Read an Excerpt
Electron Spin Resonance Volume 2
A Review of the Literature Published between June 1972 and November 1973
By R. O. C. Norman The Royal Society of Chemistry
Copyright © 1974 The Chemical Society
All rights reserved.
ISBN: 978-1-84755-335-5
CHAPTER 1
Theoretical Aspects of Hyperfine Splittings and g-Factors
BY C. THOMSON
1 Introduction
Since the first volume of this series there have been several advances in the theoretical aspects of e.s.r. spectroscopy, as well as very many uses of theoretical work in helping to interpret experimental results. We have confined this Report primarily to recent advances in the calculation of hyperfine coupling constants (hfcc) of free radicals, using both non-empirical (ab initio) and semi-empirical methods. We do, however, include a limited number of references to papers dealing with experimental results in which theoretical calculations have also been reported. We have not considered the theoretical aspects of gas-phase e.s.r., which have been reviewed recently by Brown,1 nor the theory of transition-metal e.s.r., which is dealt with elsewhere in this Report. Hyperfine coupling constants are quoted in gauss (G) (1 G = 10-4 T).
Analysis of the e.s.r. spectra of free radicals in solution gives the g-value for the radical, and the hyperfine coupling constants. We shall mainly use the notation ai(N) where this hfcc refers to the coupling between the unpaired electron and nucleus N, the subscript i referring (if necessary) to the position of the nucleus in the radical. Since the isotropic hfcc are related to the value of the electronic wavefunction at the nucleus in question, the calculation of wavefunctions for open-shell systems and the results of e.s.r. experiments are closely related. Anisotropic hfcc can be obtained from single-crystal studies, or by suitable analysis of e.s.r. spectra from polycrystalline samples, and these values are also of interest in evaluating the various methods for calculating the electronic wavefunctions of radicals.
There have been some new books and reviews of the theoretical aspects of e.s.r. since Volume 1. A useful introductory text on e.s.r. by McLauchlan has been published, and a more detailed exposition by Poole and Farach. An excellent introduction to the advanced molecular quantum mechanics necessary to understand the detailed derivation of hyperfine and other operators which occur in magnetic resonance is provided by Moss, and the MTP volume on magnetic resonance contains several articles of interest to those concerned with theory. Applications of tensor operators in the analysis of e.s.r. spectra have been reviewed by Buckmaster, Chatterjee, and Shing. An excellent monograph dealing with ab initio calculations contains several sections devoted to the problems involved in, and the results obtained from, hfcc calculations in atoms and molecules. 6 Despite early pessimism, it is apparent from the recent work reported here that reliable calculation of hyperfine coupling constants by ab initio methods is now feasible, providing large basis sets are used, but such calculations are expensive and are likely to remain so. Therefore the success of the semi-empirical INDO method for calculating hfcc, which has been amply demonstrated as shown in Section 4, means that we now have available a semi-empirical method for calculating hfcc which is more reliable than some of the previously used methods.
g-Values can be measured very accurately in careful experiments, and, although deviations from the free-spin value of ge= 2.00232 are small, such deviations are of some theoretical interest and recent work is reviewed in the next section.
2 The Theory of g-Factors
Interest in the measurement and interpretation of g-tensors has continued. Lin has extended earlier work on the question of obtaining g-tensors from e.s.r. experiments where the spin Hamiltonian is of the form of equation (1):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
Previous work, using second-order perturbation theory, resulted in expressions for the g-tensor, firstly for axially symmetric and secondly for non-axially symmetric g. In these cases the tensors were described in their principal axis co-ordinates, and were therefore diagonal. However, the tensors may not always have the same set of principal axes. Lin has therefore derived expressions, correct to second order, for the general case of an arbitrary reference frame and with non-diagonal g, D, and A tensors, which should be useful in the determination of the tensor elements gij Dij, and Aij from experimental data. Golding and Tennant 14 have independently considered the same problem and illustrated the effects of this by considering the case of the HCO radical e.s.r. spectrum, in both single-crystal and polycrystalline samples.
