771:
the cations, (b) that the structure is electroneutral when the ions carry charges equal to their valence, and (c) that all the bonds have a cation at one end and an anion at the other. If these conditions are satisfied, as they are in many ionic and covalent compounds, the electrons forming a bond can all be formally assigned to the anion. The anion thus acquires a formal negative charge and the cation a formal positive charge, which is the picture on which the ionic model is based. The electrostatic flux that links the cation core to its bonding electrons now links the cation core to the anion. In this picture, a cation and anion are bonded to each other if they are linked by electrostatic flux, with the flux being equal to the valence of the bond. In a representative set of compounds
Preiser et al. have confirmed that the electrostatic flux is the same as the bond valence determined from the bond lengths using Eq. 2.
809:
is not known, a plausible network can be created by connecting well matched cations and anions that satisfy Eq. 4. If the finite network contains only cation-anion bonds, every bond can be treated as an electric capacitor (two equal and opposite charges linked by electrostatic flux). The bond network is thus equivalent to a capacitive electrical circuit with the charge on each capacitor being equivalent to the bond valence. The individual bond capacitors are not initially known, but in the absence of any information to the contrary we assume that they are all equal. In this case the circuit can be solved using the
Kirchhoff equations, yielding the valences of each bond. Eq. 2 can then be used to calculate bond lengths which are found to lie within a few picometres of the observed bond lengths if no additional constraints are present. Additional constraints include electronic anisotropies (lone pairs and
796:
has a valence, V, that is equal to its coordination number, N, its bonding strength according to Eq. 3 is exactly 1.0 vu (valence units), a condition that greatly simplifies the model. This condition is obeyed by carbon, hydrogen and silicon. Since these atoms all have bonding strengths of 1.0 vu the bonds between them are all predicted to have integral valences with carbon forming four single bonds and hydrogen one. Under these conditions, the bonds are all single bonds (or multiples of single bonds). Compounds can be constructed by linking carbon and hydrogen atoms with bonds that are all exactly equivalent. Under certain conditions, nitrogen can form three bonds and oxygen two, but since nitrogen and oxygen typically also form
654:
they are stereoactive they are concentrated in one portion of the coordination sphere preventing that portion from forming bonds. This results in the atom having a smaller coordination number, hence a higher bonding strength, when the lone pair is stereoactive. Ions with lone pairs have a greater ability to adapt their bonding strength to match that of the counter-ion. The lone pairs become stereoactive when the bonding strength of the counter-ion exceeds twice the bonding strength of the ion when its lone pairs are inactive.
826:
only the composition. The empirical parameters of the model are tabulated and are readily transferable between bonds of the same type. The concepts used are familiar to chemists and provide ready insight into the chemical restraints acting on the structure. The bond valence model uses mostly classical physics, and with little more than a pocket calculator, it gives quantitative predictions of bond lengths and places limits on what structures can be formed.
355:, S, is defined as the number of electron pairs forming the bond. In general this is not an integral number. Since each of the terminal atoms contributes equal numbers of electrons to the bond, the bond valence is also equal to the number of valence electrons that each atom contributes. Further, since within each atom, the negatively charged valence shell is linked to the positively charged core by an
813:) or steric constraints, (bonds stretched or compressed in order to fit them into three-dimensional space). Hydrogen bonds are an example of a steric constraint. The repulsion resulting from the close approach of the donor and acceptor atoms causes the bonds to be stretched, and under this constraint the distortion theorem predicts that the hydrogen atom will move off-center.
800:, the resulting N-H and O-H bonds have valences less than 1.0 vu, leading through the application of Eq. 1, to the C-C and C-H bonds having valences that differ from 1.0 vu. Nevertheless, the simple bonding rules of organic chemistry are still good approximations, though the rules of the bond valence model are better.
