Fractional-Electron and Transition-Potential Methods for Core-to-Valence Excitation Energies Using Density Functional Theory

: Methods for computing X-ray absorption spectra based on a constrained core hole (possibly containing a fractional electron) are examined. These methods are based on Slater’s transition concept and its generalizations, wherein core-to-valence excitation energies are determined using Kohn − Sham orbital energies. Methods examined here avoid promoting electrons beyond the lowest unoccupied molecular orbital, facilitating robust convergence. Variants of these ideas are systematically tested, revealing a best-case accuracy of 0.3 − 0.4 eV (with respect to experiment) for K-edge transition energies. Absolute errors are much larger for higher-lying near-edge transitions but can be reduced below 1 eV by introducing an empirical shift based on a charge-neutral transition-potential method, in conjunction with functionals such as SCAN, SCAN0, or B3LYP. This procedure affords an entire excitation spectrum from a single fractional-electron calculation, at the cost of ground-state density functional theory and without the need for state-by-state calculations. This shifted transition-potential approach may be especially useful for simulating transient spectroscopies or in complex systems where excited-state Kohn − Sham calculations are challenging.


INTRODUCTION
−43 Each of these methods is widely used and available in standard electronic structure codes, but there are limitations.Not least among these is cost, and only the DFT-based approaches are scalable to large systems such as proteins or liquid environments.Excited-state DFT methods, which are based on finding a non-Aufbau solution to the Kohn−Sham equations, 22 require tedious state-by-state calculations if an entire excitation spectrum is desired.In contrast, TD-DFT can furnish the entire core-level spectrum in a single shot (when used with frozen occupied orbitals for the valence electrons), 20−22 but absolute errors are often >10 eV for light elements 26−29 and much larger for heavier ones. 30,31uilding on previous work, 44 we seek simplified approaches based on Kohn−Sham eigenvalues only.−50 Thus, we investigate "core-hole constraining" methods for XAS that are based on time-independent (ground-state) DFT calculations.There are several variants, 19,51 as described in Section 2, and the unifying feature of these methods is that an electron or a fraction of an electron is removed from a core orbital, and then orbital relaxation is incorporated by solving the Kohn−Sham equations in the presence of a (possibly fractional) core hole.Core-to-valence excitation energies = c c (1)   are estimated as differences between final-state (virtual) energy levels ε υ and the initial-state (core) level ε c .In some cases, a fractional electron may be placed into the lowest unoccupied molecular orbital (LUMO), as described in Section 2, but not into any higher-lying virtual MO. (2) The signature of these approaches is that no proper excitedstate calculation is performed.Instead, the requisite information for XAS is extracted from the MOs and their energy levels.
−67 Some of these methods are based on the use of fractional-electron self-consistent field (SCF) calculations. 19,51They differ in whether the virtual orbitals are probed one by one, as in Slater's original conception, or whether the virtual orbital energy levels (ε υ ) are obtained from a single calculation.The latter approach is the basis of the TPM and its variants, which afford an entire core-level spectrum (at one particular edge) from a single calculation. 19These methods have the same computational cost as a single "ΔSCF" calculation (Section 2.1) but without the need for state-by-state calculations and with less concern about variational collapse.They may hold some advantages for modeling complex systems or experiments, insofar as the spectrum is computed in a single shot and is closely tied to Kohn−Sham eigenvalue information (chemical shifts).
Like the ΔSCF approach, however, eigenvalue-based methods may depend sensitively on the choice of exchangecorrelation (XC) functional.The present work systematically investigates different approaches using XC functionals on various rungs of Jacob's ladder. 22It follows a similar investigation of fractional-electron methods for computing core-level electron binding energies, 44 as in X-ray photoelectron spectroscopy (XPS).In that previous work, we demonstrated that an empirically shifted version of STM with a single fitting parameter affords K-shell electron binding energies that are more accurate than those obtained from the best ab initio methods, including GW-type approaches. 44A similar empirical shifting procedure is introduced here, for core-to-valence excitation energies.

