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Chemistry 533

Electron Counting Transition metal and organometallic chemistry

Electron counting organometallic compounds

Organic structure and bonding

The nature of organic chemistry self-evidently confines it to the chemistry of the Main Group. In ageneral view of structure and bonding, structure and bonding in organic chemistry is determined entirelyby s and p orbitals. Classical hybridization1 is the most basic approach that offers any qualitative utility fororganic structure and bonding but is not necessarily an accurate picture of the system in question.Hybridization fails when applied to apparently hypervalent main group compounds and is useless whendiscussing transition metal structure and bonding.

Applying group theory to organic systems results in physically reasonable orbitals of mixed atomicparentage that can be stocked with the necessary eight electrons, providing a useful description of thebonding in the system which is in accord with observed measurements of the system.

There are several features of organic molecules that differ strongly from those containing transitionmetals. These include a distinct lack of high order symmetry and an almost invariable eight electron countat any atom in the system that is not hydrogen with concomitant coordinative saturation. Moreover, thenumber of relatively heavy atoms (C, N, O) is usually high and this generates a relatively high density ofelectronic states with similar energies.

These features of organic molecules dictate that a simple structure and bonding scheme is usuallysufficient for synthetic applications but transfer of such a scheme to organometallics or other transitionmetal system usually results in error. For this reason, electron counting transition metal compounds is abasic and important feature of the description of their chemistry. Appendix I gives some details of theassembly of valence bond hybrids, derived from the work of Linus Pauling.

Inorganic structure and bonding

There is a natural distinction between inorganic structure and bonding problems that involvetransition metals and those that involve the p-block metals. In the latter case, the most importantfundamental concepts are the widening ns-np gap and the alternation effect, which involves the 3d/4fcontractions and the so-called 'inert pair' effect.

The transition metals are very different, due to the presence of nd-orbitals and (n+1)s and (n+1)porbitals. Depending on the row, the electron count and the position in the transition series, a very widevariety of reactivities, formal oxidation states and so on are available for stoichiometric and catalyticreactivity.

In order to be able to account for the profusion of organometallic and coordination complexes ofthe transition metals, a most basic step is the accurate determination of the formal oxidation state and theassessment of the coordination sphere and electron count. The most satisfactory method of countingelectrons in this respect is the MLX method, which invariably gives the correct answer, without an a priori

assumption about oxidation state. For a compound such as WF6 or WMe6 , the formal oxidation state isstraightforward to assess. For a compound such as ��C7 H7 � Ti ��C5 H5 � , this is less clear, with formaloxidation states of 0 and IV being available.

1 We define 'hybridization' as the use of constructs such as sp3 sp2 and sp hybrids with unweighted orbital contributions.

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MLX classification

The MLX classification assigns the metal, M, an oxidation state of 0 and then homolytically breaksthe bonds between ligands so that all the fragments are neutral. This ensures that the ligand leaves as aneutral ligand and the metal retains its full d count.1 Simple ligands

Ligands such as phosphines, alkenes and CO leave as the neutral ligand and therefore act as two electrondonors and are designated as L. Ligands such as CF3 SO3 , Cl, OMe, H and Me leave as radicals i.e. they areopen shell and are therefore one electron donors; these are designated as X. More complex ligands areconstructed from these two simple classifications. Examples of these simple ligand types are given in Table1.

X ligands L ligands

F, Cl, Br, I, OR, R, Ar, H, CF3 SO3 , CN, N3, NO2, CO, R3P, R3N, N2, R2C=CR2, RCN, RNC, H2,

Table 1: A non-exhaustive set of examples of X and L ligands

Examples

3×CO=L3

3×H2 O=L3

[Tc �CO �3 �H2 O �3 ]+=ML6

5×Cl=X5

MoCl5=MX5

2×Ph2 PCH2 CH2 PPh2=2L2=L4

2×N2=L2

�Ph2 PCH2 CH2 PPh2�2Mo �N2�2=ML6

Aebischer, N.; Schibli, R.; Alberto,R.; Merbach, A. E. Angew. Chemie,

Int. Ed. (2000), 39(1), 254-256.

Faegri, K.; Martinsen, G.;Strand, T. G.; Volden, H. V.Acta Chem. Scand. (1993),47(6), 547-53.

Hidai, M.; Tominari, K.; Uchida, Y.;Misono, A. J. Chem. Soc. D: Chem.