An important paper by Moores and McWeeny bas dealt with the theory of spin-orbit splitting and the g-tensor in general terms, starting from the Breit-Pauli Hamiltonian. The Hamiltonian derived has explicit terms for all the internal fields and includes specific two-electron terms approximated in earlier treatments. The components of the g-tensor are expressed in terms of density functions, and in terms of a principal axis system in which the tensor is diagonal may be written:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where g e is the free-electron value (ge= 2.002 32) and dλλ and bλλ are diagonal and off-diagonal corrections; dλλ is expressed in terms of matrix elements of the spin-orbit coupling operators within the ground-state manifold and bλλ comes from off-diagonal elements connecting the ground state with excited states of the same multiplicity.
The authors have developed methods for the calculation of the matrix elements of the full spin-orbit operator by a Gaussian expansion method, and have applied the theory in an ab initio calculation of the spin-orbit splitting in NO and CH, and also in calculations of the g-values for CN and NO2. Using the full operator it was apparent that various two-electron terms were of considerable importance and were necessary to obtain the good agreement with experiment which was achieved. In particular, the observed negative value of Δgzz was predicted by the calculations, in contrast to an earlier semi-empirical calculation. However, since the many two-electron integrals are not easy to evaluate, the authors also investigated the use of the simpler (and commonly used) operator
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where [xi]N is an empirical spin-orbit coupling constant. Using values of [xi]N and [xi]o to fit the non-empirically determined g-value calculations on NO2, the same values were then used to predict the spin-orbit splitting in NO. This was successful and it seems that the use of equation (3) with relatively simple wavefunctions is a reasonable alternative to the more expensive calculations using the complete operator which will be difficult to apply to large molecules. The molecular wavefunctions used in this work were determined using open-shell SCF methods, and excited-state configurations obtained by configuration interaction using a set of single excitations.
A related ab initio study of g-values was carried out by Hayden and McCain using, however, the simplified spin-orbit Hamiltonian. Using a minimal basis set of Slater-type orbitals, and a number of excited states for each species, g-values were calculated for O3[??], NO22-, and NF2. The calculated components were in fair agreement with experiment, but it was pointed out that the experimental values are influenced by lattice effects in unknown ways. The largest error in the calculations was associated with the calculation of the excitation energies.
There have been further calculations of g-values using semi-empirical methods. An alternative to the INDO method (described below in Section 4) has been used 21 and g-values have been calculated for 15 small radicals, using the accepted values of the spin-orbit coupling constants. The agreement with experiment was good in most cases.
Turning now to experimental studies, Fassaert and de Boer have shown that the change in sign of the difference between the in-plane g-components, g1 — g2, in going from the negative to the positive ion, which is predicted by Stone's theory, is found for pyracene ions.
Studies of alkali-metal-radical ion pairs 23 have shown that quite large variations in g occur for the heavier alkali-metal cations Rb and Cs. The variations could have either positive or negative sign. A simple perturbation treatment of the problem, in which the ground-state wavefunction of the alkali-metal-radical ion pair is mixed by spin-orbit interaction with metal p-orbitals, gives negative g shifts if the singly occupied π-MO is mixed with metal π-orbitals, and positive shifts if the mixing is with doubly occupied 7T-orbitals. The mixing depends on the location of the counter ion and it is suggested that the g-factor measurements be used to give insight into the geometry of ion pairs.
3 Ab Initio Calculations of Hyperfine Splitting Constants
A general description of the relation between hyperfine coupling constants and the wavefunction corresponding to the particular state of an atom or molecule was given in Volume 1 of this series. The specific expression for the isotropic hfcc (the Fermi contact term) with Hamiltonian Hc is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [betae] and [betaN] are the Bohr and nuclear magnetons, ge and gN are the electron and nuclear g-factors, and rkN is the vector connecting electron k and nucleus N, which have spin operators Sk and IN, respectively; δ(rkN) is the Dirac delta function.
The expectation value of Hc leads to equation (5) for the isotropic hfcc:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
Ψ is the electronic ground-state wavefunction of the radical and γ(rkN) is the spin density at the nucleus. With the recent advances in computing power, and the increasing availability of sophisticated programmes to calculate ab initio wavefunctions, it seems likely that the calculation of hyperfine coupling constants by non-empirical methods will continue to attract the attention of the theoretician, in view of the high accuracy obtainable in experimental measurements. Following the work described last year, there have been several further attempts to calculate accurate ab initio hfcc.
