View of Lone Electron Pairs and Their Role in Structural Chemistry

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View of Lone Electron Pairs and Their Role in Structural Chemistry I. David Brown* Brockhouse Institute for Materials Research, McMaster University, Hamilton, ON, Canada L8S 4M1 ABSTRACT: Nonbonding valence electrons, better known as lone pairs, are found in all anions as well as in cations in their lower oxidation states. In this paper the properties of lone pairs are analyzed using a bond valence model defined in terms of a core-and-valence-shell picture, which can be reduced to the point charges of the ionic model. A bond is defined in terms of the electrostatic flux linking the atoms. When the lone pairs are inactive, they are uniformly distributed around the valence shell and the ion behaves like a main group ion that has no lone pairs. In this state it can be assigned a bonding strength that obeys the valence matching rule (for stable bonds the cation and anion bonding strengths should not differ by more than a factor of 2). However, when an ion with a lone pair has a bonding strength less than half that of the counterion, it has the flexibility to form a stronger bond by converting lone pair electron density to bonding electron density in the region where the valence shells overlap. To conserve the number of lone pairs, bonding electron density elsewhere in the valence shell is converted to lone pairs. The result is the adoption of an anisotropic coordination environment with fewer bonds but bonds whose enhanced strength matches the higher bonding strength of the counterion. This analysis raises questions about the validity of a number of traditional ideas on the nature of chemical bonds. It shows that lone pairs do not form dative bonds. In a neutral molecule, the base function that is often attributed to a lone pair is always accompanied by an acid function, and both functions must be simultaneously activated. Electron-pair bonds, which are found only around strongly bonding cations, normally result in the formation of molecules, explaining why the models developed to describe molecular structures are unable to give good descriptions of extended crystal structures.

1. INTRODUCTION The properties of nonbonding valence electron pairs, more generally known as lone pairs, and their effects on the stereochemistry of inorganic compounds, are usually explained in terms of the valence shell electron pair repulsion (VSEPR) model1 in which the valence shell electrons are treated as localized electron pairs, either bonding or nonbonding, that arrange themselves within the valence shell in such a way as to maximize the distance between them. Each bonding electron pair gives rise to a bond between the central atom and one of its ligands, but each lone pair, being nonbonding, gives rise to a ligand site that is unoccupied. This model gives a good prediction of the way ligands are arranged around an atom containing one or more lone pairs when the lone pairs are stereoactive, and it explains the deviations from the expected ideal geometry. However, the VSEPR model does not account for the many cases in which ions with lone pairs are found in high symmetry environments in which the lone pair is said to be inactive. Such an environment is found, for example, around Tl+ in TlCl, which at low temperatures has the six-coordinate NaCl structure (ICSD 61518)2 and at room temperature the CsCl structure (ICSD 44935) in which thallium lies at the center of a uniform cube of eight Cl
ligands. This is in contrast to the Tl+ coordination in Tl3BO3 (ICSD 8084) in which it forms three Tl
O bonds arranged in the form of a triangular pyramid with the Tl+ ion at its apex. The bonding around ions with stereoactive lone pairs is loosely described as covalent, while that around ions whose lone pairs are inactive is often described as ionic, but the terms “covalent” and “ionic” are not well-defined and the terminology r 2011 American Chemical Society

is more descriptive than predictive. There is a need for a more quantitative approach to lone pair chemistry, one that can, inter alia, predict when the lone pairs will be stereoactive and when they will not. This paper shows that the bond valence model meets these requirements and explores some of its consequences. This paper starts with a development of the bond valence model from a conceptual core-and-valence-shell description of the atom. It then analyses the stereochemistry of the lone pair in terms of the relative bonding strengths of the counterion. Finally, it explores some of the consequences of this model.

2. BOND VALENCE MODEL This section develops the necessary tools of the bond valence model used in this study. Further details are given by Brown.3 Valence and electron counting rules are the concepts most closely associated with models of localized chemical bonds, but electrons can only be counted when they are added to, or removed from, an atom. Once inside the atom, the electrons lose their individual identity and all that remains is an undifferentiated electron density.4 Because it cannot be used to identify either the number or location of the valence electrons, the electron density is a poor starting point for the development of Special Issue: Richard F. W. Bader Festschrift Received: April 7, 2011 Revised: June 9, 2011 Published: June 30, 2011 12638

