Table of contents

A Nobel Symposium provides an excellent opportunity to bring together a group of prominent scientists for a stimulating meeting. The Nobel Symposia are very small meetings by invitation only and the number of key participants is usually in the range 20-40. These symposia are organized through a special Nobel Symposium Committee after proposals from individuals. They have been made possible through a major grant from the Tri-Centennial Fund of the Bank of Sweden.

Our first ideas to arrange a Nobel Symposium on many-body theory of atomic systems came up more than two years ago. It was quite obvious to us that a major break-through was happening in this field.

Very accurate schemes have been available for some time for studying the static properties of small closed-shell atomic systems. By "atomic" systems we understand here atoms as well as free molecules, which can be treated by the same formalism, although the technical approaches might be quite different. The conceptual and computational developments in recent years, however, have made it possible to apply the many-body formalism also to heavier systems. Although no rigorous relativistic many-body theory yet exists, there seems to be a general agreement about the way relativistic calculations should be performed on normal atoms and molecules. Schemes based on relativistic perturbation theory as well as on relativistic multi- configurational Hartree-Fock are now in operation and a rapid development is expected in this area.

Another field of atomic theory, where significant progress has been made recently, is in the application of many-body formalism to open-shell systems. General schemes, applicable to systems with one or several open shells, are now available, which will make it possible to apply many-body formalism to a much larger group of atomic systems and, in particular, to systems of more physical interest, A number of atomic properties - not only the correlation energy - can then be compared with the corresponding experimental results, which will eventually lead to a better understanding of the behaviour of many-electron systems and possibly also of many-fermion systems in general.

In addition to the static properties of atomic systems there is nowadays a great interest in the dynamics of the excitation process, which is of fundamental importance for our understanding of photoelectron and photoabsorption spectra. The experimental data being produced in this field are enormous and many intricate physical problems appear, which can only be understood by considering the atom as a fully interacting many-body system.

All the new developments mentioned here have opened entirely new areas in atomic many-body theory, and we are evidently just at the verge of a very interesting period of rapid progress.

It is quite evident that we could have limited the Symposium to atomic problems of the type described here. However, related problems appear in atoms bound in solids and in atoms/molecules bound to solid surfaces. Therefore, we proposed to include also some aspects of these fields in our program, which brought together scientists with different backgrounds, such as atomic and molecular physicists, theoretical chemists, solid state and surface physicists as well as nuclear physicists and quantum- liquid experts. The Symposium then got a distinctive inter-disciplinary character at the same time as it was concentrated on the specific atomic many-body problem.

The response to our invitations to the Nobel Symposium was overwhelming. Many other participants were suggested and we extended the number of participants as far as we could. With the wide scope of the Symposium program and small format with regard to number, only a few representatives of each major area could be invited.

The symposium gave an excellent picture how the various areas are developing. The various methods to treat the many-body problem were thoroughly discussed and many new results were reported. The relativistic many-body problem offers many challenging problems as does the open-shell many-body formalism. Substantial progress has been made in these fields in recent years and several good reviews and status reports were given. The physics of photoionization and photoabsorbtion is also a very active area of research and it was the subject of much stimulating discussions at the Symposium. The area of atoms in solids and at surfaces covers a wide range of interesting problems and some problems of particular atomic interest were selected. This volume contains practically all papers presented at the Symposium and therefore it should give an excellent picture of the actual Symposium program. Together the articles form an impressive report of recent developments of the field of atomic many-body theory in a broad sense, reviewing as well as pointing towards new problems and approaches.

We ourselves experienced this symposium as an outstanding scientific event in the field, thanks to the excellent contributions of our participants. From the different expressions of appreciations we received after the Symposium we feel confident that our opinion was shared by many participants. We hope that these proceedings will convey to the reader something of the excitement felt by the participants during the Symposium week.

We would like to place on record our thanks to all the participants who have contributed substantially in the planning of the Symposium by making valuable comments and suggestions and not the least to the members of our research groups for carrying out all the service functions during the Symposium and doing this so well. Mrs Agneta Connant deserves our very special thanks for her excellent work for the symposium on top of all her regular duties. Finally. we wish to express our gratitude to Mrs Birgitta Parenius and all the staff members at the Aspenäsgården for all their self-sacrificing efforts to make the Nobel Symposium such a memorable event.

