Multiple Bonds Between Metal Atoms / 16-Physical, Spectroscopic and Theoretical Results

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of metal character is different. For Mo2(NH2)6, the upper eu-orbital (5eu) plays a greater role than the lower one (4eu). In Mo2(NMe2)6, the situation is reversed, with the lower orbital 10eu, being the main instrument of Mo–Mo /-bonding.

Fig. 16.23. SCF-X_-SW energy levels for Mo2(NH2)6 and Mo2(NMe2)6. Percentages give atomic sphere molybdenum contributions.

The M2X6 type molecule has also been treated by other theoretical methods, with the question of the rotational potential energy function being particularly addressed. From the SCF- X_-SW calculations just described, one would conclude that the M>M bond per se does not imply any rotational preference and that the staggered conformation invariably found in all these molecules is dictated by the nonbonded repulsive forces between the ligands.

However, an examination of this question by an essentially qualitative frontier orbital analysis was said to show that the M>M bond is inherently biased (by 45 kJ mol-1) toward an eclipsed conformation.173 It was also suggested that for an X3MMX3 molecule with small enough ligands, such a conformation would be observed,173 but this “prediction” is incapable of being experimentally proven wrong. So long as no eclipsed X3MMX3 molecule is found, it can simply be said that small enough ligands have not been used. It seems unlikely that the overall analysis is correct, since it supposes that:

1.the metal atoms form octahedral hybrid orbitals of the d2sp3 type;

2.they use a mutually cis set of three to form M–X bonds; and

3.the two X3M units then approach each other along a common threefold axis with a relative rotational relationship that maximizes the overlaps of the two sets of three

hybrid orbitals.

The overlap is maximized when the X3M–MX3 relationship is eclipsed. The validity of this analysis requires significant involvement of the metal p-orbitals in the M–M bonding, but there is little likelihood that the degree of involvement is very great, certainly not to the extent of corresponding to full d 2sp3 hybridization.

738Multiple Bonds Between Metal Atoms Chapter 16

In quantitative calculations174 on H3MoMoH3 by the Hartree-Fock method, it was found that at the SCF (i.e. single configuration) level the eclipsed conformation was favored by only 1.0 kcal mol−1 and that when CI was introduced this preference vanished and free rotation was predicted. This is equivalent to attributing the Mo>Mo bonding to pure 4d-4d overlaps with negligible 5p participation.

Other Hartree-Fock calculations175,176 have also concluded that there is no inherent rotational barrier in the M>M bond, but that the staggered conformations result essentially from repulsive interactions between vicinal metal-ligand bonding electrons.

16.3.8 Other calculations

An SCF-X_-SW calculation177 has been carried out on Mo2(HNCHNH)4 and Mo2(HNCHNH)4+. The results were similar to those for Mo2(O2CH)4, but the greater basicity of the formamidinate ligand led to understandable shifts in some orbitals.

SCF-X_-SW calculations178 on Rh2(O2CH)4(H2O)2 and [Pt2(O2CH)4(H2O)2]2+ have shown that there is extensive and complex mixing of metal–metal and metal–ligand character in nearly all the molecular orbitals, but more so in the case of the platinum compound. In each case, however, the LUMO is mainly an M–M μ* orbital and an orbital of significant β* character lies either immediately below it (Rh) or not far below (Pt). Thus, in each case the metal– metal bond may be roughly described as a single bond of μ character.

The Re2(allyl)4 molecule is an example of a μ2/4β2β*2 triply-bonded system, but of a special type both structurally and electronically. The structure, which has D2d symmetry and the nature of the C3H5 ligands introduces bonding features not encountered in molecules of the usual X4MMX4 type. SCF-X_-SW calculations179 show that the functions of the dxy and dx2-y2 orbitals are not markedly differentiated and there are, in effect, two sets of β-orbitals and two sets of β* orbitals. Moreover, there are MOs arising mainly from combinations of the /-nonbonding and /* orbitals of the individual C3H5 groups, and one of the resulting MOs, 10e, turns out to be the HOMO of the Re2(C3H5)4 molecule.

