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Many-body perturbation theory in atomic structure calculations
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Many-Body Perturbation Theory in Atomic Structure Calculations Ann-Marie Miirtensson-Pendrill
Department of Physics, University of Goteborg and Chalmers University of Technology, S-412 96 Goteborg, Sweden
Received September 16,1992: accepted September 25,1992
Abstract The path to accurate relativistic atomic structure calculations is lined with several points of interest or danger. Much of the non-relativistic many- body perturbation theory formalism, including the “coupled-cluster approach”, can be taken over, but certain problems, such as the choice of electron-electron interaction, the need for projection operators and an apparent gauge dependence, must be dealt with. Results for the ground state energy of beryllium and also 21s3P-21S, transition energies in beryllium-like Fe and MO are presented as examples, where the accuracy is such that the calculation of QED effects in many-electron systems might be tested. In addition, the application to the study of additional perturbations in alkali-like system is discussed.
1. Atomic structure calculations
“In the spectrum of one element we are given a vast amount of data which is measurable with great precision”, Condon and Shortley write in their classical textbook [l] and con- tinue: “Evidently the spectrum is somehow determined by the structure of the atom so spectroscopy stood out clearly in the minds of physicists as an important means for study- ing that structure”.
Modern workstations make possible tabletop calculations of atomic spectra in more and more detail. The gross fea- tures have, of course, long been accessible within the Inde- pendent Particle Model (IPM), including Hartree-Fock (HF) or Dirac-Fock (DF) calculations. These methods are now generally available through published programs [2-51 and widely used. Although the IPM gives a qualitatively correct description, electron correlation is essential to account for finer details in the spectra. Hylleraas-type wave- functions [ti] can give results of astonishing accuracy for two-electron systems, as demonstrated e.g. in the talk by Gordon Drake [7] at this symposium. The method has also been applied to three-electron systems [SI, but is not easily applicable for many-electrons, nor, indeed, is the relativistic generalisation obvious. Another approach with roots in the early days of quantum mechanics [9] is the method of Con- figuration Interaction (CI), widely used in quantum chem- istry [lo], although mostly in the non-relativistic framework. The wavefunction is then usually expanded in terms of an analytical basis set. In the Multi-Configuration Hartree-Fock (MCHF) [2] or Multi-Configuration Dirac- Fock (MCDF) [3-51 approaches the basis set is usually numeric and the orbitals themselves, can be varied together with the coefficients in the wavefunction expansion, leading to rather compact descriptions of the wavefunction. The MCDF approach is discussed in Paul Indelicato’s talk [ 111. Here we will concentrate on the application of many-body perturbation theory (MBPT) to atoms. It is sometimes
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advantageous to abandon the order-by-order approach to MBPT and this leads naturally to the coupled-cluster approach, which has been used in many of our calculations. The first applications were within the non-relativistic frame- work. However, in particular the studies of parity non- conservation in atomic physics, discussed elsewhere [ 121, raises demands for an accurate relativistic treatment. Much of the non-relativistic formalism can be taken over directly, but some modifications are necessary, as discussed in Section 3.
2. Many-body perturbation theory and the coupled-cluster
A many-electron system can be described by the Hamilto- nian
approach
where hnu, is a sum of the kinetic energy term for electron i and a potential energy term due to the interaction with the nucleus. The interaction between the electrons is described by vj. The Schrodinger equation HY = EY can, of course, not be solved for a many-electron system without approx- imations. One approach is Many-Body Perturbation Theory, which was introduced in atomic physics by Kelly [13-151 and provides a convenient diagrammatic form for dealing with the many-electron problem. The Hamiltonian is divided into an approximate Hamiltonian H o and a remainder term or “perturbation” KO,, . Explicit expressions for the energy contribution up to fourth order in the pertur- bation KO,, have been given by Wilson [16].
The evaluation of the wavefunction correction requires summation over virtual orbitals. Kelly obtained the wave- function corrections through an explicit summation over both bound and continuum orbitals [13, 15, 171. Similar approaches have been used also in later work, e.g. by the Novosibirsk group [18]. It is also possible to use an analyti- cal basis set, as done in most quantum chemistry applica- tions [19] or a discretized numerical basis set, without any continuum orbitals [20-223. An alternative to explicit sum- mation is to solve inhomogeneous two-particle differential equations - pair equations - for these admixtures. The sum- mation over excited states is then obtained by invoking the closure relation, giving a sum over all eigenstates. This pro- cedure was first applied in 1968 in studies of two-electron systems [23] and later applied to larger systems [24, 251 and has been used extensively in earlier non-relativistic work in our group.