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The Journal of Physical Chemistry A a bond model of chemistry, though this has not discouraged many people from trying.5
7 The concept of valence is based on the observation, fully supported by quantum calculations, that all atoms contain a small number electrons, known as the valence electrons, that are easily removed from the atom, hence can be readily counted. Their number is called the valence of the atom. The valence electrons are significantly more weakly bound than the remaining electrons, known as core electrons, hence they are more readily involved in bonding. In practice, the core electrons can be, and usually are, ignored in bond models; it is sufficient to consider only the valence electrons. The valence of an atom is the basis of the periodic table, which is widely used to understand and predict the chemical properties of atoms. The key to any bond model based on electron counting rules is the ability to distinguish between the valence and core electrons, a distinction that is, in principle, impossible to make in the undifferentiated electron density of the atom. The core-and-valence-shell model is a conceptual model with this property. The electrons in an atom are divided into two groups, the valence electrons and the core electrons, the latter usually combined with the nucleus. Lone pair electrons belong in the valence shell, but initially we assume that lone pairs are not present; they are added to the model later. The model makes no assumptions about the spatial arrangement of these two groups of electrons because their physical location is irrelevant to the properties required by the bond model, as will be shown below. Any spatial arrangement that allows the core and valence electrons to be addressed separately will work. Most people find it helpful to adopt a physical picture of the atom, and because the electron density in the physical atom is close to being spherically symmetric with the active bonding electron density near its surface, the conceptual atom can be visualized as consisting of a spherical core with the atomic nucleus at its center, surrounded by a spherical shell of valence electrons. It is convenient, but not essential, that the core and valence shell be separated by a space as this makes it easier to visualize them as separate entities. In a neutral atom, the valence shell carries a negative charge equal to the valence of the atom, and the core carries an equal positive charge. Because the valence shell and core carry equal and opposite electric charges, they are connected by electrostatic flux equal to the charge on the valence shell, that is, the flux linking the valence electrons to the core is equal to the valence of the atom. This is the essence of the model. The usefulness of this model lies in the fact that the flux depends only on the number of electrons in the valence shell. It does not depend on their location or on the way they are distributed; the flux would be the same if the core and valence electrons were represented by two point charges separated by any arbitrary distance. For this reason the core and valence shell of the model may be represented by any convenient shape. In principle, they could be mapped onto the physical electron density, if it were possible to identify the location of the electron density representing the valence electrons, but the spherical model described above is the most convenient. An important property of this model is that, because the electrostatic flux linking the core and valence shell depends only on the number of valence electrons, it contains no information about the distribution of the electron density in the atom. Conversely, the electron density distribution in the atom does not provide any information about the flux. The electron-density and core-and-valenceshell models are orthogonal and complementary. This does not

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mean that the two models are unrelated. Both are derived from the same quantum mechanical model, and both are solutions to the same problem, albeit presented in different and mutually independent forms. That said, the electron density does reflect some features of the bond models. Local concentrations of density in the neighborhood of bonds and lone pairs are revealed in the Laplacian of the electron density,8 and various methods have been suggested for showing how electrons are localized,8
10 but none of these can separate out the valence electrons or reveal the flux linking the valence electrons to the core. A bond is formed when the valence shells of two atoms overlap and the electron density in the valence shell of one of the atoms spin-pairs with the electron density in the valence shell of the other. Because it is the valence-shell electron density, not the individual valence electrons, that spin-pair, the bond is not restricted to an integral number of electron pairs, and indeed, integral electron-pair bonds are the exception rather than the rule. The electrons forming a bond are each linked to their own cores by electrostatic flux, and because spin pairing ensures that each atom contributes equal amounts of electron density to the bond, the fluxes linking the bonding electrons to their respective cores are equal, and both are equal to the number of electron pairs that form the bond as given in eq 1. This number is called the bond valence. Each atom contributes the same number of electrons to the bond: This number is equal to the flux linking the bonding electrons to their respective atomic cores and is called the bond valence; S:

ð1Þ Because the total flux received by the core is equal to the atomic valence, the sum of the valences (fluxes) of all the bonds formed by an atom is equal to the its atomic valence (charge). From this we immediately get the valence sum rule, eq 2: The sum of the valences; Sij ; of all the bonds formed by an atom i with its ligands j; is equal to the valence; Vi ; of atom i

Vi ¼

∑j Sij

ð2Þ

This equation is widely used for validating crystal structure determinations. The core-and-valence-shell model can be used to derive and justify the ionic model in the following way. We first assign all atoms to one of two classes, which are conveniently called “cations” and “anions”. At this stage these names are just labels and all the atoms are still electrically neutral. In the commonly found case where each bond has an anion at one end and a cation at the other, i.e., in the case where there are no cation
cation or anion
anion bonds, we can conceptually transfer the bonding electrons of the cation to the valence shell of the anion. As the cation and anion valence electrons forming the bond are already spin paired and located at the same point in the conceptual model, this does not involve any movement of the electrons, just a transfer from one overlapping shell to the other. Each cation now consists of just its core, together with any lone pair electrons if present. It carries a positive charge equal to its cation valence (formal charge). Similarly, each anion carries a complementary negative charge equal to its valence. If all the core and valence electrons, with the bonding electrons assigned to the anions, together with their nuclei, are shrunk to a point charge, we have generated an array of point charges; the anions are represented 12639

