# Assessment of basis sets for density functional theory-based calculations of core-electron spectroscopies

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## Abstract

The performance of Gaussian basis sets for density functional theory-based calculations of core-electron spectroscopies is assessed. The convergence of core-electron binding energies and core-excitation energies using a range of basis sets including split-valence, correlation-consistent, polarisation-consistent and individual gauge for localised orbitals basis sets is studied. For \(\varDelta \)self-consistent field calculations of core-electron binding energies and core-excitation energies of first-row elements, relatively small basis sets can accurately reproduce the values of much larger basis sets, with the IGLO basis sets performing particularly well. Calculations for the *K*-edge of second-row elements are more challenging, and of the smaller basis sets, pcSseg-2 has the best performance. For the correlation-consistent basis sets, inclusion of core-valence correlation functions is important, with the cc-pCVTZ basis set giving accurate results. Time-dependent density functional theory-based calculations of core-excitation energies show less sensitivity to the basis set with relatively small basis sets, such as pcSseg-1 or pcSseg-2, reproducing the values for much larger basis sets accurately. In contrast, time-dependent density functional theory calculations of X-ray emission energies are highly dependent on the basis set, but the IGLO-II, IGLO-III and pcSseg-2 basis sets provide a good level of accuracy.

## Keywords

Basis sets Core-electron X-ray spectroscopy## 1 Introduction

Spectroscopy in the X-ray region has become firmly established as a key technique for the study of the electronic and geometrical structure of chemical and biological systems. Furthermore, the development of X-ray free-electron lasers that can deliver short femtosecond pulses of X-rays has opened up the possibility of resolving ultrafast chemical processes at an atomic level. Recent examples of experimental work include studies on liquids [1, 2], the composition of an active site in a metalloprotein [3], the nature of bonding in metal containing complexes [4] and the real-time monitoring of bond breaking of a carbon monoxide molecule absorbed on a metal surface [5]. Computational simulations of X-ray spectroscopy often play an important role in the interpretation of experimental data. Within the context of quantum chemical calculations, core-electron binding energies (CEBEs) are most commonly computed using a \(\varDelta \)self-consistent field (\(\varDelta \)SCF) approach [6, 7, 8, 9], although an unrestricted generalised transition state method has also been proposed for the calculation of CEBEs [10, 11]. X-ray absorption spectra can be computed using the transition potential method [12], time-dependent density functional theory (TDDFT) [13, 14], Bethe–Salpeter equation [15], coupled cluster theory [16, 17], the algebraic diagrammatic construction (CVS-ADC) scheme [18] and multi-reference methods [19]. TDDFT and EOM-CCSD methods have also been used to study X-ray emission spectroscopy through the use of a reference determinant with a core–hole [20, 21, 22, 23]. More recently, resonant inelastic X-ray scattering spectra have been simulated based upon multi-reference wavefunction methods [24] and also using Kohn–Sham density functional theory with a core-excited reference determinant [25].

Common to all these approaches for simulating core-electron spectroscopies is the choice of basis set used in the calculation. In many applications, the simulation of core-electron spectroscopy can be computationally demanding, for example, in the study of large systems such as metalloprotein active sites or the study of liquids where is necessary to incorporate averaging over conformations. Consequently, it is important to understand which are the most efficient basis sets for these calculations, i.e. which basis sets provide a good approximation to the complete basis set limit with fewest basis functions. There are a number of well-established families of Gaussian basis sets, including the split-valence basis sets of Pople [26, 27, 28, 29, 30, 31, 32] and the correlation-consistent basis sets of Dunning [33, 34, 35, 36, 37, 38]. More recently, Jensen has introduced the polarisation-consistent basis sets [39, 40, 41]. A common feature of all of these basis sets is that they are designed for the calculation of properties that primarily depend on the nature of the valence electrons. Consequently, their performance for the calculation of properties that depend on the core-electrons is less well understood. In order to describe core orbitals accurately, it is often necessary to add additional tight basis functions and correlation-consistent basis sets are available with functions that describe core-electron correlation (cc-pCVXZ). The calculation of nuclear magnetic (NMR) shielding constants represents an example of a molecular property that requires a good representation of the orbitals near the nuclei, and basis sets have been designed for the calculation of NMR properties. For example, the individual gauge for localised orbitals (IGLO) basis sets [42, 43] are often used for the calculation of magnetic properties; however, they have only been defined for hydrogen and the first- and second-row p-block elements. A family of segmented contracted basis sets, denoted pcSseg-n, optimised for the calculation of nuclear magnetic shielding constants has also been reported [44].