Many-Body Perturbation Theory in Atomic Structure Calculations 103
A disadvantage of perturbation theory is sometimes its slow convergence - and it is often essential to find ways of summing infinite orders of certain terms. Examples of such summations are the VN-’ potential to describe the virtual orbitals [ 151, and shifted energy denominators including the energy corrections in addition to the unperturbed energies [13], as included already in Kelly’s pioneering work. The correction of a H F potential to a V N - l potential removes the unphysical interaction of an excited orbital with the pre- viously occupied orbital and gives a summation over higher-order terms involving single excitations without mixing different excitations. These terms will be included if single excitations are summed to all orders either within the framework of MBPT [26] or methods such as the RPA or TDHFC27-J.
The shifted energy denominators give a summation of certain two-particle diagrams to all orders [14]. Later, schemes for more complete summations have been devel- oped, e.g. by means of iterative solution of the pair equation [28]. In this way, essentially exact solutions can be obtained for a two-electron system [20-22, 281 and the MBPT gives prescriptions for extending the treatment to many electrons, including coupling between different electron pairs. The iter- ative solution of the pair equation is closely related to the “Coupled-Electron-Pair approximation” (CEPA) [29]. This scheme was put into the formal framework of the “coupled- cluster approach” for open shell systems by Lindgren [30], thereby generalizing a method with roots in both nuclear physics [31] and quantum chemistry [32]. This approach has now been applied to several atomic systems. Most applications so far have been restricted to single and double excitation clusters (CCSD). Another technique, leading to summation of a different subset of higher-order diagrams, is to use Feynman propagators, as suggested by Dzuba et al. [33]. The relation between this approach and MBPT is analysed in Ref. [34].
In cases where several energy levels of the same symmetry lie close, the convergence of the perturbation expansion is often slow. It would then be an advantage to deal with this mixing through a diagonalization, while still including the mixing with other configurations as perturbations. The “Extended Model Space” formulation of MBPT was devel- oped by Lindgren in 1974 [35] to deal with this problem. This approach turns out to be very useful for relativistic calculation as seen in Section 4.2.
Recent years have seen significant developments in the implementations of non-relativistic methods. An example is the very large scale MCHF calculations by Olsen and Sun- dholm [36, 371 which might be characterized as “Mega- Configuration HF”. This extends the applicability of MCHF and gives results of impressive accuracy. Another example is the inclusion of triple and even quadruple excita- tions in the quantum chemistry Coupled-Cluster programs [38], (although the accuracy in the results obtained so far has often been limited by the basis sets used). Similar devel- opments are likely to be necessary also in the atomic methods described below.
3. Relativistic developments
Relativistic effects must be taken into account also for small systems when accurate results are desired. For light atoms,
it may be sufficient to evaluate relativistic corrections using good non-relativistic wavefunctions, whereas for heavy atoms and ions it is essential to include relativity from the start, using relativistic wavefunctions. The dominating con- tribution is accounted for by replacing the kinetic energy term p2/2m in the one-electron Hamiltonian h,,, in (1) by its relativistic counterpart, i.e. (B - l)mc2 + ca p. This gives the “Dirac-Coulomb” equation. Programs to solve the rela- tivistic one-particle equations including the Dirac-Fock approach [3-41 were developed during the 1960’s, at about the same time as their non-relativistic counterparts [2]. Toward the end of the 1970’s, also the “ R P A equation had been generalized to the relativistic case [39] and a similar approach, “PNCHF” was suggested by Sandars [40] for the study of parity non conserving (PNC) weak interactions and implemented soon after [41]. However, the accurate experi- mental results for heavy atoms demand the calculations also of relativistic correlation effects.
