Section 8-13 ESR Hyperfine Splitting Constants

Section 8-13 ESR Hyperﬁne Splitting ConstantsFigure 8-16 Theoretical versus experimental bond lengths for phenanthrene in order of decreasing observed length: (•) experimental; (×) theoretical with s = 1.54 Å, k = 0.765; (◦) theoretical with s = 1.515 Å, k = 1.05.

8-13 π-Electron Densities and Electron Spin Resonance

Hyperﬁne Splitting Constants12

If a molecule with one or more unpaired electron spins is placed in an external magnetic ﬁeld, the interaction energy of an unpaired electron’s spin dipole with the external ﬁeld will be different depending on whether the electron’s spin is α or β. This produces two slightly different energy levels for an unpaired electron. Suppose microwave radiation is supplied in an effort to excite such electrons from the lower to the higher of these two levels. If the radiation’s energy is larger than required, no absorption occurs. But, if we gradually increase the strength of the magnetic ﬁeld, thereby increasing the energy-level 12A number of reviews on this subject have been published. See, for example, Gerson and Hammons [9].

Chapter 8 The Simple H ¨uckel Method and Applications gap, we eventually achieve a magnetic ﬁeld where absorption of the microwave energy occurs. This would give us a spectrum (absorption vs. magnetic ﬁeld) with just one line.

Now suppose that our radical has a hydrogen atom attached to it. The proton nucleus of the hydrogen can have a spin of α or β, and this can have a small effect on the energies of the electron whose spin we are trying to ﬂip. We ﬁnd now that half of the radicals absorb at a slightly different applied magnetic ﬁeld than the other half giving us a two-line spectrum: The original single line now exhibits hyperﬁne splitting due to the presence of a hydrogen atom, and the distance between the two peaks on the magnetic axis is called the hyperﬁne splitting constant, usually given in millitesla (mT), or in gauss (G).

If the unpaired electron is a π electron, HMO theory can be applied to the value of the ESR splitting constant. The interaction between an unpaired π electron and a proton (or other nucleus with nonzero nuclear spin) falls off rapidly with increasing separation.

Therefore, the hyperﬁne structure is generally ascribed to interactions involving protons directly bonded to carbons in the π system (α protons) or else separated from the π system by two σ bonds (β protons). The equilibrium position of an α proton is in the nodal plane of the π system, so it is clear that any net spin density at the proton must be only indirectly due to the presence of an unpaired π electron. This indirect effect arises because the unpaired π electron interacts slightly differently with α- and β-spin σ electrons on carbon, and so the spatial distributions of these become slightly different in the σ MOs, ultimately producing net spin density at the proton; the σ electrons are said to be spin polarized by the π electron (see Fig. 8-17).

The extent of spin polarization at a given hydrogen should depend on the percent age of time the unpaired π electron spends on the carbon to which that hydrogen is bonded. It is therefore reasonable to look for relationships between the distribution of π -spin density in a radical and the hyperﬁne coupling constants characterizing its ESR spectrum. The simplest assumption one can make in this regard, called the McConnell relation, is that the hyperﬁne splitting constant, aH , for a proton directly bonded to µ the µth carbon, is proportional to the net π -spin density ρµ on that carbon: aH = Qρ

µ

µ

(8-62)

Thus, if the unpaired π electron density is twice as great on carbon 1 as on carbon 2, the ESR splitting constant for a proton attached to carbon 1 should be twice as great as that for a proton attached to carbon 2.

Figure 8-17 The unpaired π -spin density at carbon repels both σ electrons in the C–H bond region, but does not repel them equally. As a result, slight spin imbalance due to σ electrons occurs at the proton.

Section 8-13 ESR Hyperﬁne Splitting Constants

The HMO method predicts that the spin density at the µth carbon due to an unpaired electron in the mth MO is simply c

2

µm

. This is only a ﬁrst approximation to ρµ, but a graph of observed splitting constants plotted against ρµ calculated in this way shows a respectable correlation (Fig. 8-18). The proportionality factor Q, given by the slope of the line of best ﬁt, varies somewhat depending on the type of system and the charge of the radical. Thus, in Fig. 8-18, where all the data are from aromatic fused ring hydrocarbon radical anions, the correlation is quite good. As data for nonalternants and radical cations are added (Fig. 8-19) the scatter increases.

Some ESR spectra are interpreted to be consistent with the presence of some negative spin density. That is, if the extra π electron is taken to have α spin, some of the carbons appear to have an excess of β-spin π density. This has the effect of giving us a total spin density that is greater than unity, even though we have only one electron with unpaired spin. (It sounds absurd, but read on.) Now the amount of splitting seen in ESR spectra depends only on the magnitudes of spin densities at the protons, not on whether these spin densities are α or β, and the presence of negative spin density does not produce a qualitative change in an ESR spectrum (such as a “negative splitting,” whatever that might be). Rather it leads to an increase in the total amount of spin density in the system, and this, in turn, leads to an increase in the sum of splitting constants (over Figure 8-18 ESR splitting constants aH in gauss versus HMO unpaired spin densities. The sys µ

tems are fused ring alternant hydrocarbon radical anions (naphthalene, anthracene, tetracene, pyrene).

The underlined point is thought to result from negative spin density. (Data from Streitwieser [7].)

Chapter 8 The Simple H ¨uckel Method and ApplicationsFigure 8-19 ESR hyperﬁne splitting constants in gauss versus c2 for the highest occupied MO of the radical. All data are from hydrocarbon radical anions or cations of alternant or nonalternant type. (•) Anion radical; (×) cation radical. Uncertain assignments: (◦) anion; (⊗) cation. The two correlation lines are merely sight ﬁtted to the anion and cation points separately and suggest that Q for cations should be larger than for anions. Points thought to result from negative spin density are underlined. [See Tables I, III, VIII, XII, XIII, XIV, XV, XVI, XVII, XVIII of Gerson and Hammons [9], and Table 6.2 of Streitwieser [7] for data plotted here.]

what would be predicted in the absence of negative spin density). For example, the HMO prediction for allyl radical would give a net spin density of 1 at each terminal

2

carbon and zero at the central carbon. [The unpaired electron is in the nonbonding MO

1

√ χ1 − 1 √ χ3.] However, we might imagine the situation wherein there is a net spin

2

2

of 2 α at each terminal carbon and 1 β at the central atom. This retains a net spin value

3

3

of 1α yet results in a larger splitting constant aH for every proton in the molecule.

µ

It is possible to think of a physical explanation for this kind of spin distribution. The π electrons in lower-energy, ﬁlled MOs are being spin polarized similarly to the σ electrons mentioned earlier. If an α-spin electron in effect repels electrons of β spin more strongly than those of α spin, a buildup of β spin on the central carbon of allyl radical could be expected. Because this secondary effect is expected to be fairly small,