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by a negative point charge and the cations by a positive charge. This is just the ion array of the ionic model. The electrostatic flux (bond valence) that linked the cations to their bonding valence electrons in the core-and-valence-shell model is the same flux that now links the cations to the anions in the ionic model because the valence electrons of the cation now lie on the anions to which it is bonded. This model makes no assumptions about the physical location of the spin-paired electron density that constitutes the bond. The real bonding electron density may lie on the anion as assumed in the ionic model, or it may lie in the middle of the bond as assumed in covalent models. Either way, the bond valence depends only on the number of electrons forming the bond. The only restriction on the ionic model is that every bond must link a cation to an anion. This flux provides the chemical cohesion that holds the atoms together, but in classical electrostatics, such a neutral array of point charges would collapse to a point. This does not occur in the quantum mechanical models, but to keep the ionic model classical and simple, this requirement is replaced by introducing an arbitrary short-range repulsion. In the traditional ionic model this takes the form of a repulsive potential, empirically chosen to ensure that the resulting arrangement of point charges corresponds to the observed positions of the ions in the crystal. The repulsive force prevents the ions from approaching too closely, but if the flux increases, the bond necessarily becomes shorter. The correlation between bond length, R, and bond valence, S, can be determined by calculating the electrostatic flux in cases where the observed ionic positions are known from experiment. Preiser et al.11 have shown that the correlation between the electrostatic flux (valence) of the bonds in many different compounds can be represented by eq 3. S ¼ expððRO
RÞ=bÞ

Table 1. Selected Cation and Anion Bonding Strengths cation

NA (to oxygen)

Weakly Bonding Tl+

0.11

9.0

Rb+

0.124

8.0

Na+

0.156

6.4

H+ (acceptor)

0.20

Sn2+

0.25

8

Mg2+

0.33

6.0

As3+

0.37

8

Sn4+ H+ (donor)

0.68 0.80

5.9

Strongly Bonding

ð3Þ

where RO and b are empirically fitted parameters that depend only on the nature of the two terminal ions. RO is a measure of the size of the two ions and b the softness of their interaction. Like the repulsive potential in the traditional ionic model, eq 3 provides the empirical normalization that brings the positions of the ions in the ionic model into coincidence with positions of the atoms in the real structure. A cumulative list of these parameters obtained from various sources is available on the web.12 The bond valence, S, in eq 3 is measured in valence units (vu). In the ionic model every bond is an electric capacitor, having equal and opposite charges at each end linked by electrostatic flux. The flux links the ions into a bond network, which is equivalent to a capacitive electrical circuit that can be solved using the Kirchhoff equations3 to predict the valences of the bond which, using eq 3, can be converted into bond lengths. Provided the structure experiences no steric or electronic stresses, the predicted bond lengths typically differ from the observed bond lengths by less than 0.02 Å, an accuracy similar to that achieved by quantum calculations.11 Stereoactive lone pairs are the source of an electronic stress on the system, and the present study shows how the bond valence model is able to include these effects. The topology of the bond network must be known before the Kirchhoff equations can be used; i.e., it is necessary to know which ions are bonded. In many cases this can be predicted using the bonding strengths of the ions and the valence matching rule. The bonding strength (SA or SB) is an a priori estimate of the valence of the bonds formed by the ion. It can be calculated using

SA (valence units)

Br7+

1.17

6

S4+

1.2a

3.4

C4+

1.33

3

P5+

1.33

4

Se6+

1.50

4

S6+ N5+

1.50 1.67

4 3

Cl7+

1.75

O6+

(2.00)

(3)

F7+

(2.33)

(3)

4

anions

SB

NBb

NO3


0.11

9

Cl


0.17

6

SO42
CO32


0.17 0.22

12 9

PO43


0.25

12

F


0.25

4

BO33


0.33

9

O2


0.50

4

a

Only known with a stereoactive lone pair. b The values of NB of the simple anions are assumed and are used in calculating the coordination numbers of complex anions. (Values given in parentheses are estimates.)

eq 4. SA ¼ VA =NA

ðcationsÞ

ð4aÞ

SB ¼ VB =NB

ðanionsÞ

ð4bÞ

VA is the valence of the cation, VB is the valence of the anion, and NA and NB are typical values of their respective coordination numbers, the subscripts referring to Lewis acid and Lewis base, respectively. In the absence of any better value, the typical coordination number of a cation can be obtained by averaging the coordination numbers found in a random sample of oxide structures.3 Incidentally, SA can also be used as a surrogate for the electronegativity as it leads to the same ordering of the elements. A selection of bonding strengths is given in Table 1. This table is based on those given in ref 13 and Appendix 4 of ref 3. The cations are conveniently divided into two groups, those that are weakly bonding (SA < 1) and those that are strongly bonding (SA > 1). The two groups have distinctly different properties because the weakly bonding cations have more bonds than bonding electrons while the opposite is true for the strongly bonding 12640

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cations. The presence of more electrons than bonds increases the likelihood that the strongly bonding cations will contain lone pairs. Because the bonding strengths are the expectation values of the valences of the bonds formed by the ion, the most stable bonds are expected to be formed between cations and anions that have the same bonding strengths. In practice, some latitude is found, but stable compounds are usually only formed if their bonding strengths differ by less than a factor of 2: 0:5 < SA =SB < 2:0