Similar to calculations of NMR spectroscopy, accurate calculations of spectroscopy in the X-ray region also rely on a correct description of the core orbitals. Several studies have considered the basis set dependence of CEBEs calculations using a \(\varDelta \)SCF approach in conjunction with density functional theory (DFT) or Møller–Plesset perturbation theory [9, 45, 46, 47, 48, 49, 50]. It has been reported that the core-valence correlated triple-zeta basis set (cc-pCVTZ) to be accurate and efficient compared with the cc-pV5Z basis set. It was also found that exponent scaled basis sets did not perform well [45]. In another study using a \(\varDelta \)SCF approach, it was found that CEBEs calculated with large basis sets could be reproduced to within 0.2 eV by optimising the exponents and contraction coefficients of relatively small basis sets for the core–hole state [47]. The use of Slater-type basis functions has also been explored for the calculation of CEBEs and the results indicate that polarised triple-zeta basis set of Slater-type orbitals to be adequate [48]. In more recent work, the performance of a range of basis sets and exchange–correlation functionals was investigated for the calculation of CEBEs of first-row hydrides and glycine [50]. The inclusion of polarisation and diffuse functions on the heavy atoms was found to have a significant effect for medium-sized basis sets, such as 6-311G. Several density functionals were found to perform well with large basis sets that have considerable flexibility in the core region. For example, the B3LYP5 and TPSSh functionals had a mean unsigned error of less than 0.2 eV with a fully uncontracted triple-zeta quality basis set augmented with diffuse and polarisation functions. The majority of studies of CEBEs have only considered excitations from the 1s orbitals of first-row elements. Segala and Chong assessed DFT-based calculations of CEBEs for a set of 40 sulphur and phosphorous containing molecules [51]. An additional complication for second-row elements is that relativistic effects become significant and an empirical correction to account for these effects was used. A wide variety of exchange–correlation functionals were considered, and the most accurate functional was found to be VS8 [52], with an average unsigned error of 0.43 eV. A few studies have considered core-excitation energies in addition to CEBEs. In calculations on the formaldehyde molecule, it was concluded that diffuse basis functions were important for simulating X-ray absorption spectra but less important for CEBEs and X-ray emission spectra [49]. In a study of core-excitation energies and CEBEs, it was found that using the 6-311G(d,p) basis set with uncontracted basis functions gave results that were comparable with the much larger cc-pCVQZ basis set [9]. There has been much less attention to the basis set dependence of X-ray emission energies, although one study has found that X-ray emission energies of transition metal complexes computed using TDDFT were highly dependent on the basis set used [23].

In this paper, we explore the basis set dependence of DFT-based calculations of a range of core-electron spectroscopies, including CEBEs computed using a \(\varDelta \)SCF approach, core-excitation energies computed using \(\varDelta \)SCF and TDDFT, and X-ray emission energies computed using TDDFT. We consider a set of 34 molecules that includes excitations from first- and second-row s- and p-block elements and wide range of basis sets with the goal of identifying the most efficient and reliable basis sets for calculations of core-electron spectroscopies.

