Toward a correct treatment of core properties with local hybrid functionals

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position‐dependent exact‐exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt‐LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn–Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high‐density and iso‐orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K‐shell core excitations of second‐ and third‐period systems including accurate state‐of‐the‐art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange‐energy densities, as well as by B95c meta‐GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.


| INTRODUCTION
Most density functionals are designed to provide accurate valence-shell energies, as these are needed to correctly reproduce thermochemical and kinetics data, or the energies of valence-shell excitations.The progress in accuracy on such functionals in the past decades has been remarkable.However, about six years ago, a controversial discussion has been kicked off on the development of new functionals "straying from the right path" since about the year 2000. 1 The argument made was that, while energetics had indeed been improved during this time (mainly by fitting many adjustable parameters to databases of reaction energies, barriers, and so on) the resulting electron densities actually got worse.This critique was aimed in particular at the functionals from the Minnesota group.Those and other authors pointed out, however, [2][3][4][5] that this lament had been biased by looking at total densities of atoms and thus focusing not on the valence densities relevant for most of chemical properties and processes but on the core region (but see References 6 and 7).Indeed, many of the more highly parameterized functionals give relatively good descriptions of valence densities, which are in turn most important for the mentioned energy differences.So far so good, one could say.However, the core regions near the nuclei are of course relevant for many spectroscopic properties, and their accurate description is clearly needed when aiming at calculating accurate spectra.
Examples include, without a claim for completeness, core-excitation or core-ionization energies, hyperfine couplings, the related hyperfine shifts in the NMR spectroscopy of paramagnetic species, Mößbauer isotope shifts, and nuclear quadrupole couplings.For example, we and others found highly erratic behavior of some (but not all) of the Minnesota functionals for the hyperfine couplings of 3d transition-metal nuclei, and in part for NMR chemical shifts 8,9 (note that such properties relate in a more complicated way to the core-shell densities than, for example, core-excitation energies).
3][14][15] Among the proposed remedies, ΔSCF and related state-specific approaches rather than LR-TDDFT calculations have been suggested, as this allows the core-shell relaxation to be covered. 10,16However, this then leads to other problems, as each excitation requires a new calculation of a "non-Aufbau-principle" state, with potentially difficult SCF convergence (improvements are possible using the maximum overlap method 17 or square gradient optimization 10,16 ).We mention recent attempts to combine ΔSCF and TDDFT approaches, or TDDFT for core excitations based on orbitals for the system with n À 1 electrons. 11,18,19Within LR-TDDFT proper, alternative suggestions involved functionals designed to reduce the SIE in specific core shells, for example, by adding a tailored shortrange exact-exchange (EXX) contribution to range-separated hybrid functionals. 20Unfortunately, such functionals have so far tended to perform poorly for valence properties and a different parameterization may even be required for elements from different periods or for different core shells.3][24] Very recently, Janesko used a less complicated adiabatic projection approach to achieve this goal with much less parameterization and in a computationally more convenient way. 25[28] A more natural way to vary EXX admixtures in different regions of space (core, valence, asymptotics) for LR-TDDFT is offered by local hybrid functionals ("local hybrids", LHs). 29These exhibit a position-dependent EXX admixture in real space, based on a mixing of semi-local and EXX energy densities governed by a local mixing function (LMF), thus enabling a selective modification of the core potential and thus of the core-electron densities.Indeed, we have shown that even some of our early LHs based only on local density approximation exchange and a so-called t-LMF provided large improvements for K-shell core excitations of second-period elements from Tozer's core excitation test set 20 (denoted Tozer-II in the following), while simultaneously giving accurate valence-shell and Rydberg excitations, with particularly good performance for valence triplet excitations. 21Similarly, such LHs with t-LMFs have been shown recently to give improvements on the "spin-polarization/spin-contamination dilemma" of 3d transition-metal hyperfine coupling constants (HFCs) by reducing the spin contamination caused by valence-shell spin polarization. 8This is due to larger core-shell EXX admixtures improving the required core-shell spin polarization, while the lower valence-shell EXX admixtures of an LH help to limit the spin contamination caused by valence-shell spin polarization.
Let us write an LH in the following form based on 100% EXX where the middle term may be viewed as a nonlocal correlation term, 30 g σ ðrÞ is the LMF, e sl x,σ ðrÞ a semi-local exchange-energy density, e ex 2][33][34] A semi-local correlation term, E sl c , completes the functional form.In the high-density limit, the middle term should then also scale like a proper correlation functional.Given that the exchange-energy densities in the integrand scale like ρ 4= 3 ðrÞ, this requires that the complement of the LMF, ½1 À gðrÞ, scales like ρ À n = 3 ðrÞ with n ≥ 1.However, the complement of the t-LMF does not scale at all with the electron density ρðrÞ. 29,35,36As the high-density limit becomes increasingly more important near the nuclei of heavier atoms, one thus expects a worse description of core-shell properties going down the periodic table. 35ese results show the potential of LHs for improved core properties but also the need for improved LMFs that have the correct behavior in the core region, as well as in the valence region if we aim at general-purpose functionals.The present work is thus concerned with the construction and optimization of improved LMFs for LHs that exhibit simultaneously good descriptions of core and valence properties.Notably, we start from the t-LMF-based secondgeneration functional LH20t 37 that exhibits GGA exchange (PBE 38 ) and meta-GGA correlation (B95c 39 ), together with a pig2 CF 34 for G σ ðrÞ in Equation (1).The aim is to retain or extend the excellent performance of LH20t for main-group and transition-metal energetics and valence excitations while improving substantially the treatment of core shells.We note in passing that this work concentrates on the demanding core region of the LMF, and we do not yet optimize the asymptotic behavior of the LMF for small densities far away from the nuclei, nor do we include recently constructed corrections of LHs for strong correlations 40,41 or range-separated exchange-energy densities to account for the correct long-range asymptotics of the potential. 42,43n this section, we describe the theoretical foundations of the new LH23pt model.As the distinction between common-and spin-channel LMFs is vital for local hybrid functionals, we use a spin-resolved notation, with σ, ς being general spin variables.Density matrices are denoted as D, molecular orbitals as φ and general orbital indices as i, j, ….Throughout the paper, we follow the common convention of omitting the space variable r for clarity, except in specific, essential cases.

| Theoretical constraints
The satisfaction of theoretical constraints, either exact or approximate, is important for retaining physical insight into the XC functional, as well as to reduce the degrees of freedom in the functional space, often leading toward more physically meaningful functional forms.Some degree of semi-empirical parametrization is unavoidable on rung 4 of the usual ladder hierarchy of functionals. 30Here we will make use of constraints on the core region of LHs, of desirable features of the LMF, and of a moderate number of semi-empirical parameters.
