Improving the predictive quality of time‐dependent density functional theory calculations of the X‐ray emission spectroscopy of organic molecules

The simulation of X‐ray emission spectra of organic molecules using time‐dependent density functional theory (TDDFT) is explored. TDDFT calculations using standard hybrid exchange‐correlation functionals in conjunction with large basis sets can predict accurate X‐ray emission spectra provided an energy shift is applied to align the spectra with experiment. The relaxation of the orbitals in the intermediate state is an important factor, and neglect of this relaxation leads to considerably poorer predicted spectra. A short‐range corrected functional is found to give emission energies that required a relatively small energy shift to align with experiment. However, increasing the amount of Hartree–Fock exchange in this functional to remove the need for any energy shift led to a deterioration in the quality of the calculated spectral profile. To predict accurate spectra without reference to experimental measurements, we use the CAM‐B3LYP functional with the energy scale determined with reference to a Δself‐consistent field calculation for the highest energy emission transition.


| INTRODUCTION
The availability of improved X-ray light sources has led to an increased interest in spectroscopic techniques in the X-ray region. One attraction of these techniques is that they provide an element specific, local probe of geometric and electronic structure, and they have been applied to study a wide variety of applications, ranging from biological systems to materials. [1][2][3] X-ray emission spectroscopy (XES) measures the X-ray emission following ionization of a core electron and can be performed under non-resonant or resonant conditions. XES probes the occupied molecular orbitals and is complementary to X-ray absorption spectroscopy (XAS) which is sensitive to the unoccupied orbitals. XES is a widely used technique that has been used to probe the structure of liquids [4] and the absorption of molecules on surfaces. [1] Accurate simulation of X-ray emission spectra can play a vital role in the assignment of the spectral bands and enable the full interpretation of the experimental data. A number of different theoretical approaches have been developed and applied to the study of XES. The most formally straightforward approach to determine the energy and associated oscillator strengths for the various valence to core transitions required to simulate an X-ray emission spectrum is to approximate the transitions energies using the energy difference between the orbital energies and estimate the oscillator strengths from where ϕ c is a core orbital and ϕ v is a valence orbital. This approach has been used successfully in many studies, for example, the study of valence-to-core XES of transition metal complexes. [5,6] One of the important advantages of this approach is that it allows the spectra to be determined directly from the wavefunction [7][8][9][10] or Kohn-Sham density functional theory (DFT) [5] calculations.
Following ionization of a core electron, the effective nuclear charge of the relevant nuclei increases by one, and as a consequence the molecular orbitals will adapt to the new environment. This relaxation of the electronic structure in the presence of the core hole is described explicitly in Δself-consistent field (SCF) approaches. In this approach, separate SCF calculations are performed for each coreionized final state. The transition energies are then given by the difference in the electronic-state energies and the transition dipole moments evaluated between the wavefunctions for the initial and final states. This approach has been shown to provide accurate coreionization energies [11][12][13][14][15] ; however, it is difficult to apply to large systems owing to the number of individual SCF calculations required, and the challenge of performing SCF calculations for arbitrary electronic states. This has led to the development of methods that incorporate the electronic relaxation while avoiding the practical limitations of ΔSCF. For example, in the Z + 1 approximation, an increased nuclear charge is used for the absorbing atom [16] or the transition potential approach where a half filled core orbital is used providing a balance between final and initial states. [17] More recently, an alternative approach to computing X-ray emission spectra using conventional response theory methods, such as time-dependent DFT (TDDFT), was introduced. In this scheme, the determinant for the state with a core hole is used as a reference and the transitions of interest are found as those with negative eigenvalues. [18] In this approach, the relaxation of the orbitals in the presence of the core hole is captured by the core-ionized reference determinant. This has been applied within the context of equation-ofmotion coupled cluster theory, [19] TDDFT [20] and the algebraic construction methods. [21] An advantage of this approach compared with ΔSCF is that the complete spectrum can be determined in a single calculation. The XES of transition metal systems have been studied using TDDFT, [22][23][24] and it has been shown that the experimental spectra are reproduced well although an energy shift needs to be applied to the calculated spectra to align with experiment. This compromises the predictive power of TDDFT and is a consequence of deficiencies in the exchange-correlation functional associated with the electron selfinteraction error. [20] There has been less focus on the calculation of the XES of organic molecules, although studies of water [25] and a small selection of organic molecules [20] have been reported. In the latter study, an attempt was made to parameterize a hybrid exchangecorrelation functional to describe the X-ray emission energies more accurately. However, although the systematic overestimation of the excitation energies was reduced by increasing the fraction of Hartree-Fock exchange in the functional, the calculated spectral profiles were adversely affected.