## 2 Computational details

Molecules used in the study

s-block | | | | |

| | | | |

| | | | |

| ||||

p-block | | | | |

| | | HC | |

CH\(_3\)C | | H\(_2\) | B | |

| | | | |

| | H\(_2\) | H\(_2\)C | |

H |

CEBEs were computed using a \(\varDelta \)SCF approach in conjunction with DFT using the B97-1 exchange–correlation functional [53]. The focus of this study is the variation of the computed core-electron properties with basis set, and the B97-1 functional is assumed to be representative of DFT exchange–correlation functionals. This is explored further in this work where the basis set dependence of difference exchange–correlation functionals is compared. The core-ionised states were optimised using an overlap criterion [54] to maintain the core–hole during the SCF process. In a similar manner, \(\varDelta \)SCF core-excitation energies were computed for the lowest excitation energy arising from an excitation from the relevant core orbital to the lowest unoccupied molecular orbital (LUMO). We note that simulation of X-ray absorption spectra will typically involve excitation to higher-lying orbitals that may be diffuse (Rydberg) in nature. The performance of the basis sets for these states is not assessed directly, and as noted later, the addition of diffuse basis functions would be needed to describe these states adequately. Core-excitation energies were also computed for the core to LUMO transitions with TDDFT. TDDFT can be applied to compute core-excitation energies through limiting the single excitation subspace to include only excitations from the relevant core orbital(s) [13]. It is well known that standard exchange–correlation functionals underestimate core-excitation energies when computed using TDDFT, and several groups have developed functionals specifically designed for core-excitation energies [55, 56, 57, 58, 59]. However, in this study we use the B97-1 functional throughout since we are not primarily concerned with a direct comparison with experiment. X-ray emission energies where computed for the highest occupied molecular orbital (HOMO) \(\rightarrow \) core transition by applying TDDFT to a Kohn–Sham determinant with a core hole in a procedure described in more detail elsewhere [21]. All calculations use an unrestricted Kohn–Sham formalism except the TDDFT calculations of core-excitation energies. A range of molecules (shown in Table 1) including first- and second-row s- and p-block elements was considered, with the structure of the molecules optimised at the B97-1/6-311G(d,p) level of theory. This set of molecules was chosen to include the core-excitations from the range of elements in the first and second rows of the periodic table.

*N*, where

*N*is the number of contracted basis functions for a first- and second-row p-block atom is introduced as a measure of the size of the basis set. In determining

*N*, a pure representation of the

*d*and higher angular momentum basis functions is assumed. All calculations were performed with the Q-Chem software package [65].

Basis sets used in this study

Split-valence | Ahlrichs | IGLO | ANO | Correlation consistent | Polarisation consistent |
---|---|---|---|---|---|

STO-3G | Ahlrichs VDZ | IGLO-II | ROOS aug-VTZ | cc-pVDZ | pcS-0 |

6-31G(d,p) | Ahlrichs VTZ | IGLO-III | cc-pCVDZ | pcSseg-0 | |

6-311G(2df,2pd) | cc-pVTZ | pcS-1 | |||

u6-31G(d,p) | cc-pCVTZ | pcSseg-1 | |||

cc-pVQZ | pcSseg1 | ||||

cc-pV5Z | pcS-2 | ||||

pcSseg-2 | |||||

aug-pcSseg-2 | |||||

pcS-3 | |||||

pcSseg-3 | |||||

pcS-4 | |||||

pcSseg-4 |

## 3 Results and discussion

### 3.1 \(\varDelta \)SCF core-electron binding energies and core-excitation energies

Error in calculated core-electron binding energies (in eV)