857:
Starting with
Pauling in 1947 a correlation between cation–anion bond length and bond strength was noted. It was shown later that if bond lengths were included in the calculation of bond strength, its accuracy was improved, and this revised method of calculation was termed the bond valence. These new
499:
where S is the valence and R is the length of the bond, and Ro and b are parameters that are empirically determined for each bond type. For many bond types (but not all), b is found to be close to 0.37 Å. A list of bond valence parameters for different bond types (i.e. for different pairs of cation
657:
Compounds that do not satisfy Eq. 4 are difficult, if not impossible, to prepare, and chemical reactions tend to favour the compounds that provide the best valence match. For example, the aqueous solubility of a compound depends on whether its ions are better matched to water than they are to each
512:
Eq. 2 is used to derive the distortion theorem which states that the more the individual bond lengths in a coordination sphere deviate from their average, the more the average bond length increases provided the valence sum is kept constant. Alternatively if the average bond length is kept constant,
503:
If the structure of a compound is known, the empirical bond valence - bond length correlation of Eq. 2 can be used to estimate the bond valences from their observed bond lengths. Eq. 1 can then be used to check that the structure is chemically valid; any deviation between the atomic valence and the
808:
A chemical structure can be represented by a bond network of the kind familiar in molecular diagrams. The infinitely connected bond networks found in crystals can be simplified into finite networks by extracting one formula unit and reconnecting any broken bonds to each other. If the bond network
770:
The bond valence model can be reduced to the traditional ionic model if certain conditions are satisfied. These conditions require that atoms be divided into cations and anions in such a way that (a) the electronegativity of every anion is equal to, or greater than, the electronegativity of any of
653:
Atoms with non-bonding valence electrons, i.e., with lone pairs, have more flexibility in their bonding strength than those without lone pairs depending on whether the lone pairs are stereoactive or not. If the lone pairs are not stereoactive, they are spread uniformly around the valence shell, if
581:
Since the bonding strength of an atom is the valence expected for a bond formed by that atom, it follows that the most stable bonds will be formed between atoms with the same bonding strengths. In practice some tolerance is allowed, but bonds are rarely formed if the ratio of the bonding strengths
359:
that is equal to the charge on the valence shell, it follows that the bond valence is also equal to the electrostatic flux that links the core to the electrons forming the bond. The bond valence is thus equal to three different quantities: the number of electrons each atom contributes to the bond,
825:
The bond valence model is an extension of the electron counting rules and its strength lies in its simplicity and robustness. Unlike most models of chemical bonding, it does not require a prior knowledge of the atomic positions and so can be used to construct chemically plausible structures given
795:
Structures containing covalent bonds can be treated using the ionic model providing they satisfy the topological conditions given above, but a special situation applies to hydrocarbons which allows the bond valence model to be reduced to the traditional bond model of organic chemistry. If an atom
573:
If the coordination number is not known, a typical coordination number for the atom can be used instead. Some atoms, such as sulfur(VI), are only found with one coordination number with oxygen, in this case 4, but others, such as sodium, are found with a range of coordination numbers, though most
829:
However, like all models, the bond valence model has its limitations. It is restricted to compounds with localized bonds; it does not, in general, apply to metals or aromatic compounds where the electrons are delocalized. It cannot in principle predict electron density distributions or energies
427:
A bond is formed when the valence shells of two atoms overlap. It is apparent that the closer two atoms approach each other, the larger the overlap region and the more electrons are associated with the bond. We therefore expect a correlation between the bond valence and the bond length and find
866:
It is possible by bond valence calculations to estimate how great a contribution a given oxygen atom is making to the assumed valence of uranium. Zachariasen lists the parameters to allow such calculations to be done for many of the actinides. Bond valence calculations use parameters which are
774:
The association of the cation bonding electrons with the anion in the ionic model is purely formal. There is no change in physical locations of any electrons, and there is no change in the bond valence. The terms "anion" and "cation" in the bond valence model are defined in terms of the bond
500:
and anion in given oxidation states) can be found at the web site. It is this empirical relation that links the formal theorems of the bond valence model to the real world and allows the bond valence model to be used to predict the real structure, geometry and properties of a compound.
