Spin-Contamination Dilemma of

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Quantum Electronic Structure

Lessons from the spin-polarization/spin-contamination dilemma of transition-metal hyperfine couplings for the construction of exchange-correlation functionals Caspar J. Schattenberg, Toni M. Maier, and Martin Kaupp J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00597 • Publication Date (Web): 09 Oct 2018 Downloaded from http://pubs.acs.org on October 13, 2018

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Journal of Chemical Theory and Computation

Lessons from the spin-polarization/spin-contamination dilemma of transition-metal hyperfine couplings for the construction of exchangecorrelation functionals Caspar J. Schattenberg,a Toni M. Maier,b Martin Kauppa,* a

Institut für Chemie, Technische Universität Berlin, Theoretische Chemie/Quantenchemie,

Sekr. C7, Straße des 17. Juni 135, D-10623, Berlin, Germany b

Department of Chemistry and Biochemistry, School of Advanced Science and Engineering,

Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

Abstract. Hyperfine couplings (HFCs) of open-shell transition-metal centers are known to often depend crucially on core-shell spin polarization (CSSP). The latter is typically underestimated by semi-local density functionals, while admixture of exact exchange (EXX) in (global) hybrid functionals enhances CSSP. Unfortunately, a metal-ligand antibonding character of one or more of the singly occupied molecular orbitals of the complex will cause substantial valence-shell spin polarization (VSSP), which for global hybrids with higher EXX admixtures may lead to substantial spin contamination, thereby deteriorating the overall electronic structure and the dipolar couplings. In view of this known dilemma, we use a subset of 3d complexes from an earlier study (M. Munzarová, M. Kaupp J. Phys. Chem. A 1999, 103, 9966-9983) to examine systematically a wide range of exchange-correlation functionals for metal HFCs, including highly parameterized (meta-)GGAs, global and range-separated hybrid functionals not yet available in earlier studies, as well as for the first time local hybrids with real-space position-dependent EXX admixture. Both CSSP and VSSP have been carefully analyzed in terms of their orbital contributions, both for cases dominated only by CSSP and for systems influenced crucially by VSSP and spin contamination. While some more parameterized meta-GGA functionals (-HCTH, VSXC, partially M06-L) provide surprisingly realistic CSSP, some others (MN12-L, MN15-L) and some global hybrids (M05, M06, partly MN15) exhibit dramatic shortcomings in describing the CSSP contributions. Local hybrid functionals provide a promising way of enhancing CSSP by high EXX admixture in the core region while avoiding excessive VSSP and thus spin contamination. These analyses provide important insights that

* Corresponding author, email: [email protected].

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may help to construct improved functionals for HFCs and related properties, e.g. contact NMR shifts.

Keywords. Hyperfine coupling, local hybrid functionals, range-separated hybrid functionals, meta-GGA functionals, global hybrid functionals, spin-density analyses, spin contamination, spin polarization, 3d transition-metal complexes.


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1. Introduction Hyperfine couplings (HFCs) are important spectroscopic parameters that present substantial challenges to their accurate quantum-chemical calculation.1,2,3 In particular, isotropic hyperfine couplings require, within a nonrelativistic framework, computation of the spin density specifically at the nucleus. This leaves little room for error compensation, as it requires not only an accurate description of electronic structure at this particular nucleus but frequently also of subtle spin-polarization processes involving core and valence shells. These aspects make isotropic HFCs interesting properties for the critical evaluation, and possibly for the improvement of quantum-chemical methods. In 1999, Munzarová and Kaupp4 provided such a critical evaluation of density functional (DFT) and coupled-cluster (CC) methods for the metal HFCs of a series of 3d transition-metal complexes. One crucial observation for DFT methods was, that standard local or semi-local functionals (the local spin-density approximation  LSDA  or generalized gradient approximations  GGA) severely underestimate the important coreshell spin polarization (CSSP) of the 2s and 3s core shells by the 3d-orbital-dominated singly occupied molecular orbital(s) (SOMO). As the negative contribution from the 2s polarization exceeds the positive one from the 3s shell, the overall CSSP contribution to the spin density at the metal nucleus is negative5 (as was found very early at UHF level for atomic ions6 and at MS-X level for complexes7). Its underestimate by such functionals thus renders the computed isotropic HFCs too positive.4 Similar CSSP mechanisms have been identified for 4d and 5d complexes,8,9 but can of course also be nontrivial for main-group atoms (see, e.g., ref. 10). In the 1999 paper4 and subsequent work8,11 for transition-metal complexes we found that inclusion of substantial amounts of exact (Hartree-Fock-type) exchange (EXX) in global hybrid functionals increased the CSSP contributions and thus improved agreement with experiment. However, for an appreciable number of complexes studied, too large EXX admixtures led to severe spin contamination, which in turn deteriorated both the isotropic HFCs and the HFC anisotropies.4 The latter are much less dependent on CSSP, but the “contaminated” spin densities led to large errors here as well. Closer analysis revealed, that the spin contamination was related to large valence-shell spin polarization (VSSP) in cases, where the SOMO(s) of the system exhibited substantial metal-ligand anti-bonding character, and the corresponding bonding MOs were found to be involved in the VSSP process.5 This spin contamination for larger EXX admixtures is a general observation for many transition-metal complexes that affects also a large number of other properties of a given molecule (e.g. g-tensors,8,12 but also 3 ACS Paragon Plus Environment


optical spectra,13 vibrational spectra14 or even structures14). In the context of HFCs, we thus face the dilemma that larger EXX admixtures are needed to enhance the CSSP, but too large admixtures cause exaggerated VSSP and thus spin contamination in the cases discussed. It appears virtually impossible to solve this “spin-polarization/spin-contamination dilemma” using a variation of EXX admixture in global hybrid functionals. Attempts towards improvement included Rinkevicius’ “restricted/unrestricted” approach15 and one application of constrained-DFT (CDFT).16 None of these approaches was entirely satisfactory: In the restricted/unrestricted approach, the spin polarization is added by linear response theory on top of a restricted DFT calculation. It turned out that for the very cases, where the unrestricted calculations with larger EXX admixtures led to spin contamination, the perturbational treatment underestimated both CSSP and VSSP substantially (and the approach is computationally much more demanding).15 Adding spin-contamination constraints by CDFT also does not solve the problem: restricting spin polarization for the entire system by CDFT obviously eliminates CSSP together with VSSP completely. “Switching off” spin polarization only outside a sphere around the metal center reduced the spin contamination and gave reasonable HFCs for the metal.16 However, this procedure is highly arbitrary and creates other problems. Of course, high-level post-Hartree-Fock ab initio methods can achieve an accurate treatment of both CSSP and VSSP without excessive spin contamination.2,4,17 However, the computational requirements of these methods tend to be very high, making them so far less suitable to treat the HFCs for larger systems of chemical interest (some recent developments on local correlation methods give hope that this may change18). Here we revisit the HFCs of some of the complexes from our 1999 paper, exploring a number of DFT exchange-correlation functionals that were not yet available at the time. This includes some contemporary, parameterized meta-GGA functionals, some more highly parameterized global hybrid functionals (e.g. of the Minnesota type), several range-separated hybrid functionals,19,20 where EXX admixture depends on inter-electronic distance, and in particular some of our own local hybrid functionals21,22,23,24 exhibiting a real-space position-dependent EXX admixture. The latter type of functionals should in principle be able to provide the features needed to solve the spin-polarization/spin-contamination dilemma, as they can exhibit large EXX admixture in the core region combined with lower EXX admixture in the valence region, depending on the so-called local mixing function (LMF) used. Yet so far, such functionals have not yet been scrutinized nor optimized for HFCs (but see ref. 25). Here we will study 4 ACS Paragon Plus Environment

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systematically the effect of scaling two simple types of LMF. These investigations set the stage for future improvements of DFT approaches to solve the underlying dilemma.