THEORETICAL METHODS
This section introduces eigenvalue-based methods for computing core-to-valence excitation energies based on a time-independent, ground-state DFT formalism.Conceptually, these methods are approximations to a ΔSCF calculation, which we therefore consider first.
2.1.ΔSCF Method.The ΔSCF approach for excitation energies has also been called "excited-state Kohn-Sham theory", 18,22 as it is based on finding a non-Auf bau Slater determinant to represent an excited state.The excitation energy is then simply the energy difference, where E i and E f are the total energies of the ground-state determinant and the non-Auf bau determinant, respectively, the latter of which contains a core hole.−73 The maximum overlap method (MOM) 69−71 often works well for the lowest excited state (or the lowest state of a given symmetry) and has previously been applied to core-to-LUMO excitations. 74In our experience, however, more sophisticated methods are often required to converge higher-lying excited states. 73.2.Slater-Type Methods.The use of fractional-electron SCF calculations originated with Slater, 52−54 although these are now used widely as a means to diagnose and correct problems with delocalization error and self-interaction in DFT.59,75−83 Fractional-electron calculations have also been used in conjunction with correlated wave function models.84−87 Slater's original transition method, 52−54 formulated here for a core → virtual (c → υ) excitation, is based on promoting n c = 1/2 electron into a valence virtual MO, so that n υ = 1/2.To understand why this is relevant, imagine the energy E({n i }) of a Slater determinant is expanded as a Taylor series in orbital occupation numbers {n i }, treated as continuous variables. Setting E 0 = E({n i 0 }), one may write where we have used the Slater−Janak theorem, 88 ε i = ∂E/∂n i .We wish to approximate the ΔSCF excitation energy in eq 3, which we write as Here, E i = E(1,0) and E f = E(0,1) are the initial-and final-state energies, expressed in the form E(n c ,n υ ).Choosing the reference state {n i 0 } in eq 4 to be n c = 1/2 = n υ , one obtains a leading-order approximation ΔE ≈ ΔE STM , where 22,53 c STM (6)   In this and subsequent equations, we use the notation ε r (n c ,n υ ) to mean the Kohn−Sham eigenvalue for MO ψ r , obtained from an SCF calculation that employs occupancies n c and n υ for the core and virtual orbitals in question.(We assume spin−orbitals in this notation, so 0 < n c ≤ 1 and 0 ≤ n υ ≤ 1, and all calculations are performed within a spin-unrestricted formalism.)Alternative derivations of eq 6 have also been suggested, 22,58,59 e.g., based on integration of ∂E/∂n r starting from a determinant with integer occupancies, 58 or based on cancellation of self-interaction error. 59lthough formulated above for c → υ excitation, the STM has also been used to estimate core electron binding energies (CEBEs), 44,83,89−92 via a Koopmans-style approximation but with a fractional occupancy (n c = 1/2) for the core level in question.Using a notation similar to that introduced above, this approximation is (Variants with n c = 2/3 or n c = 3/4 have also been suggested. 44,59) In a recent study of K-shell CEBEs, 44 we found that eq 7 affords an accuracy of ∼0.5 eV at the Hartree− Fock level but is significantly less accurate at DFT levels of theory.Reasonable accuracy was recovered (even for DFT) using generalized approaches that require more than one fractional-electron SCF calculation per CEBE. 44,59,60Similar ideas for excitation energies are explored below.The simple STM in eq 6 overestimates excitation energies. 44his observation motivated a generalization in which the fractional-electron SCF calculation is mixed with a ground-state eigenvalue difference, 58 a method that can be understood as a higher-order extension of Slater's original transition method. 58,89For excitation energies, this generalized (G)STM takes the form Here, ε c (1,0) and ε υ (1,0) are eigenvalues from a ground-state calculation (n c = 1 and n υ = 0), whereas ε υ (1/3,2/3) and ε c (1/ 3,2/3) come from a fractional-occupancy calculation with n c = 1/3 and n υ = 2/3.In the early days of molecular DFT calculations, the GSTM approach showed promising accuracy of ∼0.3 eV for K-shell electron binding energies, 89−91 although this was later shown to benefit from some error cancellation. 92ther schemes involving different fractional occupancies have been proposed more recently, 44,59,60 and we have elsewhere evaluated some of them for CEBEs. 44.3.Transition Potential Methods.As originally formulated (to approximate a ΔSCF calculation), the STM and its generalizations require separate SCF calculations for each excited state of interest, i.e., for each virtual level ε υ into which a fractional electron is promoted.Like the ΔSCF approach itself, this is a tedious and inconvenient way to compute an entire spectrum, and promotions beyond the LUMO are prone to variational collapse in the absence of symmetry constraints.Alternatives are to modify the core occupancy only, leaving the virtual space empty (n υ = 0) or else to promote an electron or fraction of an electron into the LUMO and then use the full spectrum of virtual eigenvalues to estimate the excitation energies (ΔE = ε υ − ε c ).The latter approximation assumes that the potential generated by the LUMO is similar to that generated by the higher-lying virtual orbitals.
The widely used TPM 19,51,61−63 corresponds to the first of these strategies, in which no electrons whatsoever are placed in the virtual space.Excitation energies within the TPM are given by For obvious reasons, this method has also been called the "half core-hole" (HCH) approach. 93A generalized (G)TPM can then be envisioned in the spirit of eq 8: Note that n c + n υ ≠ 1 for the fractional-occupation calculation that is used to obtain ε υ (1/3,0) and ε c (1/3,0), so that calculation involves a charged system.This can be a problem for DFT under periodic boundary conditions; 19,94 therefore, some charge-neutral alternatives have been explored. 51,64hese are discussed below.A summary of different approximations is provided in Table 1, in the form where F υ and F c are simple functions of ε υ and ε c , respectively, computed from one or more SCF calculations that typically involve fractional occupancies and a (fractional) core hole.In addition to the methods discussed above, the list in Table 1 also includes the FCHM approach, 19,65−67 which is the n c = 0 analogue of the TPM/HCH method in eq 9.In FCHM, a full electron is removed from the core but nothing is placed in the virtual space: As with the TPM and GTPM, this procedure creates a charged system.A charge-neutral alternative is the extended core-hole method (XCHM), 51 in which an entire electron is removed from the core and placed in the LUMO:

Excitation beyond the LUMO.
Eigenvalue-based methods can be extended to higher-lying transitions, but such calculations often suffer variational collapse or other SCF convergence issues if an electron (or a fraction of an electron) must be promoted into a virtual orbital above the LUMO.Some methods sidestep this problem by using the LUMO's potential to stand in for that of higher-lying virtual orbitals, and an especially promising protocol is the charge-neutral XCHM procedure. 51Generalizing eq 13 for virtual orbitals υ lying beyond the LUMO, we have Recently, charge-neutral variants of the TPM and GTPM have been suggested, 19,51 called the extended (X)TPM and the extended (X)GTPM, respectively.These methods correspond to the formulas and Lastly, we consider the ionization potential-corrected TPM method (IP-TPM). 19,61This approach removes n c = 1/2 from the core and uses the formula b These methods require two SCF calculations.
for the transition energies, where ΔE IP is the absolute CEBE for the occupied orbital in question, computed using a ΔSCF approach.We will also consider an alternative formulation with n c = 1/3: Physically, both of these methods include the core-hole relaxation effects in valence states, which is important for relative peak positions and intensities, while ΔE IP helps to incorporate core-hole screening and thus to provide reliable chemical shifts. 19Both IP-TPM approaches can be used for core → LUMO and higher-lying excitations.2.5.Limitations.Slater-style methods are intended as extremely simple approaches for excitation or ionization energy calculations that exploit only the information contained in Kohn−Sham eigenvalues, albeit possibly for a fictitious fractional-electron state.The simplicity of this approach may hold advantages for complex systems but also imbues these methods with significant limitations, some of which are worth pointing out.First of all, these methods do not include spin coupling, so there is no distinction between singlet and triplet excitations starting from a singlet ground state.The singlet− triplet excitation gap for c → υ excitation could be estimated as 95,96 −101 However, direct use of electron repulsion integrals is a step away from the purpose of using eigenvalue-based methods in the first place.As such, we will not attempt to compute any spin couplings in the present work.
A separate issue is that these methods might exhibit a stateassignment problem in certain cases, insofar as the identification of excited states is explicitly tied to (and therefore cannot go beyond) the MO picture.Limitations of the MO picture for both excitation 95,96,102 and ionization, 103 due to electron correlation effects, have been discussed elsewhere.In the context of ΔSCF calculations of core-excited states, this is sometimes discussed in terms of the missing "many-electron response to the core hole". 104When using the methods described herein, that response is taken into account (if at all) merely in terms of the fractional nature of the core hole.