Comm. (1969), (23), 392

1 The d count on the metal includes the s2 electrons for the neutral metal atom

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3×Ph3 P=L3

1×N2=L

1×H=X

�Ph3 P �3 Co �N2�H=ML4 X

6×H=X6

2× Pr2i PhP=L2

� Pr2i PhP �2 Os H6=ML2X6

3×H2CCH2=L3

�H2 CCH2 �2 Pt=ML3

Pu, L. S.; Yamamoto, A.; Ikeda, S.J. Am. Chem. Soc. (1968),90(14), 3896

Howard, J. A. K.; Johnson, O.;Koetzle, T. F.; Spencer, J. L.Inorg. Chem. (1987), 26(18),2930

Howard, J. A. K.; Spencer, J. L.;Mason, S. A. Proc. Roy. Soc. Lon. A(1983), 386(1790), 145

Complex ligands

Unsaturated hydrocarbons

These are ligands that donate more than one or two electrons to the metal center. They can besimilarly divided into 'compound LX' classifications as shown below in Table 2. The 'eta' notation, �n�

denotes the number of atoms bound to the central metal atom in question by the superscript employed (n).The absence of a superscript implies that the maximum number of available atoms is bonded. Under thisnotation, ferrocene is written ��Cp �2 Fe , bis(allyl)nickel is written as ��C3 H5 �2 Ni and so on.

Taking the [(allyl)Ni] fragment as an example, there are two possible resonance extrema that canexist as shown in figure 1.

The arrow represents the L interaction, whereas the single bond to Ni represents the X interaction.The accurate picture for the structure of bis(allyl)Ni shows no contribution from either resonanceextremum, but for 'bookkeeping' purposes, it is a useful tool. The allyl fragment in [(allyl)Ni] is thereforeacting as an LX ligand and ��C3 H5 is always counted as a 3 electron, LX ligand. Similarly, Cp acts as anL2X ligand when it displays ��Cp coordination.

The coordination of an unsaturated carbocycle to a transition metal can take a variety of possiblevalues, usually varying in steps of 2 as individual L interactions change due to the electronics and sterics at

3

Figure 1: Hypothetical resonance extrema for the (allyl)Ni fragment

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the metal center. Some examples of differing coordination of the same ligand are given below.

��1�Cp � M ��Cp �

��1�Cp � Be ��Cp�

Nugent, K. W.; Beattie, J. K.; Hambley, T. W.; Snow,M. R. Aust. J. Chem. (1984), 37(8), 1601-6

Almenningen, A.; Haaland, A.; Lusztyk, J. J.

Organomet. Chem. (1979), 170(3), 271-84

��1�Cp � � ��Cp � Fe �CO �2

Bennett, M. J., Jr.; Cotton, F. A.; Davison, A.; Faller,J. W.; Lippard, S. J.; Morehouse, S. M. J. Am. Chem.

Soc. (1966), 88(19), 4371-6.

��Cp �2Mo �Nt Bu �

Green, M. L. H.; Konidaris, P. C.; Michaelidou, D.M.; Mountford, P. J. Chem. Soc. Dalton Trans.(1995), (2), 155-62Green, J. C.; Green, M. L. H.; James, J. T.;Konidaris, P. C.; Maunder, G. H.; Mountford, P.J. Chem. Soc. Chem. Commun. (1992), (18), 1361-4

��Cp � ��3�Ind �Mo �Nt Bu �

Green, M. L. H.; Konidaris, P. C.; Michaelidou, D.M.; Mountford, P. J. Chem. Soc. Dalton Trans.(1995), (2), 155-62Green, J. C.; Green, M. L. H.; James, J. T.;Konidaris, P. C.; Maunder, G. H.; Mountford, P.J. Chem. Soc. Chem. Commun. (1992), (18), 1361-4

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��n�cht � M

��Cp � Ti ��cht �

Menconi, G.; Kaltsoyannis, N. Organometallics(2005), 24(6), 1189-1197.

Zeinstra, J. D.; De Boer, J. L. J. Organomet. Chem.(1973), 54 207-11

��5�cht �Fe �CO �2�SnPH3 �

Muhandiram, D. R.; Kiel, G.-Y.; Aarts, G. H. M.;Saez, I. M.; Reuvers, J. G. A.; Heinekey, D. M.;Graham, W. A. G.; Takats, J.; McClung, R. E. D.Department of Chemistry, Organometallics (2002),21(13), 2687-2704.