816:
The bond valence is a vector directed along the bond since it represents the electrostatic field linking the ions. If the atom is unconstrained, the sum of the bond valence vectors around an atom is expected to be zero, a condition that limits the range of possible bond angles.
344:, V, is defined as the number of electrons the atom uses for bonding. This is equal to the number of electrons in its valence shell if all the valence shell electrons are used for bonding. If they are not, the remainder will form non-bonding electron pairs, usually known as
574:
lie close to the average, which for sodium is 6.2. In the absence of any better information, the average coordination number observed with oxygen is a convenient approximation, and when this number is used in Eq. 3, the resulting average bond valence is known as the
666:
Several factors influence the coordination number of an atom, but the most important of these is its size; larger atoms have larger coordination numbers. The coordination number depends on the surface area of the atom, and so is proportional to r. If
867:
estimated after examining a large number of crystal structures of uranium oxides (and related uranium compounds); note that the oxidation states which this method provides are only a guide which assists in the understanding of a crystal structure.
938:
It is possible to do these simple calculations on paper or to use software. A program which does it can be obtained free of charge. In 2020 David Brown published a nearly comprehensive set of bond valence parameters on the IuCr web site.
782:
For compounds that contain cation-cation or anion-anion bonds it is usually possible to transform these homoionic bonds into cation-anion bonds either by treating the atoms linked by the homoionic bond as a single complex cation (e.g.,
188:
39:
of atoms. It is derived from the bond valence model, which is a simple yet robust model for validating chemical structures with localized bonds or used to predict some of their properties. This model is a development of
321:
336:
Although the bond valence model is mostly used for validating newly determined structures, it is capable of predicting many of the properties of those chemical structures that can be described by localized bonds
753:
gives the same ordering of the main group elements as the electronegativity, though it differs in its numerical value from traditional electronegativity scales. Because it is defined in structural terms,
775:
topology, not the chemical properties of the atoms. This extends the scope of the ionic model well beyond compounds in which the bonding would normally be considered as "ionic". For example, methane, CH
647:
854:
emanating from cations in proportion to the cation charge and ending on anions. The lines of force are divided equally between the bonds to the corners of the coordination polyhedron.
106:
743:
493:
779:, obeys the conditions for the ionic model with carbon as the cation and hydrogen as the anion (or vice versa, since carbon and hydrogen have the same electronegativity).
421:
246:
567:
787:), or by treating the bonding electrons in the homoionic bond as a pseudo-anion to transform a cation-cation bond into two cation - pseudo-anion bonds, e.g., Hg-e-Hg.
830:
since these require the solution of the
Schoedinger equation using the long-range Coulomb potential which is incompatible with the concept of a localized bond.
1438:
368:
It follows from these definitions, that the valence of an atom is equal to the sum of the valences of all the bonds it forms. This is known as the
117:
254:
1342:
Harvey, M. A.; Baggio, S.; Baggio, R. (2006). "A new simplifying approach to molecular geometry description: the vectorial bond-valence model".
671:
is the charge on the atomic core (which is the same as the valence of the atom when all the electrons in the valence shell are bonding), and N
1040:
1623:
1271:
1497:
360:
the number of electron pairs that form the bond, and the electrostatic flux linking each core to the bonding electron pair.
1102:
Preiser, C.; Loesel, J.; Brown, I. D.; Kunz, M.; Skowron, A. (1999). "Long range
Coulomb forces and localized bonds".
1618:
592:
877:
and B are tabulated in the table below. For each oxidation state use the parameters from the table shown below.
525:
can be calculated from the atomic valence, V, if the coordination number, N, of the atom is known using Eq. 3.
582:
of the two atoms exceeds two, a condition expressed by the inequality shown in Eq. 4. This is known and the
1032:
69:
810:
32:
858:
insights were developed by later workers culminating in the set of rules termed the bond valence model.
