Performance of the TPSS Functional on Predicting Core Level Binding Energies of Main Group Elements Containing Molecules: A Good Choice for Molecules Adsorbed on Metal Surfaces

Here we explored the performance of Hartree-Fock (HF), Perdew-Burke-Ernzerhof (PBE), and Tao-Perdew-Staroverov-Scuseria (TPSS) functionals in predicting core level 1s binding energies (BEs) and BE shifts (ΔBEs) for a large set of 68 molecules containing a wide variety of functional groups for main group elements B → F and considering up to 185 core levels. A statistical analysis comparing with X-ray photoelectron spectroscopy (XPS) experiments shows that BEs estimations are very accurate, TPSS exhibiting the best performance. Considering ΔBEs, the three methods yield very similar and excellent results, with mean absolute deviations of ∼0.25 eV. When considering relativistic effects, BEs deviations drop approaching experimental values. So, the largest mean percentage deviation is of 0.25% only. Linear trends among experimental and estimated values have been found, gaining offsets with respect to ideality. By adding relativistic effects to offsets, HF and TPSS methods underestimate experimental values by solely 0.11 and 0.05 eV, respectively, well within XPS chemical precision. TPSS is posed as an excellent choice for the characterization, by XPS, of molecules on metal solid substrates, given its suitability in describing metal substrates bonds and atomic and/or molecular orbitals.


Introduction
X-ray Photoelectron Spectroscopy (XPS), also known as Electron Spectroscopy for Chemical Analysis (ESCA), is an experimental technique available in many materials and surface science laboratories and facilities and is essentially used for the chemical or elemental analysis of bulk materials, but especially for surfaces because of its surface sensitivity. 1,2 In addition, XPS is a technique currently used to observe in situ the evolution of an heterogeneously catalyzed reaction, allowing for the characterization of reactants, intermediates, and products, thus serving as a powerful tool to determine the reaction mechanism. [3][4][5] The XPS performance on elemental analysis hangs on the measurements of core level electron Binding Energies (BEs). The BEs are characteristic of a given element and, more importantly, of a given element in a given chemical environment and electronic state. Thus, BEs provide quantitative information of the chemical composition and also on the physicochemical properties including insights of the electronic structure, thus providing more significant information apart from those obtained by the elemental analyses. 1-5 BEs from XPS experiments in condensed phase systems provide chemical fingerprints but also reflect the chemical properties and bonding between the species in a given system. 6 The small variations for a type of atom in different chemical environments are often referred to as BEs shifts (ΔBEs) and provide meaningful information. This is used to distinguish structural as well as oxidation state of the atom. 7 Core level BEs and their respective shifts can be accurately predicted from ab initio calculations at the well-known Hartree-Fock (HF) theory 8-10 to currently used Density Functional Theory (DFT) based methods, 11-13 especially with hybrid functionals 3 as recently illustrated for N-containing molecules in gas phase. 14,15 Nevertheless, one must admit that prediction of BEs and especially of ΔBEs will be especially relevant and highly desirable for molecules adsorbed on metal surfaces. In fact, transition metal (TM) surfaces are ubiquitous in heterogeneous catalysis as supported active phases, but also employed in chemical resolution, nanotechnology, synthesis, and other related processes of technological interest.
Unfortunately, hybrid functionals are not especially well suited for metals 16,17 because of the failure to attain the exact homogeneous electron gas limit. 18 In fact, recent studies [19][20][21] show that, among various DFT methods including Generalized Gradient Approximation (GGA) type, meta-GGA and hybrid density functionals, the GGA-type Perdew-Burke-Ernzerhof (PBE) 22 implementation of the exchangecorrelation potential provides the best description of the structural, energetic, physical, and electronic properties for all 3d, 4d, and 5d TMs. On the other hand, hybrid functionals 23 are required for a good description of the thermochemistry of main group elements. 24 In heterogeneous catalysis one has simultaneously to describe main group element molecules and transition metal surfaces which leads to a dilemma regarding the appropriate choice of the DFT method, GGA type functionals being usually selected. 25 Here it is worth to mention that recently it has been shown that the meta-GGA type Tao-Perdew-Staroverov-Scuseria (TPSS) 26 functional provides a description of the three transition metal series nearly as accurate as PBE. 19 Note also that compared to PBE and other GGA functionals, being a meta-GGA, TPSS represents an improvement in describing the thermochemistry of main group molecules. In this sense, TPSS seems to be a good choice to describe the interaction of molecules with transition metal surfaces.
In the present work we assess the performance TPSS in providing estimates of BEs and ΔBEs for a wide variety of molecules including the main group heteroatoms 4 (B→F) and encompassing possible chemical interactions by selecting a vast dataset, with the eye put on subsequent studies aimed at describing similar molecules in contact to metal surfaces. For comparison, HF and PBE are also used since HF is a standard method in molecular quantum chemistry and PBE is a prototypal functional used in theoretical heterogeneous catalysis and surface science. 25