Below we will outline in particular iso-orbital-limit and uniform coordinate scaling constraints on the LMF.We refer the reader to a comprehensive review on LHs for a wider view of applicable theoretical constraints. 29 iso-orbital regions the density of a given spin is dominated by only one occupied spatial orbital.Same-spin correlation contributions are therefore absent, and the self-interaction error (SIE) of the classical Coulomb interaction, which is a same-spin contribution, needs to be canceled fully by employing 100% EXX (as well as a selfinteraction-corrected dynamical correlation functional).We refer to this as iso-orbital constraint. 44,45A special case of iso-orbital regions are iso-electron regions, which impose the additional constraint of featuring only one electron irrespective of the spin.In contrast, if the single spatial orbital is occupied by two electrons of opposite spin, opposite-spin correlation can be present in iso-orbital regions.In isoelectron regions, 100% EXX is required to cancel the SIE (iso-electron constraint). 29For two-electron iso-orbital systems such as the helium atom or H 2 , only the exchange contribution is described exactly by 100% EXX, while the remaining opposite-spin correlation may be described by an additional correlation functional.While the latter might also be simulated via a semi-local exchange-like contribution, in the practice of LHs it seems most straightforward to aim for 100 % EXX admixture for iso-orbital regions in general.
While exact iso-orbital regions can be found only in a few systems, satisfaction of the iso-orbital constraint can be important in many more cases.Regions which are dominated by one orbital can be found either far away from or very close to the nuclei.In both cases approximately 100% EXX admixture is required to cancel the SIE.In the context of LHs, the regions far from the nuclei are sometimes termed "asymptotic regions".This should not be confused with the asymptotic behavior in inter-electronic space. 29As these regions are less important in the context of the present work, we refer to a review 29 for more details.The core electron density near the nuclei is dominated by the respective 1s orbitals.Higher-angular-momentum orbitals of the same atom do not contribute at the nuclear position due to nodal planes while the contribution of higher s-orbitals or of orbitals from other atoms tends to be comparably small.Accordingly, the nuclear positions of heavier elements should be clearly identified as effective iso-orbital regions.For lighter elements high EXX admixtures somewhat below 100% can be anticipated at the nucleus.Here the iso-orbital constraint is less strict, as contributions from other orbitals can still be non-negligible.As we go down the periodic table, we should further approach 100 % EXX admixture at the nuclear position.This is clearly important for core properties, as a minimization of SIE becomes crucial.
The second important constraint for the accurate description of core properties is uniform coordinate scaling in the high-density limit.
Uniform coordinate scaling is a powerful theoretical concept, in which the scaled space variable r !r λ ¼ λr is introduced, where λ is a scaling parameter with 0 ≤ λ ≤ ∞.In the most common approach, 46,47 the total electron number is kept fixed, which results in a scaling of the electron density of Under this condition, EXX exhibits a scaling of which is also satisfied by essentially all (semi-)local exchange functionals.The scaling under uniform coordinate scaling for the exact correlation functional is more complicated and is related to the adiabatic connection. 46,48However, in the high-density limit, that is, λ !∞, simple constraints are known that can be used for the construction of new XC functionals.The so-called weak scaling condition 49 just requires that the exchange contribution dominates over correlation in the high-density limit In this case, the exact correlation energy density is thus required to scale as On the other hand, the strong scaling condition 49 requires the correlation energy to approach a constant in the high-density limit, 47 which results in a scaling of the exact correlation energy density of In principle, the nonlocal correlation term E LH nlc in Equation ( 2) should also satisfy these scaling conditions.Since the scaling of the CF and of the exchange-energy densities is fixed to an exchange-like scaling of λ 4 , the weak and strong scaling conditions can be directly translated into scaling conditions for the complement of the LMF (i.e., ½1 À gðrÞ) in the high-density limit. 36,49,50That is, the weak scaling condition requires an LMF scaling of λ n , n < 0, while the strong scaling condition implies λ n , n ≤ À 1.An LMF scaling of λ À1 would result in a constant correlation energy in the high-density limit, while λ n , n < À 1 would give a vanishing correlation contribution.Since E LH nlc is usually combined with a semi-local correlation functional E sl c , obtaining a nonvanishing correlation contribution in the high-density limit is no hard requirement.It therefore also depends on the choice of the semi-local correlation functional.While the high-density limit may not in fact be reached in many core shells, it is still important for the accurate description of core properties.As nuclear charge increases, the core densities also increase.That is, even if the iso-orbital limit may not hold exactly except for the 1s-orbital of the He atom, in the core region exchange should dominate increasingly over correlation with increasing nuclear charge.A functional respecting the high-density limit will indeed increase EXX admixture near the nuclei with increasing nuclear charge.This should allow an accurate treatment of the core shells of different atoms without having to reparametrize the functional (the latter has been done in previous works 20,51 ).However, as the highdensity limit only provides information about the limiting case, some parametrization of the functional may still be required.
None of the LMFs reported so far for LHs thus far is able to satisfy both the constraints of the iso-orbital and high-density limits simultaneously.The LMF proposed by Kümmel and co-workers, 36 a deorbitalized version by Corminboeuf and co-workers, 52 as well as the LMF of the PSTS functional, 32 are three of the few examples that satisfy the uniform scaling constraint.However, all of them violate the iso-orbital constraint even qualitatively.[55] But it does not scale at all in the high-density limit.Other LMFs such as the s-LMF based on the reduced density gradient, 56 Johnson's LMF based on the correlation length, 57 or a related recent LMF by Holzer and Franzke 58 do not satisfy any of the two constraints even qualitatively.In fact, the latter LMF and the one by Johnson explicitly give an almost vanishing exact-exchange contribution at the nuclar positions and a decreasing contribution in the core region, which severely violates the iso-orbital constraint.The fact that reasonable HFCs can be obtained with this LH 58 suggests that the underlying semi-local exchange functional based on a density matrix expansion may perform well for core-shell spin polarization.This may require further investigation.Our goal in the present work is, however, the construction of an LMF that respects both the iso-orbital and high-density limits.

| LH23pt model
To achieve this goal, we introduce a general form we label Padé t-LMF (pt-LMF), Here, t r ð Þ is the ratio of the von-Weizsäcker and Kohn-Sham kinetic energy densities where we have used the common definitions for the spin electron density, its squared gradient, and the Kohn-Sham kinetic energy density t r ð Þ is the basis of t-LMFs (typically scaled down) [53][54][55] of LHs but also a common ingredient in meta-GGA and hyper-GGA functionals to identify iso-orbital regions, 44,45 since it approaches a value of 1 in the iso-orbital limit, while becoming 0 in homogeneous regions.While the earliest LHs with t-LMF 53,55 used a spin-channel-separated variant, it turned out that using the so-called common t-LMF featuring the same LMF value for α and β spin channels, where 0 ≤ a t ≤ 1 is an empirical prefactor, is able to provide better results for barriers and atomization energies. 54Hence, we adopt this approach by introducing and optimizing the new pt-LMF model as pure common-LMF.Conversion to a spin-channel LMF is straightforward but will not be parameterized here.In Equation (8) x is an empirical parameter introduced to enable the fine-tuning of the LMF within the optimization procedure.h r ð Þ is an auxiliary function that needs to be fixed by considering the exact constraints described above as well as by optimization.
