The Chemical Bond in Inorganic Chemistry - The Bond Valence Model - I. Brown (Oxford, 2002) WW

Russia

P. (Chairman), USA

G. India

The Netherlands

UK

USA

UK

IUCr Monographs on Crystallography

Accurate molecular structures: their determination and importance

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2 P. P. Ewald and his dynamical theory of X-ray diffraction

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Electron diffraction techniques, Volume 1

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Electron diffraction techniques, Volume

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The Rietveld method

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Introduction to crystallographic statistics

Crystallographic instrumentation

Direct phasing in crystallography: fundamentals and applications

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10 Defect and microstructure analysis by diffraction

11 Dynamical theory of X-ray diffraction

12 The chemical bond in inorganic chemistry

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Crystallographic computing 5: from chemistry to biology

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Crystallographic computing 6: a window on modern crystallography

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1 Correlations, transformations, and interactions in organic crystal chemistry

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BROWN

Department of Physics and Astronomy

McMaster University

Hamilton

Ontario

OXFORD

Preface

11 April 2001

Contents

a priori

Historical introduction

1.1 Introduction

a

posteriori

1.2 Chemical bonds

HISTORICAL INTRODUCTION

Fig.

et al. et al.

4

MODEL

U

1.3 The ionic model

ionic model

U,

1.4 Quantum mechanics

et al. et al.

1.5 The symmetry model

et al.

1.6 Topological models

1.7 Pauling's electrostatic valence model

s =

The rule of parsimony.

bond valence

bond valence model

et al. et al.

2The ionic bond

2.1 Introduction

U

et al.

BOND

Crystal energy and the Coulomb field

Eh

Ei

E^oclii

E^oclii

EitTnono

EitTnono i,

Ei

BOND

Q{,

Qt

Q{.

Every part of the electron density of the crystal must belong to at least one

atom.

The partitioning must ensure an appropriate assignment of charges, Qf.

Qh

Qt Vt.

Consistent with the above constraints, the partitioning should be chosen so as

to minimize Emu\t.

Emu\t

et al.

Vt.

2.4 The Madelung field of a crystal

et al.

i

Q{ Qt.

2

BOND

Fig.

{(x,y,z);x + y = I } . et al.

Fig.

et al.

Cy,

Py,

qt

Py,

Qf

qt. Ptj

Cy

Cy. Cy,

Cy

Qf Cy. Cy

a priori,

Cy

Cy

Bond networks and bond graphs

a network consists of an

array of nodes which are connected by links.

valences the fluxes

Na Na connectivity adjacency matrix,

Table 2.1

BlB2HIH2H3H4H5H6

Bl

00111010

B2

00110101

HI

11000000

H2

11000000

H3

10000000

H4

01000000

H5

10000000

H6

01000000

BOND

Fig. 2.4.

Fig. 2.6.

2.6 Coordination number

Definition of coordination number. The coordination number of an ion is the

number of ions to which it is linked by electrostatic flux.

et al.

tertiary bonds,5

secondary bond was

tertiary bond

THE BOND

Fig.

et al.

a c z

Conclusions

The chemical bonding can be completely described in terms of localized bonds

between neighbouring atoms.

The bonds have a cation at one end and an anion at the other.

The bond valence model

3.1 Experimental bond valences and bond lengths

Ctj,

bond strength,

3

EXPERIMENTAL BOND VALENCES AND BOND LENGTHS

Fig. 3.1.

et al.

experimental

bond valence, S.

Rtj j, Sy

R0, B,

B

RQ,

B

B

28 THE BOND MODEL

valence sum rule,

(Vf

Qf Vj)

Sy,

et al.

RQ, B,

3.2 Empirical network equations

Sy

valence sum

rule loop, equal valence, rule,

network equations.

theoretical bond valences

s.

S,

s,

unstrained

structures,

S.

strained structures

(3-

Table 3.1 s, S, R,

s S s-S

Qh Vt,

sy,

for unstrained structures at equilibrium all bonds have the same bond capacitance.

R0 B,

et al.

2

THE BOND MODEL 31

Vi

sy

the bond valence model cannot distinguish between ionic and covalent

bonding.