Core Level Binding Energy Theory Background
Core level BEs can be rather accurately predicted from ab initio calculations, such as HF or DFT based methods. [8][9][10][11][12][13][14][15] Absolute BEs are obtained from the difference between the total energies of the neutral state and the same system with a core hole configuration generated by subtracting one electron in the desired core level as in Eq. 1.
A possible way to obtain the energy values in Eq. 1 is to make use of separate Self-Consistent-Field (SCF) calculations. The resulting procedure usually referred to as ΔSCF approach 8,10 and Eq. 1 usually written as where the subindex i in ! indicates the ionized core whereas E N SCF and E i N-1 are the variationally optimized energy for the initial system with N electrons and the final systems with (N-1) electrons and the corresponding i core hole. It is customary to decompose ! into initial and final states .9,10 This, however, will not be considered here since the main goal is to assess the accuracy on the BEs rather than to analyze the initial and final state contributions. The chosen molecular data set is listed in Appendix A and it is fully described in

Results and Discussion
Let BEs is also given in Table 1  level BEs are most often consistently lower than the experimental BEs as seen in Figure   1, and thus behave similar to PBE. Note by passing that for O there are outliers, in particular, O 2 and H 2 O 2 , whose disagreement is translated to the ΔBE S , see below, although one has to keep in mind that we deal here with average behaviors, instead of particular agreements. When the BE(ΔSCF) values are analyzed for B-and Ccontaining molecules, the ME and MAE values start to show a higher disagreement with experiment; the error associated is not systematic, and this is reflected in Figure 1 by values above and below the experimental ones. Nevertheless, BEs for the 1s core of these molecules get really close to the experimental ones, note that the difference is only 0.3 eV for both HF and TPSS, close to the XPS experimental precision of ∼0.1 eV. 9 TPSS is slightly better than HF by ~0.04 eV. This better adequacy increases for N→F to improvements up to 0.31 eV.
From another point of view one can analyze BEs of molecules in terms of their respective shifts relative to a reference molecule. Figure 2 represents the ΔBE(ΔSCF) values with respect to the experimental shifts, ΔBE(Exp.), for each element using the reference molecules indicated above. The most evident difference, in comparison to BE(ΔSCF) values in Figure 1, is that there is no significant variations of the shifts with respect to the used method. As expected, the systematic errors cancel each other. To further analyze the core level ΔBEs the statistical analysis is reported in Table 2. The ME results at HF and TPSS levels of theory are, in general terms, negative, with the caveat for N 1s , which appears to show an agreement within chemical accuracy.
Curiously, this happens for all ME PBE values with the caveat of F 1s results.
Notwithstanding that, the MAE values do not follow the same trend above described.
For the three studied methods the MAEs are similar one to each other as it is shown in data are reported in Table S1 in the Supporting Information.
So far, the discussion concerned non-relativistic calculations. We indeed considered the influence of relativistic effect on the 1s BEs on the studied elements.
Relativistic (Rel.) and non-relativistic (Non-Rel.) calculations have been carried out for the B→F atoms, and the relativistic contributions and the comparison with the nonrelativistic ones are shown in Table 3. The relativistic change in the final state, including the relaxation of the electrons with a core-hole, Diff., is defined as: It has been reported 15 that the relativistic changes on the initial and final states are very close with each other, which means that the relativistic changes are dominant, already at the initial state. Since we want to establish the relativistic contribution to the absolute core level BEs, the changes were studied for the BE(ΔSCF) values. The Diff.
values in Table 3 show that the relativistic 1s core BEs are larger than the nonrelativistic values, although the relativistic effects lead to a very small increase in the core level BEs for these light atoms as expected; from 0.06 eV for B to 0.75 eV for F, thus increasing along the B→F series, as expected.
