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Electron Spin Resonance Volume 3
A Review of the Literature Published between November 1973 and June 1975
By R. O. C. Norman The Royal Society of Chemistry
Copyright © 1976 The Chemical Society
All rights reserved.
ISBN: 978-0-85186-771-7
CHAPTER 1
Theoretical Calculations of Hyperfine Coupling Constants
BY C. THOMSON
1 Introduction
In common with earlier work in Volumes 1 and 2 of this series, the present report deals primarily with recent work on the theoretical calculation of isotropic and anisotropic hyperfine coupling constants (hfcc) in free radicals. Unlike earlier reports, many routine calculations by semi-empirical methods such as INDO 3 are not dealt with here, and such work is discussed elsewhere in this volume. The majority of the calculations referred to involve the ab initio calculations of hfcc, an area which has continued to expand. We do not consider theoretical aspects of work on triplet states, transition-metal ions, nor gas-phase e.s.r. spectroscopy. Hyperfine coupling constants are quoted in Gauss (G) (1 G = 10-4 T), and we shall usually use the notation ai(N) for the hfcc between the unpaired electron and nucleus N, the subscript i referring (if necessary) to the position of the nucleus in the radical. An introduction to the subject matter has been given in earlier reports in Volumes 1 and 2 and will not be repeated here. An important new series of Specialist Reports on Theoretical Chemistry contains several articles dealing with theoretical work of relevance to e.s.r. spectroscopists interested in theory, and in particular the chapter by Richards et al. contains a critical discussion of molecular hfcc calculations, particularly unrestricted Hartree-Fock (UHF) calculations.
2 Ab lnitio Calculations of Hyperfine Coupling Constants
Isotropic hyperfine coupling constants can be calculated from the ground-state electronic wavefunction Ψ of the radical by evaluation of the expectation value of the contact Hamiltonian [??]c [equation (1)]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [betae] and [betaN] are the Bohr and nuclear magnetons, ge and gN are the electron and nuclear g-factors, rkN is the vector connecting electron k and nucleus N which have spin operators Sk and IN respectively, and δ(rkN) is the Dirac delta function. The expectation value leads to equation (2) for the hfcc:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
is the spin density at the nucleus N.
The recent advances in the evaluation of reliable wave functions 'Yby non-empirical methods have continued and since Volume 2 a number of authors have examined several different species of interest to the e.s.r. spectroscopist.
There have not, however, been any radically different theoretical approaches to the problem, and in hfcc calculations on atoms there has been relatively little new work. Burke and Cassar have computed an extended Hartree-Fock wave-function for lithium, in which two determinants were used with different spin functions derived from the αβα and βαα spin products. The wavefunction was, however, evaluated numerically, and the computed spin density at the nucleus of 2.8424/4π compares well with the experimental value of 2.9062/4π and with earlier values calculated using analytic basis sets.
Correlation effects on hfcc were considered in Volume 2, and Brown and Larsson have recently extended this work. The UHF method has been extensively used in the past in atomic and molecular calculations, both with and without annihilation of the contaminating quartet component, but there has been a great deal of disagreement as to the reliability and theoretical validity of the results (see refs. 1 — 5). Brown and Larsson have compared spin densities calculated with the spin polarized Hartree-Fock (SPHF) method (which is a close approximation to UHF) with those computed from first-order (FO) wave-functions described previously. 10 The two sets of spin densities were in close agreement, and it was concluded that correlation effects tend to reduce the spin-polarization contribution to the spin densities. However, it was also shown that the important correlation effects are not accounted for by spin-extended Hartree-Fock (SEHF) or spin-optimized Hartree-Fock (SOHF) methods and, furthermore, projection takes correlation into account in a rather arbitrary fashion. Several molecular UHF calculations are referred to later.