ð5Þ

The inequality shown as eq 5 is known as the valence matching rule. A useful theorem of the bond valence model is the distortion theorem which is easily derived from the concave shape of the correlation given in eq 3. This theorem states that If the environment of an atom is distorted by increasing the length of some bonds and decreasing the lengths of others; the average bond length increases provided the bond valence sum is held constant or; alternatively; the bond valence sum increases provided the average bond length is held constant:14

Figure 1. Schematic diagram showing the valence fluxes around a Tl+ and Cl
ion in six-coordinate TlCl. The heavy lines represent a flux of 0.50 vu; the lighter lines represent a flux of 0.08 vu. Only one ion of each type is included, but the valence flux lines to all their neighbors are shown. The cores are shown in black, the bonding electrons in red, and the lone pairs on both ions are shown in blue. The flux lines are colored according to whether they terminate on bonding electrons or lone pairs. For convenience the bonding electrons are all shown as residing on the anion.

ð6Þ

3. APPLICATION OF THE BOND VALENCE MODEL TO LONE PAIR IONS In this section the bond valence model is applied to the stereochemistry of lone pair ions. All the simple anions, i.e., anions that consist of just one atom, such as O2
or F
, have more valence electrons in their neutral atoms than the number of bonds they can form. They cannot use all their electrons for bonding. Similarly, main-group cations in lower oxidation states also contain valence electrons that are not used for bonding. Nonbonding valence electrons remain in the valence shell but form spin pairs internally, and for this reason they are not available for bonding. Such nonbonding valence electrons are usually known as lone pairs. The description of lone pairs in the bond valence model is based on the spherically symmetric core-and-valence-shell model described in section 2. Figures 1 and 2 illustrate various arrangements of the ion cores (black) surrounded by a valence shell containing lone-pair electron-density (blue) and bonding electron-density (red). The bonding electrons are shown on the anion as assumed in the ionic model, because this emphasizes the difference between the cations and anions. Valence field lines are shown emanating from the cores and terminating on the valence electrons, blue for those terminating on the lone pairs, red for those terminating on the bonding electrons. According to where the valence field lines terminate, the electron density can be labeled as bonding or lone pair. The cation and anion valence shells are shown as overlapping, illustrating how the bonding electrons simultaneously reside in the valence shells of both the cation and the anion, while the lone pairs are found only in the valence shell of their own ion. The flux linking the bonding electrons to their cores is the same for both the cation and anion and is equal to the valence of the bond. However, the lone pair is linked only to its own core, from which it receives a total of 2.0 vu of flux. Because of the strong field of the nucleus, the real electron density is distributed with approximately spherical symmetry, both around the cation and around the anion. For this reason the electron density in the valence shell of the core-and-valence-shell model is also assumed to be spherically symmetric, regardless of how it is assigned as bonding or lone pair. What changes as the

Figure 2. Schematic diagram showing the electrons and valence fluxes in Tl3BO3. Each line represents 0.33 vu; conventions are otherwise as in Figure 1.

bonding and nonbonding electrons redistribute themselves is not the electron density, but its function, and hence its ability to form bonds to the ligands. Where the lone pair function is concentrated at one point in the valence shell, no bonds will be formed because there are no electrons with unpaired spins available for forming bonds. Where the bonding and lone pair densities are both distributed uniformly around the valence shell as shown for both the Tl+ and the Cl
ions in Figure 1 the lone pairs on both ions are inactive and the ion is uniformly surrounded by ligands. Alternatively, if the bonding electron density is concentrated on one side of the ion with the lone pair density concentrated on the opposite side as shown for the Tl+ ion in Figure 2, the lone pair is said to be stereoactive. Intermediate cases are known in which strong bonds are found on one side and weak bonds on the other, as illustrated by the oxygen ions in Figure 2. Alcock15 has referred to the strong B
O bonds in this geometry as primary and the weak Tl
O bonds as secondary. Unlike the VSEPR model, which assumes that the natural state of the lone pair is to be stereoactive, the bond valence model assumes that the natural state is the one in which the lone pair is inactive, because in this state both cations and anions behave like the main group elements that have no lone pairs. In this case, the bond network can be solved using the Kirchhoff equations and the distribution of the bonding and lone pair functions of the electron density shows no anisotropy. Each cation is surrounded uniformly by as many anions as it can conveniently accommodate and the corresponding coordination numbers, NA, given in Table 1, can be used to calculate the bonding strength using eq 4a. Because the cations have relatively large coordination numbers, ions with inactive lone pairs have relatively small bonding strengths, SA, as can be seen in Table 1, and according to the valence matching rule, eq 5, they can form bonds with counterions having a similarly small bonding strength, SB. The 12641

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Table 2. Coordination Numbers around Tl+ and Valences of Tl+
O Bonds in a Variety of Tl+ Compounds Arranged in Order of Increasing Anion Bonding Strength, SBa compound

SB (vu)

N(Tl)

S (vu)

TlNO3

0.11

8

0.10
0.12

inactive

75253

TlCl

0.17

8

0.12

inactive

44935

TlCl

0.17

6

0.17

inactive

61518

Tl2SO4

0.17

9

0.07
0.14

inactive

Tl2CO3

0.28

3+4 4+1

0.08
0.26 0.12
0.28

partially active

Tl3BO3

0.33

3+0

0.28
0.35

active

lone pair

ICSD (2)

59944 260371

Figure 3. Coordination sphere around a lone-pair ion when (a) the lone pair is inactive, (b) when the lone pair is partially active showing primary and secondary bonds, and (c) when the lone pair is fully active.