3.1. Projection operators Around 1980 a black bird came hovering over relativistic atomic structure calculations. Sucher [42] reminded us that it is important to take care not to mix in negative energy states unintentionally: The Dirac one-electron equation has both positive and negative-energy solutions. Thus, for each eigenvalue E, there exists a continuous infinity of com- binations of one-electron states, where one electron is in the negative energy continuum and the other high in the posi- tive one, which have a total energy E . Each two-electron state could thus “dissolve into the continuum” if such exci- tations were allowed. The Dirac-Coulomb equation, HY = EY, has no normalizable solutions as noted already in 1951 by Brown and Ravenhall [43]. Of course these admixtures are not allowed! An excitation into a negative energy state is only allowed when preceded by an excitation out of the same state. This corresponds to the annihilation of a positron, following the creation of a virtual electron- positron pair, which is associated with an energy denomina- tor of the order 2mc2. To avoid these forbidden excitations, we may surround the many-electron Hamiltonian by projec- tion operators, A’, for positive energy states. Our goal, within the “no virtual-pair” approximation, is then to obtain solutions to a new eigenvalue equation
A + H A + Y = E A + Y (2) Alternatively a second-quantized formulation can be used to ensure a proper treatment of all states.
The black bird also flew back in time to cast doubt over the justification of the Dirac-Fock approach, which is often derived through an energy minimization without any explicit projection operators. The Dirac-Fock equation, however, did not seem to be aware of any problems, since it was quite successful in describing heavy atoms. There now seems to be consensus that there is no fundamental problem with one-particle equations [44, 451 (even if an incomplete analytical basis set may lead to problems already at the one- particle level [46]). The Dirac-Fock equations need not be derived through energy minimization, but can also be derived from the condition that single excitations vanish in the lowest order wavefunction corrections. This holds for the excitation of a virtual electron-positron pair, as well. Mittlemann [47] showed that the D F equations could also
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104 A.-M. Mirtensson-Pendrill
be derived by requiring that the energy be stationary with respect to the choice of projection operators. Grant [48] pointed out that the projection operators are implicit through boundary conditions in the generation of a basis set. This, however, will not necessarily save the many- electron problem where we need to live with projection operators ! The situation concerning the multi-configura- tional Dirac-Fock approach is also more complex, as dis- cussed in the next talk [ 111.
The choice of one-electron Hamiltonian for the definition of the projection operators becomes important in the con- struction of perturbed orbitals : In general, the perturbation can mix in negative energy states - i.e. negative energy states of the unperturbed Hamiltonian - into the orbitals [44]. These admixtures are, however, still part of the positive energy states of the perturbed Hamiltonian. Particularly for strongly relativistic operators, such as magnetic interactions and parity-non-conserving weak interactions the effects of these admixtures cannot be expected to be negligible. We also expect relatively large contributions from the Breit interaction in these cases [49, SO]. The effects included by allowing these “negative-energy” admixtures must otherwise be included by evaluating additional diagrams in pertur- bation theory, so the perturbed orbital approach is quite a powerful method.
Toward the end of the 1980s we had learned to tame the projection operators. Explicit approximate expressions for the projection operators had been attempted [42, 511. However, it seems to be more convenient to obtain a more or less “complete basis sets” in terms of known analytical functions [19] or in a numerical basis [20, 22, IS]. The dis- tinction between positive and negative energy states then becomes trivial. Such basis sets have been used to solve the relativistic no-virtual-pair pair equation, first in second order [19, 201, then the all-order equation for two-electron systems [22, 521, and finally also for larger systems [49, 53- 551.
3.2. Choice of electron-electron interaction Another problem in the relativistic generalization is the choice of the electron-electron interaction. Already in 1971, D F calculations were performed including a self-consistent treatment of the Breit interaction together with the Coulomb interaction [56]. In the years that followed, the Breit interaction was usually left out, mostly for the reason that it is very computationally demanding, but partly because of the warnings about higher-order Breit terms by Bethe and Salpeter [57] who recommend that it be evalu- ated as an expectation value between eigenfunctions of the Dirac-Coulomb Hamiltonian (sic !). However, the Breit interaction may affect the orbitals and thus also have effect on other properties studied [SO]. It is therefore advanta- geous to include it together with the Coulomb interaction already in the potential used to generate the orbitals. This leads to the Dirac-Fock Breit (DFB) approach, which is now used by several groups [58]. The justification for this approach is discussed in [SO].