2. Theoretical background Hyperfine couplings. The interaction of electronic spin with the magnetic moment of a nucleus can be described by a hyperfine coupling tensor (A).1,26,27 Using a perturbational treatment of spin-orbit (SO) coupling, based on a nonrelativistic first-order description, this 3x3 matrix can be separated into three contributions, (1)

𝐀 = 𝐴𝐹𝐶 𝟙 + 𝐓 + 𝐀 SO ,

the isotropic Fermi contact term (𝐴𝐹𝐶 ), the (symmetric and traceless) dipolar coupling term (𝐓) and the asymmetrical SO hyperfine matrix (𝐀 SO ), which includes contributions to the anisotropic part Aaniso (TSO) and to the isotropic HFC Aiso (the so-called “pseudo-contact term”, APC). Note, that the isotropic part of the full tensor is defined by its trace (𝐴𝑖𝑠𝑜 = 𝐴𝐹𝐶 + 𝐴𝑃𝐶 = 1 3

𝑇𝑟(𝐀)). We will focus mostly on a subset of complexes from ref. 4, where we know ASO to be small,

as we want to minimize complications arising from this term (for which so far implementations with some of the more recent functionals are lacking). We will thus focus on the non-relativistic (first order) contributions (see Computational Details below for SO and scalar relativistic contributions). Then Aiso = AFC, Aaniso = T. Aiso and the anisotropic matrix elements Tkl (where k and l denote the entries of the Tensor which relate to the directions of the magnetic field) are defined as 𝐴𝑖𝑠𝑜 (𝑁) = 𝑇𝑘𝑙 =

4𝜋 𝛼−𝛽 𝛽 𝑔 𝛽 𝑔 〈𝑆 〉−1 𝜌N , 3 𝑒 𝑒 N N 𝑍

1 𝛼−𝛽 𝛽𝑒 𝑔𝑒 𝛽N 𝑔N 〈𝑆𝑍 〉−1 ∑ 𝑃𝜇,𝜈 ⟨𝜙𝜇 |𝑟N−5 (𝑟N2 𝛿𝑘𝑙 − 3 (𝐫𝑁 )𝑘 (𝐫𝑁 )𝑙 )|𝜙𝜈 ⟩, 2

(2)

(3)

𝜇,𝜈

where 𝛽𝑒 and 𝛽N are the Bohr and nuclear magneton, respectively, 𝑔𝑒 corresponds to the freeelectron g-value (2.00231931),28 and 𝑔𝑁 =

𝜇 𝐼N

(with 𝜇 describing the nuclear magnetic moment

and 𝐼N the total spin of the nucleus of interest) is the g-value of the respective nucleus. < 𝑆𝑍 > is the expectation value of the z-projection of total electronic spin, 𝑟𝑁 = |𝐫 − 𝐑 N | the electron5 ACS Paragon Plus Environment


𝛼−𝛽

nucleus distance, and 𝜌N

(in short also 𝜌𝑁 ) denotes the spin-differential density at nucleus N, 𝛼−𝛽

which can be expressed in terms of the spin-differential matrix 𝑃𝜇,𝜈 𝛼−𝛽

𝜌𝑁

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𝛼−𝛽

= ∑ 𝑃𝜇,𝜈 ⟨𝜙𝜇 |𝛿(𝐫N )|𝜙𝜈 ⟩.

in the AO basis {𝜙} as (4)

𝜇,𝜈

For all nuclei with an electronic environment of at least threefold (“axial”) symmetry the anisotropic coupling tensor 𝐓 can be expressed in terms of the dipolar coupling constant Adip and may be diagonalized, resulting in the form (-Adip, -Adip, 2Adip). In our quantum-chemical approach, both quantities, Aiso and Adip, are easily accessible via the simple eigenvalue equations (2) and (3), and can therefore be used as a measure of the (property-specific) quality of the electronic structure provided by the theoretical model applied. Functionals. In addition to more recent parameterizations and forms of semi-local functionals (in particular meta-GGAs, which were not yet widely available in the initial studies) and of global hybrids, we extend the evaluations to, on one hand, range-separated hybrid functionals and to, on the other hand, local hybrids. While it is not a priori clear how addition of either long- or short-range EXX admixture should benefit HFC calculations, a screening of such functionals should provide further insights in any case. Very limited HFC studies of two range hybrids for some heavy-metal complexes have been done in a relativistic framework,29 but more systematic evaluations are lacking. We note in passing, that one study of the B2PLYP double hybrid functional (with an MP2 correlation correction) for transition-metal HFCs has been reported.30 While the large EXX admixture of 53% in the functional is well suited to give good CSSP contributions, the results exposed similar spin-contamination problems as found for global hybrids with large EXX admixtures (e.g. for MnO3 and Mn(CO)5). This is expected to hold essentially for all double hybrids reported so far, and we have thus not included this type of functionals in the present study. We note in passing that some authors prefer to use the terms “Hartree-Fock exchange” or “HFX”. We will follow our previous works and use the terms “exact-exchange” or “EXX” admixture. We feel that this more a more precise term. It denotes exchange of (exact) Hartree-Fock functional form computed with the (generalized) KS orbitals. The potential of local hybrids with position-dependent EXX admixture for HFCs is more obvious (see introduction above). Nevertheless, local hybrids had so far not been studied in detail in this context. This gap will be filled in the present work. As local hybrids are currently less well established than the other types of functionals, we briefly summarize their construction. When replacing the constant (global) fraction of exact exchange admixture by a 6 ACS Paragon Plus Environment

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real-space local mixing function (LMF, 𝑔𝜎 ), that governs the position-dependent EXX admixture,31, 32 we arrive at a local hybrid exchange correlation functional:31 𝑙ℎ 𝑒𝑥 (𝐫) 𝑠𝑙 (𝐫) 𝐸𝑋𝐶 = ∑ ∫[𝑔𝜎 (𝐫)𝑒𝑋,𝜎 + (1 − 𝑔𝜎 (𝐫))𝑒𝑋,𝜎 + 𝐺𝜎 (𝐫)]𝑑𝐫 + 𝐸𝐶𝑠𝑙

(5)

𝜎 𝑒𝑥 (𝐫) 𝑠𝑙 Here 𝑒𝑋,𝜎 denotes the (spin-resolved) exact, 𝑒𝑋,𝜎 (𝐫) a (semi-)local exchange-energy

density, and 𝐸𝐶𝑠𝑙 is some semi-local correlation energy contribution. 𝐺𝜎 (𝐫) is a so-called calibration function (CF) with the property ∫[𝐺𝜎 (𝐫)]𝑑𝐫 = 0, that accounts for the ambiguity of both the exact and semi-local exchange energy density, to deal with the so-called “gauge problem” of local hybrids.33 The latter affects, for example, weak non-covalent interactions,24,34,35 but we do not expect it to be very important for the HFCs studied here. In this initial work we thus deal, except for one functional (see below), with “uncalibrated” local hybrids, where 𝐺𝜎 (𝐫) vanishes. A reformulation (without CF) that shows best the introduction of non-dynamical correlation (NDC) by local hybrids36 is 𝑙ℎ 𝑠𝑙 (𝐫) 𝑒𝑥 (𝐫))] 𝐸𝑋𝐶 = 𝐸𝑋𝑒𝑥𝑎𝑐𝑡 + ∑ ∫ [(1 − 𝑔𝜎 (𝐫)) (𝑒𝑋,𝜎 − 𝑒𝑋,𝜎 𝑑𝐫 + 𝑬𝒔𝒍 𝑪.