When we analyze the BE(ΔSCF) and the ΔBE(ΔSCF) for the whole data set, the above obtained excellent agreement is maintained, i.e. mixing of B→F results does not disrupt the statistics. On the contrary, the overall performance is enhanced. The statistical analysis values for the overall calculations, with or without including the relativistic changes, are reported in Table 4. The absolute non-relativistic BEs and their ΔBEs are shown in Figures 3 and 4, respectively. Inspecting the data on Let us now analyze the ΔBE values for the different 1s cores. As it is shown in  Table 5. This highlights once more that these three 12 methods provide reliable estimates of the core level shifts, with very small differences with respect to the experimental values. Nevertheless, one should keep in mind that HF overestimates ΔBEs, whereas DFT underestimates them. Regarding the absolute BE(ΔSCF) values, the intercepts are much higher than for ΔBEs simply because a small deviation on the slope is magnified by the large range of energies considered, see Figure   3.
One can better determine the ideal deviation by further analyzing the offsets.
Here we gained the offsets for the non-relativistic calculations, and offsets taking into account the relativistic contributions. First, we analyze the non-relativistic offsets to give a brief idea of the cases studied and next, the study with the relativistic effects will be also discussed. For the full set, the overall obtained offsets; including or not the relativistic corrections, are always below the experiment. The non-relativistic offset results itemized for each B→F atoms studied, generally increase following the series, yet the tendency along methods is kept. Results from PBE, as previously shown, lead to results with the largest deviation respect to the linearity, from -0.7 to -1.8 eV, with a global offset of ~-1.2 eV. The results from HF and TPSS are akin to each other or, in many cases, better for TPSS. The offsets range from -0.02 to -0.8 eV, representing the closest or least deviated trend compared to ideality. When taking into account the full set, the offset is ~-0.4 eV. Including the relativistic contribution to the BEs improves the already excellent agreement with experiment. For HF and TPSS, the offsets become now -0.11 eV and -0.05 eV, respectively, well within XPS chemical accuracy.
In light of the above presented discussion, the obtained results for the three explored methods, with and without relativistic corrections, are very good when computing ΔBEs of main group organic molecules, and HF, and specially the TPSS xc 13 are excellent in predicting absolute core level BEs, when relativistic effect are considered. Thus, TPSS poses itself as an convenient and accurate method for the study of core level BEs of adsorbates of main group elements on metallic surfaces which, given its suitability in describing bulk transition metals and the thermochemistry of organic molecules, appears to be an excellent choice to study this type of systems.

Conclusions
Here we have explored the performance of HF, PBE and TPSS methods in predicting 1s core level BEs of a set of 68 molecules (185 core levels explored in total) containing a wide variety of functional groups and chemical environments for main group elements B→F. The obtained results using ΔSCF methodology have been compared to reported experimental references. This has been carried out in a nonrelativistic fashion, yet the relativistic effects have been explicitly considered on isolated atoms, and, since are known to be independent of the particular chemical environment, added a posteriori on the obtained estimates.
The analysis yields that computed absolute core level BEs are, overall, and regardless of the method, in very good agreement with the reported experimental values.
Nevertheless, it is important to highlight that TPSS values are slightly better than those predicted from HF and these two clearly better than PBE. The BEs values have also been analyzed in terms of shifts, ΔBEs, from selected references. As long as ΔBEs are concerned, the three methods yield very similar and excellent results, with mean absolute deviations of ~0.25 eV.
When relativistic effects are included BE(ΔSCF) deviations drop and estimations get much closer to the experimental values. Linear relationships have been found between experimental and estimated values, obtaining, for any method, and any set or 14 subset of data, an offset with respect to ideal linearity. Correcting the values by including the relativistic effects leads to offsets of 0.11 and 0.05 eV for HF and TPPS, respectively. Both, HF and TPSS methods slight underestimate the experimental BEs but within the accuracy of the XPS measurement. Thus, TPSS is posed as an excellent choice for the characterization, by XPS, of molecules on metal solid substrates, given its suitability in describing metal substrates bonds and atomic and/or molecular orbitals.