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Core-Level Binding Energies in FHI-aims by Occupation Constrained ΔSCF Calculations

Learn to use FHI-aims to calculate XPS core-level binding energies using the core-hole occupation constrained ΔSCF approach. The tutorial demonstrates the approach using the molecules ETFA and azulene as well as the periodic system graphite and highlights common pitfalls in the calculations.

Tutorial Objectives

  • Understanding the necessary steps and calculations to calculate core-level binding energies
  • Recognising and avoiding pitfalls in the method
  • Learn how to compare calculation results to experimental data
  • Understanding the limitations of the method

Prerequisites

  • Base-level knowledge for running FHI-aims calculations is required. We recommend the general tutorial as a good place to start
  • A compiled FHI-aims binary with version 231105 or newer
  • A suitably powerful computer. The molecular calculations in Part 1 and Part 2 can be run on a powerful desktop machine.

Tutorial Outline

Introduction

When matter is exposed to X-ray radiation with the energy h\(\nu\), emitted photoelectrons can be detected at certain kinetic energies E\(_{\mathrm{kin}}\) according to the following relationship:

\[\begin{equation} E_{\mathrm{kin}} = h \nu - E_{\mathrm{B}} - \phi \end{equation}\]

In this equation, \(E_B\) is the electron's binding energy before photoemission, and \(\phi\) is the work function of the electron energy analyser. A typical photoelectron spectrum plots the intensity of the detected electrons against the binding energy (from highest to lowest binding energy). The following schematic shows how the energy of an ejected electron is calculated by the analyzer.


Scheme 1. Schematic of how the energy of an ejected core electron is calculated experimentally in XPS at the analyzer.

The simplest approach to calculate binding energies in an ab initio calculation is the use of Koopmans’ theorem:

\[\begin{equation} E_\mathrm{B}(i) = - \epsilon_{i, \frac{\mathrm{HF}}{\mathrm{KS}}} \end{equation}\]

This formula from Hartree–Fock theory relates the binding energy of electrons occupying an electronic state with the negative eigenenergy of the relevant Hartree–Fock eigenstate. Within Density Functional Theory (DFT), this relationship holds only for the highest occupied Kohn–Sham (KS) state. However, KS energies of core levels are also often used to estimate initial state contributions, i.e., the chemical shift of the core level before the removal of the electron to the binding energy.

However, core-hole relaxation (or final state effects) cannot be neglected in predicting core-level binding energies. The ΔSCF DFT (or ΔSCF) method, which includes relaxation effects, is the most common approach to simulate core-level binding energies.

Within the ΔSCF method, the core electron binding energy is calculated as the difference between the total energies of two self-consistent KS DFT calculations. The two necessary calculations for the XP spectra simulations are the ground-state and core-hole excited calculations. In the excited calculation, the electron population of the core state is constrained to remove one electron.

\[\begin{equation} E_B = E_{N-1}^{\text{core hole}}-E_N \end{equation}\]

The big advantage of this approach is that it only uses the well-established ground-state DFT formalism and still yields comparatively good results, including both initial and final state effects.

For a more detailed discussion see B.P. Klein, S.J. Hall, R.J. Maurer, J. Phys.: Condens. Matter 2021, 33, 154005. https://doi.org/10.1088/1361-648X/abdf00.1

Limitations

General Limitations

  • It is assumed that upon photoemission of the core electron, the system ends up in the lowest energy (fully-screened) final state. Hence, conventional ΔSCF calculations do not yield information about satellite peaks in photoemission spectra. The GW implementation of XPS simulations can be used to simulate satellite features. There is already a tutorial on how to do this.

Implementation Limitations

  • This technique is the most straightforward for calculating core electron binding energies of 1s levels. Due to orbital degeneracies and spin-orbit coupling, additional complications arise when dealing with atomic shells with orbital angular momentum, e.g., 2p, 3d, etc. For literature that discusses this please refer to papers by Kahk and Lischner.2 3
  • Spin-orbit splitting is not accounted for (e.g., separate calculation of the 2p\(_{1/2}\) and 2p\(_{3/2}\) peaks)
  • Relativistic effects are only approximated with ZORA
  • There are challenges associated with electrostatic effects in periodic systems. Klein et al.1, Taucher et al.4, and Schmid et al.5 provide in-depth discussions of the topic.

Generally, it is much easier to obtain relative binding energies, both within one system and compared to similar calculations, than it is to achieve binding energies on an absolute energy scale comparable to experiments. On the absolute energy scale, minor systematic errors of the functional or implementation are enough to lead to significant deviations. In addition, for condensed matter systems, the material's work function must be considered. For most use cases, however, calculating relative binding energies and a rigid shift to match the experimental and computational energy scales is sufficient.

Automation

For all of the manual work required below, a Python wrapper to FHI-aims has been written to automate the process. For access to the code and documentation on usage, please see this GitHub Repository. The wrapper can automate the setup of the various calculations required for both basis and projector methods (see below) and run the calculations if required. It attempts to choose sane defaults for the highest likelihood of converging the core hole; however, users can still choose their defaults if desired. There is also functionality to plot the simulated spectrum and calculate the mean average binding energy of the system.

References


  1. B.P. Klein, S.J. Hall, R.J. Maurer, J. Phys.: Condens. Matter 2021, 33, 154005. https://doi.org/10.1088/1361-648X/abdf00 

  2. J.M. Kahk and J. Lischner Phys. Rev. Materials 2019, 3, 100801. https://journals.aps.org/prmaterials/abstract/10.1103/PhysRevMaterials.3.100801 

  3. J.M. Kahk and J. Lischner Faraday Discuss, 2022, 236, 364-373. https://pubs.rsc.org/en/content/articlelanding/2022/fd/d1fd00103e#!divAbstract 

  4. T.C. Taucher, O.T. Hofmann, E. Zojer, ACS Omega 2020, 5, 25868−25881. https://dx.doi.org/10.1021/acsomega.0c03209 

  5. J.M. Kahk, G.S. Michelitsch, R.J. Maurer, K. Reuter, J. Lischner, J Phys. Chem. Lett. 2021, 12, 9353-9359.https://doi.org/10.1021/acs.jpclett.1c02380