There have been a few calculations of electron density maps (summed over all occupied orbitals) and comparisons have been made with experimental results. The latter are of uncertain accuracy, but agreement has been reported180 for Cr2(O2CCH3)4(H2O)2, and Mo2(O2CCH3)4. Two dichromium compounds with very short bonds,181 Cr2[(CH2)2PMe2]4 and Cr2(mhp)4, were studied by theory and experiment, respectively, and each was found to have a buildup of electron density consistent with a μ2/4β2 bonding pattern.

16.4 Electronic Spectra

The electronic absorption spectra of compounds with M–M multiple bonds have presented some unusual and fascinating problems. Dinuclear species in which the metal atoms are strongly bonded to each other have spectral properties entirely different from those of mononuclear complexes, where many techniques of interpretation (i.e. ligand field and crystal field models) that rely on the survival of the central-field, atomic character of the metal orbitals can be employed. The simplifications and approximations that arise from the fact that the mononuclear complex can be treated as a perturbed atom, with symmetry lowered from spherical to Oh or Td, do not exist for dinuclear species with strong M–M bonds. More rigorous and complex theoretical arguments are needed. There is, however, one special advantage that the dinuclear species have, and that is the existence of a unique molecular axis of high symmetry. This can be utilized to classify orbitals, molecular states, electronic transitions, vibrations, and so on, and thus aids greatly in the interpretation of the spectroscopic data.

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The material that follows will be mainly concerned with species having μ2/4β2 configurations, with some attention also paid to the μ2/4β2β*m/*n species. There has been relatively little study, and thus little to be said here, of the electronic spectra of M2X6 type compounds.182 For all of them the lowest metal-centered, allowed transition should be a /Α/* transition and bands in the 25 500-27 800 cm-1 region have been so assigned. Strong charge transfer bands in the M2(NMe2)6 compounds (N2pΑ/*) are also observed.

While many types of electronic transition contribute to the electronic absorption spectra observed from compounds containing M2n+ cores,183,184 the one that has dominated the experimental study and the discourse involves the photon-induced promotion of an electron from a β orbital to a β* orbital. Before presenting a review of the data on such βΑβ* transitions, it will be appropriate to look carefully at the β-bond itself. In Sect. 16.4.1, a generally useful approximate way of looking at the bond in cases where there are two β electrons present in the ground state will be explained in detail. It should be pointed out that an alternative so-called valence bond (VB) approach based on resonance interaction between covalent and ionic states also provides insight and can also be computationally feasible.185,132b

16.4.1 Details of the β manifold of states

While the β2 bond is weak, because of the relatively small overlap (compared to μ and / overlaps) between the β atomic orbitals on the two metal atoms, it displays all of the basic characteristics of any two-electron bond. What makes it unique is that it is possible to obtain experimental data on the entire manifold of four states that are associated with it as the strength of the bond is changed from maximal to minimal.186,187

The idea that wave functions for the interaction between a pair of bonded atoms could be constructed as linear combinations of overlapping atomic orbitals (LCAO-MOs) was fully implemented in 1949 by Coulson and Fischer 188,189 for the μ bond in the hydrogen molecule, H2. The Coulson and Fischer treatment (a) described an entire manifold of four states, (b) showed in theory how their energies should change as the internuclear distance increased from the equilibrium value to the dissociation limit, and (c) drew attention to the critical role of configuration interaction. All of this can be done for the β bond manifold, except that for part (b), the weakening of the bond is actually accomplished experimentally, not by stretching it (which is not experimentally realizable) but by twisting it (which is).

The theoretical argument proceeds in three steps. We begin by aligning the two ends of an M2X8 type ion or molecule so the β-β overlap is maximized, as shown in Fig 1.5 (a). Designating the two atomic orbitals as α1 and α2 we write bonding, θ and antibonding, ρ, LCAO-MOs (neglecting overlap) as follows:

φ = 12 (γ1 + γ2)

χ = 21 (γ1 − γ2)

The energies of these MOs are

Eφ = φ|H|φ

= ∫γiHγidτ + ∫γ1Hγ2dτ where i = 1 or 2

= Eγ + W (W < 0)

Eχ = Eγ - W

740Multiple Bonds Between Metal Atoms Chapter 16

Since Eα is the energy of one electron in the atomic orbital α1 or α2, we may take this as the zero of energy and write

Eθ = W and Eρ = -W

If there is only one electron to occupy these MOs, we have a very simple (and very familiar) picture, in which there are only two states, θ and ρ, and only one electronic transition, namely, that from the ground state to the excited state, whose energy is exactly 2W.