Without abstract; see full text

This paper reviews the major steps in the development of many-body theory since the early 1950's. Very few systems permit an exact solution by selective diagram summation or by exact solution of a truncated Hamiltonian. Formal methods have usually had little success for real physical systems. Examples are all the quantum liquids such as nuclear matter, liquid He³, liquid He⁴, superconductors and metallic conductors. Atomic and molecular systems and finite nuclei present additional problems. Many-body theory has probably had its greatest success in the application to atomic properties and the development in recent years is reviewed.

The equations of motion method is discussed as a many-body approach to the calculation of excitation energies and oscillator strengths. Various features of the method and the results of its applications to diatomic and polyatomic molecules are discussed.

A description is provided of the common conceptual origins of many-body equations of motion and Green's function methods in Liouville operator formulations of the quantum mechanics of atomic and molecular electronic structure. Numerical evidence is provided to show the inadequacies of the traditional strictly perturbative approaches to these methods. Nonperturbative methods are introduced by analogy with techniques developed for handling large configuration interaction calculations and by evaluating individual matrix elements to higher accuracy. The important role of higher excitations is exhibited by the numerical calculations, and explicit comparisons are made between converged equations of motion and configuration interaction calculations for systems where a fundamental theorem requires the equality of the energy differences produced by these different approaches.

A brief overview concerning the motivations and the development of the coupled cluster approach to the many-electron correlation problem for both closed and open shell systems is given. The basic characteristics and applications of this approach are briefly summarized as well as its relationship to other approaches.

A series of molecular applications of many-body perturbation theory (MBPT) and the coupled-cluster doubles (CCD) model are described. Even though these methods have been available for sometime, only recently have large scale, MBPT molecular calculations become available. In the case of CCD, the results presented here are among the first obtained from a general purpose ab initio program. The intention of this paper is to present an overview of the current state of the many-body approach to ground state properties of molecules. The properties studied are correlation energies, including contributions from single, double, and quadruple excitations diagrams in fourth-and higher-order; dissociation energies; potential energy surfaces; and molecular polarizabilities and hyperpolarizabilities. Examples are taken from studies of a variety of molecules including HF, H₂O, HCO, C₆H₆, B₂H₆, CO₂, and N₂. In many cases, it is found that quantitatively accurate dissociation energies, geometries, and force constants can be obtained. In an illustration of the X¹Σ_g⁺ potential energy curve of N₂, it is shown that a single UHF or RHF reference function MBPT/CCD approach is inadequate at some internuclear separation.

The many-body perturbation theory for open-shell systems is briefly discussed, particularly with regard to the derivation of the diagrammatic expansion for the effective interaction. We then compare the application of this theory to atoms and to nuclei and observe that calculations are more difficult for nuclei. Most of the problems in the nuclear calculations arise from the complicated nature of the nucleon-nucleon (or strong) interaction. In particular the diagrammatic expansion for the nuclear effective interaction tends to diverge due to highly collective nuclear states which lie low in energy, because of the nature and strength of the nucleon-nucleon interaction. Other methods for calculating the effective interaction besides perturbation theory are also discussed. At the present time the exp (S) or coupled-cluster method appears to be the best and the most accurate method for computing nuclear spectra.

The applications of Many Body Theory (MBT) to electron-atom scattering are reviewed. Three processes are considered: elastic scattering, inelastic scattering, and ionization.

It is first recalled briefly how the functional differentiation technique of Martin and Schwinger [7] leads to a physical approximation of the opitcal potential in terms of the linear response of the target.

The case of elastic electron-Helium scattering is considered. Above the first excitation threshold, it is shown how MBT justifies the use of a particular single pseudo-state in the expansion over target eigenstates which appears in the optical potential. From a few eV up to about 200 eV, the consistently good argument of MBT with experiment is demonstrated.

For inelastic scattering, good agreement with experiment is shown for optically allowed transitions, when the first order approximation of the theory (FOMBT) is used. This leads to the formulation of the two-time model. The basic deficiencies of FOMBT are pointed out. It is shown how they are remedied formally in the second order approximation (SOMBT).

Studies of the triple differential cross-section for the ionization of helium at 80.5 eV and 256.5 eV are reviewed. The success of FOMBT is again attributed to an implementation of the two-time model.

We review calculations on parity non-conserving E1 matrix elements for transitions of experimental interest in heavy atoms. We find agreement between different methods in Cs and Tl but a more complex situation in Bi. Here there is satisfactory consistency at the single particle level using the dipole length operator but significant modifications are found in going beyond the approximation. While more calculations are required, it is felt that the outlook for a reliable calculation is good.