Nevertheless, the Breit interaction has more surprises in store: In 1987, Gorceix and Indelicato [59] discovered that the energy contributions from the transverse photons depended on the choice of gauge. The difference between the Feynman and Coulomb gauge expressions was found to be
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as large as O(a2) [60]. The cause could be traced to projec- tion operators : The difference between the two expressions can be written as a commutator with the Dirac-Coulomb Hamiltonian [60]. Normally expectation values of com- mutators with the Hamiltonian vanish. However, in the no- virtual pair approximation, the equation solved is no longer HY = EY but instead A’HA’Y = EA’Y’. From this battle the Coulomb gauge emerges as a champion: Already in the frequency-independent form, which is Breit’s original expression, it gives results correct to O(a2)Ry, whereas this does not hold for the Feynman gauge even if the frequency dependence, which makes the calculations more compli- cated, is included [61, 621. This apparent gauge discrepancy must, of course, be resolved in a proper treatment and it turns out that crossed photons give contribution of O(a2)Ry in Feynman gauge [62] but can be neglected to a good approximation if Coulomb gauge is used.
Although much of the non-relativistic development can be taken over, as we have seen in this section, some new fea- tures are demanded of the relativistic MBPT. The under- lying quantum field theory requires us to deal with projection operators and to choose a gauge for the electron- electron interaction. In addition, at some level it will become necessary to deal with corrections to the “no-virtual pair equation’’ and include the effect of virtual electron- positron pairs and of the frequency dependence in the Breit interaction, and of self-energy corrections, as discussed more in other talks at this workshop.
4. Applications
“It was argued, particularly by Max Born, that the simplest crucial test of the correctness of wave mechanics in general was to be found in its application to the helium atom - in particular to the ground state” Hylleraas tells us in his “Reminiscences from early quantum mechanics of two- electron atoms” [63]. Even today, helium has the character of a visitor centre, where scientists using widely different methods can get together and compare notes. However, like on our symposium boat trip to Bullero (“Boulder island”), the theorists are too large a group to stay together, and only some can have a very close look at the cusps and corners of the building itself [7]. For others, calculations for helium- like systems more often have the character of demonstrating the ability to treat pair correlation and the MBPT then pro- vides the formalism required to treat many electrons. The quantum chemists may join the atomic MBPT practitioners in studies of Be [lo] but are then likely to choose to go for a swim in the sea and study water molecules, maybe stop- ping first at small diatomic systems such as LiH, LiF, HF. The pioneering atomic MBPT work followed the periodic table nature trail to increasing nuclear charge 2 [13-15, 171 without fearing the complex vegetation of angular momenta. So far, the quagmire of transition elements has been untouched by MBPT and left for other methods to explore. Most applications of relativistic MBPT have chosen a climbing route to higher 2, mostly staying on the solid rocks of the alkali atoms, with an accurate calculation of parity non-conserving weak interaction effects in Cs as a tempting goal [12]. It is, of course, impossible to cover all applications of MBPT in this paper. Results for the ground state energy of beryllium ground state and also 2’p3P-2’So
Many-Body Perturbation Theory in Atomic Structure Calculations 105
transition energies in beryllium-like Fe and MO are pre- sented as examples, where the accuracy is such that the cal- culation of QED effects in many-electron systems might be tested. In addition, the application to the study of additional perturbations in alkali-like system are discussed.
4.1. The ground state of beryllium For systems like beryllium with only a few electrons, essen- tially complete non-relativistic calculations are within reach and these benchmark calculations have become important sources of information concerning the accuracy of the various algorithms, approximations and basis sets used in the calculations. However, before a comparison with experi- ment can be attempted, a number of other effects must be included, e.g. the motion of the nucleus with its finite mass and relativistic and radiative contributions. Even if these are usually considered to be small, they enter at the level of theoretical and experimental accuracy.
The most critical comparison with experiment involves the ionization energies of the outer electrons only - the energy of the helium-like system Be2+ is known better from theory [7, 641 than from experiment [65, 661. Thus, theo- reticians must learn to subtract, as emphasized by Deslattes [67]. Moreover, we must learn how to add, and also know when to add, when to subtract and when to do neither [37].
Figure 1 shows the various corrections to non-relativistic results, divided into corrections for energy of the ground state of Be2+ and to the energy difference Be-Be”. More details can be found in Refs [37, 54). Most important is the “DF-HF” contribution which arises from the use of a rela- tivistic rather than non-relativistic one-electron Hamilto- nian. “DFB-DF” gives the additional contributions that arise when the effect of transverse photons (the Breit interaction) is included in the D F calculation. The correc- tions to the correlation effects were obtained by performing analogous relativistic and non-relativistic calculations at the CC-SD level and the relativistic correlation calculations were performed with and without a self-consistent treatment of the Breit interaction [54]. The Lamb shift contributions for Be-Be2 + were estimated by multiplying accurate hydrogenic results with the ratio between the Dirac-Fock and the hydrogenic electron densities at the nucleus. The mass polarization arises from a correlation of the electronic momenta due to the nuclear motion. Its contributions for
t -3 ! , I I I I I ! -0.2
DF-HP DFB-DF COml. BreitCot~ Lambshift MassPol.