��3�cht �Re �CO �4

Muhandiram, D. R.; Kiel, G.-Y.; Aarts, G. H. M.;Saez, I. M.; Reuvers, J. G. A.; Heinekey, D. M.;Graham, W. A. G.; Takats, J.; McClung, R. E. D.Department of Chemistry, Organometallics (2002),21(13), 2687-2704.

Whereas ligands with higher potential hapticities, such as cht (cycloheptatrienyl), have theelectronic flexibility to display a number of hapticities, ligands derived from Cp tend not to do so andexamples of ��3�Cp � M are rare; an exception to this is the indenyl ligand, which can undergo a facile�5

�� 3 'ring-slippage' or can, in certain cases, form an ��3�Ind � M ground state.

Main group metals however, show completely different structural and therefore reactive trendswith potential � -ligands and the electronic reasons for this have been investigated recently.1

Polyfunctional ligands

Ligands such as Ar2 N , R 2N , ArN , RN , ArO and RO appear to be formally simple X ligands,and for the alkoxides and aryloxides, this is often though not invariably true. However, the full ligand p-orbital on the ligating atom can act as a � -base, donating the electron pair in a dative manner to the metalcenter. In this respect, it acts as L and so the formally monoanionic ligands above �Ar2 N, R2 N � , are better

1 Trends in Cyclopentadienyl-Main-Group-Metal Bonding Budzelaar, P. H. M.; Engelberts, J. J.; van Lenthe, J. H. Organometallics

(2003), 22, 1562-1576

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formulated as LX in most cases. The orbital interactions are shown in figure 2 for this particular case.

M�A

class X interaction

M �A

class L interaction

M� A

class Z interaction

Given that there are several acid and base interactions that exist in transition metal chemistry,there are a series of possible bonding scenarios that can be written under the MLX scheme. Donor andacceptor interactions can occur in both the � - and � -manifolds, as shown below, where A represents theligating atom. These interactions are detailed in Table 1

� -manifold � -manifold

X� M�A X�

L� M � A L�

Z� M �A Z�

Table 2: � - and � -interactions used in the MLX scheme

6

Figure 2: Bond scission processes for the LX amido ligand

Chemistry 533


Donor-acceptor functions

One of the most important rules for MLX counting is the assessment of donor and acceptorfunctions. The presence of a Z function on a ligand allows the metal to back-donate electron density to theligand, in the conventional Dewar-Chatt-Duncanson1 manner. The Z function is present in molecularligands and is almost always an unoccupied, antibonding orbital. Note that this function, the Z� is implicitand and is distinct from the Z� ligand classification that we encounter in the boron halides or BH 3 .

We can use the Kubas dihydrogen complexes as an example. These complexes contain the

��2�H2 � ligand and, under the DCD bonding description, there is an L interaction to the metal center andthe molecule is side-bound. The bond length in dihydrogen is ~0.75 Å and this lengthens in dihydrogencomplexes; in figure 3, the HOMO and LUMO for dihydrogen at an equilibrium bond length of 0.85 Å isshown.

Figure 3: The HOMO (left) and LUMO (right) for dihydrogen at a rHH=0.85 Å

These two orbitals can interact in the obvious manner with suitable orbitals on the metal center:

The L� interaction The Z� interaction

Donation from the HOMO to the metal and from the metal into the LUMO changes the bond lengthin the HH bond; large amounts of back-donation from the metal can clearly break the HH bond and forcethe system into a hydridic scheme. The H2 molecule is completely reduced to 2H� ions, and the formaloxidation state of the metal, defined in the MLX sense, is raised by 2. The presence of a suitable Z functionon the ligand, the LUMO, and the filling of the orbital, which requires a symmetry and energy match on the

1 Dewar, M. J. S. Bull. Soc. Chim. Fr. (1951), C71-9.

Chatt, J.; Duncanson, L. A. J. Chem. Soc. (1953), 2939-47.

See the following references for more specific and recent reviews:

Frenking, G.; Sola, M.; Vyboishchikov, S. F. J. Organomet. Chem. (2005), 690(24-25), 6178-6204; Frenking, G. Modern Coordination

Chemistry (2002), 111-122; Winterton, N. Modern Coordination Chemistry (2002), 103-110; Kubas, G. J. J. Organomet. Chem.