The neutral-current induced parity non-conserving optical rotation in Bi has been calculated using a local potential and a Dirac-Hartree-Fock potential giving the values - 18 × 10^-8 and - 10 × 10^-8, respectively for the ratio R = Im (E1)/M1. When the parity-violating parts of all orbitals were considered in the Dirac-Hartree-Fock calculation, the value - 17 × 10^-8 was obtained.

The various existing approaches for the evaluation of matrix elements of unitary group generators and their products with respect to the basis of electronic Gelfand states or the corresponding Yamanouchi-Kotani states are interrelated, and their desirable features combined, yielding a direct algorithm for the evaluation of matrix elements of products of two generators and, consequently, a simple and efficient algorithm for the calculation of two-electron matrix elements of spin-independent Hamiltonians needed in the unitary group configuration interaction (shell model) approach. Moreover, this algorithm is compatible with the efficient generation and representation scheme for electronic Gelfand states based on the distinct row table concept. Diagrammatic techniques based on the time-independent Wick theorem and graphical methods of spin algebras are used to derive the required factors for both one and two-generator (or electron) matrix elements for three different phase conventions and several possible simplifications in the evaluation of the two-electron part of the Hamiltonian matrix are outlined.

The graphical unitary group approach (UGA) has been cast into an extraordinarily powerful form by restructuring the Hamiltonian in terms of loop types. This allows the adoption of the loop-driven formulation which illuminates vast numbers of previously unappreciated relationships between otherwise distinct Hamiltonian matrix elements. Several new developments are presented and discussed. Among these developments are the use of new segment coefficients, improvements in the loop-driven algorithm, implicit generation of loops wholly within the external space adopted within the framework of the loop-driven methodology, and comparisons of the diagonalization tape method to the direct method. It is also shown how it is possible to implement the GUGA method without the time-consuming full (m⁵) four-index transformation. A particularly promising new direction involves the use of the graphical UGA methodology to obtain one-electron and two-electron density matrices. Once these are known, analytical gradients (first derivatives) of the CI potential energy can be easily obtained. Several calculations have been performed and times are compared with our preliminary implementation [J. Chem Phys. 70, 5092, (1979)] of the method. Also included is a calculation on the asymmetric 2¹A' state of SO₂ with 23 613 configurations to demonstrate methods for the diagonalization of very large matrices on a minicomputer.

A density matrix formulation is presented of the super-C I and Newton-Raphson methods in complete active space SCF (CASSCF) calculations. The CASSCF method is a special form of the MC-SCF method, where the C I wave function is assumed to be complete in a subset of the orbital space (the active space), leaving the remaining orbitals doubly occupied in all configurations.

Explicit formulas are given for all matrix elements in the super-C I method and the first and second derivatives in the Newton-Raphson formulation. The similarities between the two methods are pointed out and the differences in the detailed formulations are discussed. Especially interesting is the fact, that while the second derivatives can be expressed in terms of first and second order density matrices, the matrix elements between the super-C I states involve also the third order density matrix in some cases.

From the point of view of a dedicated configuration interaction practitioner, and for a given amount of available computer resources, the current limitations in the accuracy of atomic energy calculations are assumed to be: (i) an inadequate theoretical knowledge regarding the proper handling of slowly convergent radial and harmonic expansions in electron-pair functions, (ii) insufficient experience regarding the quantitative implications of possible systematic simplifications in the Hamiltonian matrix, (iii) tedious and empirical procedures to account for three-excited linked clusters, (iv) quantitatively uncertain methods to predict the effect of four and higher-excited unlinked clusters, and (v) a limited conceptual versatility in available computer programs. These problems are discussed vis-á-vis energy errors in contemporary calculations and new ideas to attempt increased accuracy.

A coupled-cluster procedure, applicable to closed-shell as well as open-shell atoms, is described, and results from calculations on the closed-shell atoms Be and Ne are presented. The procedure is based on numerical solution of coupled radial pair equations, which can be iterated to self-consistency. The coupling between different pair excitations is considered in two steps. The first step, called intra-shell coupling, includes the interaction between excitations differing in the final states as well as in the m_lm_s values of the electrons being excited. In the second step, the inter-shell coupling, also the residual interaction between excitations involving different nl shells is considered. While the former coupling is quite important, it has been found that the latter contributes only about 1% to the correlation energy. Single and triple excitations are neglected in the present work, but quadrupole excitations are included by means of the exponential ansatz of the coupled-cluster procedure. According to our results for Be, 98.6% of the experimental correlation energy is obtained with complete pair-pair coupling, which is consistent with the estimated value of about 1.5% for the contribution of single and triple excitations. With only intra-shell coupling the result is in almost exact agreement with the experimental value due to cancellations between single/triple excitations and inter-shell couplings. Similar results are obtained for Ne. The results of the present work are compared and discussed in relation to earlier many-body calculations on these systems.