Many-body perturbation theory calculations of hfcc continue to appear, and a recent paper by Garpman, Lindgren, Lindgren, and Morrison 11 has reviewed the effective operator form of the theory and presented results of calculations on the 2p, 3p, and 4p excited states of lithium and the 3p state of sodium. The results agreed with earlier calculations by Nesbet, Lyons, and Lunnell. Lindgren has also presented a review of the method in another paper. The hfcc in various excited S states of the Na atom have been computed by Mahanti et al. Turning now to molecular calculations, several authors have continued to perform UHF calculations, following earlier work, particularly by Claxton and his co-workers, which has been reported in Volumes 1 and 2 and also in ref. 4. These calculations all use a single-determinant wavefunction with different spatial orbitals for different spins of the form of equation (4). Since this wave-
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
function is not an eigenfunction of S2, it does not describe a pure spin state, and most previous workers have removed the major contaminating quartet using a projection operator. It is the fact that the unprojected wavefunction is obtained variationally whereas the projected wavef unction is not variationally optimized which has led to earlier criticisms of the method. The results of recent UHF calculations on atoms described in Volume 1 were very disappointing, large errors of up to 250% in the spin densities being found,17 but these numerically obtained wavefunctions were not subject to spin projection. Most previous work on molecules shows that projection makes a substantial difference to the spin densities and in the more recent work, the majority of the UHF calculations were subjected to spin projection, giving values of nearer to the value of 0.75.
The recent experimental determination of more accurate values for the anisotropic hfcc of a variety of small radicals has prompted several ab initio calculations of both isotropic and anisotropic hfcc. Almlöf et al. studied in some detail •CH3, •NH3+, •C2H5 , and •N2H4+, all π-radicals. The restricted Hartree-Fock (RHF) method, of course, predicts zero spin densities at the nuclei in such radicals. The authors used a medium-size Gaussian basis set for most calculations, comprising seven s-type functions and three groups of p-type functions. These were contracted in the usual way to four s- and two groups of p-functions. We use the standard notation (7,3) [right arrow] <4,2> to denote such basis sets. The molecular geometries of the radicals were obtained by minimization of the energy before annihilation, although complete optimization was only carried out for •CH3 and •NH3+. Both these radicals were predicted to be planar. Annihilation brings very close to the expected value of 0.75 and also reduces the magnitude of the isotropic hfcc to about one-third of the value before annihilation in most cases, the latter usually being too large. This has been previously noted and interpreted assuming the spin density is due mainly to spin polarization. (Note, however, the above results for atoms where inclusion of correlation tends to reduce further the spin polarization contribution.) In ·C2H5, however, annihilation does not improve agreement with experiment for the value of a(H). Most of the computed values are in fair agreement with experiment, although usually sma11er in magnitude. Rather surprisingly, for •CH3 and ·N2H4+, which were also investigated with a larger basis set [(10,6,1/5,1) [right arrow] <5,3,1/3,1> for •N2H4+], the hfcc were very little different from the small basis set results. This is not the case in the RHF calculations on a-radicals referred to below.
The anisotropic hyperfine coupling constants are evaluated as expectation values of the dipolar Hamiltonian [??]d (5):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
and Almlöf et al. found better agreement with experiment for these hfcc than in the case of the isotropic hfcc. It was noted, however, that spin annihilation is essential to obtain the good agreement found for the proton hfcc, but the anisotropic 13C and 14N hfcc are only slightly changed after spin annihilation.
Claxton and Overill 20 have also studied the ethyl radical by this method. A direct comparison with Almlof 's work is not possible, since Claxton and Overill did not report total energies and did not optimize the geometry. The C-C bond length assumed (1.46 A) was significantly shorter than the optimized value of 1.52 A. However, Claxton and Overill did use a larger (95) basis set which was contracted to a minimal basis. The isotropic coupling constants were in general closer to experiment than in Almlof's calculation, and the agreement with experiment for the anisotropic coupling constant is rather similar. It was observed that some earlier semi-empirical calculations were less successful in estimating the difference between A[??] and A.[perpendicular to]
Davis et al. also used the UHF method and the method of molecular fragments to study •C2H5, however, using a basis set much smaller than those used above. Spin annihilation and geometry optimization were not attempted and the results were not as good as in the case of the more extended calculations.
•CH3 was also investigated by Aarons et al., who partially optimized the geometry and found an out-of-plane angle of 1.3°, whereas Almlof et al. predicted a planar radical. Slightly different bond lengths were obtained, and with a slightly smaller basis set small differences in the hfcc were obtained. It is regrettable that several authors do not report total energies in these papers since comparison with other work is made more difficult if this is the case.
Almlöf et al. have also studied the a-radicals NO2 and CO2- by the same method, using the experimental geometry of N02 and optimizing the CO2- geometry [angle]OCO = 134.1°; R(CO)= 1.43 A]. Annihilation has rather little influence on the values of the isotropic hfcc, which is also rather insensitive to the basis set size. Indeed RHF calculations for these radicals give rather similar values. However, annihilation was essential to obtain reliable values for the anisotropic hfcc, but further work is needed to explain this difference between σ- and π-radicals in UHF calculations.