COMPUTATIONAL DETAILS
Fractional-occupancy methods have been implemented in a locally modified version of Q-Chem 105 and will be made available in v. 6.1.Although various algorithms are available to optimize a non-Auf bau determinant that contains a fractional core hole, 69−73 the calculations presented herein use either the MOM algorithm 69 or else the "initial MOM" (IMOM) algorithm. 70These differ only in whether overlaps are computed with respect to the previous SCF cycle's occupied MOs (in the MOM procedure) or else with respect the MOs at the first SCF cycle (in IMOM).
The def2-QZVPD basis set is used for all calculations, in an effort to separate basis-set errors from methodological errors.Previous results for CEBEs indicate that DFT/def2-QZVP values are converged, such that uncontracting the core functions makes negligible difference, 44 whereas uncontracting the basis set changes DFT/def2-TZVP binding energies by ∼0.4 eV. 44We add a set of diffuse functions here, in order to better describe the virtual orbitals.The mean absolute errors (MAEs) for ΔSCF calculations using SCAN, SCAN0, and B3LYP are 0.3 eV for 1s → LUMO excitation, whereas the Kshell ΔSCF ionization energy errors with these same functionals are 0.2−0.3eV. 44Addition of a second set of diffuse functions to def2-QZVPD reduces the MAE from 0.31 to 0.25 eV (see Table S3), for a data set that includes excitations beyond the LUMO, and the excitation energy differences between the two basis sets are <0.2 eV, on average.As such, we regard that the present results are well converged.
For XC functionals that are generalized gradient approximations (GGAs) or hybrids thereof, we use the SG-1 quadrature grid. 116The SG-3 grid 117 is used for meta-GGA functionals and their hybrids.However, tests using the smaller SG-2 grid 117 showed no differences, even for functionals such as SCAN with well-documented grid sensitivity. 118This suggests that SG-2 would have been sufficient for the calculations reported here.
All calculations are performed using a spin-unrestricted formalism.We apply Boys localization 119 prior to the fractional-electron or other non-Auf bau SCF calculation (including ΔSCF), in order to avoid problems in cases where symmetry-equivalent atoms give rise to delocalized core orbitals.This issue is not unique to DFT calculations, 120−123 and a detailed analysis suggests that orbital relaxation and electron correlation effects are comparable in magnitude for ionization from a delocalized core state, whereas relaxation effects dominate when localized orbitals are used. 121−126 Although Boys localization was used for all calculations reported here, spot checks suggest that its effect is practically nil for the examples that we consider, even for a molecule like ethylene with symmetry-equivalent C(1s) orbitals.The effect of localization is somewhat larger for CEBEs.
Element-specific relativistic corrections have been included in all calculations, as in previous work, 44 so that results may be compared directly to experiment.These corrections were taken from ref 127, and they are 0.14 eV for C(1s), 0.28 eV for N(1s), 0.51 eV for O(1s), and 0.85 eV for F(1s), which are close to values reported elsewhere. 42These corrections are added to the nonrelativistic excitation energy, meaning that the corrected excitation energy is larger than the nonrelativistic result, e.g., by 0.14 eV for carbon K-edge excitation energies.

RESULTS AND DISCUSSION
We survey the methods introduced above as applied to 1s → LUMO transitions, in Section 4.1, as well as higher-lying transitions (1s → LUMO+1, LUMO+2, ...) in Section 4.2.Statistical assessments in terms of MAEs are given here, but all computational results can be found in the Supporting Information.An empirically shifted approach is presented in Section 4.3, and finally we consider full XAS spectra (including oscillator strengths) in Section 4.4.