There are various advantageous properties of the pt-LMF form: it reaches a value of g σ ¼ 0 in the homogeneous limit, as long as As a consequence, the pt-LMF tends to be similar to the t-LMF in the valence region, thus potentially sharing or fine-tuning its favorable performance in that region, for example, for thermochemical and kinetic data as well as for valence excitations. 21,59The homogeneous limit is no exact constraint on the LMF, as the EXX energy density anyway is exact in the homogeneous case.
Low LMF values in that limit are nevertheless advantageous, in particular regarding the simulation of left-right correlations in bonding regions. 37The complement of the pt-LMF is As t r ð Þ exhibits a scaling of 1 with respect to uniform coordinate scaling, the required scaling in the high-density limit needs to be introduced exclusively by h r ð Þ.In the case of satisfying the weak scaling condition, this requires h r ð Þ to scale as λ n , n > 0, while the strong scaling condition would impose a scaling of λ n , n ≥ 1.Third, the iso-orbital constraint requires h r ð Þ to approach ∞ in iso-orbital regions.Essentially, the pt-LMF approach thus provides a mapping of theoretical constraints to the auxiliary function h r ð Þ.This will be important in the detailed construction of the pt-LMF described below.We note in passing that the Padé form is only one of the many possible mappings to a function between zero and one, and we have been considering others.However, the Padé form has the added advantage of being particularly straightforward.
What remains is an adequate choice of h r ð Þ that satisfies the above-mentioned constraints, (a) noninfinite values in homogeneous regions, (b) a scaling of λ n , n ≥ 1 in the high-density limit to satisfy the strong scaling condition, and (c) infinite values in iso-orbital regions.
Multiple constructions would be able to satisfy these constraints, and even application of highly sophisticated machine-learning approaches is conceivable in this context.We decided on a particularly simple definition based on powers of electron density that allows a systematic evaluation for core properties, where a, b, c, and d are adjustable parameters.Constant a helps providing the desired t-LMF-like behavior in valence regions.Indeed, in case of the other terms in h r ð Þ being zero, and x ¼ 1, the pt-LMF reduces to a simple scaled common t-LMF.The second term in Equation (15), scaled by b, is introduced to satisfy the iso-orbital limit in the asymptotic real-space region.Since the focus of the present work is on core regions, we leave an optimization of this term (which could be changed from a linear dependence on ρðrÞ in the denominator to other powers) to future work.The last term in Equation ( 15) should provide the correct scaling in the high-density limit.While c is a mere scaling parameter, choosing the exponent d ≥ 1=3 enables satisfaction of the strong scaling condition.This fulfills implicitly also the isoorbital constraint near heavy nuclei, while the precise choice of parameters c and d should provide sufficient flexibility to allow also the description of the core regions of lighter nuclei, where the highdensity limit is not reached.Overall, this definition of h r ð Þ contains one term each for describing core, valence and asymptotic regions.
Obviously, more complicated constructions are conceivable, for example, by using an expansion into a series of different powers of ρ r ð Þ, but this would of course also introduce more empiricism.We note also that different LMF mapping schemes might affect the preferred The construction of our first pt-LMF-based LH (LH23pt) follows the setup used previously for the successful LH20t functional, 37 replacing its single t-LMF prefactor by the five parameters of the current pt-LMF model.That is, we employ PBE exchange 38 with a scaled GGA correction managed by c slx and a pig2 CF 34,60 together with a B88 damping function 61 The CF exhibits the two linear parameters f 1 and f 2 and the damping function parameter β.Here, the common definitions for the reduced spin density gradient, the reduced spin density Laplacian and the reduced spin density Hessian Reparametrized B95 semi-local correlation was added, 39 where d opp and d ss are linear scaling parameters for the opposite and same-spin correlation terms, respectively, while c opp and c ss are the corresponding damping parameters.e ueg c,opp , e ueg c,ss , τ ueg σ are the oppositespin and same-spin components of the correlation-energy density and the kinetic-energy density of the uniform electron gas, respectively.In total, the LH23pt functional thus contains 13 adjustable parameters.
In contrast to highly parameterized functionals based on extensive series expansions, 62,63 the parameters of LH23pt tend to be well defined physically, and the optimization procedure (see below) accounts for the different physical meanings of the parameters.That is, the pt-LMF with its 5 parameters simply controls the local EXX admixture and is constrained to be between 0 and 1.The reoptimization of the two damping parameters of B95 correlation corresponds just to a fine-tuning relative to the linear parameters and is not expected to cause any overfitting.Indeed the parameters generally stay in a physically reasonable range.Finally, the three parameters of the pig2 CF are fixed orthogonally to the other parameters of the functional (see below) and serve to minimize unphysical artifacts due to the gauge ambiguity of exchange-energy densities.We are thus confident that LH23pt remains in the category of functionals that may be considered as "density functional theory with fine tuning" rather than "density functional fits". 64LH23pt has been implemented into TURBOMOLE 65 for ground-state SCF, 66 gradient 67 and linearresponse TDDFT excitations. 68| COMPUTATIONAL DETAILS

| Optimization procedure
For determining the free parameters of the new LH23pt functional, we extended the multilayer optimization procedure implemented in our in-house program Panda, 69 which has been used for the development of our more recent functionals, 37,40,42  properties of another subset.Accordingly, the resulting optimized LH23pt functional can be expected to satisfy all considered three properties to a similar extent.The main limitations of this approach are that each parameter is required to be assigned to just one property subset and that the individual subsets exhibit only moderate interdependencies.In fact, the three chosen property subsets in this work can be safely expected to satisfy the latter requirement, while the LH23pt model allows a clear distinction between valence, core and calibration parameters already by construction so that an assignment is straightforward.We note in passing that an extension of our multilayer ansatz to further properties such as Rydberg excitations or strong correlation is straightforward but not relevant in the context of this work.In the following, we will describe the individual optimization steps (1), (2), and (3) in more detail.
In the first step, the LMF parameters a, b and x together with c slx and the four B95 parameters are optimized with respect to the average of the mean absolute deviations (MADs) of W4-08 atomization energies 70 and BH76 barrier heights. 71,72In the first iteration, the three parameters of the pig2 CF and the core parameters c and d are set to 0. We employed a global multilayer single-linkage (MLSL) 73 optimization algorithm based on self-consistent TURBOMOLE 65 energy calculations.In addition to the MAD with respect to the W4-08 and BH76 test sets, the Panda program also requires its numerical gradient with respect to the parameters.Therefore, all TUR-BOMOLE calculations during the global optimization with the MLSL algorithm use a medium-sized def2-TZVP basis set to achieve feasible computation times (for further computational details see Section 3.2).
For optimizing the two LMF core parameters c and d, we exploited that core excitations calculated using linear-response Overview of the different elements involved in the optimization of the LH23pt functional.See text for details.
TDDFT, in contrast to ΔSCF, are known to be highly sensitive to the amount of exact exchange in core regions. 21On that account, we employed the second-and third-period 1s core-valence excitation energies of the Tozer-II test set 20 to determine c and d using the global MLSL algorithm (for further computational details see Section 3.2).We note that core-Rydberg excitations were not included in the optimization set due to the necessity of correctly assigning calculated excitations automatically, which is trivial for the first core-valence excitation in each irreducible representation but not for the higher-lying core-Rydberg excitations.Similarly, the core-valence excitation of SiH 4 needed to be excluded to avoid instabilities during the optimization caused by a state mixing with different parameter values for c and d.