(2001), 635(1-2), 37-68; Price, D. W.; Drew, M. G. B.; Hii, K. K.; Brown, J. M. Chem. - Eur. J. (2000), 6(24), 4587-4596.

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metal, causes this 2 electron oxidative addition of dihydrogen.This is a general rule; when there are the correct energetics, symmetry and orbital occupancy for a

ligand antibonding orbital and a metal orbital to interact, then

L�Z�=�X� �2where shows the net combination of the two interactions on the same ligand.

I-III oxidative addition of a diatomic is the clearest example of this effect. However, it also occursin systems where there is a degenerate orbital that can act as an acceptor.

A second example that often causes confusion occurs in cyclic unsaturated hydrocarbons with a �1

-type ground state for the radical. The most obvious examples of this are c�C3H3 and c�C7 H7

. Inthese cases, two possibilities are available:

� formal oxidation to the cation to form either [c�C3 H3 ]�

or [c�C7 H7 ]�

, which contain 2 and 6

electrons respectively � 4n�2 ;n=0, 1� and are therefore aromatic

� formal reduction to [c�C7H7]3�

or [c�C3 H3]3�

, which are similarly aromatic with 6 and 10

electrons respectively �4n�2 ;n=1, 2�

The molecular � -orbitals for [c�C3 H3 ]�

and [c�C7H7]�

are shown in figures 4 and 5.

3 nodes

2 nodes

1 node

0 nodes

Figure 4: The � -orbitals for [c�C3 H3]�

Figure 5: The � -orbitals for [c�C7H7]�

Coordination of these ligands as ��C3 H3 � and ��C7 H7 � , i.e. with maximal hapticity, usuallyoccurs for early transition metals. In the case of maximal hapticity, the bonding of these ligands is best

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described by assuming full reduction of the ligand, in which case, they act as LX or L3X ligands. For � -systems which have a general �3 -type ground state for the radical, reduction is more obvious; thearchetype here is ��C5 H5 � .

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Appendix I: Organic or classical hybrid orbitals

Rationale for the use of hybridization

In order to accurately describe the bonding in a molecule, we need to solve the Schrödingerequation for the molecule, using the potential from the nuclear positions as a starting point. Other termsthat we will need to account for are the interelectron repulsions and correlation effects. Given that onlytwo-particle systems, such as hydrogenic atoms, are soluble in an analytical sense, then the calculations areapproximate and hard and require significant computing power for physically accurate results.

While quantum chemical calculations are useful and important, there is a place for a qualitativeapproach to organic structure, bonding and reactivity that is intuitive and based simply on the structure ofthe atom. This is the role of the hybrid approach. It is very important to recognize that it does notnecessarily represent the physical nature of the bonding situation in a molecule and will not necessarilyaccount for all of the results of measurements on the system in question.

Some of the features of organic chemistry that must be explicable qualitatively by any descriptionof organic structure and bonding include the geometries of carbon atoms in alkanes, alkenes and alkynes,as well as chirality.The basis of real atomic orbitals

The basis1 set that we shall use is the valence set of orbitals on carbon. Carbon has the electronicconfiguration 1 s2 2 s2 2 p2 and we shall only use the n=2 set. These orbitals are therefore the 2 s , 2 p1 ,2 p0 and 2 p�1 orbitals. We shall also ignore the radial wavefunction as, although the radial function in

general does effect the chemistry of an element, we are more concerned with the angular shapes of theorbitals. The angular or orbital solutions for the Schrödinger equation are given in table 1.

l ml Spherical harmonics with l=0

0 0 12 � 1

�

Spherical harmonics with l=1 Cartesian form

1 -112 � 3

2�e

�i sin�

12 � 3

2�� x�iy �

r

1 0 12 � 3

�cos�

12 � 3

�

z

r

1 1 �12 � 3

2�e

i sin �

�12 � 3

2�� x �iy �

r

Table 3: Angular functions (spherical harmonics) for the hybrid basis set on carbon

A fine point, which does not affect our arguments, is that, of the p orbitals, only the p0 orbital ismathematically real. The p±1 contain the term e±i , which is the polar representation of a complexnumber, which occurs in the Cartesian form as x±iy . We require that our basis orbitals and our hybridsare real as this makes more intuitive sense, and so we take linear combinations of these mathematicallyimaginary orbitals to give a pair of mathematically real ones; we show the production of the p

x orbital as

1 The basis set is the set of orbitals that we use for solving the problem in hand. We are at liberty to choose any basis; clearly achoice that is physically reasonable is best. More generally, a basis for a calculation can be any set of chemically relevant objects.