K-Edge Transitions. Table 2 reports
MAEs versus experiment for a data set of 1s → LUMO transitions that is taken from ref 128.These data consist of excitation energies at the elemental K-edge for carbon (14 data points), nitrogen (8 data points), oxygen (11 data points), and fluorine (4 data points).
When only the elemental K-edge (and not any higher-lying states) is desired, the ΔSCF procedure is usually straightforward and ΔSCF results therefore serve as a baseline (for a given XC functional) to evaluate alternative Slater-style methods based on Kohn−Sham eigenvalues.At the ΔSCF level, several functionals afford results within ∼0.3 eV of experiment when atomic relativistic corrections are included: SCAN, SCAN0, and B3LYP.The functionals BH&HLYP, CAM-B3LYP, and ωB97X-V afford MAEs that are slightly larger but still below 0.5 eV.These errors are comparable to statistical errors in K-shell ionization energies for the same functionals. 44Whereas an early study 129 suggested that ΔSCF calculations of core-level excitation energies might be significantly more accurate than those for core ionization (due to error cancelation involving the core hole in the former case), that conclusion is not borne out for the present data set and functionals.
The LRC-ωPBE and LRC-ωPBEh functionals do not fare particularly well at the ΔSCF level yet afford the smallest MAEs for the STM approach, whereas STM errors for other functionals lie in the range 1−2 eV.Note that LRC functionals have been used in the past to improve the agreement between valence Kohn−Sham energy levels and ionization energies, 79,130,131 but these functionals are inferior to B3LYP for K-edge excitation energies in TD-DFT. 132The performance might be improved via "optimal tuning", 133 adjusting ω such that ε HOMO = −ΔE IP , but we have not pursued such a strategy, in the interest of obtaining a black-box method that does not need to be adjusted for each new molecule.−137 ) In previous work on CEBEs, we showed that STM is not competitive with ΔSCF but that some variants of the GSTM approach the accuracy of ΔSCF. 44This can be understood based on the fact that the GSTM amounts to a higher-order Taylor series approximation, 89 a higher-order quadrature scheme (for a type of thermodynamic integration), 58 or a more effective cancellation of self-interaction error. 59The same is true for these K-edge excitations, where all functionals tested afford MAEs smaller than 1 eV at the GSTM level.Using B3LYP, the GSTM and ΔSCF results are quite similar.
We next consider the transition-potential approaches: TPM (eq 9) and GTPM (eq 10).These methods modify only the core occupancy n c and do not put any electrons into the virtual space, resulting in underestimation of the electron−hole attraction and thus a shift to higher excitation energies. 19his is evident in the data presented in Table 2, where GTPM errors are generally smaller than TPM errors yet even the former are larger than 2 eV for all functionals tested.The FCHM (eq 12), which also does not place electrons into the virtual space, affords similarly large errors.Given that these methods also create a charged system, and are therefore problematic under periodic boundary conditions, neither the TPM, GTPM, nor FCHM can be recommended.−142 Results for relative peak positions in near-edge XAS spectra are better than absolute excitation energy predictions. 51he XCHM approach (eq 13) creates a charge-neutral excitation but we find that results are erratic, improving somewhat with respect to TPM-style methods for certain functionals (e.g., SCAN and CAM-B3LYP) yet seriously degraded as compared to TPM approaches when some other functionals are used, such as BH&HLYP, for which the XCHM error exceeds 7 eV.The XCHM also continues to enjoy widespread use in materials science and other condensed-phase applications, 142−150 despite its sensitivity to the choice of XC functional.
Finally, the performance of IP-TPM@1/2 and IP-TPM@1/ 3 is quite interesting.For a given functional, these methods typically afford smaller errors as compared to any of the TPMbased approaches that do not place electrons in the virtual space, and they also perform better than XCHM in many cases.IP-TPM@1/3 is consistently better than other eigenvaluebased approaches, affording MAEs below 1 eV when used in conjunction with SCAN, SCAN0, or B3LYP.For B3LYP the MAE is 0.3 eV when using IP-TPM@1/3, essentially identical to the statistical error in ΔSCF results for the same functional.
4.2.Higher-Lying Near-Edge Transitions.Table 3 reports error statistics for a data set of 20 higher-lying, dipole-allowed transitions originating from C(1s) orbitals (8 transitions), N(1s) orbitals (6 transitions), and O(1s) orbitals (6 transitions).These are "higher-lying" transitions in the sense that the final state is not the LUMO.Experimental excitation energies are taken from various sources as detailed in the Supporting Information.For each of the methods except ΔSCF, the full XAS spectrum is evaluated by populating the LUMO (only) with a fractional electron.As discussed above, this makes the eigenvalue-based methods more robust against variational collapse, as compared to the ΔSCF procedure that does involve promotion beyond the LUMO.