Table 3. Simple Anions in Decreasing Order of SAa SA (vu)

ion

VB (vu)

lone pairs

SB (vu)

(2.23) (2.00)

F O


1
2

3 2

0.25 0.50

1.75

Cl


1

3

0..17

1.67

M


3

1

0.75

1.50

S


2

2

0.33

1.50

Se


2

2

0.33

1.17

Br


1

3

0.17

9094

a

SB is the bonding strength of the anion, N(Tl) is the coordination number of Tl split into the number of primary and secondary bonds, and S is the observed range of the Tl
O bond valences. +

+

lone-pair ion Tl and the non-lone-pair ion Rb have similar bonding strengths (0.11 and 0.12 vu, respectively), and according to eq 5, both cations should be able to form stable bonds with anions having bonding strengths between 0.06 and 0.23 vu. The nitrate, chloride, and sulfate ions satisfy this condition as shown in Table 2. As expected, the coordination numbers around Tl+ are large (6
9) and the strongest observed bonds have valences close to the bonding strength of the anion. The stereochemistry of Rb+ parallels that of Tl+; it also forms a nitrate, chloride and sulfate. When the anion has a bonding strength larger than 0.23 vu, the valence matching rule, eq 5, predicts that no compound should be formed and this is indeed the case for rubidium. Rubidium carbonate can only be formed at high pressure, and Rb3BO3 is unknown. On the other hand, when Tl+ is required to form a stronger bond, the lone pair electron density already lying in the bonding region of the Tl+ ion is readily converted to bonding electron density. To compensate, the bonding electron density on the opposite side of the Tl+ ion is converted to lone pair without significantly changing the distribution of electron density. This allows Tl+ to form stronger bonds to match the stronger bonding strength of the anion, an option not open to cations like rubidium that have no lone pairs. Bonding to a stronger counterion results in the formation of stronger, and therefore fewer, bonds, leaving the coordination sphere partially empty opposite the lone pair. Table 2 shows that with the borate ion the lone pair around Tl+ is fully stereoactive. When bonded to the weaker carbonate ion the lone pair is only partially active and weaker secondary bonds are present. Although it is normal to describe this anisotropy as the result of the stereoactivity of the lone pair, in practice the lone pair plays only a passive role. The anisotropy is properly described in terms of the stereoactivity of the bonding electron density found concentrated in the region where the stronger bonds are required. The lone pair electron density merely responds by occupying regions of the valence shell where bonds are not formed. Although it would be more correct to speak of the bonding electrons as being stereoactive, the expression “stereoactive lone pair” is too well entrenched to be easily changed. This picture reveals the origin of the lone pair stereoactivity, but there is an alternative and simpler geometric description that can be useful in some circumstances. The coordination shell of the lone-pair ion can be treated as an invariant sphere with a uniform distribution of the ligands. When the lone pair is inactive, the ion lies at the center of this sphere and the lengths

a

SA is the bonding strength of the atom as a cation used here as a measure of the electronegativity, VB is the valence of the anion, and SB is the anion bonding strength. (Values in parentheses are estimates.)

of all the bonds are equal to the radius of the coordination sphere as shown in Figure 3a. If some of the ligands are strongly bonding, they draw the cation toward them so that the bonds to the stronger ligand become shorter (primary) while the bonds on the opposite side become longer (secondary). The distortion theorem states that the displacement of the central ion causes an increase in the average bond length (i.e., the radius of the coordination sphere), but for small distortions the increase is small and can to a first approximation be ignored. The radius of the coordination sphere remains unchanged even as the bonds divide themselves into primary and secondary as shown in Figure 3b. When the displacement of the lone pair becomes sufficiently large, the increase in the average bond length can no longer be ignored, but this increase arises because the secondary bonds are lengthened to the point where they are no longer considered bonding, leading to the situation shown in Figure 3c. The description so far has focused on the cations. The behaviors of anions are similar except that in the traditional ionic picture their valence shells are formally filled with eight electrons according to the octet rule. Simple anions, those listed in Table 3 consist of a single strongly bonding ion with SA > 1.0 vu. Their anionic valence is determined by the number of holes in the valence shell of the neutral atom. These holes are filled by the electrons from the overlapped valence shells of the cations and are matched (spin-paired) to an equal number bonding valence electrons from the anion. The valence shell of the anion therefore contains VB bonding electron pairs and 4-VB lone pairs. Fluorine with seven electrons and one vacancy forms one bonding electron pair and three lone pairs. Oxygen with six electrons and two vacancies forms two bonding pairs and two lone pairs. This description of the valence shell as containing electron pairs is only useful for keeping track of the numbers of electrons. As mentioned above, the individual electrons cannot be observed, the electron density is uniformly distributed around the valence shell, and the terms “bonding” or “lone pair” refers only to the way the electron density functions. As with the cations, the lone pair is stereoactive only when the counterion has a large bonding strength. 12642