The bond valence model

Principle of maximum

symmetry,

THE BOND MODEL

Rule 3.1 As far as allowed by the chemical

and geometric constraints, all atoms and all bonds in a compound will be

chemically and geometrically indistinguishable.

Rule The breaking of symmetry is always the consequence of an

identifiable chemical or spatial constraint.

Rule The sum of the valences of all the bonds

formed by an ion is equal to the valence of the ion.

Condition 3.1. The stoichiometry must obey the electroneutrality principle, namely

that the sum of all the atomic valences (formal ionic charges), having regard to

their sign, is zero.

Condition The bond graph must be bipartite as described in Section 3.5, i.e. all

of the bonds must connect a cation, e.g. Na+, to an anion, e.g. Cl~.

loop, equal valence rule.

Rule 3.4 The sum of bond valences around any

loop in the bond network, having regard to the direction of the bond, is zero.

The valence of each atom is distributed as uniformly as possible among the bonds

that it forms.

Rule Increasing

the bond valence between two ions reduces the distance between them as shown

in Fig. 3.1.

3.4 The distortion theorem

Rule 3.6 For any ion, lengthening some of its bonds and

shortening others, keeping the bond valence sum the same, will always increase the

average bond length.

AR,

34 THE BOND MODEL

AR

Rule 3.6a For any ion,

lengthening some of its bonds and shortening others, keeping the average bond

length the same, will always increase the valence sum.

Rule If an ion is placed in an environment in which the average

bond length is too long, i.e. in a cavity which is too large for the ion, the

environment will distort in such a way as to increase the lengths of some bonds and

decrease the lengths of others in order to raise the bond valence sum to the

expected value.

3.5 Bond networks with non-bipartite graphs

BOND NETWORKS WITH NON-BIPARTITE GRAPHS 35

Fig. 3.3.

Fig. 3.4.

CF3CO^

C\+

C4+Q\~]~

36 THE BOND MODEL

Fig.

Fig. 3.6.

38 THE BOND MODEL

(T

TT

a

et al.

et al.

Anion and cation bonding strengths

4.1 Bond graphs and coordination number

Rule 4.1 A bond exists between a cationand an anion if its experimental bond valence is larger than 0.04 the cation

valence.

4

BONDING STRENGTHS

ideal coordination

numbers

BONDING STRENGTH

4.2 Anion bonding strength

anion bonding strength, sa,

P-O

Fig. 4.1.

BONDING STRENGTHS

maximum and minimum anion bonding strengths.

BONDING STRENGTH 47

pKa,

pK^

Lewis base strength.

strong weak anions

4.3 Cation bonding strength

cation bonding strengths, sc,

vc

Fig. 4.3. pKa.

BONDING STRENGTHS

strong weak cation

Lewis acid

strength.

Fig. 4.4.

Fig. 4.5.

4.4 The valence matching principle

Valence matching principle,

Rule The most stable compounds are formed

between cations and anions that have the same bonding strength.

(sa = (sc =

(sc =

(sc=

BONDING STRENGTHS

4.5 Hard and soft acids and bases

hard soft,

APPLICATIONS OF THE MATCHING PRINCIPLE

4.6 Applications of the valence matching principle

Fig. 4.6.

sc/sa

BONDING STRENGTHS

s^Sn

5Liquids

5.1 Introduction

The cation and anion bonding strength of water

Fig. 5.1.

sjsa

sc/sasjsa

(sc =

Reactions of cations with water

et al. et al.

n,

n.

5.4 Reactions of unions with water

Fig. 5.4.

PO^~

PO34~

(sc =

pK^,

Aqueous solubility

Mg(H2O)6SO4

CO\

CG>\ +

Fig.

(sc Sg)

A=

5.6 Aqueous solutions of soft ions

(sc =

Non-aqueous solutions and melts

(sc = Q.Q6

Fig. 5.6.

likeweak

Cation coordination number

6.1 Introduction

ab initio

mutatis mutandis,

6.2 Anion-anion repulsion

Fig. 6.1.

Fig. a.

Table 6.1

et al.

Fig. 6.3.

a,

s'',

s

COORDINATION

RQ = b =

s'.

s'

Sh s'.

Rmin s' >

Rmin. Sh

Rmin.

Sh,

s', Sh s'

Fig.