The accuracy for these higher-lying transitions is not as good as what we reported for 1s → LUMO transitions, and this conclusion holds across a variety of XC functionals.For the best-performing functionals (SCAN, SCAN0, and B3LYP), the accuracy of the ΔSCF procedure is only slightly worse than it was for the 1s → LUMO transitions, and the MAEs are smaller than 0.5 eV even for the higher-lying transitions.Perhaps surprisingly, the ωB97X-V functional exhibits a MAE of 1.2 eV for this data set, as compared to 0.4 eV for the K-edge transitions.
It has been noted that transition-potential methods do not exhibit the correct asymptotic electron−ion potential for highlying Rydberg states, 151 which may place some limitations on the accuracy of higher-lying excitation energies.Regarding the eigenvalue-based approaches examined here, the widely used XCHM affords MAEs larger than 1 eV for every functional that we tested, including very large MAEs of 5.0 eV for XCHM-SCAN0 and 2.8 eV for XCHM-B3LYP.The XGTPM is the best-performing eigenvalue method, with MAEs of 0.7−0.8eV when used with either SCAN, SCAN0, or B3LYP.To reduce these errors, we turn to an empirical shifting scheme that proved quite successful for core-level electron binding energies. 44.3.Empirically-Shifted Method.In previous work, 44 we demonstrated that introduction of a single, functional-specific shifting parameter turned the primitive STM approach into the most accurate electronic structure method for K-shell CEBEs, outperforming not only ΔSCF calculations but also more expensive methods including variants of the GW approach.In a similar spirit, we introduce an empirically shifted version of XTPM, Here, ΔE XTPM is the XTPM excitation energy defined in eq 15, and the shift δ υ is given by where β is an empirical parameter.The shift corrects for excitation energies that are overestimated by XTPM.Note that eqs 20 and 21 can be rewritten in the form XTPM (22)   which demonstrates that the shifted XTPM approach can also be viewed as a weighted average of the original XTPM excitation energy (ΔE XTPM ) and the unrelaxed orbital energy difference, ε υ (1,0) − ε c (1,0).Unlike the shifted-STM approach for CEBEs that was introduced in ref 44, for excitation energies the shift δ υ depends on the virtual MO ψ υ , even for a given X-ray edge (corresponding to a given occupied MO, ψ c ).To determine β, we use the same data set of 20 higher-lying excitation energies used to obtain the error statistics in Table 3. Fitted values of β for several different XC functionals are listed in Table 4 along with MAEs for the corresponding shifted-XTPM approach defined by eq 20.(The best-fit values do have some basis-set sensitivity.) For most of the XC functionals considered here, this empirical shift considerably improves the accuracy of the XTPM approach, although BH&HLYP, B3LYP, and LRC-ωPBE are exceptions that show little improvement.MAEs of 0.6−0.7 eV are obtained from shifted XTPM calculations using any of the functionals SCAN, SCAN0, B3LYP, or CAM-B3LYP.
To put this level of accuracy in context alongside other stateof-the-art approaches, Figure 1 illustrates the errors alongside some of the competing alternatives.The data set consists of 29 K-edge and near-edge excitation energies for which BSE@ G 0 W 0 results are available, 42 and the latter method is compared to ΔSCF and also to four different eigenvalue-based methods, using the B3LYP functional for all but the BSE@G 0 W 0 calculations.We also juxtapose results in which a TD-DFT excitation spectrum is shifted so that its lowest excitation energy matches the 1s → LUMO excitation energy from a ΔSCF calculation. 24This shift serves to eliminate what are otherwise rather large 26−31 (but systematic 152 ) errors in TD-DFT excitation energies.
Several alternative methods are able to match or outperform BSE@G 0 W 0 , for which the MAE is 0.67 eV. 42Even without empirical shifting, the eigenvalue-based XTPM performs almost as well (MAE = 0.76 eV), and even better with empirical shifting (MAE = 0.57 eV), while XGTPM performs about as well as BSE@G 0 W 0 (MAE = 0.64 eV).The accuracy  of both methods is comparable to that of TD-DFT with a ΔSCF shift (MAE = 0.53 eV). 24The ΔSCF approach remains the most accurate (even as compared to BSE@G 0 W 0 ), with a MAE of 0.31 eV, but requires a separate calculation for each state.
Finally, we examine the performance of the shifted-XTPM approach for 1s → LUMO excitations that were not included in the training set that was used to determine the β parameter.For this, we use the same data set as in Table 2, with results summarized in Table 5 for a variety of XC functionals.The empirical shift significantly narrows the accuracy gap between different functionals, all of which afford overall errors smaller than 1 eV.Importantly, for SCAN, B3LYP, and CAM-B3LYP the MAEs are ≲0.7 eV even when the higher-lying transitions are considered.The shifted-XTPM approach using SCAN or B3LYP is therefore recommended for full-spectrum XAS calculations, as a convenient and robust alternative to state-bystate techniques such as ΔSCF.Like XGTPM (eq 16), the shifted-XTPM approach requires two different SCF calculations (from which a full spectrum is obtained), yet the latter is somewhat more accurate and also more consistent (or perhaps less erratic) across XC functionals.