1592:
958:
693:
1280:
1193:
434:
1183:
Urusov, V. S. (2003). "Theoretical analysis and empirical manifestation of the distortion theorem".
1524:
1344:
1148:
1104:
985:
24:
378:
224:
1140:
847:
839:
41:
1522:
Zachariasen, W. H. (1978). "Bond lengths in oxygen and halogen compounds of d and f elements".
531:
1390:
1361:
1316:
1296:
1248:
1240:
1165:
1121:
1081:
1036:
28:
1532:
1447:
1418:
1382:
1353:
1324:
1288:
1232:
1201:
1157:
1113:
1071:
1063:
994:
1269:
Brown, I. D. (2011). "View of Lone
Electron Pairs and Their Role in Structural Chemistry".
1571:
1410:
1185:
36:
1501:
1141:"Relationship between bond valence and bond softness of alkali halides and chalcogenides"
1284:
1197:
1225:
Acta
Crystallographica Section B: Structural Science, Crystal Engineering and Materials
1076:
1050:
1025:
843:
1612:
1536:
851:
797:
1550:
1220:
111:
The individual bond valences in turn are calculated from the observed bond lengths.
1478:
1466:
1374:
1205:
683:
is proportional to the electric field at the surface of the core, represented by S
1422:
1314:
Brown, I. D.; Skowron, A. (1990). "Electronegativity and Lewis acid strength".
1051:"Recent developments in the methods and applications of the bond valence model"
1357:
1236:
1161:
1117:
1055:
999:
980:
183:{\displaystyle v_{\text{i}}=\exp \left({\frac {R_{0}-R_{\text{i}}}{b}}\right)}
1372:
Zachara, J. (2007). "Novel approach to the concept of bond-valence vectors".
1244:
206:
is a tabulated parameter expressing the (ideal) bond length when the element
981:"The automatic searching for chemical bonds in inorganic crystal structures"
316:{\displaystyle v_{\text{i}}=\left({\frac {R_{\text{i}}}{R_{0}}}\right)^{-6}}
1394:
1365:
1300:
1252:
1169:
1125:
1085:
215:
1451:
1436:
Pauling, L. (1947). "Atomic Radii and
Interatomic Distances in Metals".
1328:
1386:
1292:
1067:
23:
or mean method (or bond valence sum) (not to be mistaken for the
428:
empirically that for most bonds it can be described by Eq. 2:
953:
951:
521:
If the structure is not known, the average bond valence, S
56:
of an atom is the sum of the individual bond valences
696:
595:
534:
437:
381:
372:, Eq. 1, which is central to the bond valence model.
257:
227:
120:
72:
1031:. IUCr Monographs in Crystallography. Vol. 12.
1408:Bragg, W. L. (1930). "The structure of silicates".
675:
is the corresponding average coordination number, V
1024:
737:
641:
561:
487:
415:
315:
240:
182:
100:
1471:O in crystal structures determined by X-rays"
8:
1498:"kristall.uni-mki.gwdg.de/softbv/references"
1264:
1262:
870:For uranium binding to oxygen the constants
838:The bond valence method is a development of
504:bond valence sum needs to be accounted for.
1551:"www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown"
642:{\displaystyle 0.5<(S_{1}/S_{2})<2.0}
1500:. Kristall.uni-mki.gwdg.de. Archived from
1097:
1095:
1075:
998:
729:
720:
714:
701:
695:
624:
615:
609:
594:
551:
539:
533:
513:the more the bond valence sum increases
474:
436:
404:
380:
304:
292:
282:
276:
262:
256:
232:
226:
214:is an empirical constant, typically 0.37
164:
151:
144:
125:
119:
89:
71:
1572:"www.ccp14.ac.uk/solution/bond_valence/"
1439:Journal of the American Chemical Society
1027:The Chemical Bond in Inorganic Chemistry
879:
1018:
1016:
1014:
1012:
1010:
947:
850:could be represented by electrostatic
821:Strengths and limitations of the model
101:{\displaystyle V=\sum (v_{\text{i}})}
52:The basic method is that the valence
7:
979:Altermatt, D.; Brown, I. D. (1985).