S^ Q.4vu,

S^>Q.2vu,

The normal hydrogen bond

normal hydrogen bond

STRONG HYDROGEN BONDS

Strong hydrogen bonds

Strong hydrogen

bonds

Fig. 7.6.

Weak hydrogen bonds

Rmin,

(Sh 3~,

r, 6)

6 r

D.

9,

9.

al.

et al.

8.3 Transition metals

TRANSITION

8.3.1 Jahn- Teller distorted cations

Fig. 8.6.

dx2_y2

dx2_y2

x-y

8.3.2 Transition-metal cations with empty or near-empty d shells

TRANSITION

et al.

et al.

Fig. 8.9.

TRANSITION

Fig. 8.10.

Table

8.4 Conclusions

9Physical properties of bonds

9.1 Introduction

9.2 Bond lengths and bond angles

R0,

B

q,

V, k

BOND LENGTHS AND BOND

RQ.1

k

k

RQ,

RQ

TT

R0

RQ

TT

TT

TT

RQ

TT RQ et al.

et al.

k k

et al.

x = [(Si + S2)/2V]

Si S2, 0

V/4, V

(S =

BOND LENGTHS AND BOND

9.3 Bond force constants and thermal vibrations

Fig. 9.1.

G,

S.

S> a b,1 1 1

a = 1 1 b = 1 .

A, G,

(A2),

Fig. 9.2.

T,

k A.

(A2}

U,

(A2) U

(A2)

et al.

(A2),

U,

(A2)

d(A2)/dT

dUuncoi/dT

THERMAL

Fig. 9.3.

9.4 Thermal expansion

et al.

R&.

114 PHYSICAL PROPERTIES OF BONDS

R'

(R) (R) Re SR

S',

SR/B SR/B

SR

S' S

Re.

S

(R) Re = AR

AR

(SR2),

(SR2)

(A2)

B = 37 G

Fig. 9.4.

G pS, p S

R S Rp

et al.

(A*),

R,

Fig. 9.5.

R',

(A2)/2R.

Gt

Fig. 9.6.

9.5 The variation of R0 with temperature

RQ,

AR.

R + AR R%,

R0 T,

T dR/dT

S

Solids

10

Space and space groups

10.1 Introduction

SPACE AND SPACE GROUPS

10.2 The crystal lattice and translational symmetry

R(l, m, n),

b, c

m, n)

b unit cell

R(l, m, n)

R(l,m,ri)

THE CRYSTAL SYMMETRY

Fig. 10.1.

Fig. 10.2.

124 SPACE AND SPACE GROUPS

Fig. 10.3.

Fig. 10.4.

10.3 Space groups

asymmetric unit

G,

space group.

x, y, z x, y, z.

p/n p ri)

G,

ofmG R,

mG mG =

10.4 Special positions

mG

Fig.

CO\

mG =

ofmG/2 =

non-translational unit. mN,

mG,

Fig. 10.6. nis mw

(ms 2,mwT). ms6, mw

ms3, mwT), ms 2, mw

ms mw

mN.

mN

S

W, S

mw mw\&

S, ms,

S W* S= N,

Rule 10.1. In a given space group, all the Wyckoff positions with the same multi-

plicity, mw, have site symmetries of the same order, ms.

Fig. 10.7. ms,

Rule 10.2. The order of the site symmetry of any Wyckoff position is in inverse

proportion to its multiplicity.

S,

mw,

10.5 Matching the special positions to the chemistry

his fundamental law of crystal chemistry

Rule 10.3. Since all the atoms in the chemical formula must exist on one or other

of the Wyckoff positions of a crystal, the multiplicity of an occupied Wyckoff

position must correspond to the frequency with which the corresponding atom

appears in the chemical formula, and the site symmetry of the Wyckoff position

must correspond to a possible symmetry of the atom's bonded environment.

spectrum

mN,

10.6 The symmetry of bonded neighbours

inter alia

ms = 4%,

ms = 24).

(D^k)

ms=

ms=8)

ms =

ms=6.

Table 10.1

m3

m+12

Table 10.2

m6

26

10.7 Summary

Modelling inorganic structures

11.1 The problem of a priori modelling

topology structure,

geometry,

relative

stability.