When there are two electrons, we must write determinantal wave functions for the four states that can arise. If both electrons occupy the θ MO, to give a full bond, we have

			+		-
+	-	1	|φ (1)		φ(1)|
ψ1 = |φ	φ| =		+		-
		2 |φ (2)			φ(2)|
	+	-		-	+
= 12 [φ (1) φ(2) –				φ (1) φ(2)]

After separating orbital and spin functions, using _ (S = ½) and ` (S = −½) for the latter, we obtain

ψ1= 12 φ(1)φ(2)[αβ - βα]

where the antisymmetrization required by the Pauli principle is accomplished by the spin function. We could also place both electrons in the ρ MO and get an analogous expression,

ψ4= 12 χ(1)χ(2)[αβ - βα]

Both of these represent spin singlet states.

When we develop the corresponding expressions for the states arising from placing one electron in θ and the other in ρ, the Pauli principle no longer restricts us to antisymmetrizing the wave function by way of the spins. Antisymmetrization can also be done if both electrons have the same spin by way of an antisymmetric orbital (i.e., spatial) function, giving a triplet state. Altogether, we have the following four states in what is called the bond manifold:

ψ1=	12	φ(1)φ(2)[αβ - βα]
		[αα]
ψ2=	12	[φ(1)χ(2) − φ(2)χ(1)]{12 [αβ[ββ]+ βα]
ψ3=	12	[φ(1)χ(2) + φ(2)χ(1)][αβ - βα]
ψ4=	12	χ(1)χ(2)[αβ - βα]

The two-term orbital factors in σ2 and σ3 arise because of the indistinguishability of electrons; we cannot assert that electron 1 is in θ and electron 2 in ρ rather than the reverse, so we must give both assignments equal weight.

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For β bonding in a unit such as Mo2Cl84- or Re2Cl82-, where the symmetry is D4h, the symmetries of these four wave functions, and the corresponding MO configurations are as follows:

σ1 (1ββ)

1A1g

σ2 (3ββ*) 3A2u

σ3 (1ββ*) 1A2u

σ4 (1β*β*) 1A1g*

To obtain the energies of the four states in the manifold, the following equations, obtained by inserting the wave functions into the wave equation, E= 0σnΝ σn*dο, must be solved:

|2W + Jθθ - E		K	| = 0
|	K	-2W + Jρρ - E |

where E- = E1 and E+ = E4

E2 = Jθρ − K

E3 = Jθρ + K

Note that σ1 and σ4 have the same symmetry, and the energies E1 and E4 cannot be obtained independently because σ1 and σ4 interact to give the off-diagonal matrix elements, K.

In these equations, ± W has the same meaning as before, namely, it is the energy by which θ or ρ, as a one-electron orbital, is lowered or raised, respectively, from their average value. Jθθ, Jρρ, Jθρ are Coulomb integrals, inherently positive, and represent the repulsive interaction between the charge clouds of two electrons that are either in the same orbital (Jθθ, Jρρ) or in different orbitals (Jθρ). Finally we have K, the exchange integral, which is simply half the energy required, for two atoms, X, infinitely far apart, to convert from X + X to X+ + X-.

The approximation of neglecting the small β-β overlap was used to write the bonding and antibonding LCAO wave functions with which we began. It may, consistently, be invoked once more190 to simplify the energy equations by assuming that Jθθ and Jρρ (which are, of course, equal) are about equal to Jθρ Since all the J’s are additive to the En values, they may all be omitted and the energies will come out as shown in Fig. 16.24. The large magnitude of K relative to W is a consequence of the small β-β overlap. In Mo2Cl84- 2K/W is about 4.