Basis set | s-block | p-block | Total | MAE | |
---|---|---|---|---|---|

\(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | |||

STO-3G | 3.00 (4.83) | 5.69 (6.68) | 4.69 | 29.68 (PF\(_3\)) | 14 |

pcS-0 | 6.86 (6.24) | 5.13 (3.89) | 5.77 | 14.40 (MgCl\(_2\)) | 22 |

pcSseg-0 | 9.09 (8.54) | 4.64 (4.19) | 6.29 | 18.80 (NaH) | 22 |

Ahlrichs VDZ | 13.16 (8.42) | 6.03 (5.21) | 8.68 | 23.78 (MgCl\(_2\)) | 22 |

6-31G(d,p) | 9.09 (8.47) | 5.93 (5.50) | 7.10 | 20.35 (NaH) | 32 |

cc-pVDZ | 10.14 (8.53) | 6.51 (5.77) | 7.10 | 20.60 (NaH) | 32 |

Ahlrichs VTZ | 3.92 (2.65) | 2.43 (4.37) | 2.98 | 18.10 (PF\(_3\)) | 37 |

pcS-1 | 1.30 (0.40) | 2.86 (3.25) | 2.28 | 11.94 (PF\(_3\)) | 38 |

pcSseg-1 | 1.98 (0.56) | 2.23 (2.53) | 2.14 | 12.38 (PF\(_3\)) | 38 |

u6-31G(d,p) | 7.50 (7.78) | 5.99 (6.73) | 6.55 | 24.48 (H\(_2\)CS) | 42 |

cc-pCVDZ | 2.24 (1.62) | 2.25 (3.91) | 2.25 | 18.04 (PF\(_3\)) | 45 |

IGLO-II | – | 0.89 (1.85) | – | 8.47 (H\(_2\)CS) | 57 |

cc-pVTZ | 6.42 (6.88) | 2.99 (3.62) | 4.20 | 15.30 (NaF) | 64 |

6-311G(2df,2pd) | 0.58 (1.00) | 1.42 (2.43) | 1.11 | 9.02 (H\(_2\)CS) | 68 |

pcS-2 | 0.86 (0.92) | 1.09 (2.23) | 1.00 | 2.23 (H\(_2\)CS) | 68 |

pcSseg-2 | 0.89 (0.76) | 1.05 (0.82) | 0.99 | 4.34 (H\(_2\)CS) | 68 |

IGLO-III | – | 0.89 (1.80) | – | 8.15 (H\(_2\)CS) | 79 |

ROOS aug-VTZ | 5.49 (6.23) | 3.16 (3.80) | 4.03 | 14.05 (NaH) | 96 |

cc-pCVTZ | 0.69 (1.00) | 0.88 (1.78) | 0.81 | 8.05 (H\(_2\)CS) | 102 |

aug-pcSseg-2 | 0.59 (0.98) | 1.32 (2.48) | 1.05 | 10.56 (H\(_2\)CS) | 105 |

cc-pVQZ | 5.72 (6.66) | 3.13 (3.88) | 4.09 | 14.97 (NaF) | 114 |

pcS-3 | 0.66 (1.17) | 0.79 (1.79) | 0.74 | 8.26 (H\(_2\)CS) | 150 |

pcSseg-3 | 0.68 (1.17) | 0.15 (0.12) | 0.35 | 4.32 (MgCl\(_2\)) | 150 |

cc-pV5Z | 1.11 (1.47) | 0.30 (1.12) | 0.60 | 5.20 (H\(_2\)CS) | 186 |

pcS-4 | 0.93 (1.38) | 0.93 (1.74) | 0.93 | 7.78 (H\(_2\)CS) | 244 |

Error in core-electron binding energies for row 1 and row 2 nuclei (in eV)

Basis set | Row 1 | Row 2 |
---|---|---|

\(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | |

6-31G(d,p) | 1.98 | 13.53 |

cc-pVDZ | 2.72 | 14.52 |

Ahlrichs VTZ | 2.31 | 4.22 |

pcSseg-1 | 1.64 | 2.81 |

u6-31G(d,p) | 1.28 | 13.23 |

cc-pCVDZ | 1.54 | 3.19 |

IGLO-III | 0.06 | 2.23 |

cc-pVTZ | 0.58 | 7.26 |

6-311G(2df,2pd) | 0.59 | 1.75 |

pcS-2 | 0.61 | 1.53 |

pcSseg-2 | 0.99 | 1.02 |

IGLO-III | 0.04 | 2.29 |

cc-pCVTZ | 0.09 | 1.74 |

cc-pVQZ | 0.34 | 8.89 |

pcSseg-3 | 0.06 | 0.72 |

Variation in basis set error for different exchange–correlation functionals (in eV)

Molecule | Experiment | Method | pcSseg-4 | pcSseg-3 | cc-pCVTZ | IGLO-III | pcSseg-2 | pcSseg-1 |
---|---|---|---|---|---|---|---|---|