The effective-operator form of many-body theory is reviewed and applied to the calculation of the effective interaction of electrons in an open-shell atom. Numerical results are given for the 1s²2s²2p² configuration of carbon. The effect of correlation upon the interaction of the 2p electrons of this configuration is represented by effective two-body operators of the form Σa_kT^k(1) · T^k(2). These operators are evaluated using angular-momentum diagrams and solving numerically a two-particle equation for the linear combination of excited states which contribute to the Goldstone diagrams. The effect of the operators of even rank is to depress the values of the two-electron Slater integrals F^k(2p, 2p) below their Hartree-Fock values. The two-body operator of odd rank does not appear in the Hartree-Fock theory. Our second-order values of the Slater integrals agree quite well with experiment but the value which we obtain of the coefficient of odd rank is much too small. This is partly due to a large cancellation which occurs for the contribution of the outer 2s², 2s2p, 2p² pair excitations. In order to study the convergence properties of the theory and to obtain more accurate values of the interaction integrals, we consider the higher-order terms in the perturbation expansion. An important family of two-particle effects is included to all orders by solving the pair equations iteratively until self-consistency is achieved. A more accurate description of the electron-electron interaction is obtained in this way. There are three additional families of wave-operator diagrams which can have an important effect. One family has an additional open-shell line which polarizes a closed-, open-, or excited orbital. There are also the coupled-cluster diagrams and a family of diagrams involving two polarizing open-shell lines, which appears first in fourth order. All of these diagrams can be included in our iterative scheme and they include all possible two-particle effects to self-consistency.

We report here numerical perturbation calculations on Be and C²⁺ starting from a model space consisting of the two strongly interacting configurations 1s²2s² and 1s²2p². We use numerically represented radial pair functions which are solutions of a system of coupled differential equations obtained by a finite difference method. By iterating the system of pair equations the most important correlation effects are included to all orders. This is demonstrated for C²⁺, where excitation energies for the 2p² levels are obtained with an accuracy better than 1% or 0.2 eV. For Be only second-order results are reported. Here the iterative scheme does not converge, probably due to the presence of "intruder states" of the type 2sns¹S, which lie between the two ¹S states originating from the model space. The second-order calculation with the two-configurational model space and orbitals generated in the 1s² Hertree-Fock core yields 93.6% of the correlation energy, compared to 80.9% for a similar calculation using a model space with only the ground-state configuration 1s²2s² and orbitals generated in the Hartree-Fock potential of that configuration.

A diagrammatic many-body perturbation theory applicable to arbitrary model spaces is presented. The necessity of having a complete model space (all possible occupancies of the partially-filled shells) is avoided. This requirement may be troublesome for systems with several well-spaced open shells, such as most atomic and molecular excited states, as a complete model space spans a very broad energy range and leaves out states within that range, leading to poor or no convergence of the perturbation series. The method presented here would be particularly useful for such states. The solution of a model problem (He₂ excited Σ_g⁺ states) is demonstrated.

Through the multiconfigurational Dirac-Fock method, an ab-initio evaluation of an off-diagonal atomic field-isotope-shift parameter in (5d + 6s)^N has been obtained. It appears impossible to relate unambiguously this parameter to the relaxation of definite atomic subshells.

A brief outline is given for the use of the so(4, 2) algebra in perturbation theory problems. In particular, the case of the hydrogen atom in a homogeneous magnetic field is considered.

We present some preliminary results of the application of the Schwinger variational principle to electron-molecular ion scattering. The results of this application to e-H₂⁺ scattering in the static-exchange approximation are encouraging.

A graphical procedure is presented for calculating first order transition matrices for a general (open-shell) atom. The first order transition matrix may be used to calculate matrix elements of a general one-body operator of rank λ in orbital space and σ in spin space. In the random phase approximation we obtain a set of N + N' coupled differential equations for N final state radial functions and N' initial state radial functions which completely determine the first order transition matrix for an atomic system having N final state channels. (The relation of N' to N is dependent on the atomic system studied.) These N + N' differential equations reduce to familiar forms in the following cases: (1) When initial state correlations are ignored, we obtain the N coupled differential equations of the Close-Coupling Approximation; (2) When the atom has only closed subshells we obtain N' = N and the 2N coupled differential equations are those obtained in the Chang-Fano version of the Random Phase Approximation.