Let us now return to the wave functions previously written for the four states and see what they tell us about the electron distribution in each state. If we take the state wave functions and substitute in the LCAO expressions for θ and ρ, we obtain the following results:

	Ionic		Covalent
σ1	= [α1(1)α1(2) + α2(1)α2(2)]	+	[α1(1)α2(2) + α2(1)α1(2)]
σ2	=		[α1(1)α2(2) + α2(1)α1(2)]
σ3	= [α1(1)α1(2) + α2(1)α2(2)]
σ4	= [α1(1)α1(2) + α2(1)α2(2)]	−	[α1(1)α2(2) + α2(1)α1(2)]

742Multiple Bonds Between Metal Atoms Chapter 16

Fig. 16.24. Energy level diagram for the states of the β manifold when two electrons are present. ¨W = Eρ − Eθ.

σ1 to σ4 here correspond to those numbered E1 to E4 in Figure 16.24. We see that σ2 and σ3 which are the actual wave functions (so long as we treat the β manifold alone), are, respectively, purely covalent and purely ionic. On the other hand, σ1 and σ4 both have half covalent and half ionic character. These are not credible wave functions as they stand. It is not, for example, believable that in the 1A1g state there are two electrons on one atom half the time. The ionic distribution must be of much higher energy than the covalent one and, accordingly should contribute mainly to the 1A1g* state, while the 1A1g ground state should be mainly covalent. This is, in fact, exactly what occurs. The wave functions σ1 and σ4 are not really the orbital wave functions for the 1A1g and 1A1g* states; through the off-diagonal element, these two orbital wave functions are mixed (configuration interaction ) and the true orbital wave functions for these two states are given by

σ(1A1g) = σ1 - ησ4

σ(1A1g*) = σ4 + ησ1

If we examine the expressions for σ1 and σ4 given above we see that as η increases, σ (1A1g) becomes more covalent and σ (1A1g*) becomes more ionic. This mixing contributes to the stability of the 1A1g ground state and raises the energy of the 1A1g* state.

With the LCAO picture of the fully developed β bond (i.e. an eclipsed molecule with a β2 configuration and an empty β* orbital) we may inquire how the β manifold will evolve as we weaken the bond. As has been shown in Sect. 16.1.2, with an increase in the torsion angle ρ from 0° to 45°, the β-β overlap decreases, linearly with cos 2ρ. In a series of actual molecules of the type Mo2X4(PR3)4 and Mo2X4 (diphos)2 (see Sect 4.3.4) the angle ρ has been found to vary from 0° to ~ 40°, and for each of these molecules the energy of the 1A1gΑ1A2u transition (which is commonly called “the βΑβ* transition”, but more precisely, the β2Αββ* transition), has been measured.191 These energy differences which have already been shown in Fig 16.4. may be replotted as shown in Fig. 16.25 (a). Since we are interested only in energy differences within the manifold, it is convenient to keep the energy of the A2u1 state horizontal.

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Fig. 16.25. The experimental evolution of the manifold of states for the β bond in Mo2Cl4(diphos)2 molecules. (a) The 1A1g and 1A2u states. (b) Adding the 3A2u state.

(c) Adding the 1A1g* state.

Theory leads us to expect that in addition to the 1A1g and 1A2u states which are shown in full lines in Fig 16.25 (a) there should be 3A2u and 1A1g* states as shown by the broken lines. Experimental data to support the theory has been obtained. In principle, spectroscopic observations of the 1A1gΑ3A2u transitions could provide verification of the position of the line for the 3A2u state in Fig 16.24, but such transitions are too weak to be observed. The problem of measuring the 1A1gΑ3A2u gap was solved by a non-spectroscopic method. When the torsion angle is in the range of 20-40°, the gap is of the order kT at and below room temperature. Therefore, there is enough thermal population of the 3A2u state, following a Boltzmann distribution, to cause a measurable change in the chemical shift of the 31P resonance, without making the line too broad for accurate measurement. This NMR method was used192 for several of the Mo2Cl4(P-P)2 compounds to afford 1A1gΑ3A2u energy gaps for six compounds of the Mo2Cl4(P-P)2 type, with ρ values of 20.6, 24.7, 25.5, 30.5, 40.0 and 41.4°. These data define a line that is parallel to the one for the 1A2u state and separated from it by 10,400 ± 200 cm-1, which is the value of 2 K. In this way we proceed from Fig 16.25 (a) to Fig. 16.25 (b).