848:Pauling's electrostatic valence rule
1221:"A note on the distortion theorem"
14:
738:{\displaystyle S_{E}=V_{E}/N_{E}}
1465:Donnay, G.; Allmann, R. (1970).
1272:Journal of Physical Chemistry A
1219:Nickolsky, M. S. (2017-10-01).
488:{\displaystyle S=exp((Ro-R)/b)}
340:In the bond valence model, the
1467:"How to recognize O, OH, and H
630:
602:
482:
471:
456:
453:
410:
397:
95:
82:
1:
1206:10.1524/zkri.218.11.709.20301
199:is the observed bond length,
1537:10.1016/0022-5088(78)90010-3
758:is the preferred measure of
416:{\displaystyle V=sum(S_{j})}
241:{\displaystyle v_{\text{i}}}
804:Predicting bonding geometry
762:in the bond valence model,
210:has exactly valence 1, and
1640:
1423:10.1524/zkri.1930.74.1.237
1593:"Bond Valence Parameters"
1574:. Ccp14.ac.uk. 2001-08-13
1358:10.1107/S0108768106026553
1237:10.1107/S205252061700912X
1162:10.1107/S0108768101003068
1118:10.1107/S0108768199003961
1000:10.1107/S0108768185002051
959:"Bond valence parameters"
562:{\displaystyle S_{a}=V/N}
517:The valence matching rule
31:) is a popular method in
1033:Oxford University Press
811:Jahn-Teller distortions
1624:Coordination chemistry
934:Doing the calculations
739:
643:
563:
508:The distortion theorem
489:
417:
317:
242:
184:
102:
63:surrounding the atom:
33:coordination chemistry
1049:Brown, I. D. (2009).
1023:Brown, I. D. (2002).
740:
644:
584:valence matching rule
564:
490:
418:
318:
243:
185:
103:
694:
593:
532:
435:
379:
364:The valence sum rule
255:
248:has also been used:
225:
221:Another formula for
118:
70:
1525:J. Less Common Met.
1452:10.1021/ja01195a024
1345:Acta Crystallogr. B
1329:10.1021/ja00165a023
1285:2011JPCA..11512638B
1279:(45): 12638–12645.
1198:2003ZK....218..709U
1149:Acta Crystallogr. B
1105:Acta Crystallogr. B
986:Acta Crystallogr. B
749:Not surprisingly, S
25:valence bond theory
1139:Adams, S. (2001).
791:The covalent model
735:
639:
559:
485:
413:
357:electrostatic flux
342:valence of an atom
313:
238:
180:
98:
1387:10.1021/ic7011809
1381:(23): 9760–9767.
1317:J. Am. Chem. Soc.
1293:10.1021/jp203242m
1068:10.1021/cr900053k
1062:(12): 6858–6919.
931:
930:
760:electronegativity
662:Electronegativity
353:valence of a bond
298:
285:
265:
235:
174:
167:
128:
92:
29:quantum chemistry
1631:
1619:Chemical bonding
1604:
1603:
1601:
1600:
1589:
1583:
1582:
1580:
1579:
1568:
1562:
1561:
1559:
1558:
1547:
1541:
1540:
1519:
1513:
1512:
1510:
1509:
1493:
1487:
1486:
1475:
1462:
1456:
1455:
1433:
1427:
1426:
1417:(1–6): 237–305.
1405:
1399:
1398:
1369:
1352:(6): 1038–1042.
1339:
1333:
1332:
1323:(9): 3401–3402.