Recent results concerning the consistency of the random phase approximation and its generalizations are reviewed. Particular attention is given to the ground state correlation energy problem as reflected in the calculation of chemical bond energies.

Electron propagator or Green's function methods are discussed using the algebraic methods of superoperators, inner and outer projections on a linear space of fermionlike electron field operator products, partitioning, and perturbation theory. The problems of accurate determination of molecular electron binding energies from the core through the valence region are addressed. The complications of diagonalization of very large matrices in the self-energy expressions are pointed out and various diagonal approximations are discussed with the use of renormalization theory.

Simple applications to the photoelectron spectra of water and acetylene are presented for illustrative purposes.

Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP (E_AGP) in a Brueckner orbital representation can be used to generate a manifold of excited states|K⟩ = O_K⁺|AGP⟩, O_K|AGP⟩ = 0 with the excitation operators, O⁺_K, and killers, O_K, being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as

⟨⟨X, Y⟩⟩_E = - ⟨AGP|XT₀(-E + E_AGP)Y + YT₀(E + E_AGP)X|AGP⟩

in terms of the resolvent T₀() = P( -PHP)^-1 with P spanned by the states|K⟩⟨K|, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.

The density functional formalism provides a framework for including exchange and correlation effects in the calculation of ground state properties of many-electron systems. The reduction of the problem to the solution of single-particle equations leads to important numerical advantages over other ab initio methods of incorporating correlation effects. The essential features of the scheme are outlined and results obtained for atomic and molecular systems are surveyed. The local spin density (LSD) approximation gives generally good results for systems where the bonding involves s and p electrons, but results are less satisfactory for d-bonded systems. Non-local modifications to the LSD approximation have been tested on atomic systems yielding much improved total energies.

It is well known from density functional theory that the ground-state electron density ρ(r) in an atom or molecule can be calculated exactly in principle, from a one-body potential energy V(r) into which exchange and correlation interactions are incorporated. Therefore, it is of considerable interest to set up a differential equation which will enable ρ(r) to be calculated directly from V(r) for atoms and molecules with many electrons, without recourse to wave function determination.

The problem will be tackled as follows:

(i) A careful discussion will be given of the differential equation in one-dimension. Its relation to local density approximations will be demonstrated and it will be shown that the linear harmonic oscillator is a case when the local density equation becomes exact.

(ii) A treatment of a closed shell atom with many-electrons will be presented. Here, central field separability is utilized, and two methods are developed. In the first, a separate differential equation is to be solved for each value of the orbital angular momentum l. The second, and presently less accurate, approach solves directly for the total density ρ(r).

(iii) For three-dimensional potentials V(r), with arbitrarily low symmetry, as in the general molecular problem, the route to ρ(r) via a generalization of the partition function Z(r, β) with β = (k_BT)^-1 is then explored. In particular, the effective potential U(rβ) which is essentially - k_BT ln Z(rβ) is the tool employed. Such a route is first shown to afford an exact formulation of (i) and (ii) above, which are essentially one-dimensional problems. For a general V(r), a generalization which is approximate is proposed, based on a linearized equation for the effective potential U. This approximation can then be refined by (a) use of the lowest bound state wave function or (b) incorporating non-linear gradient terms in V(r).

The relativistic random phase approximation RRPA is developed from linearized time-dependent Hartree-Fock theory. Applications of the resulting relativistic many-body equations to determine excitation energies and oscillator strengths along the He, Be, Mg, Zn and Ne isoelectronic sequences are discussed and compared with other recent experimental and theoretical work. The multi-channel RRPA treatment of photoionization is described and applications are given to total cross sections, branching ratios, angular distributions and to autoionizing resonances.

A set of programs developed in the department for the study of atomic structure problems is briefly described. Its core is a relativistic multiconfiguration Dirac-Fock self-consistent field program which, in various versions, can handle up to 36 orbitals and 60 configuration state functions (CSF). The system can deal with very general open-shell configurations, the necessary Racah algebra being handled automatically. The full transverse electron pair interaction, correct to O(1/c²), can be computed if desired, along with estimates of the lowest order electron self-energy and vacuum polarization corrections.

Typical applications to the calculation of X-ray levels and to the interpretation of 5p excited spectra in atomic barium are briefly described.

Some recent calculations using the relativistic random phase approximation are briefly described. The ¹P and ³P energy levels of several highly stripped helium-like ions are tabulated and photoionization cross sections of neutral magnesium and zinc and of magnesium-like and zinc-like molybdenum ions are presented.