(6)

𝜎

Here the middle term on the R.H.S. may be interpreted as NDC term. The above formulation of LH functionals (eq. (5)) uses spin-resolved quantities only. But the interpretation of eq. (6) formally allows consideration of opposite-spin terms in 𝑔𝜎 (𝐫), introducing the concept of “common-LMFs”, which are constructed from properties of the total density or kinetic-energy density (e.g. 𝜌𝜎 (𝐫) is substituted by 𝜌(𝐫) = 𝜌𝜎 (𝐫) + 𝜌𝜎′ (𝐫) and 𝑔𝜎 (𝐫) → 𝑔(𝐫) = 𝑔𝛼 (𝐫) = 𝑔𝛽 (𝐫)). This interpretation helps explain why functionals constructed from common LMFs, rather than those with LMFs that depend separately on  or  spin-channel properties (“spinchannel LMFs”) are expected to account for some further NDC contributions (cross terms) and generally exhibited improved performance.23 A detailed discussion of various forms of local hybrids that have been suggested over the past 15 years can be found in a recent comprehensive review.37 The specific first-generation uncalibrated local hybrids based on LSDA (Slater) exchange used here may be distinguished by their LMF. The first one is the so-called t-LMF (given here in its spin-channel variant), 𝑔𝜎 (𝐫) = 𝑎 ∙

𝜏𝑊,𝜎 (𝐫) 𝜏𝜎 (𝐫)

,

(7)



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which uses a scaled ratio of the von-Weizsäcker kinetic energy density 𝜏𝑊,𝜎 (𝐫) = 𝛾𝜎𝜎 8𝜌𝜎 (𝐫) 1 2

(𝛾𝜎𝜗 (𝐫) = ∇𝐓 𝜌𝜎 (𝐫)∇𝜌𝜗 (𝐫)) 2

∑𝑜𝑐𝑐. 𝑖𝜎 |∇𝜑𝑖𝜎 (𝐫)|

and

the

KS

kinetic

energy

density

𝜏𝜎 (𝐫) =

(the unscaled version had already been proposed in ref. 32). A

thermochemical optimization with VWN5 correlation38 for 𝐸𝐶𝑠𝑙 gave a = 0.48.22 We will call this functional Lh07t-SVWN in the following. Using the common-LMF variant instead provided even better thermochemistry and barriers with a = 0.534 (Lh12ct-SVWN).23 Modifications of 𝐸𝐶𝑠𝑙 by introducing range separation and self-interaction corrections into the (PW9239) local correlation functional gave two further functionals with promising performance: Lh12ct-SsifPW92 introduces a full elimination of short-range self-correlation and features a = 0.709 (and thus overall more EXX admixture).23 A partial elimination of short-range selfcorrelation provides the Lh12ct-SsirPW92 functional with a = 0.646.23 As the only GGA-based local hybrid, we will also look at a local hybrid with spin-channel t-LMF (a = 0.50) that is based on PBE exchange and correlation and uses a CF to account for the gauge problem (Lh14tcalPBE).24 We also evaluate one LSDA-based local hybrid with an s-LMF (Lh07s-SVWN):21 𝑔𝜎 (𝐫) = erf(𝑏 ∙ 𝑠𝜎 ),

(8)

1/2

where 𝑠𝜎 (𝐫) =

1 𝛾𝜎𝜎 (𝐫) 4/3 𝑘 𝜌𝜎 (𝐫)

is the reduced spin density gradient (in the original definition of ref. 21

with 𝑘 = 2(3𝜋 2 )1/3 and the empirical parameter b = 0.22).

3. Computational Details We will concentrate only on a subset of the complexes from ref. 4. The reason is, that the HFCs for some of the systems studied there depend strongly on spin-orbit corrections or on environmental effects, both of which are difficult to include meaningfully into the present analysis. While, in contrast to the time of ref. 4, we do have good linear-response DFT implementations of spin-orbit corrections to HFCs40 (or two- and four-component relativistic approaches8,9), these exhibit a dependence on the functional that is distinct from that of the nonrelativistic contributions. As the availability of functionals for such SO corrections and relativistic approaches is currently limited, we will exclude complexes, for which these contributions are important (e.g. [Co(CO)4] or [NiH(CO)3])4 and will concentrate on systems 8 ACS Paragon Plus Environment

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with small SO effects (see below). Furthermore, in [Mn(CN)5NO]2- we observe severe spin contamination already for (m)-GGA functionals, which may suggest that this molecule is rather a multi-reference case and thus not treatable suitably with simpler approximate DFT methods (see Supporting Information). Charged complexes may be problematic due to counter-ion effects. We will include a few of such charged complexes but have to carefully monitor the importance of the environment. Some of the investigated complexes exhibit very similar behavior as others in our selection, and including them does not add much new insight. The selection of test systems will thus be based on a classification with respect to the importance of CSSP, VSSP, and spin contamination. We distinguish systems with large CSSP but small VSSP and negligible spin contamination (e.g. [Mn(CN)4]2-, but the isoelectronic [Cr(CO)4]+ gives comparable results), complexes with intermediate VSSP and spin contamination (e.g. MnO, TiN) and those with large spin contamination (here we will concentrate on the neutral MnO3 but will also touch the charged [Mn(CN)4N]-). Indeed, the focus in the main text will be just on four manganese complexes exemplifying these cases, namely [Mn(CN)4]2-, MnO, MnO3, and [Mn(CN)4N]-, whereas data for the remaining species will be provided in Supporting Information. Structures have been taken from ref. 4 and are provided in Figure S1 in Supporting Information. While the accuracy of the structures is also a parameter of the computations, our test calculations indicate that their influence is much smaller than that of the functional that this study focuses on. The specifically designed metal 9s7p4d basis sets used here have emerged from an extensive basis-set study (with a more limited set of functionals) in Ref. 4. We found them to exhibit some beneficial error compensation and to thereby provide HFCs agreeing well (typically within a few MHz for Aiso up to some tens of MHz for cases like MnO3) with much larger basis sets including very tight s-exponents. We expect these observations to hold also for the additional functionals studied in this work and have confirmed this by test calculations. IGLO-III basis sets have been used for the ligand atoms. Calculations using local hybrid functionals and a systematic variation of EXX admixture for global hybrids based on LSDA were done with TURBOMOLE 7.2.41 The other functionals studied, including the Minnesota-type global hybrids and various range-separated hybrid functionals, were used as implemented in the Gaussian09 program (release D01),42 and in case of some recent functionals (MN15, MN15-L, and MN12-L) in the Gaussian16 program (revision A03).43 All calculations use unrestricted Kohn-Sham methods. 9 ACS Paragon Plus Environment