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Table 4. Lone Pairs in Oxidesa compound aSA (vu)

lone pair

N(O)

structure type

ICSD (2)

Na2O

0.16

inactive

8

antfluorite unstable

60435

MgO

0.33

inactive

6

NaCl

52056

3

rutile

SnO2

0.68

inactive

H2 O

0.80

partially active 2 + 2 ice, liquid

64772

9163

SiO2

1.00

active

2

quartz

93093

P2O5

1.25

active

1, 2

molecule reactive

40865

CO2

1.33

active

1

molecule

SO3

1.50

active

1

molecule reactive

a

SA is the bonding strength of the cation; N(O) is the coordination number of the oxygen.

Oxygen is normally found with coordination numbers between 2 and 6. Choosing 4 as a typical value gives oxygen an anion bonding strength (SB) of 0.50 vu, meaning that oxygen can bond to cations with bonding strengths (SA) in the range 0.25
1.00 vu without the need for its lone pairs to become stereoactive (eq 5). Some examples are given in Table 4. Na+ with a bonding strength of 0.16 vu is too weak to form a stable bond to oxygen and Na2O absorbs water and carbon dioxide from the air. Mg2+ with a bonding strength of 0.33 vu forms the stable compound MgO with an inactive lone pair on oxygen, as does Sn4+ in SnO2. The lone pair on oxygen starts to become active as the bonding strength of the cation approaches 2SB = 1.00 vu when its coordination number drops to 2. Hydrogen acts simultaneously as both a strong acid (donor: SA = 0.80 vu) and a weak acid (acceptor: SA = 0.20 vu) and causes the lone pair of oxygen to be partial active, while the cations Si4+, P5+, and C4+, which are all strongly bonding, convert the lone pair in the bonding region into bonding electrons, thereby forcing it to be fully active. Table 4 shows only the neutral oxides, but the large valences of the strongly bonding cations such as carbon produce bonds that are so strong that just one bond saturates the valence of the oxygen anion. This keeps the coordination number of the central atom low, leaving space for further oxygen ions to be attached. The neutral molecules tend to be reactive, picking up additional oxygen ions from water to create complex anions. The molecule SO3 adds a fourth oxygen ion to form the complex anion SO42
. The S
O bond of 1.50 vu no longer saturates the valence of the oxygen so its lone pair is only partially active, as shown in Figure 4. The oxygen ion forms one primary bond of 1.50 vu to sulfur, leaving 0.50 vu of residual valence to form secondary bonds with weaker cations such as sodium. Assuming a coordination number of four for the oxygen ions, the sulfate ion is expected to form 12 external bonds, giving it a bonding strength of 2/12 = 0.17 vu. The secondary bonds then link the complex anions through a weak cation such as sodium (SA = 0.16 vu) into a three-dimensionally connected structure. In the sulfate ion the oxygen lone pair is distributed around its valence shell with 0.75 lone pairs on the side of the primary bond and 1.25 lone pairs on the side of the secondary bonds, as can be seen in Figure 4.

4. IMPLICATIONS OF THIS MODEL FOR OTHER ASPECTS OF CHEMISTRY This section points out some of the consequences of this model for the traditional descriptions of chemical bonds.

Figure 4. Core and valence shells in the SO42
ion. Conventions are as in Figure 1, but the flux lines are not shown.

There are bonds, particularly in transition metal complexes, where the lone pair of an ion such as nitrogen in NH3 is sometimes shown as forming a dative bond, that is, a bond in which both bonding electrons are supplied by the lone pair of the nitrogen ion while the transition metal makes no contribution. While this appears to be a convenient way to draw the Lewis structure, it violates the normal valence rules that are based on the assumption that each ion contributes equal numbers of electrons to the bond (eq 1). Further, a lone pair is by definition nonbonding and so cannot form a dative bond without the ion changing its oxidation state. The ionic valence of nitrogen becomes undefined; it is not clear whether it should be taken as
3,
4, or
5. Using the lone pair as an explanation obscures the true origins of the dative bond and leads to a misunderstanding of the chemistry. The classic example of dative bonding is the N
B bond in H3N 3 BCl3. It is difficult to explain the formation of this bond in any other way as long as one focuses on the isolated molecule. Both H3N and BCl3 are known and are conventionally described in terms of electron-pair (i.e., single) bonds linking the cations (H+, B3+) to their respective anions (N3
, Cl
). In both compounds the valence sum rule is fully satisfied and their structures are the ones predicted by both the VSEPR and the bond valence model: BCl3 is a planar molecule and H3N has a stereoactive lone pair on nitrogen because each hydrogen forms a strong bond of 1.0 vu. There is no valence left for the formation of the N
B bond. H3N is a Lewis base, which means that it has an associated negative charge, but because the ammonia molecule is electroneutral, any negative charge associated with the nitrogen ion must be balanced by an equal positive charge distributed over the hydrogen ions, that is, the Lewis base function of nitrogen must be accompanied by a Lewis acid function of the hydrogens. Similarly, the Lewis acid function of boron in BCl3 requires that the chlorine ions act as Lewis bases. In both cases the stoichiometry ensures that the primary functions of the nitrogen and boron are 3 times stronger than the secondary functions of the hydrogen and chlorine. When H3N and BCl3 are brought together, the two molecules will simultaneously form an acid
base N
B bond and, to compensate, three intermolecular N
H 3 3 3 Cl hydrogen bonds. The effect is to weaken both the H
N and B
Cl bonds, thereby freeing bonding electrons to form the N
B and the H 3 3 3 Cl bonds. Application of the valence sum rule shows that the valence of the N
B bond is 3 times that of the individual H 3 3 3 Cl bonds, making it comparable in strength to the H
N and B
Cl bonds 12643