CH\(_4\) | 290.91 | B97-1 | 291.13 | + 0.08 | + 0.03 | − 0.02 | + 1.04 | + 1.28 |

PBE0 | 290.31 | + 0.08 | + 0.03 | − 0.02 | + 1.01 | + 1.23 | ||

M06 | 290.30 | + 0.00 | − 0.09 | − 0.07 | + 0.94 | + 1.17 | ||

HF | 290.71 | + 0.09 | + 0.01 | − 0.04 | + 0.99 | + 1.21 | ||

NH\(_3\) | 405.56 | B97-1 | 406.87 | − 0.06 | − 0.04 | − 0.02 | + 1.09 | + 1.20 |

PBE0 | 406.01 | − 0.05 | − 0.04 | − 0.02 | + 1.06 | + 1.16 | ||

M06 | 404.90 | − 0.10 | − 0.04 | − 0.07 | + 1.05 | + 1.33 | ||

HF | 405.30 | − 0.02 | − 0.05 | − 0.04 | + 1.06 | + 1.21 | ||

H\(_2\)O | 539.90 | B97-1 | 539.96 | + 0.09 | − 0.09 | − 0.06 | + 1.13 | + 1.19 |

PBE0 | 538.79 | + 0.09 | − 0.07 | − 0.06 | + 1.12 | + 1.15 | ||

M06 | 539.15 | + 0.04 | − 0.14 | − 0.09 | + 1.10 | + 1.40 | ||

HF | 539.28 | + 0.11 | − 0.03 | − 0.03 | + 1.11 | + 1.24 | ||

PH\(_3\) | 2150.69 | B97-1 | 2151.40 | + 0.33 | − 0.04 | − 1.44 | + 0.76 | + 2.06 |

PBE0 | 2150.52 | + 0.24 | − 0.04 | − 1.55 | + 0.66 | + 1.89 | ||

M06 | 2152.46 | + 0.20 | − 0.04 | − 1.60 | + 0.66 | + 1.80 | ||

HF | 2154.04 | + 0.22 | − 0.05 | − 1.50 | + 0.68 | + 2.13 | ||

H\(_2\)S | 2478.58 | B97-1 | 2478.60 | + 0.37 | − 1.50 | − 1.54 | + 0.89 | + 1.98 |

PBE0 | 2477.73 | + 0.37 | − 1.51 | − 1.55 | + 0.89 | + 1.95 | ||

M06 | 2479.67 | + 0.35 | − 1.68 | − 1.55 | + 0.88 | + 1.98 | ||

HF | 2481.56 | + 0.35 | − 1.48 | − 1.52 | + 0.93 | + 2.19 |

The large basis sets pcS-2, pcSseg-2, pcS-3 and pcSseg-3 all have overall MADs of 1 eV or less with a well-balanced performance between s- and p-block elements. For the pcSseg-2 basis set, the errors for the large majority of the molecules lie in the range of 0–1.5 eV with a few molecules showing significantly larger errors with a largest error of 4.3 eV for H\(_2\)CS. The inclusion of diffuse basis functions in aug-pcSseg-2 leads to a decrease in the accuracy of the calculated CEBEs compared with pcSseg-4. However, assessment of this basis set relative to a very large basis set that includes diffuse functions might be more appropriate. The pcSseg-3 basis set has an overall error of 0.35 eV with an error of 0.15 eV for the p-block elements which represents a very good level of accuracy. The IGLO-III basis set also performs well, but it is surprising that IGLO-III shows no improvement over IGLO-II. The results also show that the basis set errors for the pcSseg-n basis sets are smaller than for their counterpart pcS-n basis sets, with the exception of pcS-0 and pcSseg-0. This and the good performance of the IGLO basis sets demonstrate that basis sets designed for the prediction of NMR also perform well in the calculation of CEBEs. One feature of the data is the poor performance of the standard correlation-consistent basis sets. Similar behaviour is observed for the ANO basis set. In general, the correlation-consistent basis sets perform poorly, particularly for the s-block elements. For these basis sets, large errors remain for the cc-pVTZ and cc-pVQZ basis sets and this is significantly reduced for the cc-pV5Z basis set. This trend is also observed in the MADs evaluated relative to the cc-pV5Z values, with an error of over 4 eV for the cc-pVQZ basis set relative to the cc-pV5Z basis set. In comparison, the pcSseg-2 and pcSseg-3 basis sets have overall \(\varDelta ^\mathrm{MAD}\) of 1.19 and 0.61 eV relative to the cc-pV5Z basis set. There is a substantial improvement in performance of the correlation-consistent basis sets with core-valence correlation functions. The cc-pCVTZ basis set has an error of <1 eV for s- and p-block elements. This basis set has been observed to be accurate in previous studies [45].