All nonrelativistic HFC calculations were performed with the MAG-ReSpect program,44 based on MOs obtained from TURBOMOLE or Gaussian using tight SCF convergence criteria and extended integration grids (energy convergence 10-8 a.u. and int=ultrafine option in Gaussian, energy convergence 10-9 a.u. and gridsize m5 in TURBOMOLE), with subsequent transfer of the unrestricted Kohn-Sham orbitals by interface routines to MAG-ReSpect. Furthermore, we have employed UHF stability analyses of the calculated wave functions in all cases, either by escf calculations in TURBOMOLE, or by calculations employing the stable keyword in Gaussian. Compared to the more limited set of GGA and GH functionals evaluated in ref. 4, here we add several more recent GGA and meta-GGA functionals into the evaluations, such as TPSS,45 VSXC,46 HCTH,47,48,49 τ-HCTH,50 M06-L,51 MN12-L,52 and MN15-L.53 We also add newer (and sometimes also more highly parameterized) GH functionals, such as τ-HCTH-hyb,50 BMK,54 B97-1,47 and the Minnesota series M05,55 M05-2X,56 M06,57 M06-2X,57 M06-HF,58,59 MN15,60 as well as MPW1K,61 and PW6B95.62 For reasons of comparison with previous work we also include TPSSh,63 B3LYP,64 BHLYP65 and PBE066 into the set of GH functionals. Furthermore, we study the range-separated hybrid functionals CAM-B3LYP,67 LCPBE,68,69,70 LC-BLYP,71 the long-range-screened HSE-0672 and the ‘middle-ranged’ HISS,73,74 together with representatives of the B97 family (B97 and B97X-D).75 Additionally, we investigate our own LHs (Lh07s-SVWN,21 Lh07t-SVWN,22 Lh12ct-SVWN23 Lh12ct-SsifPW92,23 Lh12ct-SsirPW9223 and Lh14t-calPBE24), available in TURBOMOLE since version 7.2 (Lh12ct-SVWN is not yet accessible directly via keywords). We furthermore vary the EXX admixtures in a series of GHs and LHs to aid in the understanding of the spinpolarization processes. Where available, functionals have been tested in both program packages, but since the results are very similar, in these cases we show just the Gaussian results. As mentioned above, we focus on complexes for which SO and scalar relativistic effects are expected to be small, in particular for the four central Mn species. We have nevertheless estimated relativistic contributions, so that we know the uncertainties arising in a direct comparison between nonrelativistic computations and experiment for both Aiso and Adip. Therefore, we have estimated ASO by second-order Breit-Pauli perturbation theory, using standard functionals (PBE and PBE0) and the coupled-perturbed KS implementation of ref. 40. These calculations used fully uncontracted versions of the abovementioned NMR_9s7p4d and


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IGLO-III basis sets (for comparison, the nonrelativistic calculations were also repeated with the uncontracted basis sets). Both SO and scalar relativistic effects are included in relativistic four component calculations using the matrix Dirac-Kohn-Sham (mDKS) approach76 with restricted kinetically balanced basis sets,77 as implemented in the ReSpect program, version 4.0.078 (see our previous work on HFC computations at this level9). These calculations were also performed using the PBE and PBE0 functionals, with the same uncontracted basis sets, an adaptive Lebedev grid for the angular grid points,79 and 60 and 80 radial grid points for main-group and 3d-transition metal elements, respectively. To separate SO and scalar relativistic effects at this fourcomponent level, we also carried out computations where the SO effects were scaled to essentially zero within Dyall’s definition for the separation80 (SOSCALE keyword in ReSpect), providing a scalar relativistic description. Overall, relativistic effects, and in particular SO effects, have been found to be small for the complexes central to this study. Results are provided in Tables S1, S2 in the Supporting Information. We will therefore directly compare the nonrelativistic computations with various functionals to experiment, keeping in mind the possible magnitude of relativistic corrections. Another aspect in the comparison with experimental data is the neglect of environmental effects, in particular for charged complexes. To roughly estimate the sensitivity of the computed HFCs to environmental effects, we have evaluated the effects of polarizable dielectric continuum solvent models. We used the conductor-like screening model (COSMO)81 with TURBOMOLE standard settings, and the closely related CPCM implementation with Gaussian09 and Gaussian16.82,83 In both cases, large dielectric constants, e.g. those of water, were used. However, the effects on the HFCs in these tests were generally also small (further information is provided in Tables S3–S5, which also provide dielectric constants).

4. Results and Discussion 4.1 Comparison of different types of functionals. As the amount of overall data compiled is very large, the overall numerical data are provided largely in the SI (Tables S6-S8), including all density functionals and complexes evaluated. While Table 1 summarizes some of the Aiso results for the four most important manganese complexes of this study and selected functionals, Figure 1 gives a compact representation of 11 ACS Paragon Plus Environment


deviations of results from the experimental reference values for the same functionals and complexes, keeping in mind the abovementioned influence of relativistic and environmental effects as well as possible small errors arising from the input structures and basis sets. The four manganese complexes represent different limiting situations. That is, [Mn(CN)4]2- is a system with dominant CSSP, smaller VSSP, and thus negligible spin contamination, MnO exhibits intermediate CSSP and VSSP, whereas MnO3 and [Mn(CN)4N]- are striking examples with large VSSP and thus a large sensitivity to spin contamination, in particular for higher EXX admixtures.4 Of these two complexes, MnO3 furthermore is distinct in having an sdz2-type SOMO with Mn-O -antibonding character, while [Mn(CN)4N]- is an example where the spin contamination arises mainly from spin polarization of -type MOs and is thus somewhat separate from the VSSP contributions to Aiso. Further details on the mechanisms of spin polarization and HFCs are analyzed further below. On the left-hand side of Figure 1, the deviations of the spin density at the manganese nucleus from the reference values are plotted. This provides a better overview for complexes with different overall spin multiplicity than plotting deviations for Aiso. In the middle, deviations of Adip from the reference values are plotted. These depend relatively little on CSSP but are known to deteriorate for large spin contamination (Adip is zero due to symmetry for the tetrahedral [Mn(CN)4N]-).4 On the right-hand side, the spin contamination is evaluated directly as deviation from the nominal expectation value of the KS determinant for the given spin state. While this relates to the non-interacting reference system of KS theory rather than to the real system in case of semi-local functionals, differences compared to alternative evaluations tend to be small,84 and experience shows such comparisons to reveal faithfully problems for the HFC computations.4,5 We expect thus, that the middle and right panel of Figure 1 signal largely problems with exaggerated VSSP and spin contamination, whereas the left panel reflects the more complicated interplay between CSSP and VSSP for the overall particularly difficult isotropic HFCs.


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RS Hybrids

Hybrids

(meta-)GGA

Table 1. Performance of selected functionals for the isotropic metal HFCs (in MHz) of four manganese complexesa

Local Hybrids

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[Mn(CN)4]2-

[Mn(CN)4N]-

MnO3

MnO

Func

S = 5/2

S = 1/2

S = 1/2

S = 5/2

BP86

-98.6

-169.0

2013.7

527.8

TPSS

-111.8

-184.1

1928.1

515.6

HCTH

-152.8

-272.9

1861.2

311.2*

τ-HCTH

-190.4

-308.2

1873.5

243.5*

M06-L

-117.0

-312.6

1670.4

391.6*

MN12-L

-419.3

-522.6

1632.8

234.5

MN15-L

-363.4

-768.6

1158.7

224.9

VSXC

-199.1

-322.9

1810.8

433.3

TPSSh

-127.1

-231.0

1782.7

504.1

τ-HCTH-hyb

-157.7

-287.0

1805.0

489.5

B3LYP

-116.8

-250.1

1735.7

523.4

M06

42.9

-36.3

1896.3

627.6

M05

86.4

98.8

2076.0

621.1

BMK

-216.5

-479.7

1390.4

476.9

MN15

-222.3

-392.3

1350.7

384.7

M06-2X

-303.9

-883.2

968.0

280.9

LC-ωPBE

-97.4

-216.8

1759.1

539.1

ωB97

-140.3

-265.6

1806.2

502.5

ωB97X

-127.0

-281.8

1723.7

525.6

ωB97XD

-133.2

-300.5

1696.9

527.6

CAM-B3LYP

-123.8

-294.1

1649.6

528.4

HISS

-127.6

-317.3

1472.7

531.0

HSE-06

-115.3

-264.6

1614.5

518.1

Lh07s-SVWN

-119.2

-264.8

1778.2

516.7

Lh07t-SVWN

-126.1

-279.0

1775.1

525.2

Lh12ct-SVWN

-166.3

-307.4

1728.8

470.5

Lh12ct-SsifPW92

-185.2

-366.8

1604.5

440.9

Lh12ct-SsirPW92

-178.3

-343.5

1651.5

451.7

Lh14t-calPBE

-136.5

-280.9

1723.8

511.1

Experimental

-199

-276

1613

479.861

a

Values marked with a star (*) indicate convergence to an erroneous orbital configuration at the given level, those

marked with a hash (#) indicate states, where the correct occupation was identified as an excited state.