1311:
1305:
1304:
1266:
1257:
1256:
1216:
1210:
1209:
1180:
1174:
1173:
1145:
1136:
1130:
1129:
1099:
1090:
1089:
1079:
1046:
1030:
1020:
1005:
1004:
1002:
976:
970:
969:
967:
966:
955:
880:
744:
742:
741:
736:
734:
733:
724:
719:
718:
706:
705:
648:
646:
645:
640:
629:
628:
619:
614:
613:
576:bonding strength
568:
566:
565:
560:
555:
544:
543:
494:
492:
491:
486:
478:
422:
420:
419:
414:
409:
408:
370:valence sum rule
322:
320:
319:
314:
312:
311:
303:
299:
297:
296:
287:
286:
283:
277:
267:
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233:
189:
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175:
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169:
168:
165:
156:
155:
145:
130:
129:
126:
107:
105:
104:
99:
94:
93:
90:
37:oxidation states
35:to estimate the
1639:
1638:
1634:
1633:
1632:
1630:
1629:
1628:
1609:
1608:
1607:
1598:
1596:
1591:
1590:
1586:
1577:
1575:
1570:
1569:
1565:
1556:
1554:
1549:
1548:
1544:
1521:
1520:
1516:
1507:
1505:
1495:
1494:
1490:
1473:
1470:
1464:
1463:
1459:
1435:
1434:
1430:
1411:Z. Kristallogr.
1407:
1406:
1402:
1371:
1370:
1341:
1340:
1336:
1313:
1312:
1308:
1268:
1267:
1260:
1218:
1217:
1213:
1192:(11): 709–719.
1186:Z. Kristallogr.
1182:
1181:
1177:
1143:
1138:
1137:
1133:
1101:
1100:
1093:
1048:
1047:
1043:
1022:
1021:
1008:
978:
977:
973:
964:
962:
957:
956:
949:
945:
936:
890:
883:Oxidation state
875:
864:
862:Actinide oxides
840:Pauling's rules
836:
823:
806:
793:
786:
778:
768:
766:The ionic model
757:
752:
725:
710:
697:
692:
691:
686:
682:
678:
674:
670:
664:
620:
605:
591:
590:
535:
530:
529:
524:
519:
510:
433:
432:
400:
377:
376:
366:
334:
329:
288:
278:
272:
271:
258:
253:
252:
228:
223:
222:
205:
198:
160:
147:
146:
140:
121:
116:
115:
85:
68:
67:
62:
50:
42:Pauling's rules
12:
11:
5:
1637:
1635:
1627:
1626:
1621:
1611:
1610:
1606:
1605:
1584:
1563:
1542:
1514:
1488:
1468:
1457:
1446:(3): 542–553.
1428:
1400:
1334:
1306:
1258:
1231:(5): 874–878.
1211:
1175:
1156:(3): 278–287.
1131:
1112:(5): 698–711.
1091:
1041:
1006:
993:(4): 244–247.
971:
946:
944:
941:
935:
932:
929:
928:
925:
922:
918:
917:
914:
911:
907:
906:
903:
900:
896:
895:
892:
888:
884:
873:
863:
860:
852:lines of force
844:Lawrence Bragg
835:
832:
822:
819:
805:
802:
798:hydrogen bonds
792:
789:
784:
776:
767:
764:
755:
750:
747:
746:
732:
728:
723:
717:
713:
709:
704:
700:
684:
680:
676:
672:
668:
663:
660:
651:
650:
638:
635:
632:
627:
623:
618:
612:
608:
604:
601:
598:
571:
570:
558:
554:
550:
547:
542:
538:
522:
518:
515:
509:
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496:
484:
481:
477:
473:
470:
467:
464:
461:
458:
455:
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446:
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440:
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424:
412:
407:
403:
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387:
384:
365:
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333:
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328:
325:
324:
323:
310:
307:
302:
295:
291:
281:
275:
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261:
231:
203:
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1599:2020-12-17
1578:2012-11-19
1557:2012-11-19
1508:2012-11-19
1496:Adams, S.
1056:Chem. Rev.
965:2012-11-19
943:References
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