In this review we discuss relativistic calculations of atomic and molecular structure with respect to the two following questions. First: is there any unexplained discrepancy between experiment and theory? And second: are relativistic effects necessary to understand some chemical properties?

Evidence is presented that many-electron effects in nuclear volume isotope shift can make contributions that are not readily apparent in current descriptions based on evaluating the electron density at the nucleus. Estimates of IS can be consistently computed by differencing SCF energy values directly. Implications of these studies for methods of assigning configurations to atomic levels which use measured isotope shifts are discussed.

The use of many body perturbation theory to calculate the interaction of atoms with electromagnetic radiation is discussed. Particular attention is given to atomic photoionization cross-sections including effects of electron correlations, resonance structure, and relaxation. Double photoionization in which two electrons are ejected is reviewed. Application to inner-shell transitions and to transitions involving parity nonconservation are also considered.

We discuss the use of the multiconfiguration Hartree-Fock (MCHF) to obtain initial state wave functions for photoionization calculations. An extension of the MCHF, a multiconfiguration V^(N-1)LS, can be used to obtain final state wave functions such that most intrachannel corrections to these wave functions vanish. Application of these techniques to the rare gases and the Cl are discussed. In the case of Cl, the strong interchannel interactions are included through use of the K-matrix. Excellent agreement between theory and experiment is obtained for the rare gases.

As a first step in an attempt to carry out a systematic study of the effect of the strong configuration interaction to the atomic transition in a many-electron system, we have developed a new theoretical approach to study the simultaneous photoionization and photoexcitation process of the He atom. Detailed discussion will be given to the physical interpretation of a few new aspects which have emerged from our study.

A multiconfiguration Hartree-Fock procedure for excited states is described and applied to states of the Rydberg series up to 3s 8f³F⁰ in Mg I, Al II and Si III. Energies relative to the ionization limit agree with observation to within 1%, an exception being the case of near degeneracy in Al II where a discrepancy of 2.5% occurs. Systematic errors are observed. Some possible sources of errors are considered.

This paper discusses the photoemission from the valence orbital of an atom adsorbed on a solid surface, including the emission of a surface plasmon.

A model Hamiltonian is used in which strong hole delocalization in the valence orbitals modifies the elastic and inelastic photoemission. Special attention is given to the continuous transition between adiabatic photoemission near threshold and the sudden limit at high outgoing kinetic energy.

An infinite summation method is used to obtain the intrinsic photoemission spectrum of a conduction electron in the sudden limit. It is predicted that an electron at the bottom of the conduction band should have an asymmetry index comparable to that of a core electron and that the strengths of its plasmon satellites should be similar to those of a core electron. The width of their plasmon satellites, however, should become smaller under the influence of recoil.

In essence, our results show that the exchange-correlation hole left after a photoemitted conduction electron has a very similar effect as the core hole left after a photoemitted core electron. Use of synchrotron radiation gives high enough a resolution to make possible the verification of these effects.

The spectral function of ionization is discussed for the whole energy range. It is found that different energy regions are likely to exhibit different types of many-body effects. For the ionization out of an outer valence orbital most of the intensity appears in one main line. The many-body effects explain the additional satellite lines and, in addition, can lead to an ordering of main lines which is different from the ordering determined from one-particle calculations. For the ionization out of an inner valence orbital the intensity may be distributed over several lines and in many cases it is not possible to identify any of these lines as the main line representing the orbital. Such a breakdown of the quasiparticle picture of ionization is stressed to be a common phenomenon. Ionization of core orbitals can usually be viewed within a quasiparticle picture, i.e., the process leads to a main line accompanied by satellite lines. In cases, however, where the creation of the core hole leads to a strong charge transfer, the shake-up energies can become small or even negative and the quasiparticle picture may break down. The origin of the above effects is discussed and typical examples are presented.

The concept of diabatic and resonance molecular states is discussed. A recent definition of diabatic states is presented. Current investigations of resonance states of molecular hydrogen are summarized and detailed calculations of the dissociative photoionization of H₂ are reported.

The time-independent multichannel scattering theory is formulated in a way which makes it appropriate for the study of the decay of autoionizing and inner-shell vacancy states. A general interpretation of the parameters in the Fano profile is given and the concept of decay is introduced for both autoionizing and inner-shell vacancy states in a transparent manner. The theory is applied to a study of the influence of photon-electron coupling effects on the radiative and nonradiative decay probabilities. A generalization which includes post-collision interactions in inner-shell ionization is also presented.