In total, we thus considered 9 and 5 core valence excitations of second-and third-period elements, respectively.To properly account for relativistic effects, accurate relativistic shifts for nonrelativistic excitation energies have been computed, including scalar-relativistic two-electron and quantum-electrodynamics terms.In particular, XCfunctional-dependent relativistic shifts have been computed initially for each excitation using the optimized functional of step (1).Afterwards, relativistic shifts have been kept fixed during the current optimization step, but have been recalculated in the next iteration of step (2).Details can be found in Section 3.3.We further note that the decision for the Tozer-II test set in favor of other more recent test sets such as the one used by Jin and Bartlett 51 or XABOOM 74 is intentional.Most importantly, a proper parametrization of the core part of the LMF, in particular regarding the correct scaling behavior, inherently requires a broad variety of different elements, which is not the case in the two mentioned sets.In the case of XABOOM, also the fc-CVS-EOM-CCSD reference values are not accurate enough in absolute terms to be usable for parameter optimization, since systematic element-specific errors can differ significantly between linearresponse TDDFT and EOM-CCSD due to fundamental methodological differences.A semi-empirical shift of the reference data, while highly accurate, does not seem appropriate in the present context.
In the last step of the multilayer optimization procedure, the three CF parameters f 1 , f 2 and β are determined by minimizing the spurious nondynamical correlation in noble gas dimer dissociation curves, in particular Ne 2 and Ar 2 .Since this optimization step is similar to the approach suggested earlier, 34 and used with LH20t, 37 we refer to those works for details.The differences between the reference dissociation curve for Hartree-Fock plus B95 correlation and that of the calibrated local hybrid are minimized using the MLSL algorithm. 37e convergence criteria for the difference between the curves and for the parameters was set to 0:001 a 0 Á kcal=mol and 0.001, respectively.
After going through the optimization cycle of steps (1), (2), and (3) once, the obtained CF parameters are in fact optimal.However, this is not true for the remaining parameters optimized in steps (1) and ( 2), since these have been optimized with different parameter settings.Accordingly, these parameters need to be reoptimized.
Hence, we repeated the optimization cycle until self-consistence of the parameters has been reached.Since changes of the parameters can be expected to be relatively small compared to the first optimization cycle, a local BFGS algorithm has been used for subsequent optimization steps.To reduce the influence of basis set errors on the optimized parameters, in this local reoptimization the larger def2-QZVPPD basis set was used for step (1).In fact, convergence has been achieved already after 3 iterations.A more detailed depiction of the optimization process can be found in Figure S1 in Supporting Information.We refer to the resulting local hybrid functional as LH23pt.
To enable a detailed analysis of the LH23pt model, we have opti- DFT-D4 corrections 75,76 have been added on top of the final LH23pt using the stand-alone DFT-D4 program, version 3.5, [75][76][77] and following the usual optimization procedure based on the NCIBLIND10, 78 S22 Â 5, 79 and S66 Â 8 80 test sets.

| Computational settings
Parameter optimization.All nonrelativistic DFT and TDDFT calculations for parameter optimization have been performed using a modified version of TURBOMOLE V.7.6.540, 65,81extended by the LH23pt implementation.All parameter optimizations have been performed using our in-house program Panda. 69The optimization procedure involves several layers of parallelization, including a parallel calculation of all molecules in a given optimization test set and a tailor-made SMP parallelization of the individual TURBOMOLE calculations, and it is thus well suited to be carried out efficiently on supercomputers.
The W4-08 and BH76 calculations in step (1) (see Section 3.1) were done using def2-TZVP basis sets 82 for the global optimizations (MLSL) and the larger def2-QZVPPD basis sets 83 for the local optimizations (using a BFGS algorithm).All calculations were done using the TURBOMOLE grids of gridsize 3 and an SCF convergence criterion of 10 À7 Hartree. 65,84In case of def2-QZVPPD basis sets, more diffuse grids were used by applying the TURBOMOLE setting diffuse 2. For the TDDFT calculations of the core-valence excitations of the Tozer-II test set in step (2), pcseg-3 basis sets 85,86 have been used for nonhydrogen atoms and the pc-2 basis set for hydrogen atoms.The numerical grid has been set to gridsize 3 with an increased number of radial grid points by setting radsize 10.The convergence criteria for the SCF method and the linear-response TDDFT calculations were set to 10 À9 a.u. and 10 À7 a.u., respectively.Details of the relativistic corrections are described below (Section 3.3).For calculating the dissociation curves of the neon and argon dimers in step (3) (relative to HF+B95 energy curves using the set of B95 parameters 37 from step (2), the def2-TZVP basis set was employed.The convergence criterion for the SCF procedure has been set to 10 À7 a.u.To avoid numerical instabilities for larger interatomic distances, a large numerical grid has been employed (TURBOMOLE settings gridsize 7, radsize 20 and diffuse 2).
Calculations for evaluation of the functional(s).8][89][90][91][92][93][94][95][96] In particular, we obtained 1s core ionization potentials as minus the 1s core orbital energies, which have been taken directly from the relativistic ground-state DFT calculations required to compute the relativistic corrections for the 1s core excitation energies.Accordingly, results include the same level of relativistic correction as the core excitation energies and no additional calculations needed to be performed.
To examine the performance for main-group thermochemistry, kinetics and noncovalent interactions, we employed the GMTKN55 test suite. 97Calculations follow the setup of the original work, that is, def2-QZVP basis sets and gridsize m4.Extending the evaluations to transition-metal organometallic systems, the MOR41 set of closed-shell reaction energies, 98 the ROST61 set of open-shell reaction energies, 99 as well as the modified MOBH28 version of the MOBH35 barrier height set 100,101 have been used, with settings as in our previous work. 102ere available, we used the updated transition-state structures from Furthermore, the energetics and structures of gas-phase mixedvalence oxo species have been investigated by using the MVO-10 test set. 105In accordance with our original work, def2-TZVP basis sets 82 and gridsize 3 have been applied, with the exception of V 4 O À 10 , for which full SCF convergence was only achieved with gridsize 7.
To investigate the performance for valence excited-states within linear-response TDDFT, we employed the test sets by Tozer and coworkers 106 (Tozer-I in the following), the Thiel test set, [107][108][109] as well as excitations from the QUEST database. 110For the Tozer-I set, gridsize 3 and cc-pVTZ basis sets were used, 111,112 except for Rydberg excitations, for which more diffuse d-aug-cc-pVTZ basis sets have been employed.