Error in calculated \(\varDelta \)SCF core \(\rightarrow \) LUMO excitation energies (in eV)

Basis set | s-block | p-block | Total | MAE |
---|---|---|---|---|

\(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | ||

STO-3G | 2.18 (2.54) | 4.48 (3.38) | 3.69 | 11.51 (HCl) |

pcS-0 | 7.00 (5.74) | 5.42 (4.29) | 6.01 | 14.09 (MgF\(_2\)) |

pcSseg-0 | 9.15 (8.83) | 4.96 (4.54) | 6.52 | 19.47 (Mg\(_2\)H\(_4\)) |

Ahlrichs VDZ | 13.08 (7.81) | 6.68 (5.75) | 9.06 | 22.20 (MgCl\(_2\)) |

6-31G(d,p) | 8.79 (8.55) | 6.88 (5.93) | 7.59 | 20.14 (NaH) |

cc-pVDZ | 10.15 (8.74) | 7.46 (6.35) | 8.46 | 20.39 (NaH) |

Ahlrichs VTZ | 3.83 (2.63) | 0.65 (1.53) | 1.83 | 10.67 (Be\(_2\)H\(_4\)) |

pcS-1 | 1.29 (0.56) | 2.99 (2.69) | 2.36 | 6.95 (F\(_2\)) |

pcSseg-1 | 2.05 (0.64) | 1.77 (1.13) | 1.87 | 6.27 (F\(_2\)) |

u6-31G(d,p) | 7.57 (7.95) | 5.41 (6.39) | 6.21 | 17.10 (NaH) |

cc-pCVDZ | 1.95 (1.67) | 1.11 (0.64) | 1.42 | 5.37 (MgF\(_2\)) |

IGLO-II | – | 1.10 (1.51) | – | 6.87 (F\(_2\)) |

cc-pVTZ | 6.46 (7.04) | 3.27 (3.52) | 4.45 | 15.01 (NaF) |

6-311G(2df,2pd) | 0.73 (1.54) | 2.04 (4.03) | 1.55 | 17.49 (Si\(_2\)H\(_2\)) |

pcS-2 | 0.62 (0.27) | 1.00 (1.47) | 0.86 | 6.87 (F\(_2\)) |

pcSseg-2 | 0.65 (0.47) | 1.12 (1.22) | 0.95 | 6.20 (F\(_2\)) |

IGLO-III | – | 0.61 (0.62) | – | 1.57 (AlH\(_3\)) |

ROOS aug-VTZ | 5.40 (6.04) | 2.97 (3.68) | 3.87 | 13.94 (NaH) |

cc-pCVTZ | 0.88 (1.49) | 0.78 (0.83) | 0.82 | 5.47 (Mg\(_2\)H\(_4\)) |

aug-pcSseg-2 | 0.73 (1.52) | 0.85 (1.43) | 0.81 | 6.86 (F\(_2\)) |

cc-pVQZ | 5.54 (6.64) | 3.56 (4.11) | 4.30 | 14.67 (NaF) |

pcS-3 | 0.36 (0.42) | 0.45 (0.55) | 0.42 | 1.25 (HCl) |

pcSseg-3 | 0.11 (0.17) | 0.18 (0.12) | 0.15 | 0.65 (Mg\(_2\)H\(_4\)) |

cc-pV5Z | 1.16 (1.43) | 0.10 (0.11) | 0.49 | 3.99 (MgF\(_2\)) |

pcS-4 | 1.06 (1.87) | 0.77 (0.78) | 0.88 | 6.67 (Mg\(_2\)H\(_4\)) |

*N*) for the basis sets with \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) < 3 eV (Fig. 1).

*N*is the number of contracted basis functions for a first- and second-row p-block atom and provides a rough measure of the size of the basis sets. The graph shows that, in general, the error arising from the basis set decreases as the size of the basis increases. However, it does highlight basis sets that show good or poor performance in relation to their size. For the correlation-consistent basis sets without core-valence correlation functions, the large cc-pV5Z basis set is required to achieve a reasonable level of accuracy with respect to the basis set limit. However, the use of such a large basis set is not practical for the majority of calculations.