We give a brief review of the theories describing collective dynamics of atoms, and comment on the hydrodynamic approach as well as the manybody approach based on RPA. In particular we put forward a unified approach, combining the hydrodynamical and single-particle aspects in a single theoretical framework. Some general aspects of the theory are discussed as well as a simple application to the oscillations of a spherical shell.

The p⁶ → p⁵d and d¹⁰ → d⁹f transitions to bound and continuum states are considered and it is shown that the major features in the photoionization of the neutral rare gases are present in the photoabsorption to bound states in the isoelectronic ions. This is a consequence of the collapse of d and f orbitals between the neutral and the ionized members of the particular isoelectronic sequences. It is shown that Hartree-Fock calculations which take into account the proper angular momentum coupling of the final state can account for the gross structure of the photoionization as well as the photoabsorption spectrum to within approximately a factor of 2. Much stronger correlation effects exist in bound spectra. The identification of collective resonances, which should correspond to strong correlation effects, in the continua of the neutral rare gases by other workers is shown to rely on a doubtful analogy with the use of the Random Phase Approximation to interpret the giant dipole resonance in nuclei and on the neglect of exchange in the early calculations of the photoionization cross-sections. Exchange turns out to be extremely important for the ¹P terms in p⁵d and d⁹f which are the only ones that can be reached by an electric dipole transition from the ground state. It is concluded that it is unnecessary to invoke collective effects to explain the photoionization cross-sections of the rare gases.

We present a brief review of variations of bremsstrahlung and characteristic soft X-ray emission yields near core level excitation thresholds in metallic La (4f-metals) and the 3d-metals treated by the author in collaboration with Wendin. We extend the discussion in this paper to include processes neglected in the above mentioned earlier work, the emphases being laid on the relative importance of resonant bremsstrahlung (fluorescence) versus characteristic emission and the importance of localised states versus extended states, for different core level excitation thresholds.

In this paper we present a report on the effect of correlation in the 3dⁿ-core on the transition probability of the resonance transition in Cu II, 3d¹⁰¹S → 3d⁹4p¹P. The f-value has been calculated using the MCHF method. Three different approximations have been used with varying amounts of correlation: (i) no correlation, (ii) correlation in 3d¹⁰ of the initial state, (iii) as in (ii) as well as 4p correlation with 3d⁹ in the final state.

This paper is concerned with information about electronic structure and bonding in solids which can be derived from valence band and core level photoelectron spectroscopy. The role, in research of this kind, of the excitation processes discussed elsewhere in this symposium is of concern. A number of factors contribute to the apparent chemical shifts of core levels and valence bands in solids. These factors are considered and then results for gold alloys are taken as an example of the bonding information which can be derived. Angle resolved photoelectron spectroscopy provides information about wave function symmetry as well as energy. This is reviewed and one-electron energy vs. k results for Ni metal are inspected. Such results provide the best experimental test yet devised of energy band theory and of the potentials employed.

The availability of tunable synchrotron radiation has made it possible to perform systematic X-ray diffraction studies in regions of anomalous scattering near absorption edges, e.g., in order to derive phase information for crystal structure determination. We give an overview of recent experimental and theoretical work and discuss the properties of the anomalous atomic scattering factor, with emphasis on threshold resonances and damping effects. The results are applied to a discussion of the very strong anomalous dispersion recently observed near the L₃ edge in a cesium complex [11]. We also discuss briefly other systems where similar behaviour can be expected. Finally, we discuss the influence of solid state and chemical effects on the absorption edge structure.

The intensities of certain lines in the absorption spectra of rare-earth or actinide ions are very sensitive to the neighboring ligands. The contributions to these intensities coming from vibronic transitions (combined electronic and vibrational transitions) are calculated by tensorial techniques with particular reference to octahedral complexes. The expansions that are used possess leading terms that indicate that the strongest vibronic lines should be associated with electronic transitions that satisfy the selection rules on J (the total angular momentum) that are identical to those for quadrupole radiation. General agreement is obtained with the recent work of Faulkner and Richardson on the transition ⁷F₀ → ⁵D₂ of Eu³⁺, though some of the approximations and assumptions, as well as much of the mathematics, are different.

Ab initio SCF calculations of X-ray emission chemical shifts are presented for the molecules TiH₄, TiH₃F, TiF₄, and TiF₂. The calculated Ti Kα_1,2 shifts parallel 3d orbital populations through a coupling via the radial extension of the 3p and 3s orbitals. Anomalous results are obtained for TiF₂, which was calculated in a closed-shell configuration differing from its ground state. Potential models for chemical effects in core level spectra are shown to violate the atomic virial theorem but have still value as semiempirical tools.