Fig. 1. Corrections to non-relativistic results divided into contributions for helium-like beryllium (open squares, axis on the left) and to the energy difference between Be and helium-like beryllium (filled diamonds, axis on the right). The values are given in millihartree = a.u.
the first two ionization energies were taken from experiment [68] and previous calculations [69], respectively. Subtrac- ting these corrections leads to an “experimental non- relativistic energy” of - 14.667 353(2) (the values in this section are given in Hartree atomic units).
Already the Hartree-Fock energy - 14.573 023 0 repro- duces more than 99% of the experimental result. Using Lowdin’s definition [70] of the correlation energy as the dif- ference between the H F value and the experimental result, we find a non-relativistic correlation energy of -0.094 33q2). Figure 2 shows a comparison between several attempts to calculate the correlation energy for beryllium. More detailed description of most of the calcu- lations can be found in Ref. [37].
The second order energy contribution (MBPT-2) is -0.076358, changed to -0.085225 by third order terms (MBPT-3) [71]. Kelly in his early work [13] obtained the result - 0.091 in a second-order correlation energy calcu- lation including also certain higher order terms through modification of the energy denominators and estimated enhancement factors arising from repeated summations of certain interactions. In a subsequent work, he also intro- duced the “VN-”’ potential for excited states, in order to improve convergence. This led to a somewhat lower energy, -0.092 Cl5-J. In a now classic CI calculation, which for a long time has served as the benchmark, Bunge [72] got a correlation energy of -0.094 305. Restricting the CI calcu- lations to single and double excitations (CI-SD) gave a much smaller correlation energy, -0.090 218 [72]. The coupled-cluster approach including doubles excitations (CCSD) or singles and doubles (CC-SD) gives significantly better results, as seen from Figure 2 and a large part of the remaining discrepancy is removed by adding the triples con- tributions estimated by Adamowicz [37] (CC-SD + T). An even better result is obtained in the case of Be by using the “Independent Electron Pair Approximation” (IEPA). This model is not always quite so successful, but works particu- larly well for Be where the 1s’ and 2s’ pairs are, indeed, quite well separated.
The triple excitations tend to be quite important for Be, but a large part of them can be ascribed to double excita- tions out of the ls22p2 configuration which is strongly
-100 -80 -60 -40 -20 0 - F 1 .o
Fig. 2. Comparison between different evaluations of the correlation energy for the beryllium ground state. The calculations are described in the text
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106 A.-M. Miirtensson-Pendrill
admixed in the ground state of Be [73]. It is therefore tempting to include this mixing through diagonalization, using the “extended model space” formalism developed by Lindgren [35]. The first application was done by Salomon- son et al. [74], giving a correlation energy of -0.0881. More recently, Liu and Kelly [75] performed a relativistic coupled cluster calculation for Be, starting from an extended model space, but treating the excitations out of the ls22p2 only to second order, leading to a correlation energy of -0.0951, denoted “CC-ext” in Fig. 2. This calculation was relativistic, but, as seen from Fig. 1, the relativistic contribu- tion to the correlation energy is very small [54], and this number can be compared directly to the non-relativistic correlation energies.
Recently several very large non-relativistic calculations for Be have been performed, made possible by the develop- ment of more powerful computers, and giving very accurate results. Very large MCHF calculations have been performed ,y Froese-Fischer (“MCHF”) [76] and by Olsen and Sun- lholm (“MegaCHF”) [37, 381. Davidson et al. [77], have
applied the CI approach including single and double excita- tions out of a multi-reference initial function (MR-CI) con- sisting of all configurations (ls2~2p3s3p3d)~. Very recently Chung et al. [78] presented calculations using “full core plus correlation” (FCPC) giving a correlation energy of -0.094 325 4. At this level of accuracy, the combination of theory and experiments probe the screening of the Lamb shift: Our estimate for the Lamb shift contribution to the energy difference Be-Be2 + was 1q1) pH.