THE TOPOLOGY 135

11.2 Determining the topology

Principle of electroneutrality

Rule 11.1 Since the sum of all atomic valences is

zero, the sum of the atomic valences of the cations is equal to the sum of the atomic

valences of the anions.

Coordination number rule

Rule 6.1 Since each bond starts on a cation and ends

on an anion, the sum of the coordination numbers of the cations equals the sum ofthe coordination numbers of the anions, and both are equal to the total number of

bonds in the formula unit.

Rule 11.2 The ratio of the average coordination number of the

cations, (Na), to the average coordination number of the anions, (Nx), is the same

as the ratio of the number of anions, x, to the number of cations, a.

Rule 11.3 Compounds that have a high anion content will stabilize

high cation coordination numbers.

Principle of maximum symmetry

Rule 3.1 As far as allowed by chemical and

geometric constraints, all atoms and all bonds in a compound will be chemically

and geometrically indistinguishable.

Principle of close packing

Rule 11.4 Like ions tend to lie on close packed

lattices, since this arrangement minimizes their repulsive energy when they are

confined to a fixed volume.

Shubnikov's fundamental law of crystal chemistry

Rule 10.3 Atoms will occupy Wyckoff positions

in the crystal that are compatible in both multiplicity and symmetry with the

bond graph.

11.2.1 Space-based approaches

THE TOPOLOGY

Random structure approaches

et al.

et al.

Lattice models

THE TOPOLOGY

Fig.

HCP

FCC

et al.

139

et al.

11.2.2 Chemistry-based approaches

THE TOPOLOGY 141

Rule 11.5 When generating a chemical structure, the

strongest bonds are formed first, followed by the others in decreasing order of their

valence.

a priori

Creating the bond graph

Fig. 11.2.

THE TOPOLOGY 143

Fig. 11.3.

THE TOPOLOGY 145

Sy

Sy

Fundamental building blocks

Fig. 11.4.

THE TOPOLOGY 147

OPb64+.

et al. et al.

Polyhedral linkage

Fig. 11.5.

Fig. 11.6.

THE TOPOLOGY 149

et al.

Fig. 11.7.

THE TOPOLOGY 151

The space group method

ms =

ms=

ms=

ms =

ms=

ms =

V2

ms =

P4i23,

'a).

Weakly bonded structures

a priori

11.2.3 Valence maps

Fig. 11.8.

pt

pt

THE TOPOLOGY 159

Fig. 11.9.

c

in ph pf

pt

c

et al.

Fig. 11.10.

11.3 Refining the geometry

et al.

et al.

et al.

et al.

11.4 Modelling defect structures

et al.

et al.

11.5 Modelling glasses

al.

11.6 Summary

Lattice-induced strain

12.1 The origins of lattice-induced strain

Fig. 12.1.

Fig. 12.2.

et al.

lattice-induced strains

12.2 Structures with lattice-induced strain

bond strain index

et al. cr3

S

s

global instability

index et al.

Gil

Gil

Gil

et al.

et al.

Fig. 12.4.

I992a).

12.3 Relaxation of lattice-induced strain

12.3.1 Relaxation of the geometry

(S= (S =

12.3.2 Relaxation by defects

Fig. 12.5.

et al.

12.3.3 Electronic relaxation

8.

12.3.4 Relaxation of symmetrydisplacive phase transitions

12.3.5 Changing the bond graphreconstructive phase transitions

Fig. 12.6.

12.4 Incommensurate structures

et al.

RQ = 2l4pm

STRUCTURES 175

Fig. 12.8.

et al.

Fig. 12.9.

et al.

et al.

et al. et al.

et al.

12.5 Summary

s s

13

Applications

13.1 Introduction

13.2 Crystallography

13.2.1 Structure solution

ab initio,

ab initio et al.

et al.

(p\

V\ YiS\ ^S2

V\

V2

S,

andp2,

i p

i.

et al.

et al. et al.

p\

et al.

et al.

et al.

13.2.2 Analysis of crystal structures

primae

facie

et al.

bond valences (vu) bond valence sums (vu)

Sum

0.30

0.26, 0.25

0.24

1.05

0.28, 0.27

0.31

1.12

S

1.54

1.48

1.49

1.49

6.00

0.15

0.04

(0.76)

(1.00)

0.03

(0.82), 0.15

(1.00)

Sum

1.99

2.07

2.00

2.08

2.04

13.3 Physics

13.3.1 Perovskite-related solids

Fig. 13.1.