[Mn(CN)4]2-, a pure CSSP case. For this system, VSSP is comparably small (see below), and deviations of from the nominal 8.75 for a sextet state are negligible (black circles in 13 ACS Paragon Plus Environment


Figure 1). The reason is, that the five SOMOs have almost pure metal 3d-character, limiting the spin polarization of the bonding valence levels. Starting with the semi-local functionals, we see that all GGA and older meta-GGA functionals give clearly too positive spin densities at the Mn nucleus, in agreement with previous evaluations.4,5,8,11 The -HCTH and VSXC meta-GGAs represent notable exceptions, which give appreciably better isotropic HFCs for this CSSP case, comparable with the best hybrid functionals (see below). This observation extends also to other dominant CSSP cases, such as [Cr(CO)4]+ (cf. Table S6 in SI). For both of these meta-GGA functionals, the role of the -dependence in the exchange functional in mimicking non-locality has been discussed extensively in the literature.46,50,85 However, this does not explain why other -dependent meta-GGAs (TPSS, M06-L in Table 1 and Figure 1; PKZB11) are not able to give the necessary large CSSP contributions, calling for further analysis (see below). We also note that HCTH performs appreciably better than the other two GGAs. Finally, the two recent, closely related Minnesota meta-GGAs MN12-L and MN15-L provide much too negative spin densities at the metal nucleus, both for [Mn(CN)4]2- and for [Cr(CO)4]+ (Table S6, MN12-L and MN15-L results are included in Figure 1). These data are completely out of line with all GGAs or meta-GGAs, more similar to GHs with very large EXX admixtures (see below), and our first indication that some, but by no means all, of the Minnesota functionals exhibit erratic behavior for the isotropic metal HFCs (see discussion below). Notably, MN15-L even exhibits some spin contamination here, in contrast to all other functionals of all types. Turning to GHs, we see a wide spread of results among the screened functionals, ranging from a strikingly large overestimate (M05, M06) to an equally large underestimate (M06-2X) of the spin density at the Mn nucleus in [Mn(CN)4]2- (Figure 1). Among the selected GHs, BMK and MN15 come closest to the reference value. -HCTH-hyb and B97-1, which are closely related to BMK but have much lower EXX admixtures of 15% and 21%, respectively, rather than 42%, give a somewhat too high Aiso but not dramatically so. Except for the latter three related functionals, and for the series of the closely related M06, M06-2X, and M06-HF (cf. Table S6), we cannot identify a clear correlation with EXX admixture (found previously for simpler functionals4). For example, BMK and MN15 with 42% and 44% EXX, respectively, give close results for [Mn(CN)4]2-, whereas MPW1K (42.8%) and BHLYP (50%) give a much less negative value. This suggests that other aspects of the construction of the functional, e.g. the presence or absence of a -dependence (see above) and the specific, in some cases highly parameterized functional form, are also important. We keep the extreme deviations from the 14 ACS Paragon Plus Environment

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reference value for some of the Minnesota functionals in mind for further analysis, both for the M05 and M06 GHs and for the abovementioned MN12-L and MN15-L functionals. Most of the RSHs perform comparably to standard GHs with medium EXX admixture, such as B3LYP (Figure 1), with a notably larger total spin density for LC-PBE. Maybe this should not be too surprising in case of long-range corrected functionals. For example, CAM-B3LYP has a similar EXX admixture (19% vs. 20%) as B3LYP at short range with an enhancement to 65% only at long range. The latter has some influence on CSSP, giving somewhat lower spin densities at the Mn nucleus, but the effect is not very large. Neither long-range screening (HSE06) nor the intermediate-range HISS functional (having somewhat less than 40% EXX at middle-range73) lead to notable improvements.

Figure 1. Evaluations of a selection of different functionals, grouped into (meta)-GGAs, GHs, RSHs and LHs for four selected complexes (see legend at top of right panel). Left panel: deviations of spin 𝛼−𝛽

densities at the metal nucleus (𝜌𝑁

) from experiment. Middle panel: deviations of dipolar HFCs from

experiment. Right panel: deviations of expectation value from the exact value for the given spin state. Open red diamonds indicate calculations on MnO, which converged to an excited spin state. See Tables S6-S8 in SI for numerical data, including a more complete screening of complexes and functionals.

Finally, local hybrids show a clear dependence of the pre-factor a of the t-LMF (eq. 7). While a spin-channel t-LMF with a-values of around 0.5 (Lh07t-SVWN, Lh14t-calPBE) perform only somewhat better than B3LYP, a common t-LMF (a = 0.534, Lh12ct-SVWN) provides already 15 ACS Paragon Plus Environment


notable improvement. An increase of the pre-factor above 0.6 made possible by modified correlation functionals (Lh12ct-SsifPW92, Lh12ct-SsirPW92; cf. Theoretical Background) enhances the CSSP further and thus leads already to excellent agreement with the reference value. The Lh07s-SVWN functional with spin-channel s-LMF has the worst performance among the LHs, comparable to that of B3LYP. However, its results are not too different from those of Lh07t-SVWN, in spite of the s-LMF exhibiting a negative cusp at the nucleus, where it brings EXX admixture down to zero.21 This indicates already that the EXX admixture directly at the metal nucleus is less important for the spin density at this point and thus for Aiso than the behavior in the spatial region between core and valence shell, where we expect CSSP to originate from. MnO, a moderate VSSP case. Semi-local functionals provide small spin contamination for MnO, with deviations from the nominal value (8.75) below 0.06 (Figure 1, Table S7), except for MN15-L (almost 0.10), which again stands out. In three cases (HCTH, -HCTH, M06-L), the computations converge to an MO occupation that differs from that of the correct ground state (violating the aufbau principle), thereby explaining the far too negative spin densities at the Mn nucleus (while Adip is not affected much). In these cases, it has not been possible to converge to the correct occupation (similar observations hold for the related TiN where, however, the correct occupation sometimes may be identified as excited state; Tables S6, S7). The two remaining GGAs over- and the VSXC meta-GGA underestimates Aiso somewhat, while underestimating the relatively small Adip. The latter point is notable, as we might have expected to get almost perfect reproduction of Adip in the absence of spin contamination. Neither relativistic corrections (cf. Tables S1, S2) nor environmental effects (cf. Tables S3-S5) or inaccurate input structures explain the small but notable deviations at (meta-)GGA level. Similar observations hold for MnO3 (see below), where Adip is overestimated at semi-local levels. It seems possible, that delocalization errors are responsible for these moderate deviations in the dipolar HFCs. However, the deviations for MnO pertain also to the various hybrid functionals, while spin contamination will increase them for MnO3 (see below). Finally, the more recent MN12-L and MN15-L functionals both give far too low Aiso, confirming their erratic behavior for the above “pure CSSP case”. Interestingly, in spite of its notably larger spin contamination, MN15-L provides a “good” Adip, whereas that for MN12-L, which shows essentially no spin contamination, is too negative like with all other functionals. The GHs exhibit a wide range for Aiso, with Minnesota functionals again providing the most extreme overestimate (M05, M06) and underestimate (M06-2X, M06-HF). Notably, the 16 ACS Paragon Plus Environment