Aarons et al.24 have also studied several other tetra-atomic radicals using the same basis sets, namely •CF3, •CC13, •SiH3, •SiC13, •PH3+, •PF3+, and •PC13+. As was found in several RHF calculations to be described below, the value of a(19F) in CF3 was about one-half the experimental value but in •SiF3 this coupling constant was overestimated. In general, the trends in the planarity, AH3 > ACl3 > AF3 and CX3 > PX3+ > SiX3, are correctly reproduced. Inclusion of d-functions for •SiH3 does not make much difference to the hfcc. Comparison with several RHF calculations on the same species shows that similar results are obtained and suggests spin contamination to be unimportant in determining the geometry of the δ-radicals.
Hinchcliffe has earlier reported hfcc calculations on benzyl, and in a more recent paper has used a much more extensive basis of approximately double-zeta quality. Comparison calculations were also made on the isoconjugate anilino and phenoxy radicals, assuming a regular hexagonal geometry and typical bond lengths. However, the value of (S2) was still seriously in error after annihilation, and the proton hfcc were not very much better than the earlier calculation, but were in reasonable agreement with experiment. Geometry optimization is also probably essential for these radicals (see later).
Hinchcliffe and co-workers have also studied some larger radicals using the UHF method. A particularly interesting study is that of the pyrazine anionlithium cation ion pair. Very little theoretical work on such species has been carried out (references to earlier work are given in ref. 30), and the ab initio study using a small OTO basis set was performed by varying the Li+ position. The lowest energy conformation was found to be with the lithium occupying a position in the plane of the pyrazine ring, along the C2 axis passing through the nitrogen atoms.
Hfcc were calculated both for the pyrazine anion and also for the complex. For the anion, the 13C and 14N hfcc were in good agreement with experiment, but the proton hfcc were rather far off. For the complex, quite poor agreement was obtained. However, the spin contamination was rather large, and, in view of the size of the molecule and the limited basis set, better agreement could hardly have been expected.
Hinchcliffe and Cobb have also carried out UHF calculations on the pentadienyl radical, but in view of the above results the poor agreement with experiment was not surprising.
The radical NF2 has a 2B1 ground state and has been the subject of a careful study by the UHF method by Brown et al. Four different basis sets were used, a minimal STO basis, two minimal GTO bases, and a GTO basis set of roughly double-zeta quality. An assumed geometry of R(NF)= 1.37 A, θ = 103° was used. This is, however, slightly different to an optimized geometry calculated by Thomson and Brotchie of R(NF) = 1.44 A, θ= 101.6°.34 Of particular interest in this study was the evaluation of both isotropic and anisotropic hfcc. Annihilation reduces the spin density at each nucleus by about one-third, as was also found in AlmlOf's work.18 The STO basis set gave the best agreement with experiment, but the anisotropic hfcc (which were not affected so much by annihilation as the isotropic hfcc) were in better agreement with experiment at both N and F atoms, particularly with the largest GTO basis. The isotropic coupling constants were not in good agreement, especially a(14N). Analysis of the spin-density changes in terms of spin-polarization and spin-delocalization effects was made.
More recently, yet another UHF study of NF2 has appeared. The angle was varied, assuming R(NF) = 1.37 A (as in N2F4), and a minimum energy obtained for a bond angle of ca. 100°. However, although a medium-sized basis set was used, both with and without d-functions, Table 1 shows that the energy obtained is substantially higher than Brown et al. obtained except for one of their basis sets, and is also higher than Thomson and Brotchie's results. Since the latter have shown that R(NF) is fairly sensitive to the basis set, it is clear that none of the above authors have calculated the hfcc at the optimum geometry. Higher quality calculations on this species should be very interesting and are under way. The results of Hinchcliffe and Cobb, however, seem to indicate a lack of sensitivity of the hfcc to [theta. d-Orbitals added to the basis set gave a better value of a(F) but it was still only about one-half the experimental value, and it seems that this is common to both RHF and UHF calculations on AB2 radicals.
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Excerpted from Electron Spin Resonance Volume 3 by R. O. C. Norman. Copyright © 1976 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
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