Other Applications.
As illustrative applications, we use several different methods to compute carbon, oxygen, and nitrogen K-edge excitation energies for the thymine and oxazole molecules.Errors (relative to experiment) are listed in Table 6 for a variety of methods, all based on the B3LYP functional.The transitions in question represent the lowest dipole-allowed excitation from each indicated 1s orbital.In the case of thymine, the shifted-XTPM approach affords a MAE of 0.5 eV across the 8 K-edge excitations that are considered.This compares well to ΔSCF results, for which the MAE is 0.4 eV.(The IP-TPM@1/3 method also performs very well but is not charge-neutral and thus not preferred.)Similar trends among methods are observed for oxazole, although the MAEs are slightly larger.
Oscillator strengths for the eigenvalue-based methods have been implemented based on eq 2. For methods such as the GSTM that involve more than one fractional-electron SCF calculation, Nakajima et al. 60 suggest weighting the transition intensities I v→c with the same coefficients that are used to combine the eigenvalues.However, we find that the shifted-XTPM approach works somewhat better than GSTM and requires the same number of SCF calculations so we have not computed any spectra using GSTM.For the shifted XTPM approach, we use oscillator strengths corresponding to the unshifted method.

Journal of Chemical Theory and Computation
shifted and unshifted XTPM procedures using either SCAN or B3LYP.Empirical shifting corrects the peak positions for either functional, leading to better absolute agreement with experiment, while the peak spacing is scarcely affected.The shifted-XTPM approach may therefore be a useful alternative to TD-DFT for low-lying core-to-valence XAS.