To verify the remaining broken line (for the 1A1g* state) by conventional spectroscopic measurement is also impossible because of the weakness of a one-photon, two-electron transition. However, the 1A1gΑ1A1g* transition is allowed in the two-photon absorption spectrum.193,194 Here, two β electrons are promoted to the β* level by the simultaneous absorption of two photons whose energies sum to the energy required. Because we can estimate the 1A1gΑ1A1g* transition energy from that of 1A1gΑ1A2u, it follows that the two exciting photons must be in the near-infrared frequency range. The simultaneous absorption of two photons is an unlikely event, but the probability increases with the square of the intensity of the absorbing light, so the flux of the exciting photons must be intense.

These demanding conditions of intense and tunable near-infrared photons can be satisfied with the output from optical parametric oscillators. But providing the necessary laser excitation source constitutes only one half of the experimental problem. There is also the question of how to show that the 1A1gΑ1A1g* transition is occurring. To measure the transmittance is impractical for a two-photon experiment and especially so when the spin-allowed transitions are weak, as is the case within the β manifold. Instead one can monitor a fluorescence intensity that is dependent on the population of the 1A1g* state. Although 1A1g* is sure to be photonsilent, its neighboring 1A2u excited state may be emissive for selected quadruple bond metal complexes. Because the 1A1g*Α1A2u conversion is fully allowed, 1A1g* may internally convert

744Multiple Bonds Between Metal Atoms Chapter 16

to 1A2u on a much faster time scale than that associated with emissive decay from the 1A2u state. Therefore, as the two-photon laser excitation frequency is tuned into the 1A1g* excited state, emission from 1A2u can be observed. Conversely, no 1A2u-based luminescence will be generated when the two near-infrared photons are off resonance from the 1A1gΑ1A1g* transition. In this manner, the absorption profile of the 1A1g* state was mapped out (at twice the excitation frequency) by monitoring the laser-induced fluorescence from the 1A2u excited state as the nearinfrared spectral region is scanned.

Three points have been obtained to establish experimentally the energies of 1A1g* states and when these are introduced we obtain Fig 16.25 (c). It is clear that the classic theoretical picture of a two-electron bond and its manifold of four states is quantitatively borne out by experiment for metal-metal β-bonding.

It must be understood, however, that it is one thing to see that theoretical concepts can be combined with experimentally determined numbers to provide a complete quantitative picture of the β manifold, as just shown. However, the problem of making quantitative a priori calculations of the numbers is quite another problem for which no entirely satisfactory solution has yet been found.

16.4.2 Observed βΑβ* transitions

While the electronic absorption spectra of M2n+ complexes afford a plethora of observed transitions, the greatest attention has been directed to those which may be described generically as βΑβ* type transitions. There are three subclasses, depending on the number of electrons present in β and β* orbitals in the ground state:

1.β2Αββ*

2.βΑβ*

3.β2β*Αββ*2

The transition in subclass (1) is the 1A1gΑ1A2u transition discussed in the previous section.

In all three subclasses the transition is orbitally allowed with z polarization (i.e. along the M–M axis). In class (1) the singlet - singlet transition is spin-allowed, while the singlet-triplet transition is spin-forbidden and not observed. As explained in Section 16. 4.1 the energies of the class (1) transitions are not simply related to orbital energy differences. On the other hand the class (2) and class (3) transitions both have energies that are equal to the difference in the energies of the β and β* orbitals. This is immediately obvious for class (2); for class (3) it results from the fact that interelectronic interaction energies are about the same in the ground and excited states and thus cancel out.

All three types of βΑβ* transitions are of low intensity, despite being dipole allowed. The reason for this is that the overlap of the two d-orbitals that form the β bond is quite small.195 Moreover, as Mulliken showed196 many years ago, oscillator strength in a transition of this nature is approximately proportional to the square of the overlap integral. Thus, inherently low intensity is a straight-forward consequence of the weakness of the β bond. However, the intensities actually observed are all somewhat greater than the inherent or intrinsic intensities because both the β and β* orbitals mix with ligand orbitals, but to different degrees. This has the effect of giving βΑβ* transitions some charge transfer (usually LMCT) character. It is because of the variability of such mixing from one compound to another that βΑβ* transitions, while always weak vary considerably in their intensities. For example, in the series of molecules Mo2X4(PMe3)4, the energy range is only from 15 700 cm−1 (X = I) to 17 000 cm-1 (X = Cl) but the intensity changes by nearly a factor of 2, as the XΑM LMCT transitions approach the βΑ β* transitions in energy and therefore mix in more strongly.197 With ligands such as RCO2-,

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SO42- or H2O mixing of β or β* orbitals with ligand orbitals is very small and the intensities become extremely low, with ϒmax values of c. 102 instead of 103.