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extreme behavior of these latter functionals is not reflected in the Adip or in . That is, the spin contamination of M06 or M05 is in the range expected for GHs with about 25-30% EXX admixture, while that for M06-2X or M05-2X (54% and 56% EXX, respectively) is even somewhat smaller than for BMK (42%) or BHLYP (50%, Table S7). This suggests, that the erratic behavior of some of the Minnesota functionals is somehow related to CSSP (see below). BMK and B97-1 are the GHs providing the best spin density at the Mn nucleus for MnO (followed closely by -HCTH-hyb, PBE0, and PBE0-1/3; Tables 1, S6). MN15 stands out by giving a much lower Aiso than any other GH with comparable EXX admixture, in spite of exhibiting only moderate spin contamination. This is the first indication of unexpected behavior for this more recent GH (see below). As found above for the pure CSSP cases, RSHs improve negligibly upon GHs with intermediate EXX admixture (e.g. B3LYP), with the exception of B97, which gets closest to the reference value for Aiso. All RSHs provide good Adip values, consistent with an only moderate spin contamination (Table S8). Interestingly, the middle-range functional HISS nevertheless exhibits the largest spin contamination, followed by the screened hybrid HSE-06 (B97 has the smallest value). This indicates that short- and middle-range EXX admixture affects VSSP and thus spin contamination significantly more than enhanced EXX at long range. In fact, the performance of HISS, both in terms of spin contamination and HFCs, is closely comparable to that of GHs with about 30-35% EXX admixture (e.g. BLYP35, PBE0-1/3; Tables S6-S8). This is consistent with an overall middle-range EXX admixture of about 35-40% in this functional.73 As in the pure CSSP cases above, LHs with spin-channel LMFs and moderate pre-factors for a t-LMF give still somewhat too positive spin densities at the Mn nucleus, providing essentially B3LYP quality. In contrast, common t-LMFs with a larger pre-factor a (Lh12ctSsifPW92 and Lh12ct-SsirPW92) undershoot even slightly, while exhibiting even smaller spin contamination than the other LHs (Figure 1, Tables 1, S6, S8). Interestingly, the LH with spinchannel s-LMF (Lh07s-SVWN) provides even a slightly lower spin density at the Mn nucleus than the one with the spin-channel t-LMF (Lh07t-SVWN). This confirms (see above) that the EXX admixture directly at the nucleus is not very important, as CSSP originates mainly in the middle and outer core regions, i.e. in the 2s and 3s shells.4 In summary, a number of GHs, one RSH (B97), and two LHs (Lh12ct-SsifPW92 and Lh12ct-SsirPW92) perform rather well for the intermediate VSSP case MnO, while the good 17 ACS Paragon Plus Environment


performance of the meta-GGAs -HCTH and VSXC for the above pure CSSP cases cannot be harvested here, due to convergence to a wrong ground state. MnO3, a strong VSSP/spin-contamination case with an sdz2-type SOMO. While for the previous cases spin contamination was not a large issue, this changes drastically for MnO3, as had already been found previously.4 Standard semi-local functionals give too positive spin densities at the Mn nucleus (Figure 1). The Minnesota functionals M06-L, MN12-L, and MN15-L are notable exceptions: for this complex, M06-L and MN12-L are the semi-local functionals exhibiting the best agreement for Aiso (Table S6). While MN12-L appears to benefit from error compensation with an erroneous CSSP description (see below), M06-L exhibits nonnegligible spin contamination, thus behaving more like a GH with moderate EXX admixture (but the Adip value is lower and thus in better agreement with experiment). MN15-L exhibits even larger spin contamination than M06-L, comparable to the M06 or PBE0 hybrids, and it undershoots the spin density at the Mn nucleus dramatically, even comparably to or more so than GHs with more than 30% EXX admixture (see below). As both MN12-L and MN15-L perform well for Adip, the results suggest again that the large variations in Aiso arise from the description of the core shells (see analyses below). The other meta-GGAs are intermediate between M06-L and the GGAs, both regarding Aiso and spin contamination (Tables 1, S6, S7). In case of GHs, spin contamination increases dramatically with increasing EXX admixture, as observed previously for a much smaller set of functionals.4 This is accompanied by an increasing overestimate of Adip (Figure 1) with larger EXX admixture. The more parameterized Minnesota functionals exhibit in each case a slightly smaller spin contamination than simpler GHs with comparable EXX admixture (Table S7), and the same holds for BMK, likely due to the effects of the -dependence and parameterization. These regular interdependencies between EXX admixture and performance do, however, not carry over to the spin density at the Mn nucleus and thus to Aiso. M06 and particularly M05 deviate more (in positive direction) from the reference value than other GHs with lower EXX admixture (even TPSSh with 10% EXX gives a lower value), consistent with the peculiar behavior of these two functionals for Aiso in all complexes. Many other GHs with about 25-42% EXX admixture do already overshoot in the negative direction (e.g. PBE0, BLYP35, PBE0-1/3, BMK, MN15, MPW1K, but not PW6B95), and M06-2X (54%) does so even much more than BHLYP (M06-HF even more so; Table S6).


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Long-range corrected RSHs exhibit somewhat larger spin contamination than B3LYP, except for B97 and LC-PBE, whereas HSE-06 and particularly HISS suffer even more from spin contamination (the latter again performs closely comparable to GHs with about 35% EXX admixture, reflecting the magnitude of its middle-range contribution; see above). The latter fact is reflected also in the Adip values, which are strongly overestimated (Figure 1, Table S8). As discussed already above for MnO, the short- and middle-range EXX admixture is more important for VSSP and thus for spin contamination than the long-range behavior. Thus, B97 and LC-PBE, which interpolate between 0% EXX admixture at short range and 100% at long range, exhibit somewhat lower spin contamination errors than functionals that have short-range EXX contributions. Nevertheless, several of the RSHs with short-range EXX contributions provide relatively good performance for the spin density at the Mn nucleus (e.g. B97X-D, CAM-B3LYP, and also HSE-06; HISS overshoots). This confirms that the good performance of some of the GHs with moderate EXX admixtures of about 20-25% for Aiso of VSSPdependent systems like MnO3 involves likely error compensation and occurs partly with the help of some spin contamination. Local hybrids exhibit spin contamination and Adip errors comparable to those of GHs with moderate EXX admixture, e.g. B3LYP, slightly larger for Lh12ct-SsifPW92 and Lh12ctSsirPW92. The latter two functionals provide the best results for Aiso, but the other LHs also perform well. Local hybrids with common t-LMFs are thus promising in this context, as they can clearly improve the treatment of CSSP and thus of Aiso without enhancing spin contamination and thus the deterioration of Adip. They do thus go a way towards improving on the spin-polarization/spin-contamination dilemma, but they do not completely escape a moderate influence of spin contamination on Adip. [Mn(CN)4N]-, another strong VSSP/spin-contamination case. While this larger, anionic, square pyramidal complex also has a d1 doublet ground state, such as the neutral, trigonal MnO3, its mechanisms of VSSP are distinctly different (see below), partly due to the presence of the strong axial -donor nitride ligand, partly due to the strong -antibonding of the SOMO in MnO3. Nevertheless, the expectation values for the two complexes for a given functional are rather similar (Figure 1, Tables S7, S8). That is, remains below 0.9 for most semilocal functionals (except MN15-L), increases dramatically for GHs with increasing EXX admixture, is moderate (0.9-1.1) for long-range corrected RSHs but larger for HISS, and moderately large for the LHs. While the Adip values follow the trend of the spin contamination, 19 ACS Paragon Plus Environment