Error in core \(\rightarrow \) LUMO excitation energies for row 1 and row 2 nuclei (in eV)

Basis set | Row 1 | Row 2 |
---|---|---|

\(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | |

6-31G(d,p) | 2.21 | 15.06 |

cc-pVDZ | 2.69 | 16.45 |

Ahlrichs VTZ | 2.27 | 1.64 |

pcSseg-1 | 1.70 | 2.08 |

cc-pCVDZ | 1.37 | 1.33 |

IGLO-II | 0.94 | 1.35 |

cc-pVTZ | 0.77 | 9.28 |

6-311G(2df,2pd) | 0.78 | 2.82 |

pcS-2 | 0.98 | 0.71 |

pcSseg-2 | 1.21 | 0.58 |

IGLO-III | 0.10 | 0.87 |

cc-pCVTZ | 0.30 | 1.55 |

cc-pVQZ | 0.45 | 9.54 |

pcSseg-3 | 0.11 | 0.23 |

The IGLO basis sets show the best performance of the smaller basis sets, and IGLO-II is a particularly cost-effective basis set. Unfortunately, these basis sets are not available for the s-block elements. The pcS-2 and pcSseg-2 are the smallest basis sets that perform well for both s- and p-block elements (an error of about 1 eV or less), and the large split-valence basis set also does quite well in this regard. To achieve higher levels of accuracy, it is necessary to use the cc-pCVTZ or pcSseg-3 basis sets, with an overall error of less than 0.5 eV for the pcSseg-3 basis set. To put this in context in terms of the computational cost, a single point energy calculation for PF\(_3\) with the pcSseg-2 and pcSseg-3 basis sets take about 1% and 6% of the time for the pcSseg-4 basis set when run on a single processor.

### 3.2 TDDFT core-excitation and emission energies

Error in calculated TDDFT core-excitation energies (in eV)

Basis set | s-block | p-block | Total |
---|---|---|---|

\(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | |

6-31G(d,p) | 0.50 (0.27) | 1.01 (0.66) | 0.82 |

cc-pVDZ | 0.47 (0.33) | 0.87 (0.47) | 0.72 |

Ahlrichs VTZ | 2.47 (2.35) | 0.50 (0.31) | 1.23 |

pcSseg-1 | 0.39 (0.35) | 0.32 (0.32) | 0.34 |

cc-pCVDZ | 0.26 (0.27) | 0.49 (0.50) | 0.41 |

IGLO-II | – | 0.28 (0.32) | – |

cc-pVTZ | 0.48 (0.48) | 0.50 (0.40) | 0.50 |

pcSseg-2 | 0.09 (0.06) | 0.10 (0.12) | 0.10 |

IGLO-III | – | 0.14 (0.16) | – |

Table 8 shows \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) for the core \(\rightarrow \) LUMO transition energies computed using TDDFT. These calculations are performed for a subset of the basis sets. The variation in the computed excitation energies between the different basis sets is considerably smaller for the TDDFT calculations compared to the \(\varDelta \)SCF calculations. As a consequence, very large basis sets are not necessary for TDDFT calculations of core-excitations. With the exception of the Ahlrichs VTZ basis set, all of the basis sets shown have an overall error of less than 1 eV, and this includes the 6-31G(d,p) and correlation-consistent basis sets. The pcSseg-1 and pcSseg-2 basis sets reproduce the values for the larger basis set very well, and for the p-block elements, the IGLO basis sets also perform well. As discussed earlier, the large error associated with the basis set in \(\varDelta \)SCF calculations arises predominantly from the calculation of the core-ionised or core-excited state. The TDDFT calculations are based upon the ground-state molecular orbitals which are described relatively well by the small basis sets with the result that the basis set is a less crucial factor.

Error in calculated TDDFT emission energies (in eV)

Basis set | s-block | p-block | Total |
---|---|---|---|

\(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | \(\varDelta ^\mathrm{MAD}\)(pcSseg-4) | |

6-31G(d,p) | 18.38 (17.28) | 14.09 (12.81) | 15.69 |

cc-pVDZ | 18.79 (17.54) | 13.79 (12.03) | 15.64 |

Ahlrichs VTZ | 5.21 (3.46) | 1.61 (1.98) | 2.94 |

pcSseg-1 | 3.75 (1.123) | 3.52 (1.43) | 3.60 |

cc-pCVDZ | 4.37 (3.60) | 2.45 (1.74) | 3.16 |

IGLO-II | – | 1.67 (1.92) | – |

cc-pVTZ | 11.89 (12.84) | 4.72 (7.11) | 7.38 |

pcSseg-2 | 1.21 (0.65) | 2.05 (1.33) | 1.74 |

IGLO-III | – | 1.27 (1.41) | – |

pcSseg-3 | 0.36 (0.38) | 0.58 (0.72) | 0.50 |

Basis set dependence of the computed oscillator strengths for core \(\rightarrow \) LUMO transitions