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an example. We rely on the fact that any linear combination of solutions to the Schrödinger equation isalso a solution. Given that

p�1=�34

sin�e�i

p1=�34

sin�ei

then, expanding the exponential term using standard mathematical identities,

p�1=�34

sin�e�i =�3

4sin��cos �isin �

p1=�34

sin�ei =�3

4sin�� cos �i sin �

Taking linear combinations of p�1�p1 we have

p�1�p1=�34

sin��cos �isin ��34

sin �� cos � isin �=2�34

sin�cos� .

Given that in spherical polar coordinates,

x=r sin �cos�

y=r sin� sin�

z=rcos�

then

p�1�p1=2�34

sin�cos =2�34

x

r�x

So we identify which we identify the combination p�1�p1 with the px orbital. The other linear

combination, p1�p�1 will generate py . The real orbitals that result from this process are shown in figure

6.

pz=p0 p

x~ p�1�p1 p

x~ p1�p�1 s

Figure 6: Real hydrogenic functions for the valence orbitals of carbon

It is important to realize that the orbitals in figure 1 are not hybrids; they are perfectly respectablesolutions to the Schrödinger equation and are not hybrid orbitals in the sense that we are exploring here.With a set of real hydrogenic orbitals in hand, we can proceed to examine some of the properties of thehybrids that we will require.

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Generalities and constraints for the hybrid orbitals

There are several constraints that we shall use to aid our definitions of the hybrids. We requirethat

� all the orbitals are identical overall in atomic composition,

� they contain only s and p orbitals� they should be orthonormal1 with respect to each other� they are consistent with the structural and reactive properties that we observe experimentally.

In CH4 , all the bonds are identical. This automatically presents a problem with respect to usingthe atomic orbitals to describe the bonding. The three p orbitals will overlap with the hydrogen sorbitals and will be bonded at 90° to each other with the last hydrogen bonding with the carbon s orbital.A molecule of methane bonded in this manner would have the structure shown in figure 7.

Figure 7: The bonding orbitals of methane using a pure atomic basis (left) and the hydrogenic wavefunction

for the p�s interactions (right). Note the 90° angles between the H atoms, which is not found in nature. The

s�s interaction is not shown.

This structure is not observed in nature – the H��C�H angles are known to be identical and equal

to the tetrahedral angle, arccos ��13 �=109.47 ° .2 For alkenes and alkynes, attempts at using a purely

atomic basis become even less accurate.The key is to construct orbitals that are linear combinations of the basis orbitals, oriented in a

natural way with respect to the symmetry of the carbon atom and so the problem is one of calculating thecoefficients of the constituent atomic orbitals. There is a rigorous method for determining thesecoefficients, which we give we give in Appendix II.

The geometries of carbon atoms fall into three broad categories – four coordinate (CN 4), threecoordinate (CN 3) and two coordinate (CN 2). CN 4 carbon has tetrahedral coordination, CN 3 carbon hastrigonal coordination and CN 2 carbon is linear. Although bond lengths will vary with substituent, whichwill primarily depend on the radial functions, the shape of the molecule and the geometry will depend onthe angular functions. Our task is therefore to find a set of distributions using the chosen basis that

1 An orthonormal orbital, � is one that gives the correct number of electrons when we determine the electron density

�=�� *d � and that has no interaction with orbitals in the basis; the former requirement is termed normalization and the latter istermed orthogonality.

2 We prove this in Appendix II

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reproduces these experimental geometries as accurately as possible.

The hybrids will form the � -orbital framework; we also need to account for the existence of oneor two � -bonds that are known to occur in ethylene in the former case and acetylene in the latter. Inorder to do this, we shall exclude one or two p functions from the basis. We do this because, from thestudy of diatomic molecules such as N2 , the existence of molecular � -orbitals from combinations ofatomic p orbitals is well-understood. The hydrogenic � -orbitals for N2 are shown in figure 8

Figure 8: The two � -orbitals in N2

Our bases for each coordination geometry of carbon are given in table 3. We show the hybrid basisand the orbitals that we will exclude for � -bond formation.