their deviations from the reference values are appreciably less pronounced here than for MnO3, in spite of a similar absolute magnitude of Adip (with opposite sign, Tables S7, S8). For this system, we cannot completely exclude deviations arising from the neglect of environmental effects, as the experimental data pertain to frozen acetonitrile solution,86 where a weak axial solvent coordination to Mn seems possible. Another notable difference compared to MnO3 is, that now the deviations from experiment for the spin density at the Mn nucleus, and thus for Aiso, are much smaller (Figure 1). This reflects the much smaller absolute spin-density values. The experimental Aiso for MnO3 of ca. 1600 MHz should be compared to about ca. -280 MHz for [Mn(CN)4N]- (Tables 1, S6). This has to do with the partial 4s-character of the SOMO and the resulting very large direct, positive SOMO contribution in the former complex (analysis given below). The overall dependencies of the isotropic HFC on the functional for the two complexes have similarities and differences: Most of the GGA functionals (e.g. BP86, BLYP, PBE) give insufficiently negative Aiso (Table S6). This extends also to the TPSS meta GGA. Interestingly, the HCTH GGA is close to the reference value, whereas the remaining meta-GGAs overshoot either somewhat (HCTH, M06-L, VSXC) or notably so (MN12-L, MN15-L). This suggests a mixed CSSP/VSSP origin of the spin density at the nucleus (see below). GHs cover again a wide range, with moderate EXX admixtures (e.g. -HCTH-hyb, B3LYP, but also PW6B95) providing Aiso values closest to the experimental value, whereas large EXX admixtures give clearly too negative values, in some cases dramatically so. The erratic behavior of M05 and M06 for isotropic HFCs is observed also for this system: they exhibit very small negative or positive Aiso values, thus clearly deviating dramatically from experiment in an unexpected direction when one looks at their EXX admixtures (28% for M05, 27% for M06). Interestingly, for this complex MN15 shows a less negative Aiso than GHs with comparable EXX admixture (e.g. BMK, MPW1K). This is accompanied by lower spin contamination and a better Adip. This functional thus sometimes agrees with and sometimes deviates from expectations for a GH with its EXX admixture in a seemingly nonsystematic way. The RSHs provide a comparably smaller range of values and relatively small deviations from the reference value in this case, again in spite of moderate but notable spin contamination (see above). HISS gives the most negative deviation, LC-PBE the largest positive one. The LHs, which exhibit comparably moderate spin contamination in this case, perform overall well, and again give better Aiso than GHs with comparable spin contamination. However, 20 ACS Paragon Plus Environment

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those two functionals (Lh12ct-SsifPW92 and Lh12ct-SsirPW92), which provided the best performance for the other three complexes discussed above, do now already overshoot notably, whereas the others give results close to experiment.

4.2 Orbital analyses of CSSP, VSSP, and SOMO contributions To better understand the dependencies of Aiso on the different functionals, we now analyze 𝛼−𝛽

the spin density at the metal nucleus, 𝜌𝑁

in terms of the SOMO, core, and valence MO

contributions (Figure 2). The core contributions are usually dominated by a (larger) negative 2s contribution and a (smaller) positive 3s contribution, leading to an overall negative CSSP contribution, as had been analyzed previously.4,5,6,7 These two contributions are also indicated by open symbols. In contrast, only the sum of the valence (VSSP) contributions is shown. When the latter is dominated by only one or two valence MOs, this will be pointed out in the text. We concentrate on the same four prototypical complexes used in the systematic screening of functionals in Figure 1 above. Orbital analyses for [Mn(CN)4]2- with various functionals. For this “CSSP case”, SOMO contributions are zero due to the Td symmetry and the resulting pure 3d character of the five SOMOs. The dependence on the functional is clearly dominated by the CSSP contributions (upper left panel of Figure 2), but there is also some variation of the VSSP contributions. The latter arise largely from an MO that represents a totally symmetrical (a1) Mn-CN -bonding linear combination made up from the Mn 4s orbital and -type hybrids on C and N. The precise composition and polarization of this MO seems to reflect to some extent also the mutual orthogonality of the radial parts5 of the Mn 4s and 3s shells. Thus, the 3s contribution to CSSP and the contribution of the abovementioned MO to VSSP are coupled to some extent. However, it is important to note that this type of VSSP does not lead to notable spin contamination (see above), likely due to the relatively low energy of the dominant MO compared to the Fermi level, as spin contamination is typically connected to the highest-energy occupied MOs. The most notable variations of VSSP contributions are larger values for some of the more highly parameterized semi-local functionals (HCTH, -HCTH, and particularly M06-L, MN12L, and MN15-L; see also Table S10), and particularly small values for some GHs (M05, M052X, BMK) and RSHs (B97 and B97X). Nevertheless, the CSSP contributions dominate the overall trends. The erratic behavior of MN12-L and MN15-L for Aiso noted above arises from 21 ACS Paragon Plus Environment


a dramatically exaggerated CSSP contribution. Closer analysis shows that the too large CSSP for MN12-L is due to a large negative 3s contribution, which is completely at odds with results for all other functionals (except MN15, see below). In contrast, the 2s contribution almost vanishes, and the 1s contribution is also very unusual (Table S10). In contrast, for MN15-L the overall too large CSSP contributions occur in spite of a reasonable relative magnitude of the 1s, 2s, and 3s contributions: now the absolute magnitudes of all three contributions are drastically overestimated. These analyses confirm that both MN12-L and MN15-L provide erroneous descriptions of CSSP. They do so in a completely different way regarding the individual coreshell contributions. This is remarkable given that the functional form of MN12-L and MN15-L is the same, and the latter just was optimized against a different database and with some smoothness constraints.53 More negative CSSP contributions compared to simpler GGAs, but nowhere near the erratic behavior of MN12-L or MN15-L, are seen with HCTH and also for some more parameterized meta-GGAs, explaining why the latter (-HCTH, VSXC) perform well for this CSSP case (the performance of M06-L is hampered somewhat by the rather positive VSSP contribution and a very positive 3s CSSP contribution). While increasing EXX admixture in GHs is expected to enhance the negative CSSP contribution, obviously the -dependence in some functionals (compare, e.g., TPSSh or HCTH-hyb against B3LYP) also affects the core contributions, consistent with the observations for some meta-GGAs (see above). The good performance of BMK appears to be due in part to the relatively small VSSP contribution (here M05-2X performs similarly, whereas M06-2X overshoots overall; Table S10). In contrast to the too large CSSP contributions for MN12-L and MN15-L above, the erratic behavior of M06 and M05 clearly arises from far too small CSSP contributions (small but still negative for M06, overall positive for M05), apparently caused by strongly underestimated 2s contributions (compared to all other functionals, even semi-local ones). In spite of their moderate EXX admixtures, we thus see yet a different failure for the description of CSSP. In view of the large number of empirical parameters of this type of functional, it seems difficult to pinpoint the origin of this behavior more closely. Even MN15, which seemed to perform much more similarly to GHs with comparable EXX admixture, with some exceptions (see above), exhibits an unusual pattern of the CSSP contributions: the 2s contribution is negative, as it should be, but much smaller than with most other functionals. This is partly compensated by an unexpectedly negative 3s contribution, leading to an overall 22 ACS Paragon Plus Environment