Molecule | Method | pcSseg-4 | pcSseg-3 | cc-pCVTZ | IGLO-III | pcSseg-2 | pcSseg-1 | cc-pVDZ |
---|---|---|---|---|---|---|---|---|

CO | TDDFT | 0.031 | 0.031 | 0.034 | 0.032 | 0.032 | 0.033 | 0.033 |

\(\varDelta \)SCF | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.018 | 0.017 | |

HCN | TDDFT | 0.037 | 0.038 | 0.047 | 0.041 | 0.042 | 0.045 | 0.049 |

\(\varDelta \)SCF | 0.025 | 0.025 | 0.026 | 0.025 | 0.026 | 0.026 | 0.024 | |

H\(_2\)O | TDDFT | 0.006 | 0.006 | 0.013 | 0.009 | 0.008 | 0.010 | 0.013 |

\(\varDelta \)SCF | 0.006 | 0.006 | 0.007 | 0.006 | 0.006 | 0.006 | 0.007 | |

AlH\(_3\) | TDDFT | 0.004 | 0.004 | 0.004 | 0.005 | 0.004 | 0.004 | 0.004 |

\(\varDelta \)SCF | 0.004 | 0.004 | 0.004 | 0.004 | 0.004 | 0.005 | 0.003 | |

H\(_2\)S | TDDFT | 0.002 | 0.003 | 0.004 | 0.004 | 0.003 | 0.004 | 0.005 |

\(\varDelta \)SCF | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.002 |

## 4 Conclusion

The basis set dependence of DFT-based calculations of spectroscopy in the X-ray region has been assessed. \(\varDelta \)SCF calculations of CEBEs and core \(\rightarrow \) LUMO excitation energies, and TDDFT calculations of core-excitation and emission energies have been studied for a range of molecules including excitations from the core orbitals of elements from the first and second rows of the periodic table. A range of widely used basis sets have been used with the aim of identifying relatively small basis sets that perform well for these calculations. For \(\varDelta \)SCF calculations at the *K*-edge of first-row elements, relatively small basis sets can accurately reproduce of core-electron binding energies and core-excitation energies of much larger basis sets. The IGLO-II, IGLO-III and cc-pCVTZ basis sets perform well, with IGLO-II being a particularly cost-effective basis set. However, application of the IGLO basis sets is limited to p-block elements, and for s-block elements, the cc-pCVTZ or large split-valence (at least 6-311G(d,p) quality) is recommended. Calculations for second-row elements are more challenging, and the pcSseg-2 basis set has the best performance of the smaller basis sets, with pcSseg-3 required for a greater level of accuracy significantly less than 1 eV. The standard correlation-consistent basis sets have large errors. However, the core-valence correlation versions of the basis sets (cc-pCVXZ) are much more accurate but less accurate than pcSseg-n basis sets of comparable size. For these calculations, the basis set error is predominantly associated with the calculation of the core-excited or core-ionised state. This suggests that many of the smaller basis sets lack the flexibility to adjust for the change in effective nuclear charge associated with removing a core-electron. Furthermore, smaller basis sets provide a poor description of the radial behaviour of the 1s orbital and some correlation between the quality of the description of the core orbital and size of the basis set error. The TDDFT calculations of the core-excitation energies show less sensitivity to the basis set used, and relatively small basis sets reproduce the excitation energies of the larger basis sets well. For these calculations, the pcSseg-1 and pcSseg-2 perform well, although the versions augmented with diffuse basis functions may be more appropriate for some applications. In contrast, TDDFT calculations of X-ray emission energies show a high dependence on the basis set used. This can be rationalised by the fact that TDDFT calculations of excitation energies are based upon the ground-state Kohn–Sham determinant, while TDDFT calculations of emission energies used the core-ionised determinant. For these calculations, the IGLO-II, IGLO-III and pcsSeg-2 basis sets provide a good level of accuracy for the *K*-edge of both row 1 and row 2 elements.

## Notes

### Acknowledgements

This work was supported by the Leverhulme Trust under Grant (RPG-2016-103).

## Supplementary material

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