13.3.2 Electrical properties

et al.

APPLICATIONS

et al.

Fig. 13.2.

V(O)

= = = = = = =

et al.

13.3.3 Magnetic properties

Fig. 13.3.

et al.

et al.

13.3.4 Grain boundaries

et al. et al.

et al.

et al.

et al.

13.4 Mineralogy

et al.

13.4.1 Soil chemistry

et al.

et al.

et al.

et al. (I997a,b,c)

et al. (1997 a,b)

et al.

et al.

13.4.2 Zeolites

CHEMISTRY 197

13.4.3 Glasses

et al.

13.5 Chemistry

et al.

13.5.1 Nuclear magnetic resonance

et al.

et al.

fy

A M

et al.

et al.

NMR

et al. NMR

13.5.2 Transition-metal complexes

et al. et al.

TT

TT

et al. et al.

et al.

13.5.3 Heterogeneous catalysis

s, E:

CHEMISTRY 201

et al.

et al.

13.5.4 Esterification and hydrolysis

Fig. 13.4.

Table 13.3

Fig. 13.5.

BIOLOGY 203

13.6 Biology

13.6.1 Enzymes

et al. et al. et al. et al.

et al.

et al.

et al. et al.

13.6.2 Calcium and sodium binding by proteins

BIOLOGY 205

et al.

et al.

et al.

(cis-

13.7 Databases

et al.

et al.

et al.

et al.

14

Chemical implications of the bond valence model

14.1 Why is the bond valence model so robust?

14.1.1 The attractive force

WHY IS THE BOND MODEL SO ROBUST? 209

14.1.2 The repulsive force

R0 B

TT

TT

RQ TT

14.2 Two-body potential models

14.3 The properties of the bond graph

ELECTRON-PAIR MODEL 211

14.4 The Lewis electron-pair model

Fig. 14.1.

WHY ARE FROM

et al.

14.5 Why are cations different from anions?

Fig. 14.2.

14.6 Orbital models

ad hoc

PO^"

14.7 Electron density models

et al. et al.

a, b, c d

Fig. 14.3.

et al. et al.

14.8 The topology of the Madelung field

1

14.9 Conclusions

et al.

et al.

Bond valence parameters

Al.l Introduction

et al.

S

S = R

A1.2 Determination of bond valence parameters

Table Al.l

o2-o2-o2-F-

o2-cro2-o2-

B(pm)

(Continued)

RQ, B N,

S, s,

B

B N) RQ.

B R0

RQ B N),

B N),

RQ.

R0 B

B

B

1

RQ B

RQ

i.

RQi, RQ V

B

228 APPENDIX 1

RQi B

Rw

Rw B

RQi RQ

R0.

RQ.

B RQ.

(R) et al.

RQi

RQ

a b RQ

RQ

BOND PARAMETERS 229

RQ R0

R0

rt ct Cj

RQ RQi

et al.

R00, R0^, Ros

R00 R0^.

RQ

I997a,b,c; I999a,b; et al.

R0

RQ

et al. et al. R0

R0

230 APPENDIX 1

RQ

et al.

et al.

B

et al.'s

RQ B,

A1.3 Other bond valence expressions

rc,

a, b, c, d

e

BOND PARAMETERS 231

S

g,

S = 0 R e/c.

R e/(l +g).

e = et al.

e =

et al.

a (= RI

av

232 APPENDIX 1

R SQ,

B

Space group spectra

mw,

Table A2.1

Table

Table A2.3

8

234 APPENDIX 2

Table A2.4

Table (Continued)

111

236 APPENDIX 2

Table

Table (Continued)

238 APPENDIX 2

Table A2.9 (Continued)

1 1

12,2,2,

2,1

* *

2,1

2,1

2,1

* * 2,1

* *

2,1

Table A2.10

P2l

Appendix 3

Solution of the network equations

N&+1

Nb

Nb Nb s.

M:

s NA +

M

et al.