For the 103 singlet and 63 triplet valence excitations of small organic molecules of the Thiel set, gridsize 3 and def2-TZVP basis sets 82 were used, together with the theoretical best estimates for vertical excitation energies as reference values. 107,108For both test sets, SCF convergence 10 À9 a.u. and TDDFT convergence 10 À7 a.u. was used.Following the selection process of Reference 113, pairs of matching 99 singlet and 99 triplet valence excitations from the QUEST database 110 have been used with aug-cc-pVTZ basis sets 111,112 and gridsize 7, enhanced by setting radsize 40.SCF and TDDFT convergence were set to 10 À9 a.u. and 10 À8 a.u., respectively.[116] To enable a consistent and concise comparison, we selected a few representative XC functionals to compare to the new LH23pt model.Apart from the LH20t local hybrid 69 and the PBE0 global hybrid, 38,117 which have been used for all test sets, this includes PW6B95 118 as one of the best performing XC functionals in the case of ground-state valence properties as well as LH12ct-SsifPW92 54 and ωB97X-D 63 as well-performing functionals for the valence excitedstate test sets. 21,113,119For core properties, BHLYP has been used as additional reference functional, as it is known to provide reasonable 1s core-excitation energies for second-period elements. 20,120Results for further XC functionals can be found in Supporting Information.

| Relativistic corrections
Relativistic corrections to 1s core orbital energies Δ mol E have been calculated individually for each molecule and XC functional as the sum of various correction terms, where Δ mol 1e À E and Δ mol C E describe one-and two-electron-relativistic corrections stemming from the use of the Dirac-Coulomb Hamiltonian, [121][122][123][124][125] respectively, and Δ mol B E the additional two-electron-relativistic corrections from the perturbational Breit Hamiltonian. 126Δ at VP E, Δ at MS E and Δ at SE E are additional atomic quantum electrodynamical corrections for the vacuum polarization, the mass shift and the self-energy, respectively. 127lativistic DFT calculations of the individual molecules based on the Dirac-Coulomb Hamiltonian have been performed using the scalarrelativistic infinite-order two-component (IOTC) method, 124,125,128 as implemented in the RAQET program package. 129In particular, Δ mol 1e À E has been determined as difference between 1s orbital energies of the one-electron IOTC calculation and the corresponding nonrelativistic calculation.Similarly, Δ mol c E is calculated as difference between the two-and one-electron-relativistic IOTC calculations.However, in contrast to the straightforward relativistic transformation of the oneelectron Hamiltonian, the relativistic transformation of the two-electron integrals within a DFT framework consists of several individual steps.
Apart from the conventional relativistic transformation of the Coulomb integrals, 125 this includes the picture-change transformation (PCT) of all ingredients of the XC functional. 130While the PCTs of the electron density and its gradient are straightforward, 131,132 an adequate definition of the picture-change-transformed kinetic energy density additionally requires enforcing the iso-orbital limit for the relativistic kineticenergy density, since most XC functionals beyond the GGA level are built upon the satisfaction of this constraint. 44In fact, relativistic DFT calculations without enforcing the iso-orbital constraint have been shown to result in significant methodological errors in the context of local hybrid functionals. 133Furthermore, the relativistic transformation of the two-electron integrals within a DFT framework requires the use of relativistic XC functionals. 133,134As our relativistic DFT calculations are based on the Dirac-Coulomb Hamiltonian, only longitudinal contributions have been considered.However, adequate relativistic corrections are just known for a few exchange functionals, in particular for an exact formulation for Slater-Dirac exchange 135 and empirical parametrizations for PBE and B88 exchange. 136As the present local hybrid model is based on PBE 38 exchange and employs B88 61 damping in the CF, this is nonetheless sufficient to construct a fully relativistic exchange functional. 133Relativistic variants of correlation functionals are virtually nonexistent, including the employed B95 functional, so we use their nonrelativistic variants.Associated errors are expected to play only a minor role compared to the other contributions.Accordingly, we have used relativistic variants of: BLYP, 61,137 PBE, 38 B3LYP, 61,137,138 PBE0, 38,117 BHLYP, 61,120,137 LH12ct-SsirPW92, 54 LH12ct-SsifPW92, 54 LH20t, 37 and of the newly optimized functionals.
All DFT calculations of relativistic corrections with RAQET have been performed using fully decontracted pcseg-3 basis sets 85,86 to provide sufficient flexibility to accurately describe differences of 1s orbitals regarding different levels of relativistic treatment (nonrelativistic, oneelectron IOTC, two-electron IOTC).Differences to UGBS 139 results have been found to usually be less than 0.01 eV even for heavier elements, indicating that results are sufficiently close to the basis-set limit.Tight convergence criteria for the self-consistent-field (SCF) method have been applied, 10 À9 a.u.for total energies and 10 À4 a:u: for density matrices.Also, accurate integration grids of predefined size 7 have been used (the definition within RAQET is similar but not the same as in TURBOMOLE) 130 and all integral thresholds have been disabled, including those for analytical as well as numerical and semi-numerical integration. 130While global hybrid calculations employ analytical exchange integrals, semi-numerical integration within the modified COSX 130 method has been used for calculating local hybrid exchange integrals.To speed up the computation of relativistic XC integrals, enable the direct-SCF method in the two-electron-relativistic IOTC calculations as well as relativistic local hybrid calculations, the PCT of the two-electron contributions has been performed by a PCT of the density matrix. 130e QED corrections as well as Breit contributions to 1s orbital energies have been calculated using the GRASP2018 program package for relativistic atomic structure calculations. 140These corrections were obtained for the free atoms.In contrast to Δ mol 1e À E, the two-electron-relativistic corrections Δ mol C E have been found to be virtually independent of the molecular surroundings of a given atom type.
Errors compared to molecular contributions are therefore expected to be negligible.Since the Breit term and the QED corrections are not included in the SCF procedure in GRASP2018 but added perturbationally, 1s orbital corrections are calculated by a ΔSCF approach.The influence of using ΔSCF compared to orbital energies, while being significant for total energy differences, can be expected to have only a negligible effect on relativistic and QED shifts, as the corrections are small in absolute value.In particular, we calculated the correction as total energy difference between the Hartree-Fock ground state and the corresponding 1s-core-excited state switching on the respective correction terms.While the three QED correction terms are directly used at the atomic Hartree-Fock level, assuming that the influence of the XC functional is negligible, the dependence of the Breit term on the employed XC functional, that is, the transversal part of the XC functional, is certainly not negligible.As the accurate calculation of transversal contributions using local hybrid functionals is not possible with RAQET (or with any other currently available relativistic quantum chemistry program), we employ the following approximation to adjust the Breit correction to the XC functional That is, we assume that the

| The optimized functional
As described in more detail in Section 3.1 (cf.Section 2.2 for the setup), we have developed five different pt-LMF-based local hybrid functionals.In particular, this includes the noncalibrated and calibrated functionals without the core term in the LMF, LH23pt*-nc and LH23pt*, respectively, as well as the corresponding two functionals including an optimized core term, LH23pt-nc and LH23pt, respectively.The LH23pt-IP variant, where the core-LMF has been optimized for core-ionization potentials rather than for core-excitation energies, will be discussed further below.Optimized parameters for all functionals are summarized in Table 1.We first note that the t-LMF underlying the LH20t functional, although providing the correct behavior (except for the down-scaling) in the asymptotic limit, features vanishing LMF values farther away from the molecule when evaluated in nodal planes.As discussed in detail by Schmidt et al., 141 this can be either interpreted as a feature or an incorrect behavior of the iso-orbital indicator t.In the pt-LMF model, a similar behavior can be observed but to a much lesser extent due to the ρ À1 term in the h function (see Section 2.2).We emphasize that we have so far not attempted to optimize that part of the functional, so no final conclusion can be drawn as to whether this feature of the t-LMF and the pt-LMF is desirable or not.Before adding and optimizing the core term of the LMF (LH23pt*-nc , LH23pt*), the core part resembles that of a t-LMF but with a smaller prefactor.One would therefore expect a worse description of the core region compared to LH20t (see below).Adding the core term and optimizing parameters c and d in Equation 15 (LH23pt-nc , LH23pt) changes the core part fundamentally.Now one can clearly see that EXX admixture near the nuclei increases with increasing nuclear charge.For example, the EXX admixture at the sulfur nucleus in CS is 95%, that at the carbon nucleus only 83%.This is the qualitative behavior expected from both the iso-orbital limit (where the sulfur 1s orbital dominates the density at the nucleus more than it does in the case of carbon) and  from the high-density limit.Addition of a CF in going from LH23ptnc to LH23pt changes the LMF parameters somewhat (Table 1), while the appearance of the LMF in the core region changes very little (Figure 2).Larger differences of the LMF in the core region when going from LH23pt*-nc to LH23pt* highlight that this part of the LMF is poorly defined when core properties are not adequately considered in the optimization procedure.Based on the more reasonable behavior in the core region than the conventional t-LMF, LH23pt is expected to provide an improved description of core properties.