Hybrid � -bond basis � -bond basis

CN 2

CN 3

CN 4 -

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Hybrids for CN 2

We use one p , the px in this case, and the s orbital to form these hybrids and we take linear

combinations of the orbitals so that they are normalized and orthogonal (orthonormal) to each other,

which is the source of the coefficient, �22

in this case. The hybrids are CN 2=�2

2s�

�2

2p

x and

CN 2=�2

2s�

�2

2p

x and are shown in figure 4. Because we use one p and the s orbitals only, we call

these hybrids sp hybrids

CN 2=�22

s��22

px

CN 2=�22

s��22

px

Figure 9: The hybrid � -orbitals for CN2 carbon

Hybrids for CN 3

A combination of two p orbitals and the s orbital gives the correct geometry for a three-coordinate carbon atom. The linear combinations of these orbitals so that they are orthonormal are

CN 3=�6

6� s �2�2 p

x � , CN 3=�6

6� s �2�p

x� p

y �3 � and CN 3=�6

6� s �2�p

x�p

y �3 � . We use two p

orbitals and one s orbital overall and so we term the hybrids for three-coordinate carbon sp2 hybrids.

CN 3=�66

� s�2�2 px � CN 3=

�66

� s �2� px� p

y �3� CN 3=�66

� s �2� px�p

y �3�

Figure 10: The hybrid � -orbitals for CN 3 carbon

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Hybrids for CN 4

We use all of the basis orbitals in this case. The hybrids are given by CN 4=1

2� s�p

x� p

y� p

z � ,

CN 4=1

2� s�p

x� p

y�p

z � , CN 4=1

2� s�p

x� p

y�p

z � and CN 4=1

2� s�p

x�p

y�p

z � ; figure 6 shows these

hybrid orbitals. These hybrids are termed sp3 hybrids.

CN 4=12

�s�px� p

y�p

z� CN 4=12

�s�px� p

y�p

z � CN 4=12

� s�px� p

y�p

z � CN 4=12

� s�px� p

y� p

z �

Figure 11: The hybrid � -orbitals for CN 4 carbon

Physical interpretation of hybrid orbitals

Hybrid orbitals are a method of rationalizing the observed structures and reactivity of organicspecies and the approach arises from the work of Linus Pauling in 1931.1 It is rooted in the early attempt toinclude the early results of quantum physics into the description of the chemical bond and molecularstructure. It is not meant to be a physical description of the distribution of the electron density in amolecule. A better description of this aspect of the hybrid orbital approach is given by Hinshelwood:

“Once again, it must be emphasized that we are here dealing not with a phenomenon predicted by

quantum mechanics, but with a convenient mode of description, in terms of approximations, of

matters to which those approximations ought really never to have been applied. That they have been

so applied in an imperfect is a necessity imposed by an absence of methods that are at the same time

precise and manageable.”2

However, the hybrid approach, despite the inherent imprecision is an extremely effective andconcise way for the rationalization for which it was developed. Outside of this application, however, it fails.

1 Pauling, L. J. Am. Chem. Soc. (1931), 53 1367-1400.

2 The Structure of Physical Chemistry, C. N. Hinshelwood, Oxford University Press, ISBN: 0198570252 (2005) p. 248 et seq.

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Appendix II: The tetrahedral angle

A tetrahedral is related to a cube and can be constructed by choosing four vertices of the cubewhich mark the ends of the cube diagonals, so that any diagonal has only one vertex chosen. A tetrahedronis shown below, with the cube.

The vectors from the origin at the cube center are shown as a, b, c and d. The dot product of avector is defined as xy=�x��y�cos�=r

xr

ycos� where �x� is the length of x, r

x is the length of x and �

is the angle between x and y.

In the case of the tetrahedron, all the lengths are the same and we can set them equal to 1; � isthe tetrahedral angle that we are going to determine.

The sum of the vectors a, b, c and d is zero

a�b�c�d=0

and if we calculate a�a�b�c�d� , then

a�a�b�c�d�=aa�ab�ac�ad=0 .

Using our definition of the dot product and recalling that the length of the vectors is the same,

aa�ab�ac�ad=rar

acos�0 ��r

ar

bcos��r

ar

ccos��r

ar

dcos�=0 .

cos�0�=1 and ra=r

b=r

c=r

d , so

aa�ab�ac�ad=r2�r2 cos��r2cos��r2 cos�=r2�3r2cos �=0 .

Setting r=1 then

1�3cos �=0

and so, as advertised, cos�=�13

which means that �=arccos ��13 �=109.47 ° QED

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