Calculations of various electronic excitation energies for the rare earth and actinide metals are discussed. Relativistic Hartree-Fock computations for atomic configurations most closely approximating those of the metal are initially performed, and crystal potentials are constructed by means of the renormalized atom method. Relativistic band calculations are iterated to crude self-consistency, and excitation energies derive from differences between total band energies. Results include (1) occupied and unoccupied 4f level positions in the lanthanide metals and associated values of the 4f coulomb interaction energy, U; (2) 3d levels in the rare earths; and (3) 5f excitation energies and coulomb terms for the actinide metals. There is excellent agreement between these results and X-ray photoemission and bremsstrahlung isochromat spectroscopy measurements for the rare earths, while corresponding experimental information for the actinides is as yet unavailable.

Shake-up satellites are treated in terms of a simple model. High intensities occur only in the case of weak bonding. A Hartree-Fock calculation for CuCl₄^2- is discussed and found to support the suggested model. The role of many-body effects is discussed.

Many-body description of the X-ray photoemission process from solids is discussed qualitatively, and also formulated in terms of a correlation function that contains the effects of inelastic scattering of both the excited photoelectron ("extrinsic") and photohole ("intrinsic"). The dynamical concepts, like the sudden and adiabatic limits, are discussed and obtained as limiting cases of the full description. The spectral sum rules are shown to be modified due to the extrinsic scattering. The emphasis is on the quantitative description of the inelastic effects characteristic of the solid state photoemission, and several possible scattering mechanisms are analyzed in more detail.

The general problem of boson renormalization of a quasi-localised hole state created in a photo-ejection process from a molecule or solid is considered. A quasi-localised hole can hop from the initial site of creation and this introduces the possibility of recoil when the hole interacts with the boson field.

As an illustrative example, the problem of the photoionization spectrum for a molecule adsorbed on a metal surface is considered. Hopping from the admolecule hole state into the substrate is the delocalization mechanism and intra-molecular vibrational modes are the bosons. Recoil induced interferences between the two processes lead to a more narrow lineshape than would be inferred from a straightforward convolution-of-independent-events line.

The core spectrum of an atom or a molecule absorbed on a surface is calculated for a model which includes screening of the core hole both by charge transfer to the adsorbate and by creation of a surface charge due to the excitation of surface plasmons. For a strong absorbate-substrate coupling (strong chemisorption), the spectrum has an asymmetric peak on the high energy side. When the coupling is reduced, weight is transferred to satellites on the low energy side, and simultaneously the surface plasmon screening mechanism becomes more important.

We have previously shown that one-particle calculations of X-ray spectra in simple metals give very different results depending on whether the core hole is taken into account or not. Guided by experiment and the dynamical theory of X-ray spectra developed by Nozières and DeDominicis (ND) we have, however, also established the rule that rather realistic spectra can be obtained from a one-particle calculation provided final state wavefunctions are used in the transition matrix elements. In this paper we report on direct numerical evaluations of the dynamical ND theory for several different cases including the case where a bound state appears. In all cases the results support our final state rule. We also give arguments in support of the procedure used to extract threshold exponents and asymmetry indices from X-ray absorption and emission spectra and X-ray photoemission spectra.

The local-density approximation by Hohenberg, Kohn, and Sham produces errors in the total exchange and correlation energies of atoms which are typically of the order of 10% and 100%, respectively. The errors are due to the rapid variations in the charge density of atoms. The density of the outer valence electrons has, however, a much slower variation than the total density and for these electrons the corresponding errors are only 3% and 30%. This is demonstrated here by using local-density theory to calculate the multiplet splittings for atoms with incomplete shells. The splittings are almost entirely due to exchange and correlation effects in the outer valence shell. The original density functional theory only describes ground state properties and we have therefore generalized it to be able to describe "mixed-symmetry" states. The multiplet splittings are then obtained from the energies of these "mixed-symmetry" states.

Many-electron interactions within one atomic shell can cause a breakdown of the quasi-particle picture of a core hole by causing strong fluctuations. Typical examples are 4s and 4p XPS spectra and L_γ2,3(L₁N_2,3) X-ray emission spectra of the elements ₄₆Pd to ₅₄Xe. In these cases quasiparticle picture of 4p core hole breaks down due to very strong dynamical dipolar fluctuations and associated decay processes of the 4p core hole. The theory presented agrees well with experimental data.