The choice of Gaussian basis set has emerged as another important factor in the accuracy of these calculations. [26] The higher effective nuclear charge results in a contraction of the electron density in the core-ionized state. Small or moderately sized basis sets can provide a poor description of the core-ionized state because they do not have the flexibility to describe the more compact electron density. This is particularly a problem for the study of XES at the K-edge of heavier nuclei. One solution is to simply use very large basis sets such as large Dunning basis sets with core-correlation functionals, such as cc-pCVTZ and cc-pCVQZ, which have been shown to be effective. [23] However, this can severely restrict the size of system that can be studied. Alternative basis sets that achieve a balanced description of the ground and core-ionized states have been proposed including using exponents for the basis functions midway between the element and the element with a nuclear charge of one higher [27] and explicitly including basis functions for the Z + 1 element in the basis set. [28] In this study, we investigate the accuracy of TDDFT calculations for a range of organic molecules and explore the dependence of the

| METHODOLOGY
X-ray emission spectra were computed with TDDFT within the Tamm-Dancoff approximation [29] using a protocol described previously. [20] Following a Kohn-Sham DFT calculation for the ground state, a further DFT calculation is performed for the core-ionized state with the maximum overlap method [30] used to maintain the core-hole during the SCF procedure and in a subsequent TDDFT calculation using the core-ionized state as a reference the emission energies appear as negative eigenvalues. Calculations were performed for the following molecules: methanol (carbon K-edge), ammonia (nitrogen N-edge), acetone and dimethyl sulfoxide (DMSO) (oxygen K-edge), fluorobenzene (fluorine K-edge), H 2 S (sulfur K-edge), CH 3 Cl, and CF 3 Cl (chlorine K-edge) and compared with experimental data adapted from ref. [31][32][33][34][35][36] The structures of the molecules were optimized at the B3LYP/6-311G** level of theory. Relativistic effects can also be significant, and these effects are accounted for by applying an energy shift to the computed transition energies. The magnitude of this shift is determined from the difference in the 1s orbital energy between relativistic and nonrelativistic HF/cc-pCVQZ-DK [37] with the relativistic effects modeled using the Douglas-Kroll-Hess Hamiltonian. This gives values of +0.13, +0.25, +0.44, +0.47, +0.74, +10.01, +13.08, and +13.08 eV for methanol, ammonia, acetone, DMSO, fluorobenzene, H 2 S, CH 3 Cl, and CF 3 Cl, respectively. All calculations were performed with the Q-CHEM software package. [38] Computational spectra were generated by convoluting the computed energies and oscillator strengths with Gaussian functions with full width at half maximum of 0.25 eV.

| RESULTS AND DISCUSSION
We begin our analysis by demonstrating the accuracy that is obtained using TDDFT with a standard hybrid functional to simulate X-ray emission spectra before proceeding to explore how various approximations affect the quality of the computed spectra and the dependence on the exchange-correlation functional. The accuracy that can be obtained from TDDFT calculations of XES is represented in Figure 1, which shows a comparison between the computed and experimental spectra for a range of molecules at the K-edge involving a variety of nuclei. The spectra are computed with the cc-pCVQZ basis set and the B3LYP exchange-correlation functional, and an energy shift has been applied to each spectra so that the most intense peak aligns with the experimental data. The magnitude of these shifts is given in the figure caption, and these shifts are in addition to the energy correction due to relativistic effects. The cc-pCVQZ basis set is large and has been shown to give emission energies that are converged with respect to the basis set. [26] The computed spectra reproduce the experimental spectra remarkably well. For all of the molecules, the computed spectra predict the peak heights and their relative energy spacing with a high degree of accuracy. In some spectra, most notably ammonia and H 2 S, the experiment has bands at higher emission energy which can be associated with elastic scattering and multielectron transitions. [33,34]  intensities of the bands, and as shown in Appendix S1 for ammonia, averaging over structures from an ab initio molecular dynamics simulation of the core-ionized state does result in the higher energy band becoming relatively more intense than the lower energy band.