We note that the optimized exponent d of the core LMF part tends to converge to values close to 0.5.This is above 1 = 3 and thus clearly fulfills the strong scaling condition for the complement of the LMF (Section 2.1).As the final optimized a and x parameters of the LMF are both close to one (Table 1), we may expect that the valenceshell behavior of LH23pt should not differ very much from that of the well-performing LH20t (see Section 4.3).Indeed, while the EXX admixture in the bonding region changes somewhat for the different functionals, differences are rather small (Figure 2).We also note, however, that the increase toward the asymptotic region with the pt-LMFs is initially slower than with the scaled t-LMF before increasing quickly toward 1.0 further away from the nuclei.These features are expected to affect, for example, Rydberg excitations (see Section 4.3).

| Core properties
Table 2 summarizes mean and mean absolute deviations for secondand third-period species for 1s core-valence (CVal) and core-Rydberg (CRyd) excitation energies from the Tozer-II test set.Full data are provided in Table S4 in Supporting Information.We note that the CVal data are largely those used to optimize the core LMF (see Section 3.1), except for the added SiH 4 data, while the CRyd data serve as external validation set.As The deviations averaged over the CVal subsets are provided graphically in Figure 3.
As mentioned in the introduction, t-LMF-based functionals like LH20t do already perform better than any other general-purpose functional for the 1s core excitations of the second-period elements.This is borne out by the data, where even compared to BHLYP the deviation is reduced by a factor of 3 for the CRyd subset.As expected from the above discussion, LH23pt* without core LMF part performs worse, even in comparison with BHLYP, and it gives a clearer underestimate.Looking at the mean deviations, LH23pt improves slightly over LH20t for the CVal excitations, more notably so for the CRyd excitations of the second-period elements (see Table 2), and it gets close to an overall threshold of 1 eV.Looking at specific elements, we see that the pt-LMF improves upon the carbon and nitrogen results but increases the positive deviations for oxygen and fluorine (Figure 3 and Table S3 in Supporting Information).The latter behavior is suspected to be caused by a too large increase of the EXX admixture in core regions when going from nitrogen to oxygen to fluorine.In fact, this might be a limitation of the simple core term in the pt-LMF, as it only contains two parameters to describe light and heavier elements.
We note, however, that the design of more complex core terms is not straightforward.For example, our initial attempts to improve the model within the first iteration of the optimization cycle included: (a) the sum of different constant powers of the density, (b) explicitly density-dependent powers of the density, as well as (c) enforcing a λ À1 scaling of h to give a nonvanishing energy contribution in the high-density limit.However, none of these approaches improved significantly upon the simple model suggested here.
It is for the third-period elements, where we see the most notable changes with LH23pt: improvement over LH20t by more than an order of magnitude is found, and we are now in a similar range as for T A B L E 2 Mean deviations (MDs) and mean absolute deviations (MADs, in parentheses) of K-shell core-Rydberg (CRyd) and core-valence (CVal) excitations for second-and third-period systems in eV from the Tozer-II test set for several DFT functionals (corrected for relativistic effects).the second-period elements.While LH20t underestimated both CVal and CRyd excitation energies for these elements by 15 to 25 eV (comparable to BHLYP), we are now in a range of at most a few eV.As expected, LH23pt* without core LMF performs worse than LH20t.
These results are clearly consistent with the core-LMF behavior for the different elements (cf. Figure 2).
Note that these results for 1s core-excitation energies depend critically on including the relativistic corrections outlined in Section 2.
It is worthwhile to inspect the individual terms of the relativistic corrections in somewhat more detail.Complete data can be found in Tables S1 and S2 in Supporting Information, only the major trends will or QED contributions would perform even worse than standard oneelectron treatments often used, for example, for many X2C calculations. 123For atoms like carbon, the combined Breit and QED corrections are of comparable magnitudes as the Dirac-Coulomb contributions.For elements like Cl, the Breit+QED contributions may still account for a reduction by about 30 %, or 2 À 3 eV.(c) The dependence of the relativistic corrections on the XC functional can be up to 0.7 eV (HF vs. BLYP for Cl 2 ), comparable to an earlier finding for the two-electron contributions. 133r the optimization of functionals for core properties this means that a neglect of QED effects and of the interdependence with the XC functional would lead us astray by several eV for the third-period elements and still might cause errors of 0.2 eV for, for example, carbon.This shows that some earlier evaluations of functionals for core excitations without accounting for these higherlevel contributions 20 should be taken with a grain of salt.These errors could of course be absorbed in empirical, atom-or coreshell-specific shifts that are frequently employed in the field.But if we aim at arriving at accurate core excitations without such shifts, taking all relativistic contributions into account is mandatory.The present treatment provides thus a good basis for optimizing the core potential of LH23pt and of subsequent functionals, including further local hybrids.