4.2. Excited states in 4-electron systems To reveal its spectrum, an atom cannot remain in its ground state. In a recent work Lindroth and Hvarfner studied 2 l* 3 P , + 2lS, transitions in Be-like systems [79]. The energy was evaluated relative to the common unperturbed Is2 core. The energy for the 2s’ ground state is then given by the sum of the one-electron energies and an effective two- body interaction involving the 2s electrons, evaluated much in the same way as for He-like systems. The one electron energies and the effective two-body interactions are, of course, influenced by correlation which was accounted for in the CC-SD framework including a self-consistent treatment of the Breit interaction also in the solution of the pair equa- tion.
The 2s2p states are more complex: Whereas the J = 0 and J = 2 states arise from the configurations 2s2p1,, and 2s2p3,, , respectively, the J = 1 states are a mixture between these two configurations, which are degenerate in the non- relativistic limit. To avoid treating this mixing as a pertur- bation, they can both be included in an “extended model space” [3,5]. All excitations out of this model space are then accounted for by perturbation theory, whereas the excita- tions to any of the model configurations are excluded from the summation over intermediate states in the perturbation expansion. A diagonalization of the effective interaction then gives the mixing between the two 2p states, essentially determined by the relation between the Slater integral and the spin-orbit splitting of the 2 p orbital. As expected, the mixing is less important for Mo3*+ where it could possibly have been treated by perturbation theory since the fine structure in the 2p one-electron energies starts to over- shadow the two-body interactions.
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Table I. Comparison between theoretical [79] and experi- mental [SO] results for the 2l. 3P1 + 2lS, transition energies in Be-like Fe and MO. The theoretical error bars are due to uncertainties in the contribution of higher 1-values and in the screening of the Lamb shift, respectively. The values are given in cm-’
Table I shows a very good agreement between theoretical [79] and experimental results [80] for the transitions studied so far. The theoretical uncertainty is dominated by the approximations involved in the calculations .of the screening of the Lamb shift [8l] and these systems could thus be used to test improved methods of calculation.
4.3. Additional perturbations A spectrum results not only from unperturbed systems, and we are often interested in finer details, resulting from small perturbations which carry information about, e.g. nuclear properties. Hyperfine interactions, in particular in alkali-like systems, are an old playground for MBPT with many accu- rate experimental results available for comparison [82], giving an indication of the accuracy to be expected. Other perturbations we may wish to study are e.g. the interaction with the electric or magnetic field of a photon, the effect of the nuclear size and/or the nuclear motion, and, not least, parity non-conserving weak interactions, which are dis- cussed in more detail in several talks during this meeting.
Evaluating the effect of an additional perturbation on the valence orbital alone often gives a qualitatively correct description. A Dirac-Fock valence orbital, e.g., gives about 60-70% of the ground state hyperfine structure in the alkali atoms. For the isoelectronic singly ionized alkaline earth elements, the agreement is, of course better, about 70-80%. Since the additional perturbation affects each electron, this leads to a modification of the potential felt by the other electrons. Keeping terms to lowest order in the external per- turbation in the construction of the Dirac-Fock (or Hartree-Fock) potential leads to a set of coupled equations which can be solved by iteration [26, 831 in procedures closely related to the RPA. The results obtained in this way are also closely related to the “Unrestricted” H F or D F approaches [83]. For the ground states in the alkalis, these RPA-type calculations [83, 841 reproduce about 75-85% of the experimental values, improved to about 90% for the alk- aline earth ions [85]. The remaining discrepancy must be accounted for by correlation effects. The dominating corre- lation effect is often the modification of a valence orbital to an approximate Brueckner orbital [86]. Partly, this can be accounted for by using “core polarization” potentials, which may explain the frequent success of semi-empirical approaches. Combining a lowest order evaluation of the Brueckner orbital corrections with the “RPA” calculation gives agreement 3% for the Cs hfs [87]. This value is not
Many-Body Perturbation Theory in Atomic Structure Calculations 107
necessarily improved by a direct evaluation between coupled-cluster wavefunctions [55, 88, 891. Dzuba et al. [33] have summed terms by defining a screened Coulomb interaction, using Feynman propagator techniques or by including a screening factor. In this way they claim an accu- racy of about 0.1% for the Cs ground state. In very com- plete calculation for Cs Blundell et al. [90] have added Hermitian conjugate terms [9l] in the coupled cluster approach, and also included the potential corrections from the perturbed orbitals. In this way they were able to obtain agreement within 0.3% (which does not seem to be improved by rescaling the Coulomb matrix elements). For higher angular momentum states and more complex systems, an even more detailed treatment of pair correlation effects may become essential, such as, e.g. in the ingenious T1 calculation by Liu and Kelly [92]. Work has also been initi- ated to include three-particle excitations in the non- relativistic coupled-cluster approach, and initial computations with a subset of the triple excitation diagrams for Na show encouraging results [93], although in general it is likely to be essential to include all triple excitation when very accurate results are required.