Vjv

V v

a, b, c, d, e,

242 APPENDIX 3

d=\-a,

e=l-2b.

c = Ib,

f=(lb-\}/2.

b = 1/21 =

c =

f=a =

e =

d =

5/27 = 011/27 =21/54 =13/27 =33/54 =

.18vu,0.41 vu,0.39 vu,0.48 vu,0.61 vu.

Appendix 4

Cation and anion bonding strengths

Table A4.2

Table A4.3

Appendix 5

References to the ICSD and the CSD

Table A5.1

et al.

Acta Cryst.

Acta Cryst.

Acta

Cryst.

Acta

Cryst.

Can. J. Chem.

Solid State Chem.

Zeit. Kristallogr.

Proc. Japan. Acad.

Acta Cryst.

C. R. Acad. Sci.

Inorg. Chem.

Mater. Res. Bull.

Amer. Miner.

Zeit. Kristallogr.

Acta Cryst.

Acta Cryst.

Ferroelectrics

Table A5.1 (Continued}

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Acta Cryst. 18,

Acta Cryst. 18,

Dokl. Akad. Nauk

SSSR239,

Zeit. Anorg. Allgem. Chem.

Ann. Acad. Sci. Fenn. Ser.

A6 Physica 313,

Nature

Acta Cryst. 26,

Zeit. Anorg. Allgem. Chem. 480,

Solid State Chem.

Acta Cryst. B26,

Amer. Miner.

60,

Zeit. Kristallogr.

90,

Acta Cryst.

Acta Cryst.

Acta Cryst.

Amer. Chem. Soc.

C. R. Acad. Sci.

et al. Physica Statu Solidi

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Acta Cryst.

Naturwissenschaften

Acta Cryst.

Ark. Kemi.

Zeit. Kristallogr.

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Tijdskr.

Chem. Phys.

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Australian

J. Chem.

Acta Cryst.

Kobutsugaku

ZasshilS,

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Acta Cryst.

Zeit.

Electrochem.

Zeit. Electrochem.

Adv. X-ray Anal.

Chem. Phys.

Acta Cryst.

Zeit. Kristallogr.

Table A5.1 (Continued}

Phys Rev. B

Solid State

Chem.

Zeit. Kristallogr.

Phys. Rev.

Miner. J. (Japan)

Acta Cryst.

Phys. Stat. Solidi

Acta

Cryst. B46,

Solid State

Comm.

Amer. Miner.

Amer.Miner.

Solid State Chem.

Powder Diffraction

Acta Cryst.

Mat. Res. Bull.


Physica C

Physica C 235,

Acta Chem. Scand. 148,

Zeit. Kristallogr. 145,

Zeit. Kristallogr.

156,

Solid State

Comm. 24,

Carnegie Inst. (Washington)

Yearbook


Phys. C

Phys.

C

Solid

State Chem. 61,

Amer. Chem. Soc. 109,

252 APPENDIX 5

et al.

Curr. Sci.

Proc. Roy. Soc.

Land.

Can. J. Chem.

Acta

Cryst.

Acta

Cryst.

Zeit. Kristallogr.

References

Ada Cryst.


Acta Cryst.

Solid State Ionics 105,

Phys. Rev. Leu.

Phys.

Rev. B63,

Acta Cryst. B49, Adv. Inorg.

Rad. Chem. 15,


Mol. Struct. (Theochem) 496,

Amer. Chem. Soc. Ill,

et al.

Acta

Cryst.

Monatshefte Chem. 106,

Phys.

Rev. B

Acta Cryst. B41,

Solid State Chem.

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List of symbols

B

NaNbR

R0

r

s

s'

S

V

f

Index

see

see

see see

see

see

see also

see also

see also

see also

(cont.):

see also

see also

see

see

see also

see

Cr(H2O)l+

see

see also

see

see also

see also

see

see

see

see

see

see

see

(A3B2(XO4)3)

see

see also

(cont.):

see

see

see

see see

see

see

see

see

(A2MN2O1) 111

see also

see

see also

see

see also

see

see

see also

see

see also

see

see also

see

see

(cont.):

see

see also

see also

see

see

Documents

The Chemical Bond in Inorganic Chemistry - The Bond Valence Model - I. Brown (Oxford, 2002) WW