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The Journal of Physical Chemistry A (say around 0.6
0.7 vu). The lone pair on nitrogen remains intact, but with the larger coordination number around nitrogen and the correspondingly weaker N
H and N
B bonds, it becomes inactive. The structure of H3NBCl3 has not been determined but the structures of H3NBH3 (ICSD 159679) and (CH3)3NBCl316 have been reported. In both cases the molecules pack in such a way that the Lewis acid hydrogen ions of ammonia or trimethylammonia are surrounded by the Lewis base hydride or chloride ions of the borane in positions that are consistent with the formation of N
H 3 3 3 H and C
H 3 3 3 Cl hydrogen bonds, respectively. A similar situation is found in the formation of the ammonium ion, NH4+, from ammonia. The addition of hydrogen to ammonia is formally similar and could be described as a dative bond. Here, clearly the N
H bonds have a valence of only 3/4 = 0.75 vu because each hydrogen ion has to contribute 0.25 vu to the external bonds of the ammonium cation. Because the valence of the nitrogen in the ammonium ion is still only
3, the lone pair is still present and still nonbonding, though, being inactive, its presence is not visible. The seductive but incorrect explanation that the lone pair provides the Lewis base function of the ammonia molecule is enhanced by the fact that Lewis bases have an excess of electrons and are therefore likely to have stereoactive lone pairs. These are present in the chlorine ions of BCl3 as well as the nitrogen in ammonia. It is tempting to associate the base function with the lone pair, but when the acid
base bond is formed, the electron density that represents the lone pair in the uncomplexed molecule changes its function from lone pair to bonding electron density and the lone pair function is transferred to other parts of the valence shell. The base function of a neutral molecule can only become active if its complementary acid function also becomes active, a process in which the lone pair plays only a passive role. It moves away from where it is not needed and remains nonbonding and inert. Although lone pairs are, by definition, nonbonding, it is possible under some extreme conditions for a cation to use its lone pair as a Lewis base to form a secondary bond to another cation. A stereoactive lone pair can be thought of as a divalent pseudoanion, E2
. For example, the lone-pair cation Sn2+ can be conceptually separated into a four-valent tin cation, Sn4+, bonded to a lone pair, E2
, as shown in Figure 5a. The lone pair is treated like any other terminal ligand, forming just one bond of 2.00 vu to the core of the cation to which it belongs, i.e., Sn2+ can be written as Sn4+
E2
. Unlike a regular anion, however, E2
lacks a positive core. In most cases describing Sn2+ as Sn4+
E2
adds nothing to our understanding of the chemistry, because both the Sn4+ core and the lone pair belong to the same ion and the valence flux linking them remains entirely within the Sn2+ cation. But in compounds where the cations would normally form more bonds than the anions are capable of forming, i.e., where SB . SA, the lone pair can be forced to accept a secondary bond from another cation in addition to the primary bond it forms to its own core. If the secondary bond has a valence of, e.g., x, it reduces the valence of the Sn
E bond from 2 to 2
x vu. One consequence is that, to maintain the valence sum around the Sn4+ core, the valence sum of the regular bonds must be increased from 2 to 2 + x vu, as shown in Figure 5b. The effect of the lone pair forming a weak external bond is to strengthen the already strong primary bonds formed by the cation. Such lone pair bonds are found in alkali-rich compounds such as Na4SnO3 (ICSD 49624), which is unstable in air. In a normal

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Figure 5. (a) Sn2+ ion shown (in the circle) as composed of an Sn4+ core together with a lone pair (E2
) acting as an anion of valence
2.00 vu. (b) Sn2+ cation using its lone pair to form weak bonds to other cations in Na4SnO3.

