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Electron Spin Resonance Volume 1
A Review of the Literature Published between January 1971 and May 1972
By R. O. C. Norman The Royal Society of Chemistry
Copyright © 1973 The Chemical Society
All rights reserved.
ISBN: 978-1-84755-334-8
CHAPTER 1
Theoretical Aspects of Hyperfine Splittings and g-Factors
BY C. THOMSON
1 Introduction
The fundamental theory of e.s.r. spectroscopy has been well reviewed in recent books, and will not be dealt with here. This report is concerned, furthermore, with organic free radicals and does not deal with the theory of transition-metal ion e.s.r., nor with the theory of gas-phase e.s.r. We are concerned primarily with recent advances in the calculation of hyperfine coupling constants (hfcc) of free radicals, both semi-empirically and non-empirically. A limited number of references to experimental papers containing theory which is used in the interpretation of experimental results will also be made. Hyperfine coupling constants are quoted in gauss (G).
The analysis of the e.s.r. spectra of free radicals in solution gives the magnitude of the hyperfine coupling constants in the radical. We shall use the notation aNi where this hfcc refers to the coupling with nucleus N, and the subscript i refers to the position of this nucleus in the radical. Since the hyperfine coupling constants are related to the value of the wave-function at the nucleus in question, the study of the molecular electronic structure of free radicals and the interpretation of the experimental e.s.r. parameters are closely related. The absolute signs of the aNi cannot be obtained directly from the e.s.r. spectrum, although they may be obtained by observation of line width variations in the spectra. The signs are important, however, in theoretical work, as will be seen. Determination of the total hyperfine tensor from single crystal studies also yields the signs of the components, but few radicals are stable enough to be studied in this way.
A second parameter which is obtained from an e.s.r. spectrum is the g-factor in the equation for the resonance condition
hv = gβH (1)
In equation (1), h is Planck's constant, v is the resonance frequency, β is the Bohr magneton, and H is the external magnetic field. The para-magnetism of organic free radicals is almost exclusively due to the spin of the unpaired electron, so that the g-value differs only slightly from the value for the free electron ge = 2.00232. However, in careful experimental work, g-values can be measured to ± 0.000005 and the theoretical interpretation of the small variations in g-values is of some interest. Recent work in this area is reviewed in the next section.
2 The Theory of g-Factors
A general theory of the g-tensors of molecules has been given by Stone, who derived an expression for the principal g-values of a molecule in an orbitally non-degenerate state. The theory could then be applied to hydro-carbon radicals and to semiquinone radical ions, when some simplifications in the equations had been made, with the equation for the g-factor shift being
Δg = g - ge = b + [Lambda]c (2)
where λ is the coefficient of the resonance integral β in the energy of the Hückel molecular orbital containing the unpaired electron, and b and c are semi-empirical constants, with values b= 24.7 x 10-5, c = -19.3 x 10-5. In the case of aromatic even-alternant hydrocarbons, the difference between the in-plane g tensor components was equal in magnitude but opposite in sign for the anion and cation radicals. Careful studies by Fassaert and de Boer, in which linewidth variations were used to estimate the g-shifts, gave good qualitative agreement with Stone's theory. However, there are significant quantitative discrepancies, i.e. the predicted decrease in (g1 - g2) going from anthracene± to tetracene± is not reflected in the observed values. Also for C10H8[??]", (g1 - g2) is predicted to be -9 x 10-4, whereas it is found to be essentially zero. The deviations may be due to approximations in the Stone theory, or to the assumption of isotropic rotational motion in the line width analysis.
Moss and Perry have extended the Stone theory to degenerate radicals, where there is an extra term involving the orbital angular momentum, and in estimating this term, vibronic coupling must also be taken into account.
Thermal averaging over the possible vibronic states was carried out, and the computed g-value for C6H6[??] was in qualitative agreement with the measured value. However, there were a number of approximations made which may affect the numerical results, such as the neglect of possible solvent effects and the use of data from the neutral molecule, but the general conclusions should apply to other degenerate anions of unicyclic polyenes and account for the major difference between the g-factors of degenerate and non-degenerate radicals.
