
1448:
1441:
1438:
1433:
1429:
1425:
1421:
1417:
1413:
1406:
1404:
1400:
1395:
1389:
1385:
1380:
1379:
1370:
1367:
1362:
1358:
1354:
1350:
1343:
1340:
1335:
1331:
1327:
1323:
1319:
1315:
1308:
1305:
1299:
1296:
1290:
1288:
1286:
1283:
1280:
1277:
1273:
1269:
1263:
1261:
1257:
1253:
1249:
1245:
1241:
1236:
1232:
1228:
1224:
1220:
1210:
1203:
1201:
1198:
1194:
1192:
1188:
1184:
1179:
1163:
1151:
1146:
1142:
1124:
1120:
1107:
1103:
1099:
1078:
1064:
1058:
1056:
1040:
1037:
1025:
1015:
1008:
1003:
993:
981:
972:
934:
915:
913:
909:
892:
882:
869:
865:
850:
839:
833:
824:
808:
804:
795:
791:
787:
769:
754:
749:Fourth moment
748:
746:
727:
723:
712:
701:
687:
676:
670:
661:
645:
641:
632:
628:
624:
606:
591:
575:
562:
558:
543:
530:
526:
514:
509:
505:
482:
470:
460:
453:
448:
438:
426:
417:
385:
371:
363:
361:
357:
337:
312:
308:
281:
269:
250:
246:
240:
236:
221:
208:
204:
198:
194:
185:
180:
177:
174:
170:
166:
154:
140:
137:
133:
125:
123:
119:
117:
113:
109:
105:
101:
97:
94:bond and two
88:
86:
84:
80:
77:
73:
69:
64:
62:
56:
54:
50:
46:
41:
39:
36:are known as
35:
31:
27:
22:
18:
2021:
2015:
1990:
1986:
1980:
1947:
1943:
1937:
1894:
1890:
1856:
1852:
1792:
1788:
1754:
1750:
1708:
1704:
1662:
1658:
1629:
1615:cite journal
1590:
1586:
1580:
1555:
1551:
1545:
1520:
1516:
1510:
1485:
1481:
1475:
1450:
1446:
1440:
1415:
1411:
1377:
1369:
1352:
1348:
1342:
1320:(1): 55–62.
1317:
1313:
1307:
1298:
1281:
1275:
1264:
1259:
1255:
1251:
1247:
1243:
1239:
1219:valence bond
1216:
1199:
1195:
1185:, which are
1180:
1065:
1062:
919:
910:
755:
752:
592:
367:
358:
270:
141:
129:
120:
108:wavefunction
106:. The total
103:
99:
95:
92:
75:
65:
57:
42:
16:
15:
1912:11336/74084
1711:(9): 4916.
1227:aromaticity
1059:Pipek-Mezey
364:Foster-Boys
2045:Categories
1291:References
1268:lone pairs
130:Localized
114:such as a
79:lone pairs
76:rabbit ear
72:bent bonds
1802:1306.6884
1279:lone pair
1213:cartoons.
1121:∑
1104:∑
1089:⟩
1082:^
1073:⟨
1038:−
1019:→
1009:−
997:→
976:^
950:⟩
938:^
928:⟨
896:⟩
875:⟩
866:ϕ
854:^
837:⟨
834:−
828:^
805:ϕ
801:⟨
792:∑
785:⟩
773:^
763:⟨
733:⟩
724:ϕ
705:⟩
691:^
674:⟨
671:−
665:^
642:ϕ
638:⟨
629:∑
622:⟩
610:^
600:⟨
568:⟩
559:ϕ
547:→
527:ϕ
523:⟨
506:∑
464:→
454:−
442:→
421:^
395:⟩
389:^
380:⟨
341:^
309:ϕ
285:^
256:⟩
247:ϕ
237:ϕ
225:^
205:ϕ
195:ϕ
191:⟨
171:∑
164:⟩
158:^
149:⟨
43:Standard
2007:27164445
1964:23553349
1921:16470667
1873:96731740
1835:17717923
1827:26583402
1771:26580041
1687:23248994
1537:26605725
1502:97499582
1467:26592274
1334:95235964
1272:carbenes
1131:orbitals
962:, where
407:, where
1972:2219961
1929:3659974
1807:Bibcode
1713:Bibcode
1667:Bibcode
1595:Bibcode
1560:Bibcode
1420:Bibcode
2028:
2005:
1970:
1962:
1927:
1919:
1871:
1833:
1825:
1769:
1685:
1535:
1500:
1465:
1390:
1386:–308.
1332:
271:where
1968:S2CID
1925:S2CID
1869:S2CID
1831:S2CID
1797:arXiv
1655:(PDF)
1498:S2CID
1330:S2CID
1254:<
1114:atoms
104:other
96:other
2026:ISBN
2003:PMID
1960:PMID
1917:PMID
1823:PMID
1767:PMID
1683:PMID
1621:link
1533:PMID
1463:PMID
1388:ISBN
370:Boys
100:same
19:are
1995:doi
1952:doi
1907:hdl
1899:doi
1861:doi
1815:doi
1759:doi
1721:doi
1675:doi
1663:137
1603:doi
1568:doi
1525:doi
1490:doi
1455:doi
1428:doi
1384:304
1357:doi
1353:152
1322:doi
1178:.
908:.
102:or
2047::
2001:.
1991:12
1989:.
1966:.
1958:.
1948:34
1946:.
1923:.
1915:.
1905:.
1895:27
1893:.
1881:^
1867:.
1855:.
1843:^
1829:.
1821:.
1813:.
1805:.
1791:.
1779:^
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1531:.
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1426:.
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1414:.
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1221:/
1094:PM
1055:.
945:ER
780:FM
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617:FB
356:.
268:,
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63:.
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2034:.
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1974:.
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1931:.
1909::
1901::
1875:.
1863::
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1837:.
1817::
1809::
1799::
1793:9
1773:.
1761::
1727:.
1723::
1715::
1689:.
1677::
1669::
1623:)
1609:.
1605::
1597::
1574:.
1570::
1562::
1539:.
1527::
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1457::
1451:9
1434:.
1430::
1422::
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1363:.
1359::
1336:.
1324::
1260:n
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1125:i
1108:A
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1079:L
1041:1
1033:|
1026:2
1016:r
1004:1
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982:=
973:L
935:L
893:i
889:|
883:4
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870:i
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840:i
825:x
819:(
815:|
809:i
796:i
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770:L
728:i
719:|
713:2
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702:i
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386:L
338:L
313:i
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241:i
232:|
222:L
215:|
209:i
199:i
186:n
181:1
178:=
175:i
167:=
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