Isotope shifts measurements have been performed for several chains of radioactive isotopes, and contain informa- tion about the changing nuclear charge distribution. Atomic structure calculations can assist in the interpretation of these data. In some recent calculations we have performed analogous calculations of hyperfine structures and isotope shift constants. The detailed comparisons provides insight into the semi-empirical procedures used. The calculations performed so far give consistently smaller field shift con- stants than semi-empirical values traditionally used and thus leads to revised values for the extracted d(r2) [55, 941. Knowing d(r2) for the Cs isotopes may, in fact, be of some importance [95] for the interpretation of planned experi- ments to study parity non-conservation for several Cs iso- topes in a trap [96].
The bariumion Astrid [97] and her sisters have been studied in great detail in ion trap experiments [97, 981. Recently, our group has performed a relativistic coupled- cluster calculation for Ba’, isoelectronic with Cs. The wave- function has been used to calculate hyperfine structure, isotope shifts [55] and also gj factors for the Zeeman effect C491.
5. Challenges ahead
Atomic structure calculations help us understand the struc- ture behind the spectra we observe. The comparison between theory and experiment confirms the validity of the calculations and of the approximations involved. Indica- tions of the accuracy are particularly important when the atomic calculations are used to predict parameters for inter- pretation of experimental data. Examples are the measure- ments of isotope shifts, quadrupole hyperfine interactions and, not least, the effects of parity non-conserving weak interactions in atomic physics. Over the last decade, the accuracy of theoretical predictions, in particular for heavy atoms, has improved drastically, but is still insufficient in many cases.
Many-body perturbation theory and the coupled-cluster approach usually give good descriptions of correlation
effects involving one or more core electrons, whereas the correlation between valence electrons is often better described by multi-configuration or CI methods. Allowing the orbitals to include the additional perturbation in a multi-configurational approach would lead to inclusion of important effects without any increase in the number of con- figurations required (although the resulting equations in general will be quite complicated). A combination of the approaches, by starting from model functions with many configurations could take full advantage of the complemen- tarity.
For several systems the error bars of the experimental values and of the theoretical many-body results are now smaller than the uncertainty in the QED evaluations. There may soon be a need to treat QED and correlation effects in an integrated approach. As we have seen during this work- shop, several groups are working on QED calculations using the same basis sets as used also in applications of per- turbation theory. With the close relation between MBPT and QED, we might expect to see productive cross- fertilizations.
Acknowledgements Over many years, the applications and foundations of MBPT have been a source for lively discussions in the Goteborg group under Ingvar Lindgren. Of the many other people who have contributed to the work presented here, I mention only a few: Sten Salomonson’s stringent and careful approach assures as far as possible the correctness of programs and formal- isms. Anders Ynnerman has developed the programs for evaluating matrix element between coupled-cluster wavefunctions, and has also devoted a lot of time keeping programs and computers, as well as our abilities to handle them, in running order. Eva Lindroth, continuously in search for new applications, has extended the capabilities of our programs in several aspects. Financial support for this work from the Swedish Natural Science Research Council (NFR) is gratefully acknowledged.
Dedication
This paper is dedicated to the memory of Hugh P. Kelly, whose death coincided with the opening day of this Nobel symposium. The sad news of his death reached us towards the close of the symposium. Through the grief shine memor- ies of many happy moments with a warm, generous and enthusiastic person. My personal memories start with a little group going for long morning swims at a summer school in Carry-le-Rouet in 1975. Later, he was the external examiner at my thesis defence - probably too kind an “opponent”, but such were his ways. The atomic physics community will miss his catching enthusiasm, but the intellectual joy in his work - from his pioneering introduction of many-body per- turbation theory in atomic physics to its recently published virtuoso application to the EDM in T1 - will continue to live with us.
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