What follows are some general suggestions for practical calculations, aimed to provide a helpful guide for non-experts. Note that the list is, naturally, incomplete, and unavoidably include some of the authors’ biases.

Precision versus accuracy#

Because of the many difficulties in modeling X-ray spectra (relativity, relaxation, self-interaction, high energies, etc.), even very sophisticated (and expensive!) methods often struggle in getting good absolute energies. These cannot be expected to be as good as for valence excitations, and we argue that the focus should be more on properly reproducing relative rather than absolute trends. These can be formulated as focusing on precision over accuracy, where,

  • Accuracy corresponds to the deviation from the absolute energies and intensities, or the systematic error

  • Precision focuses on the reproducability of relative energies and intensities, or (relative) error distribution

We have shown that good accuracy does not necessarily correspond to good precision [FBV+21], and our philosophy is that obtaining good trends and relative errors is typically sufficient for practical studies (provided we have some understanding of the source of the absolute error, and good reasons to think that it will not differ between compounds and structures). Absolute errors can then be corrected by a mere absolute shift in energy.

This is particularly the case when TDDFT is used, where most attempts to improving the accuracy in practise correspond to a cancellation of lack of relaxation weighted against the self-interaction error. We argue that this approach is equally valid as introducing a rigid shift, obtained either by comparison to experiment or based on, e.g., \(\Delta\)SCF.

Electronic stucture method#

Single determinant wave function methods can include relaxation by the use of (at least) doubly excited configurations. These methods tend to yield relatively consistent description of the processes, and are thus the recommended reference to be used for establishing the reliability of more approximate methods. As their high computational cost makes them hard to apply for large molecules, they provide instead a good benchmark for the computationally less expensive methods.

Families and recommendations include:

  • Coupled cluster:

    • CCSD has been noted to provide good precision and reasonable accuracy for XAS and XES [Bes12, VFE+19]

    • CC3 has been shown to improve XAS results, and can be used in a multilayer manner [PMK21]

    • If computational cost is a concern, CC2 has been shown to work quite well [FBV+21]

  • Algebraic-diagrammatic construction scheme:

    • ADC(2) yields good accuracy for XES, and works quite well for the valence states in XAS (the shift to higher-lying features are likely to be compressed, however) [FBV+21, FD19]

    • ADC(2)-x has been noted to yield best results for XAS and high precision for XES, despite being less reliable for valence excitations [WHWD15]. This has been identified as a result of balancing effects for capturing the strong relaxation in core transitions, yielding good results for X-ray spectroscopies

    • ADC(3/2) has been noted to yield a poorer description of both XAS and XES, despite its higher level of theory and higher computational cost [FBV+21, FD19, WHWD15]. However, for X-ray emission spectra of systems in an environment, double-excitations may interfere with the emission lines for ADC(2) and ADC(2)-x, and ADC(3/2) may then be most suitable

With TDDFT (or CPP-DFT), the absolute error of any exchange-correlation functional depends on the edge and the element, as the exact balance of relaxation and self-interaction varies significantly [BA10]. Good absolute energies are often obtained by balancing the error of the lack of relaxation versus the self-interaction, which is achieved by changing the amount of HF exchange. Focusing on the second row elements, we have seen that [FBV+21]:

  • CAM-B3LYP (potentially using 100% asymptotic HF) yields good relative energies (precision), but poor absolute energies (accuracy). Intensitites are also quite good, so a mere absolute shift often yields good agreement with experiment.

  • SRC2-R1, part of a family of short-range correlated functionals and formulated for good absolute XAS energies, performs very well in both absolute and relative terms (note that different functionals in this family are used for different rows in the periodic table)

If static correlation becomes substantial, methods based on a single Slater determinant reference will become unsuitable. ADC will be worst off, on account of the MP reference, with CC being slightly more successful for these cases. Spin-flip ADC and CC can improve the situation for certain few-reference systems, and TDDFT will be a bit more capable of addressing these cases compared to pure CC and ADC. However, systems with strong static correlation should be considered carefully, and multi-reference methods are likely the safest way forward.

Transition metal (TM) complexes often have strong static correlation, which would point towards using multi-reference methods. TDDFT can also work quite well, and in some cases ADC or CC provide reliable results too. Care needs to be taken if the latter methods are used. Furthermore, TM complexes may require the inclusion of (spin-orbit) relativistic effects and may involve quadruple-allowed transitions.

A plethora of other methods for considering X-ray properties are available, for which guides and recommendations may be found elsewhere (e.g. [ND18]).

Basis set selection#

As previously discussed, standard basis sets often include a minimal, or close to minimal, description for the core region, as computational chemistry usually focuses on the valence region. For core spectroscopies, the core region thus often needs to be augmented. This can be done by decontracting the core basis functions, by including core-polarizing functions, by augmenting the basis set with basis functions for \(Z+1\) or \(Z+1/2\), or some other method [ADJ21, HHGB18]. It can be noted, however, that this augmentation is not always necessary for high precision, provided that the lacking description of the core region yields similar errors for different states and atomic sites [FBV+21]. The lack of the additional flexibility in the core region is then likely to yield absolute energies away from the complete basis set limits, but reasonable relative energies.

Nevertheless, a general recommendation would be to use cc-pCVTZ (or similar) for the atoms probed, and cc-pVTZ for the remaining atoms. The Dunning basis set 6-311G** has also been shown to perform quite well, and the core basis functions can here be decontracted for u6-311G**. Hydrogen atoms and atoms away from the probed core orbitals can likely be dropped to a double-\(\zeta\) desciption. A double-\(\zeta\) (e.g. cc-pVDZ/cc-pCVDZ or 6-31G*/u6-31G*) can be used with some care for larger systems and more approximate methods. More specific recommendations include:

  • XPS: If a \(\Delta\textrm{SCF}\) method is used, a flexible description of the core and valence region is desirable

  • XAS: Probes unoccupied states, and thus requires augmented basis sets to describe this region. For the valence-type final states just adding more diffuse functions, as in aug-cc-pCVTZ or 6-311++G**, is likely sufficient. Core-polarizing functions or similar may not be necessary, and extra diffuse functions on hydrogen are typically not needed. However, if Rydberg/mixed-Rydberg states are of interest, augmentation with additional (Rydberg-like) diffuse function can become necessary.

  • XES: Largely has similar requirements as XPS, albeit may need a better description in the (outer) valence region

  • RIXS: Probes both the occupied and unoccupied states, and thus needs a good (and balanced) description of both. The combined recommendations from XAS and XES thus transfer over to this spectroscopy

Note that for heavier atoms the core region contains several shells. As the core-polarizing functions are formulated to capture core-valence correlation, they will then tend to improve the description of the outer core region, but not necessarily of the inner core region (as the outer core couples more strongly to the valence region. As such, additional (inner) core basis functions would be needed for approaching the basis set limit of these atoms. Still, with the discussion of precision above, the weak interaction between the inner core and valence region means that core-polarization may anyway just yield an absolute error correction.

Relativistic effects#

If the K-edge is considered, only scalar relativistic effects are present in the transition, the inclusion of which can be done with X2C or Douglas-Kroll-Hess Hamiltonians. Alternatively, it is often sufficient to just add an absolute scalar shift in energy for a non-relativistic calculation (such a shift is likely needed anyway, as absolute energies are notoriously hard to reproduce). This absolute shift can be estimated by simple MO energy calculations..

If transitions from shells with l>0 are considered, spin-orbit splitting effects will usually be substantial. The inclusion of spin-orbit coupling effects is beyond the scope of this tutorial.