As further tests on core properties, Table 3 provides mean deviations also for the 1s core ionization potentials of the Tozer-II test set molecules, and also for the HBr molecule, that is, for a fourth-period element.Notably, these energies are computed directly from the negative of the corresponding 1s orbital energy using the generalized Kohn-Sham version of Koopmans' theorem, rather than using ΔSCF computations.Full data are provided in Tables S5 and S6  The larger overshooting by LH23pt of core-ionization potentials for third-period elements (Table 3) compared to the 1s core excitations discussed above (Table 2) points to further effects not considered within the present methodology.For example, employing LR-TDDFT core excitation energies in the optimization procedure effectively results in a fit of the XC response kernel contribution together with the underlying XC potential.On the other hand, extracting core-ionization potentials from the negative of the HOMO energies only depends on the XC potential.Accordingly, the optimized core parameters of LH23pt in principle might be biased, resulting in a better performance for core excitations compared to the respective ionization potentials.To identify the influence of our choice on the optimized parameters of the core-LMF, we have reoptimized parameters c and d of the original LH23pt functionals with respect to the T A B L E 3 Element-or period-specific mean deviations (MD) of 1s core-ionization potentials in eV for the molecules of the Tozer-II test set 20  and so are changes in the resulting core-ionization potentials.This highlights that choosing core-excitation energies rather than coreionization potentials for the optimization does not introduce a significant bias into the obtained core-LMF parameters.On the other hand, the larger deviations of the third-period 1s core-ionization potentials, even after reoptimization, suggest some specific beneficial error compensation for the core-excitation energies.While the XC kernel in LR-TDDFT is clearly a possible source of these differences, other aspects cannot be ruled out.An insufficient flexibility of the currently used core-LMF may clearly be an issue, but this has to be examined together with other parts of the functional, including the CF and the choice of the semi-local XC ingredients.These difficult analyses are outside the scope of the present work.Here we can state in any case that LH23pt improves significantly over LH20t for both coreexcitation and core-ionization energies, in particular for third-and fourth-period elements.

| Valence properties
As we aim at a functional that does not only improve upon core properties but also retains the accuracy of LH20t for regular valenceshell energetics, we also evaluate LH23pt for such data sets, starting with the large GMTKN55 test suite of main-group thermochemistry, kinetics and noncovalent interactions.Table 4 summarizes the WTMAD-2 statistics values for the entire test suite and the usual subcategories (Table S7 in Supporting Information includes additional functionals).We note that before adding the D4 dispersion corrections, LH23pt improves over LH20t for all subcategories except for intermolecular noncovalent interactions (NCIs).The D4 corrections are not as effective for LH23pt as for LH20t, so that LH23pt-D4 also falls somewhat behind LH20t-D4 for the intramolecular NCIs, leading to an overall slightly larger overall WTMAD-2 than for the earlier functional (Table S8 in Supporting Information lists the performance of the LH23pt for all GMTKN55 subsets).Overall, LH23pt thus retains the excellent performance of the related LH20t for main-group energetics.
We had recently shown that LH20t is also among the top of rung-4 functionals regarding the energetics of transition-metal organometallic complexes (the MOBH28 set of barrier heights (cf.Table S10), the MOR41 set of closed-shell reaction energies (cf.Table S11) and the ROST61 set of open-shell reaction energies (cf.Table S12)). 102cently, the ωLH22t range-separated LH was found to perform similarly well. 42 comparable to the other best rung-4 functionals evaluated so far (note that the simpler PBE0-D4 performs comparably well here but is far inferior for GMTKN55). 42,102e particular success of LH20t has been its ability to balance excellently the requirements of low self-interaction errors and good treatment of left-right correlation for mixed-valence (MV) systems, 37 as exemplified by the MVO-10 set of gas-phase MV oxide species. 105 particular, LH20t has been suggested to be the first functional of any rung that simultaneously reproduced the localized triple-well potential-energy surface for the ½Al 2 O 4 À oxyl radical anion and the delocalized metal-centered structure of the ½V 4 O 10 À anion.We found that with an increased integration grid ( gridsize 7), the symmetry breaking for the vanadium anion with LH20t unexpectedly increases somewhat compared to the earlier values, even though the energy difference is still almost in the range allowed by the gasphase vibrational spectra (Table S13).Here we note that the energy differences obtained with LH23pt are generally somewhat smaller than with LH20t, thereby improving the performance even further.Now the energy difference for ½V 4 O 10 À indeed coincides with the lower bound suggested by the maximally 50 K internal temperature estimated for the infrared multiple photon dissociation measurement. 105,142A B L E 4 WTMAD-2 values (in kcal/mol) for GMTKN55 and its categories (basic properties and reactions of small systems, isomerizations and reactions of large systems, barrier heights, intermolecular and intramolecular NCIs).We first note that LH23pt gives statistical measures extremely close to LH20t for the singlet and triplet excitations from both QUEST and Thiel test sets.In particular, the exceptionally good accuracy for triplet valence excitations found already for first generation LHs with common t-LMF like LH12ct-SsifPW92 (cf.Table 5) is retained with the common pt-LMF.This is consistent with the relatively small differences between pt-and t-LMF in the valence region and the otherwise similar setup of the two functionals (see above).
The Tozer-I CT subset data with LH23pt are also close to the LH20t results.These excitation energies, while reasonable, are not reproduced with an as high accuracy as by the ωB97X-D range-separated hybrid (RSH).This is due to the fact that the LHs do not exhibit the correct asymptotics of the potential, while the long-range corrected RSH does (as does the recent range-separated LH ωLH22t 42 ).
This starts to matter more with increasing long-range CT character of an excitation. 143e most notable change with LH23pt relative to LH20t pertains to the Rydberg subset of the Tozer-I set, where LH23pt performs clearly worse than the older functional.As we had found earlier, 21 these low-lying Rydberg states are strongly influenced by how fast the LMF increases in the intermediate region between valence and asymptotics.Newer results with the range-separated local hybrid ωLH22t further suggests less dependency on the actual long-range asymptotics of the energy density. 42Here, the current pt-LMF initially increases more slowly than the scaled t-LMF of LH20t (see Figure 2 above).This apparently deteriorates the performance for the Rydberg subset, similarly to earlier observations for a so-called s-LMF. 56The latter also goes to 1.0 far away from the nuclei, as does the pt-LMF, but it also increases more slowly in the intermediate region.Note, however, that we have made no attempt to optimize that part of the pt-LMF at all.Clearly, an adjustment to appropriate data sets containing Rydberg excitation energies will be useful to improve upon that part of the LMF.This will be pursued in future work.The resulting LH23pt functional keeps the already good performance of the LH20t functional for excitations out of 1s orbitals of second-period elements but improves decisively upon the core regions for elements from the third period, and also tentatively for core properties of elements from the fourth period.The common pt-LMF underlying the LH23pt functional has been designed to retain the excellent performance of the common t-LMF in the valence space, as confirmed by a wide evaluation on ground-and excited-state quantities.Only Rydberg excitations tend to be less well described than with LH20t.This is likely due to the fact that we have not optimized so far the low-density part of the pt-LMF.