The theories developed above have been largely applied, with several approximations, to large aromatic radical ions. Applications to some small radicals were reported some years ago by Glarum, who carried out calculations on CH3, NH2, and CH2. However, some multicentre integrals were approximated in this work, and their influence is not negligible. In order to provide a test of the different approximations used in calculations on larger molecules, the theory of the g-tensor in one-electron systems has been studied for H2+.15 It was concluded that even for very simple molecules, some of the approximations made by Stone are not quantitatively very good, particularly the neglect of the two-centre terms. This work therefore shows that more accurate calculations on the larger systems with fewer approximations are needed.
The INDO (intermediate neglect of differential overlap) molecular orbital method which is proving very useful in the calculation of approximate hyperfine coupling constants (see Section 4) has recently been applied to the calculation of g-tensor components. The applications, within the approximations of Stone's theory, were to the radical H2NO,16 and also to a variety of σ-radicals. This work follows earlier calculations with the CNDO (complete neglect of differential overlap) method. In the INDO study, a variety of excited configurations based on the INDO orbitals was taken into account. Good agreement with experiment was found ; for example, in the case of NO2 , where gas-phase measurements are available gxx= 1.9908, gyy= 2.0024, and gzz= 2.0067 compared with the experimental values of gxx= 1.9910, gvv= 2.0020, and gzz= 2.0062. The agreement with experiment using the INDO method was substantially better in the case of H2NO than with the CNDO/2 calculations. It was pointed out, however, that the results are quite sensitive to the assumed geometry, which should be that which minimizes the energy. The anomalously low g-factors for diphenylacetylene and diphenyldiacetylene anions can be explained on the basis of a strong interaction between the phenyl groups and the acetylene p-orbitals.
3 Ab lnitioCalculations of Hyperfine Splitting Constants
There are two main contributions to the hyperfine interaction in free radicals. The first of these is the anisotropic hyperfine interaction with the Hamiltonian Hd
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where βe and βN are the Bohr and nuclear magnetons, ge and gN are the electron and nuclear g-factors, rkN is the vector which connects electron k and nucleus N, having spin operators Sk and JN, respectively. This interaction vanishes for radicals in solution in which there is rapid re-orientation of the radicals, and although the elements of the anisotropic coupling tensor may be obtained from single-crystal studies, there have until recently been rather few theoretical studies of this contribution (see later p. 16).
The second contribution is the hyperfine splitting caused by the Fermi contact term with Hamiltonian Hc
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where the symbols are as defined above and δ(rkN) is the Dirac delta function. The Fermi contact splitting for nucleus N can be calculated as the expectation value of Hc which leads to the hyperfine coupling constant for nucleus N:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
In equation (6) Ψ denotes the exact ground-state wavefunction, the other symbols are defined above, and the summation is over all the electrons. γ(rN) is the spin density at the nucleus, or the net density of electron spin at nucleus N, i.e. the number of electrons (Bohr)-3 with spin α minus the number of electrons (Bohr)-3 with spin β. For a strict comparison with experimental data the computed splittings should be averaged over the thermally populated vibronic states. Most computations, however, refer to the evaluation of aN using Ψ = Ψel where Ψel is the electronic part of the wavefunction in the Born-Oppenheimer approximation.
Semi-empirical calculations of aN for various magnetic nuclei have been widely used in the past, and recent work is discussed in Section 4. However, it is clearly desirable to calculate hyperfine coupling constants non-empirically in order to assess the quality of molecular wavefunctions for open-shell species, and to test the validity of the various approximate calculations of aN. Until relatively recently such calculations were not practicable without many approximations, but with the availability of more powerful computers there have been many recent studies of hyperfine splittings in small molecules. Before discussing these calculations it is appropriate to consider the calculations for atoms, since it is on atoms that the various theoretical methods are usually tested. In view of the large literature on the subject, we have confined our attention to those results which bear directly on the molecular calculations.