CONCLUSIONS
The performance of various XC functionals has been tested for K-edge excitation energies (1s → virtual) involving 1s orbitals of second-row elements.These methods include both ΔSCF and also fractional-electron approaches that originate in Slater's transition approximation, which use Kohn−Sham eigenvalues (only), from one or more modified SCF calculations, in order to obtain an excitation spectrum.The overall conclusions are as follows.
• For ΔSCF calculations, the SCAN, SCAN0, and B3LYP functionals are recommended.Each exhibits a statistical accuracy (for the absolute excitation energy) of 0.3 eV when an atomic relativistic correction is used.In contrast, the LRC-ωPBE functional that is widely used in TD-DFT calculations and the SRC1-r1 functional that was specifically parametrized for K-edge TD-DFT exhibit MAEs greater than 1 eV and are not recommended for ΔSCF calculations.• For the same benchmarks, a generalized version of Slater's method (GSTM) is also useful, if the functional is carefully selected.In conjunction with either B3LYP or LRC-ωPBEh, GSTM affords an absolute accuracy of 0.3−0.4eV.This method requires two fractionaloccupancy SCF calculations per elemental edge but not state-by-state calculation of the higher-lying excitations, and it should therefore be more robust against variational collapse as compared to ΔSCF calculations.• Other methods including the TPM, GTPM, FCHM, and XCHM afford larger errors and cannot be recommended, despite their continued widespread use.For example, the best we are able to do with XCHM is an overall error of about 1.6 eV when used with the SCAN functional; this is much larger than the ΔSCF error (0.3 eV) obtained using the same functional.For TPM, the smallest overall error is 2.9 eV (using B3LYP), for GTPM it is 2.2 eV (again using B3LYP), and for FCHM it is 1.8 eV (in conjunction with BH&HLYP).Each of these errors is significantly larger than the ΔSCF error for the same functional.• The IP-TPM@1/3 protocol affords errors of about 0.3 eV when used with B3LYP, which is comparable to the ΔSCF accuracy.However, this method requires the creation of a charged excitation, which is not well suited for periodic calculations.• For higher-lying excitations (involving virtual orbitals beyond the LUMO), errors are larger for the eigenvaluebased methods although XTPM-B3LYP and XGTPM-B3LYP afford absolute accuracies of 0.7 and 0.8 eV, respectively.B3LYP-based ΔSCF calculations afford an accuracy of 0.3 eV for the same data set.• To improve XTPM, which is based on a single fractional-occupancy SCF calculation, we introduce a simple shifting procedure.When used with SCAN or B3LYP, this approach achieves an accuracy of 0.3 eV (SCAN) or 0.5 eV (B3LYP) for 1s → LUMO transitions.For higher-lying excitations, the MAE is 0.6 eV for both functionals and this is as good as far more expensive many-body techniques such as BSE@ G 0 W 0 .When combined with oscillator strengths based on transition dipole moments between MOs, reasonable XAS spectra are obtained for several molecules.Overall, the shifted-XTPM approach is competitive with the best ab initio techniques for K-edge core-to-valence transition energies, just as the shifted-STM procedure is one of the most accurate methods for K-shell CEBEs. 44Both methods are based on an easy-to-converge fractional-electron procedure that does not require promotion of any electrons beyond the   4 Calculations were performed using the SCAN-and B3LYP-based XTPM procedures and their empirically shifted analogues, with Gaussian broadening (σ = 0.3 eV) to obtain spectra from transition energies and oscillator strengths.Experimental spectra were digitized from the original references.

Figure 1 .
Figure 1.MAEs for a data set of 29 C(1s), N(1s), and O(1s) core-tovalence excitation energies, using various approaches.Methods in blue are eigenvalue-based.All calculations use the B3LYP functional except for the BSE@G 0 W 0 values, which are taken from ref 42 and use LRC-ωPBEh.The TD-DFT+ΔSCF results are from ref 24, where a ΔSCF calculation of the 1s → LUMO excitation energy is used to shift the spectrum from a TD-DFT calculation.Numerical values of the MAE (in eV) are shown for each entry.

Table 2 .
Error Statistics (Versus Experiment) for 1s → LUMO Transitions a Mean Absolute Error (eV) b a Data set consists of 37 K-edge transitions from ref 128, and all theoretical values include an atomic relativistic correction.b The smallest MAE in each row is presented in boldface and the largest is underlined.c For core → LUMO excitation, STM is equivalent to XTPM.

Table 3 .
Error Statistics (Versus Experiment) for Higher-Lying K-Edge Transitions a Data set consists of 20 transitions from C(1s), N(1s), and O(1s) orbitals, and all theoretical values include an atomic relativistic correction.b The smallest MAE in each row is presented in boldface and the largest is underlined. a

Table 4 .
Errors for Higher-Lying K-Edge Transitions, a Using the Shifted-XTPM Approach (Eq 20) a Same data set as in Table3, def2-QZVPD basis set.

Table 5 .
Error Statistics (Versus Experiment) for 1s → LUMO Transitions Using Shifted-XTPM a Mean Absolute Error (eV) b Same data set as in Table 2, including atomic relativistic corrections, and with values of β from Table 4. b The smallest MAE in each row is presented in boldface and the largest is underlined. a

Table 6 .
Errors in the Lowest Dipole-Allowed Transition Energies Using B3LYP-Based Methods a Error defined as theory minus experiment, including atomic relativistic corrections.b From ref 153.c From ref 154.d From ref 155.