As noted above and in the figure caption, an energy shift is required to align the computed spectra with experiment. The calculations systematically overestimate the emission energies. This is in contrast to TDDFT calculations of X-ray absorption spectra where the transition energies are underestimated. However, both of these errors are a manifestation of the self-interaction error. [18,39] In the context of X-ray absorption calculations, the self-interaction error leads to the energy of the occupied core orbitals being too high resulting in an underestimation of the core ! virtual orbital energy differences. We note that electron correlation is also important and neglect of electron show little resemblance to experiment. This is consistent with previous work on transition metal complexes that also found TDDFT spec- with experiment. [22] Also shown in Figure 2 are spectra computed based upon the Kohn-Sham ground-state orbitals using Equations (1) and (2). This approach to computing the spectra also does not incorporate any orbital relaxation in the core-ionized state but in contrast to the TDDFT calculations provides an accurate representation of the experimental spectra. This suggests that there is some cancellation of errors between approximating the transition as We investigate next the role of the basis set on the calculated spectra. Figure 3 shows the calculated spectra for a range of basis sets of varying quality for DMSO and CF 3 Cl, and the energy differences with respect to the cc-pCVQZ basis set for all of the molecules are given in Table 1. For these spectra, no energy shift has been applied. Bold font denotes the atom that is core-ionized.
K-edge. This is consistent with the findings of previous studies. [26] It is of interest to have small basis sets that are able to replicate the results of the much larger basis sets at a vastly reduced computational cost.
In this regard, the IGLO-II basis set performs well, and this has been recognized previously in the literature. [40,41] The (Z + 1)6-31G* basis set also performs well. This is a nonstandard approach that was introduced recently, [28] and involves supplementing the 6-31G* basis set  PBE, [42] CAM-B3LYP, [43] CAM-QTP(01) [44] and SRC1-R2 [45]  shift is required to align the calculated spectra with experiment. Bold font denotes the atom that is core-ionized.
energies further requires modification of the short-range attenuation parameter. This leads to a similar problem to the hybrid functionals, that is, the quality of the computed spectra becomes poor.
The fundamental difference between the design of functionals for XAS and XES is that for X-ray emission the relevant core orbital is unoccupied. This means that the properties of this orbital are being influenced indirectly through orthogonalisation with respect to the occupied orbital space rather than directly for occupied core orbitals, which is the case for X-ray absorption. This could underlie the need for functional parameters that lead to a poor description of the occupied orbitals.
To generate spectra that requires no information from experiment we adopt a strategy that is often used in the calculation of X-ray absorption spectra, where an energy shift that is applied to the TDDFT computed spectra. [46,47] This shift is determined from a ΔSCF calculation, more specifically, here an extra calculation to determine the groundstate energy of the cation is performed. The energy shift is then determined from the difference between the energy of the highest energy transition in the TDDFT calculation and the corresponding transition energy from a ΔSCF (E core-ionized -E cation ) calculation. The resulting spectra for the CAM-B3LYP/cc-pCVQZ calculations are shown in Figure 6.
The position of the spectral bands agrees well with experiment for the molecules considered here. Although further testing would be necessary to determine how well this approach performs for heavier nuclei. There remains an overestimation of the transition energies of about 2 eV for the two chlorine containing molecules. This error may in part be associated with estimation of the relativistic correction which becomes increasingly significant for the chlorine K-edge. However, the calculations do show that TDDFT approach does provide a relatively computationally inexpensive method that can be readily applied to give X-ray emission spectra that form a reliable basis to interpret and assign experimental spectra. Spectra can be calculated by using a ΔSCF calculation of the highest energy transition as a basis to determine the energy shift that needs to be applied. This leads to calculated spectra that provide a reliable description of experiment.