compound one would expect four sodium atoms to form a total of 24 bonds and the stannite ion only nine. In a real structure these two numbers must be the same. The compromise found in this crystal is 14 bonds, giving the sodium atoms an average coordination number of only 3.5, much smaller than the 6.4 normally found. To bring the coordination number of sodium up to its normal minimum of 4, two of the sodium atoms form a total of three additional weak bonds with a total valence of 0.24 vu to the lone pair on tin. The valence of the Sn
E bond is reduced to 1.76 vu, which is compensated by increasing the valence of each of the three Sn
O bonds from the nominal value of 2/3 = 0.67 to 0.75 vu. This change is reflected in the average observed Sn
O distance of 2.02 Å, which is significantly shorter than the bond length of 2.08 Å otherwise expected.17,18 Most ions form more than one bond, leading to the formation of extended (infinite) bond networks such as those found in inorganic crystals. A molecule has a finite bond network, which can only be formed if the ions on the periphery of the molecule terminate the network, normally by forming only one bond. In organic molecules this function is mostly performed by hydrogen, but there are certain lone pair ions that can also terminate the bond network. For example, a monovalent anion such as fluorine bonded to a strongly bonding cation can lead to the formation of a molecule. Because strongly bonding cations have a bonding strength greater than 1.0 vu, they tend to form bonds with valences equal to or greater than 1.0 vu. On the other hand, a monovalent anion cannot form bonds with a valence greater than its anion valence of 1.0 vu. Because the valence of a bond between fluorine and a strongly bonding cation, such as carbon or sulfur, cannot be greater than 1.00 vu, nor can it be less, the only possibility is for the valence of the bond to be exactly 1.00 vu. With no residual valence, the fluorine ion cannot form a second bond and it necessarily terminates the bond network. The result is the formation of molecules such as CF4 and SF6. This is the reason why strongly bonding cations form electron pair bonds with monovalent anions, and why such bonds result in the formation of molecular compounds. Oxygen ions can also be terminal as the examples listed in Table 4 show, but the larger valence of O2
results in a smaller coordination number around 12644

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The Journal of Physical Chemistry A

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the cation, causing the molecules to react with water to form complex anions as discussed above. It is a curious accident of history that chemists based many of their bonding models on the structures of molecules rather than on the more commonly found infinitely connected bond-networks in crystals. Because molecules so frequently contained single bonds, it was reasonable to suppose that an electron pair lies at the heart of every covalent bond. This assumption is the basis of many theories of chemical bonding from the Lewis and valence-bond descriptions to the VSEPR models. While these models are successful in reproducing the structures of molecules, they do not properly describe the structures found in the majority of stable inorganic compounds. By focusing only on this one class of compounds, the authors of these models failed to appreciate that the electron-pair bond is mostly restricted to molecules. The archetypal covalent bond they describe is an accidental consequence of the electron-counting rules of the ionic model, rules that apply to all types of localized bond, ionic or covalent, whether found in molecules or in crystals.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT I acknowledge stimulating discussions held with the late Friedrich Liebau and Yurij Mozharivskyj. ’ REFERENCES (1) Gillespie, R. J. Molecular Geometry, Van Nostrand Reinhold Co.; London, 1972. (2) Fiz Karlsruhe. ICSD + Collection number. In The Inorganic Crystal Structure Database.Belsky, A.; Hellenbrandt, M.; Karen, V. L.; Luksch, P. Acta Crystallogr. 2002, B58, 364. (3) Brown, I. D. The Chemical Bond in Inorganic Chemistry, The Bond Valence Model; Oxford University Press; Oxford U.K., 2002. (4) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clarendon Press; Oxford, U.K., 1990. (5) Gibbs, G. V.; Downs, R. T.; Cox, D. F.; Ross, N. L.; Prewitt, C. T.; Rosso, K. M.; Lippmann, T.; Kirfel, A. Z. Kristallogr. 2008, 233, 1. (6) Mohri, J. Acta Crystallogr. 2000, B56, 626. (7) Howard, S. T.; Lamarche, O J. Phys. Org. Chem. 2003, 16, 133. (8) Bader, R. F. W.; Gillespie, R. J.; McDougal, P. J. J. Am. Chem. Soc. 1988, 110, 1729. (9) Becke, A. D.; Edgcombe, K. E. J. Chem. Phys. 1990, 92, 5397. (10) Schmider, H. L.; Becke., A. D. J. Mol. Structure (THEOCHEM) 2000, 527, 51. (11) Preiser, C.; Loesel, J.; Brown, I. D.; Kunz, M.; Skowron, A. Acta Crystallogr. 1999, B55, 698. (12) Brown, I. D. 2009 http://www.iucr.org/resources/data/datasets/bond-valence-parameters. (13) Brown, I. D. Chem. Rev. 2009, 109, 6858. (14) Urusov, V. S. Z. Kristallogr. 2003, 218, 709. (15) Alcock, N. W. Adv. Inorg. Rad. Chem. 1972, 15, 1. (16) Hess, H. Acta Crystallogr. 1969, B25, 2338. (17) Liebau, F; Wang, X; Liebau, E. Chem.—Eur. J. 2009, 15, 2728. (18) Brown, I. D. Acta Crystallogr. 2009, B65, 684.

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