Chapter 73 Chapter 73
12. Concluding remarks
Surveying the history of the theory of optical lanthanide spectroscopy, we can discern several main features: the usefulness of Lie groups, following their in- troduction by Racah (1949); the relevance of the method of second quantization, as demonstrated by the use of annihilation and creation operators for electrons; and the inability of the H a r t r e e - F o c k method and its various elaborations to provide accurate values (say to within 1~o) of such crucial quantities as the Slater integrals
F k (4f, 4f) and the Sternheimer correction factors R,~ for a free ion. The success of the tormal mathematics is in striking contrast to the failure of the machinery of computation. This turn of events has happened over a period of time when
hesitations to develop the mathematics have been interspersed with assurances of the imminent arrival of a golden age of computation. The remark of C o n d o n and Shortley (1935) that they could manage without group theory (mentioned in section 2.2), which seems so mistaken today, was gracefully put in the context of its time by Condon and Odabasi (1980, p. 307). They drew a parallel with the classic work of Jeans (1920) on electricity and magnetism, which made no use of vector analysis at a time when it would diminish rather than improve the accessibility of the text.
Slater's reluctance to use Lie groups was matched by his feeling towards second quantization, which to him was 'a more complicated method of treating matters which could be more easily taken up without their use' (Slater 1968). It would have been interesting to see how he would have handled Downer's fourth-order analysis of two-photon absorption in Gd 3 +, described in section 9.1.
The formal mathematics has some way to go. The two-electron operators needed to describe residual crystal-field effects have not yet been firmly established, and a complete Lie-group classification of the four-electron electrostatic operators for the f shell has still to be done. New multi-electron operators will undoubtedly be required for other purposes as the analysis is pushed to higher orders of perturbation theory.
However, the shortcomings of the H a r t r e e - F o c k method seem the most in need of attention at the present time. They are also the ones that will probably get the least of it, since the problem of extending the H a r t r e e - F o c k method to the myriad configurations required to give good accuracy for atoms as heavy as the lanthanides is very hard indeed. The question of incentive has to be faced here. The need to know the quadrupole moments of the lanthanide nuclei is no longer a strong motivation for calculating the Sternheimer correction factors. As for the Slater integrals F k (4f,4f), spectroscopists have learned how to adjust the H a r t r e e - F o c k values by empirical correction factors, so that the absence of precise numerical values is not an overwhelming hindrance. If accurate ab initio calculations for the lanthanides are to be done at all, they will probably come about as applications of techniques that have been worked out primarily for other purposes. The effects of parity non-conservation in heavy atoms can only be properly handled if accurate atomic wavefunctions are known, and this has stimulated interest in improving the methods for taking relativity and correlation into account. The papers presented at the Atomic Theory Workshop at the National Bureau of Standards in 1985 give a good indication of where current interest lies. Much of the discussion turned on heavy atoms that possess very few electrons; but the contribution of Dietz (1985) on the g-Hartree method broke new ground for many-electron atoms. Considerations of a field-theoretic nature lead to equations involving a parameter g which can be fixed either by appealing to some appropriate experimental result (thus weakening its ab initio character) or by demanding that the correlation energy (that is, the correction to the standard H a r t r e e - F o c k energy) to a specified order of perturbation theory be set equal to zero. Dietz (1985) concludes that 'the o-Hartree method has shown, we think, important advantages, sometimes even its superiority, over many of the current approaches for treating the inhomogeneous many-electron problem'. Whether this assessment is relevant to the lanthanides is unclear. At least it avoids the predictions of lightning-like speed and unparalleled accuracy that
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 187 t h o s e a t t e n d i n g scientific c o n f e r e n c e s h a v e p a t i e n t l y l i s t e n e d to for t h i r t y y e a r s o r m o r e . N o d o u b t we shall e v e n t u a l l y be a b l e to c a l c u l a t e m u c h of w h a t we w a n t with a h i g h d e g r e e of a c c u r a c y . T h a t d a y h a s n o t yet a r r i v e d .
Note added in proof
By d r a w i n g a c o r r e s p o n d e n c e b e t w e e n n u c l e a r states classified b y i s o s p i n a n d a t o m i c o p e r a t o r s classified by q u a s i - s p i n , L e a v i t t (1987) h a s p r o v i d e d a c o m p l e t e g r o u p - t h e o r e t i c a l l i s t i n g for t h e f shell of o r t h o g o n a l o p e r a t o r s s c a l a r with r e s p e c t to S a n d L.
Acknowledgements
P a r t s of t h e m a n u s c r i p t were r e a d b y Drs. I.E. H a n s e n , W.C. M a r t i n , D.J.
N e w m a n , M . F . R e i d a n d J. S u g a r , w h o a r e t h a n k e d for t h e i r c o m m e n t s .
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