T A B L E 5 Mean absolute deviations and mean deviations (in parentheses) in eV for excitation energies for the singlet/triplet subset 113 of the QUEST database, 110 the singlet and triplet subsets of Thiel's test set, [107][108][109] and for the Rydberg and CT subset of the Tozer-I set. 106Data for the functionals LH12ct-SsifPW92, PBE0, and ωB97X-D are from References 21 and 113.The strong scaling condition and the iso-orbital limit are exact constraints that in the context of the core shells of atoms, molecules or solids hold only in extreme cases, that is, close to the nuclei of very heavy atoms.These constraints do not determine how to approach by an additional layer for optimizing the core part of the LMF (i.e., parameters c and d in Equation 15).That is, the applied optimization cycle (Figure1) now consists of three individual optimization steps that are performed in the order shown until self-consistency is achieved: (1) optimization with respect to valence properties, (2) optimization with respect to core properties, (3) calibration to minimize gauge artifacts.In particular, only a specific subset of functional parameters is optimized in each step while all other parameters are kept fixed.In this way the optimizations of the three parameter subsets in (1), (2), and (3) are partially disentangled, which has the substantial advantage that no arbitrary weighting factors between properties of differing dimensionality need to be chosen, and that properties of one subset are not sacrificed in favor of mized the three further local hybrids LH23pt-nc, LH23pt* and LH23pt-nc.For LH23pt*-nc, neither the CF nor the core part of the LMF have been optimized.That is, LH23pt*-nc is simply the functional obtained after step (1) of the first optimization cycle for LH23pt.For LH23pt*, we performed the same multilayer procedure as for LH23pt, with the exception that step (2), the optimization of the core part of the LMF, is skipped in all optimization cycles.Parameters c and d have been set to 0. Accordingly, the optimization procedure for LH23pt* is the same as for LH20t.For LH23pt-nc, the calibration step (3) has been skipped, thus yielding a noncalibrated variant of LH23pt, which still features an optimized core LMF part.Below we will also discuss a variant, LH23pt-IP, where c and d have been optimized for core-ionization potentials rather than for core-valence excitation energies.
For evaluating the new LH23pt functional, we compare its performance to several existing successful XC functionals for a number of established test sets.For core properties, the full set of 15 core-valence and 23 core-Rydberg excitations of the Tozer-II test set 20 was used.Medium-sized diffuse grids (gridsize 3 and diffuse 4) and aug-pcseg-3 basis sets have been used to enable a technically accurate description of core-valence and core-Rydberg excitations.Convergence criteria for SCF and linear-response TDDFT were set to 10 À8 a.u. and 10 À9 a.u., respectively.Calculated nonrelativistic 1s core-excitation energies are corrected individually for each XC functional (Section 3.3).In addition to excitation energies, 1s core-ionization potentials for the molecules of the Tozer-II test set as well as for the HBr molecule were calculated, resulting in a total of 18 values.Experimental reference values have been taken from the literature (see Table

Reference 103 together
with the reworked reference energies from Reference 104 (LNO-CCSD(T)-based extrapolation scheme) for the MOBH28 test set.Due to the lack of reference energies at the same computational level, reactions 17-20, 24, and 25 from the original MOBH35 set have been excluded.Following the recommendation in Reference 104, we also excluded reaction 9, because of the sizable amount of static correlation.Weighted mean absolute deviations (WTMADs) for the latter three sets have been obtained by applying the formula Figure S2 in Supporting Information.

2
Comparison of the pt-LMFs for LH23pt and for intermediate functionals to the t-LMF of LH20t along the bond axis in the CO molecule (A) and CS molecule (B).See Figure S2 in Supporting Information for further plots.

F I G U R E 3
Comparison between LH23pt and LH20t of the mean deviations in eV of the core-valence excitations for a given element from the Tozer-II test set.Si is excluded (see Section 3.1).
be discussed here: (a) Inclusion of only the one-electron IOTC corrections would overshoot the final relativistic corrections systematically, up to more than 100 % for the second-period atoms.For the heavier atoms like Cl, the relative importance of going beyond the oneelectron corrections decreases, but now the absolute deviations can amount to more than 2 eV.(b) Including only the two-electron corrections based on the Dirac-Coulomb Hamiltonian even increases the overestimate, while Breit and QED terms lead to a reduction of the final values.That is, two-or four-component results without Breit in Supporting Information.When first just taking a bird eye's view on the overall second-period data, we see comparable results as for the prior excitation-energy tests.That is, LH20t does already improve substantially over BHLYP, and LH23pt overall retains this improvement (while LH23pt* without core LMF performs substantially worse).But we also see a clear shift of the LH23pt results compared to LH20t when looking at the individual elements: values of the MD for C and N are lowered, those for O and F increase.This change is not reflected in the overall MD for the second-period elements.It may signify that the form of the core LMF is still not sufficiently flexible to improve simultaneously for all elements.Turning to the third-period elements, we see again a dramatic improvement with LH23pt compared to LH20t, with positive deviations of around 3-6 eV compared to the usually much larger negative deviations.This reflects the increase of the core LMF values with nuclear charge.We also have bromine as an example for a fourthperiod element, bringing us more clearly outside the range of data used to optimize the core LMF parameters.Here the deviations are much larger for all functionals, but again LH23pt gives much smaller and positive deviations.This suggests that we may overshoot somewhat the increase of the EXX admixture with nuclear charge with the current pt-LMF.

5 |
CONCLUSIONSLocal hybrid functionals (LHs) with position-dependent admixture of exact exchange offer more flexibility than other exchangecorrelation functionals for providing accurate DFT calculations of properties depending on different spatial regions in an atom, molecule or crystal.So far, the potential of LHs in this context has been exploited only to a very limited extent.Here we have reported a step forward for properties related to regions close to the nuclei by devising an improved local mixing function for LHs that has been developed with particular emphasis on the core regions.We have shown that (a) the form of the LMF can be improved regarding the high-density uniform scaling limit and the iso-orbital limit, and (b) an optimization of the core-LMF part for core-excitation and energies in a linear-response TDDFT context requires the careful inclusion of relativistic corrections.
these limiting cases and what should be the behavior for lighter atoms and/or for core shells further away from the nucleus.The Padé-form of the LMF and the specific form of the h-function of LH23pt are likely only a first approach toward achieving the required flexibility to be able to describe the correct behavior in diverse core shells of arbitrary atoms.Possible future extensions and improvements may involve an expansion of h into different powers of ρðrÞ or different types of mapping functions for the LMF of local hybrid functionals.Of course such improved descriptions of the core regions may then also be combined with other developments such as (globally or locally) range-separated local hybrids, corrections of the LMF for regions far away from the nuclei, or strong-correlation terms.ACKNOWLEDGMENTS This work has been supported by Deutsche Forschungsgemeinschaft (DFG) via project KA1187/14-2.The authors gratefully acknowledge the computing time granted by the Resource Allocation Board and provided on the supercomputer Lise and Emmy at NHR@ZIB and NHR@Göttingen as part of the NHR infrastructure.Calculations in Berlin were conducted with computing resources under HLRN project bec00223.We thank Hiromi Nakai for providing access to the Raqet source code.Open Access funding enabled and organized by Projekt DEAL.
Parameter sets for LH23pt and for four related functionals obtained in the multistep optimization procedure (see Section 3.1).
calculated with different XC functionals from negative 1s coreorbital energies using Koopmans' theorem (with relativistic corrections, cf.text).potentials of the Tozer-II set.Relativistic corrections of the original LH23pt have been retained.The resulting parameters are included in Table1, and the MAE for core ionization potentials is given in TableS6in Supporting Information.Changes of parameters c and d compared to the original LH23pt are very small,