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# Correlation and Dynamical Effects

Especially since the advent of synchrotron radiation, characterised by high intensity and tunablility, X-ray absorption spectroscopy has become a powerful tool for electronic and structural characterisation of matter. Probably the most interesting part of an X-ray absorption spectrum is the region from an edge to a few tens of an electron Volt above, which is commonly called X-ray absorption near-edge structure (XANES). It corresponds to electron excitation from an inner atomic shell to a conduction state with small kinetic energy. It depends on both local and global properties of the system, a fact that makes it a rich source of information but which also enormously complicates the theoretical description. The standard methods for XANES calculation, which use the independent electron approximation, are often insufficient to explain the experimental spectra of strongly correlated electron systems. We are working on an improvement of XANES theory that should account for electron correlation and dynamical aspects of the photoabsorption process. Our approach is based on the multichannel generalisation of multiple scattering theory developed by C. R. Natoli * et al.* [1]

## What happens in a photoabsorption process from an inner atomic shell?

Let us first describe the absorption process in the independent particle picture, i.e. let us assume that only the inner shell (``core'') electron participates in the absorption process, while the state of all the other electrons remains unchanged. The measured quantity in an absorption experiment, the cross section, is essentially given by the transition amplitude from the core state to an excited state, squared and summed over all exited states allowed by energy conservation and symmetry considerations. The transition amplitude is a space integral over both the core and the excited state wavefunctions times the dipole operator. Since the core wavefunction is very localised and vanishes outside the absorption site, the photoabsorption cross section roughly measures the amplitude squared of the photoelectron wavefunction at the absorption site (more precisely the local density of unoccupied states).

When the core electron absorbs the photon, its energy is suddenly raised above the Fermi level. In this energy range, the stationary states are delocalized over the whole system (as molecular orbitals in molecules and band states in solids). Therefore, the photoelectron wavefunction spreads out from the absorption site and is scattered at the potentials of the surroundings atomic sites. The scattered waves are in turn scattered from all the other sites. The amplitude of the photoelectron wavefunction at the absorption site (and thus the absorption cross section) is determined by the interference of these multiply scattered waves. The mathematical formulation of these concepts leads to multiple scattering theory, which is physically intuitive, applicable to all kinds of materials from molecules to amorphous and crystalline solids, and sufficiently flexible to incorporate systematic improvements.

## Why does the theory need to be improved?

Let there be N electrons in the system. Standard multiple scattering theory uses the independent particle approximation which means that only one electron (the core electron becoming photoelectron) is involved in the photoabsorption process. The state of the other N-1 electrons is assumed to remain unchanged. These electrons participate in the whole process only by contributing to the * static* crystal potential.

In reality, the N-1 other electrons do react both to the creation of the core hole (i.e. missing electron) and to the photoelectron as it moves through the matter. This gives rise to so-called * final-state effects,* which may show up as pronounced structures in the absorption spectrum, as for example the * multiplet structure* observed at the M_{IV,V}-edges of rare earths compounds. Such spectra are usually calculated using atomic multiplet theory in which interaction between all the electrons (of the absorbing atom) and the core hole is taken into account by calculating N-electron wavefunctions independently for the initial and the final state of the photoabsorption process. For strongly correlated systems, atomic multiplet calculations agree much better with experiment than standard multiple scattering or band structure calculations. This might seem surprising since the atomic multiplet calculations neglect the delocalized nature of the photoelectron wavefunction (as well as of all other valence states), which, according to standard multiple scattering theory, is the main reason for the intensity variations observed in XANES. It shows that intra-atomic electron correlation can by no means be neglected for XANES of strongly correlated systems and should be taken into account in calculations based on multiple scattering theory.

If we go further away from the edge, the photoelectron will have higher kinetic energy and will spend less time at the absorption site. The core hole potential in the final state will then be less effectively screened by the photoelectron and thus will also be screened by the valence electrons of the surrounding atoms. Such extra-atomic screening is a well known phenomenon in core-level photoemission, where it gives rise to asymmetric lineshapes or additional peaks in the spectrum, so-called ``charge transfer satellites'' (in 2**p**-level photoemission of 3**d**-transition-metal oxides, for instance). The latter can be reproduced with semi-empirical calculations based on the Anderson impurity model, in which both the valence electrons' interaction with the core hole and their delocalization is taken into account. Such model calculations lack, however, the accuracy of * ab initio* calculations (multiple scattering or band structure calculations on the one hand and atomic multiplet calculations on the other hand).

C. R. Natoli * et al.* [1] have presented a generalised multiple scattering theory for photoabsorption and photoemission such that electron correlation can be incorporated from some underlying microscopic theory such as atomic multiplet theory. It is a multichannel scattering theory that naturally accounts for both elastic and inelastic scattering events. (Historically, multichannel scattering theory was developed for inelastic scattering.) Our current research project consists of developing a new computational scheme for XANES that combines multiple scattering and atomic multiplet theory by using Natoli's approach. Roughly speaking, one first makes a photoemission calculation within atomic multiplet theory, calculates from that the on-site interchannel scattering potential and finally solves the multichannel multiple scattering equations. By joining the virtues of multiple scattering and atomic multiplet calculations, such a theory treats band structure and intra-atomic electron correlation features on an equal footing. In contrast to impurity model calculations it does not rely on empirical parameters, but has the accuracy of an * ab initio* method. Moreover, Natoli * et al.* showed that it naturally accounts for dynamical effects of the photoabsorption process and goes beyond the commonly used sudden approximation.

## References

**1**- C. R. Natoli, M. Benfatto, C. Brouder, M. F. Ruiz López, and D. L. Foulis, Phys. Rev. B
**42**(1990), 1944.