For atoms in other than 2S states, the restricted Hartree-Fock method (RHF), with a wavefunction Ψ0 [(equation (7)] which has k - 1 doubly occupied orbitals and where the odd electron occupies orbital φk, gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
zero spin densities at the nucleus, and it is necessary to include excited configurations Φij ...ab ... of the same symmetry, in which electrons have been excited from spin orbitals a, b, ... to spin orbitals i,j.... Then the complete configuration interaction (CI) wavefunction can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
The summations run over all possible values of abc ..., ijk ..., and configurations may be defined as singly excited Φia, doubly excited Φijabetc. with respect to Ψ0. It is easy to construct excited configurations which are eigenfunctions of S2 and S2, but it is not obvious which configurations are the most important in the CI expansion, and even with extensive CI the resulting aN are not always in good agreement with experiment. To allow for spin polarization of the inner shells by the unpaired electron, one can allow different spatial orbitals φi, φ'i for different spins in a single determinant. In this case, ΨuHF is called an unrestricted Hartree-Fock (UHF) function but is no longer an eigenfunction of S2. This deficiency
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
can be remedied using a projection operator, but the projected UHF wavefunction (PUHF) is no longer a single determinant. Furthermore, the optimal UHF orbitals are not optimal for the PUHF function. The extended Hartree-Fock (EHF) method optimizes the orbitals after projection but has only recently become feasible in the form known as the GP method, although other authors have used essentially equivalent methods. Other sophisticated methods have been used, such as the Bethe-Goldstone method, and the method of Meyer, to which we refer later.
For molecular calculations, it is not feasible to obtain numerical approximations to the orbitals φi, and these are usually expanded in a basis set of M basis functions
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
and the coefficients Cij obtained variationally. Since the Slater orbitals (STO) have the correct asymptotic behaviour at the nuclei, until recently these have been the most widely used basis functions, and calculations for atoms using these basis sets will now be discussed.
One of the problems associated with the theoretical calculation of aN is that it depends on the electron spin density at a particular point and is therefore very sensitive to the wavefunction. However, the wavefunctions are obtained via the variation method, and depend on evaluation of the integrals over all space, and wavefunctions which give very good energies may give very poor values of the spin density at the nucleus. Slater basis functions have been used in calculations with the methods detailed above, with basis sets ranging from minimal STO to basis sets of close to Hartree-Fock quality. The recent configuration interaction calculations of Schaefer and co-workers, using first-order wavefunctions, give results for the spin densities in error by ca. 25%, and this is typical of the accuracy attainable with CI calculations, although they account for a large fraction of the correlation energy.
On the other hand, the UHF method has probably been the most widely used, and comparisons with numerical UHF calculations have been carried out by Bagus, Liu, and Schaefer. These authors found that there were very large errors even with numerical UHF wavefunctions, although these were usually within about 20% of the results obtained for analytic expansions to UHF orbitals. Their conclusions were thus rather pessimistic regarding the reliability of calculated UHF spin densities. In an important paper, Meyer 31 showed that the errors in the UHF procedure are due to a partial and incorrect inclusion of pair correlation effects. He presented a version of the extended Hartree-Fock (EHF) method 23 which uses the natural orbitals from UHF calculations, but which used a Gaussian function basis (GTO). We return to this method later. Projected (PUHF) wavefunctions sometimes give better agreement with experimental than UHF, but may give worse agreement. One problem, however, is the difficulty of carrying out the projected calculations without further approximations.
EHF calculations by Kaldor, and the similar calculations by the GF method of Goddard have also given quite good agreement in some cases, but again the results are very dependent on the basis set size. One interesting conclusion from the latter calculations is that the core polarization contribution is larger than the experimental value, and therefore correlation effects significantly decrease the electron spin density at the nucleus. However, in the usual UHF method the orbitals are allowed to be different but are still required to be symmetry functions. Goddard et al. have recently shown that relaxation of this constraint improves the anisotropic contributions to hyperfine interaction but makes little difference to the spin densities and therefore the isotropic contribution.
A similar method to the EHF, the spin-optimized (SO-SCF) method has been developed by Kaldor and Harris, in which optimum spatial orbitals and spin functions are determined. This method gives much improved agreement with experiment. An interesting application to the carbon atom 13C coupling constant in the sp3(5S) state has been used to examine the assumptions made in semi-empirical methods such as INDO. Extension of this work to molecules, however, seems to be rather difficult.
(Continues...)
Excerpted from Electron Spin Resonance Volume 1 by R. O. C. Norman. Copyright © 1973 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
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