A representative class (1) transition is found in K4[Mo2(SO4)4]·2H2O.198 As shown in Fig. 16.26, the β2Αββ* band at about 19 x 103 cm-1 narrows and the peak height increases on lowering the temperature from 300 K to 15 K, but the integrated intensity does not change, which is appropriate for an orbitally allowed (as opposed to a vibronically allowed) transition. Moreover, when the orientation of the Mo–Mo bonds relative to the crystal axes is taken into account (23.7˚ angle with the c axis), the relative intensities of the peak in the two spectra are in quantitative agreement with what would be expected for a z-polarized transition.

Fig. 16.26. Polarized crystal spectra of K4[Mo2(SO4)4]·2H2O.

The case of K4[Mo2(SO4)4]·2H2O is exceptional in that the absorption band shows no vibrational structure, even at 15 K. In other cases such structure is seen at low temperatures and occasionally even at room temperature. An example of detectable structure even at 300 K is provided by K3[Tc2Cl8],199 as shown in Fig. 16.27. It should also be noted that this is a class

(3) transition and is at much lower energy than the class (1) transitions shown in Figs. 16.26 and 16.28. On the other hand, the situation in K4[Mo2Cl8]·2H2O, due to a β2Αββ*,200 shown in Fig. 16.28, is more typical in that the vibrational structure is observed only at the lower temperature. In each of these cases the resolved vibrational structure, consists of a single series of equally spaced components. In the β2Αββ* transition only one internal coordinate (the M–M distance) is expected to change very much on going to the excited state. The molecule therefore goes from the vibrational ground state to a series of states in which the totally symmetric vibration corresponding to this internal coordinate has various degrees of excitation, and a progression in ι' (M–M) (i.e. ι', 2ι', 3ι', etc.) in the excited electronic state is seen. Since the M–M bond is weaker in the electronically excited state, this frequency (ι') is lower (by c. 30 cm-1) than that (ι) in the ground state. We shall discuss these questions in more detail in Sections 16.4.6 and 16.6.1.

746Multiple Bonds Between Metal Atoms Chapter 16

Fig. 16.27. The β2Αββ* transition in the [Tc2Cl8]3- ion at 300 K and 3.7 K.

Fig. 16.28. The β2Αββ* transition in the [Mo2Cl8]4- ion at 300 K and 3.7 K.

In the case of the [Re2Cl8]2- ion 201-203 the β2Αββ* transition contains two progressions, one being in the ι' (Re–Re) vibration, as expected. The other involves the totally symmetric Re–Re–Cl bending mode β', the progression being β', β' + ι', β' + 2ι', etc. This type of participation by two totally symmetric vibrations is not unusual. The band intensity shows no temperature dependence and is z-polarized. Thus, its assignment to the β2Αββ* transition is completely secure. Similar results were reported for the [Re2Br8]2- analog.203

Further support for the assignment of the β2Αββ* transition to the weak (ϒ 5 103) band at 14,500 cm-1 is provided by a Raman excitation profile study204 which also supports this assignment for similar bands at 17,900, 13,700 and 13,000 cm-1 in [Re2F8]2-, [Re2Br8]2- and [Re2I8]2-, respectively.

In keeping with the expected relationship between class (1) and class (3) transitions, twenty β2β*Αββ*2 transitions in a variety of Re25+ compounds including Re2Cl83- 205 are found206 in the range 6500 - 7600 cm-1, as compared to the β2Αββ* transition in Re2Cl82- at about 14,700 cm-1. Similarly, comparisons of class (1) and class (2) transitions also show the expected relationship.207 In [Re2Cl8]1- the class (2) transition occurs at 4700 cm-1.

For complexes of the Mo24+ core numerous other observations of the β2Αββ* transition are scattered throughout the literature. There would be little point in attempting to collect