The many different theoretical methods used 6 to construct the wave-functions γ are almost always tested on atoms, and the present year has seen extensions of earlier work by Lunnell using the spin-optimized self-consistent-field (SO-SCP) and unrestricted Hartree-Fock (UHF) methods. The former method, in which the coefficients of the linearly independent spin functions as well as the orbitals themselves are optimized, was previously shown to give good values for the Fermi-contact term for the 22P state of the Li atom, and Lunnell has extended this approach to some other 2P states of Li. Considerable care was taken in the choice of the basis functions, and the wavefunction obtained was used to calculate the orbital (l) and spin dipolar (d) contributions to the hyperfine interaction, as well as the Fermi contact term (f) where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Y20(i) is the spherical harmonic Y20(θ, φ, N is the number of electrons, and the total spin is S. The expectation value is taken over the total wave-function with ML= L, Ms= S, and MJ = J= L+ S. Agreement with experimental values was quite good.
The linearly independent spin functions for the doublet state of three electrons are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
and in the SO-SCP method the orbitals and the coefficients of the spin functions are both optimized. For the 2S ground state, such calculations give a value of the Fermi-contact term of 2.85 a.u., only 2% smaller than experiment (Table 1). Good agreement for the 22P state was also found in the earlier work, and in the present work the 32P and 42P states of Li were studied. The expectation values of the various operators were calculated from the natural spin orbitals (NSO). The basis sets used were made to satisfy the cusp condition, as this is necessary if reliable convergence in f is to be obtained. An analysis of the difference between the spin-polarized Hartree-Fock (SPHF) and the (SO-SCF) methods shows that basis set dependence of f should be much less in the latter method than in the U HF or SPHF methods. The values obtained were in good agreement with experiment, although it was pointed out that the relationships between the hyperfine parameters derived from the ordinary one-electron theory, which are conventionally used in analysing the experiments, contain errors of the order of 10%. Discussion of methods of overcoming this problem was given.
A different type of calculation on the 22P state of Li was carried out by Ahlenius and Larsson, who obtained the best wavefunction yet obtained from a Hyleraas expansion. The resulting value of f = -0.2162 a.u. compares with an experimental value of -0.2128 a.u. and the value of -0.2132 a.u. calculated by Lunnell. However, the energy in Ahlenius' calculation is lower by ~0.03 a.u. There is also excellent agreement for the calculated value of l in this work. These very accurate calculations thus demonstrate that it is possible, though difficult, to calculate atomic hyperfine splitting constants with quite high accuracy. However, extension of these methods to molecules seems at present to be rather unlikely.
There have been other different studies of correlation effects on atomic hfcc by two groups. Bagus and Bauche have used multiconfiguration Hartree-Fock (MCHF) wavefunctions in a study of the hfcc in the ground states of the 2p-series of atoms (B, C, O, and F). The principal aim of this work was the calculation of the orbit-dependent hfcc (l and d above) for which there are available accurate experimental values. The radial wave-functions were obtained by direct numerical integration of the MCHF integro-differential equations, thus avoiding the problem of optimizing non-linear parameters which arises if the solutions are obtained in analytic expansion approaches. Only those excitations were included which give rise to non-zero off-diagonal matrix elements of Hhf with the ground-state configuration. These excitations were of the type
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
The calculations described in this work are related to those of Schaefer 32 et al., who obtained 'polarization' and 'first-order' wavefunctions, but they do not include all the excitations used in the latter work, which also determined the wavefunctions by CI methods. However, the MCHF wavefunctions are more compact. Comparison with experiment shows that the MCHF values for the orbit-dependent terms are in good agreement with experiment, though Kelly's many-body perturbation theory (MBPT) results for O are in better agreement. The calculation of the contact term was investigated also, and it was shown that third-order contributions are of essential importance. This result was also found by Kelly in his work on O. These third-order contributions are non-core-polarization contributions. A related study using the MC-SCF method of the contact term in Li (2S and 22P states) has been carried out by Ishida and Nakatsuji. They considered seven configurations for the 2S ground state and three for the 22P state. The physical mechanism for providing contributions to hfcc was discussed in this paper. The agreement with experiment was excellent for the 22P state and almost as good for the 2S state. A comparison of the various recent calculations on the ground-state 2S and the 22P excited state of Li is given in Table 1.
(Continues...)
Excerpted from Electron Spin Resonance Volume 2 by R. O. C. Norman. Copyright © 1974 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