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not too unusual CSSP contribution (the 1s contribution is also unusual by being positive). As a result, the overall performance of MN15 for CSSP contributions and thus for Aiso and the spin density at the Mn nucleus is not that exceptional, but the individual contributions deviate from those of all other functionals. Indeed, the pattern of the 1s, 2s, and 3s CSSP contributions for the three closely related functionals MN12-L, MN15-L, and MN15 seems completely different. The fact that all of the RSHs are still above the reference value arises from generally still underestimated CSSP contributions, even though some RSHs feature relatively small VSSP. This holds in particular for B97 and B97X, which exhibit particularly small CSSP and VSSP terms. The small CSSP values seems to be due to a particularly positive 3s contribution, which is consistent with the abovementioned coupling of these two contributions (CSSP and VSSP) in this case. This is likely due to the 4s/3s radial orthogonality, which is strictly obeyed only for the spherical atom but to some extent survives in the complex.5 Finally, the LHs perform overall well, apparently due to a reasonable balance between the various contributions. Here the common-LMF-based Lh12ct-SsifPW92 and Lh12ct-SsirPW92 (see also Table S5) benefit particularly from relatively small 3s-CSSP and VSSP contributions (and small negative 1s contributions). Figure 2 (right panels; cf. Table S10) also provides the ratio between the 3s and 2s CSSP contributions, which in the absence of abnormal effects, such as spin contamination, has previously been found to be almost independent of the chemical environment or spin state for a given manganese complex, while changing with metal nuclear charge.5 The ratio obviously also depends on the functional, helping to partly explain the above trends in the CSSP contributions. In particular, except for MN15-L, the 3s/2s ratios for those Minnesota functionals that we identified as “erratic” regarding Aiso (MN12-L, M05, M06, and even MN15) are completely outside the usual range. The “expected behavior” is exemplified by the observation that both at UHF level6 and at LSDA7 or GGA5 levels, the absolute magnitude of the (negative) 2s contribution is expected to be larger than that of the (positive) 3s contribution, suggesting the ratio to be somewhere between a small negative value for early 3d elements down to about -1.0 for Cu, with values for Mn expected somewhere around -0.5.5 While this expectation is fulfilled by most functionals, the observed variations give hints on the origins of failures or particularly good performance for our CSSP-dominated case:



Figure 2. Individual contributions to the spin density at the Mn nucleus (and thus to Aiso) for different functionals. In the left and middle panels, CSSP (green circles), VSSP (blue triangles), SOMO (inverted red triangles) contributions and the total spin density (black squares) are shown. Individual contributions that are close to zero for all functionals are neglected for clarity (this pertains in particular to 1s CSSP contributions, and to the SOMO contributions for the two cyano complexes). The CSSP contributions are further broken down into their dominant 2s and 3s parts (open circles and triangles, respectively). The reference value is shown as solid vertical line. The right panels show the 3s/2s ratio of the CSSP contributions (values outside the range of the plot are given as numbers). Underlying numerical data, including those for further functionals, are provided in Tables S10-S13 in SI, results for [Cr(CO)4]+ in Table S14.

As MN12-L has a negative 3s contribution and a much too small 2s one, its 3s/2s ratio is positive and completely off the scale (Table S10). MN15-L gives overall too much CSSP contributions, but their relative magnitude appears “normal”. M05 provides a very unusual ratio of -1.55, as the positive 3s contribution is larger in absolute value than the unusually small negative 2s one (Figure 2). This explains why this functional fails not only in this case but also in all other examples studied here. The failure of M06 is less extreme, which is indicated by a less negative but still unusual 3s/2s ratio of -0.78 (still within the plot in Figure 2, cf. Table 24 ACS Paragon Plus Environment

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S10), again indicative of a clearly too small 2s contribution. In spite of its overall reasonable CSSP contribution, MN15 exhibits a positive 3s/2s ratio, as both 2s and 3s contributions are negative. This also puts this functional outside the range of a “normal” behavior. Interestingly, the other Minnesota functionals (M06-L, M05-2X, M06-2X, M06-HF; Table S10) exhibit 3s/2s ratios much closer to those of the other functionals. Leaving the outliers among the Minnesota functionals aside, the overall least negative 3s/2s ratios (above -0.4) are seen with TPSS, HISS, Lh12ct-SsifPW92, and Lh12ct-SsirPW92. The latter two LHs suggest that such values do not prohibit an overall good performance for Aiso. Further examples of values below -0.55, apart from the abovementioned outlier cases, include M06-L, BHLYP, B97, B97X, and B97X-D, also without any clear-cut effect on the overall performance. Orbital analyses for MnO with various functionals. We keep the analysis for this “intermediate VSSP case” briefer than for the previous “CSSP example”, highlighting only the most notable aspects. We exclude from the closer discussion those semi-local functionals, which converge to the wrong ground state (HCTH, -HCTH, M06-L). MN12-L, MN15-L, M05, M06, and MN15 exhibit the same erratic CSSP behavior described above, with the same characteristics (and with the same error compensation for MN15). Leaving these outlier cases aside, we see relatively large core contributions with VSXC, leading to an overall low Aiso. The 3s/2s ratios for this complex are generally somewhat more negative for a given functional than in the previous case, while the trends appear to be closely similar (Figure 2, right panel). Major differences compared to the previous complex are (Figure 2, Table S11): a) The SOMO contributions are large and positive but depend only moderately on the functional; b) the VSSP contributions are negative and depend substantially on the functional. It has been found previously that partial 4s character of the SOMO causes overall negative contributions of VSSP to Aiso (see also below for MnO3).5 The variations of the VSSP contributions with the functional reflect, on one side, spin contamination for the largest EXX admixtures of GHs (see, e.g., M06-2X, M05-2X, M06-HF, Table S11). On the other side, we also see larger negative VSSP contributions with some of the more highly parameterized semi-local functionals (HCTH, -HCTH, M06-L), exceeding to some extent even those of some of the GHs or LHs and all RSHs (Table S11). Interestingly, the VSSP and SOMO contributions of MN12-L, MN15-L, M05, M06, and MN15 seem comparably unremarkable, reinforcing the conclusion



that the different erratic behaviors identified above are restricted largely to the CSSP contributions. Orbital analyses for MnO3 with various functionals. Moving to the strong VSSP/spincontamination cases, MnO3 represents another case with large positive SOMO contributions and notably negative VSSP parts. Both observations reflect the appreciable 4s admixture to the SOMO. The large SOMO contribution, together with the much smaller negative CSSP and VSSP contributions, ensures an overall large positive Aiso (Figure 2, Table S12). Here the semi-local functionals clearly exhibit lower (positive) SOMO and (negative) VSSP contributions than the hybrid functionals, likely reflecting the spin contamination found for the latter (see above). While the CSSP contributions are overall smallest, they become more negative for the more parameterized functionals (HCTH, -HCTH, M06-L, VSXC), moving the overall Aiso closer to the reference value. M06-L, which has the most negative VSSP contribution (consistent with non-negligible spin contamination, see above), gives the lowest value in this group. Positive SOMO and negative VSSP contributions clearly increase with EXX admixture in the GHs, again reflecting the effects of spin contamination. CSSP contributions also grow with increasing EXX admixture, except again for the extreme outliers (MN12-L, MN15-L, M05, M06, MN15): here the same deficiencies in the description of CSSP occur as identified previously (including error compensation for MN15), with the slight difference that now the 3s/2s ratio for MN12-L is negative (as the 2s contribution is very small but positive). Disregarding again these outliers, moderate EXX admixture provides a good compromise to get close to the reference Aiso value, whereas the large spin contamination makes the GHs with large EXX admixture unsuitable in this case. Notably, the deterioration in the 3d spin population for such functionals also affects the CSSP contributions adversely.5 Consistent with the comparably moderate spin contamination we found above for long-range corrected RSHs, they exhibit generally contributions comparable with those provided by GHs with low EXX admixture. The middle-range HISS functional behaves more like a GH with larger EXX admixture of 35-40%, also consistent with its larger spin contamination (see above). Finally, the various contributions for the LHs also follow from the relatively moderate spin contamination (see above). The overall best-performing LHs with common t-LMF (Lh12ctSsifPW92, Lh12ct-SsirPW92) combine SOMO and VSSP contributions comparable to GHs


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having low EXX admixture with the somewhat enhanced negative CSSP contributions due to larger core-region EXX admixtures. Orbital analyses for [Mn(CN)4N]- with various functionals. Curiously, in spite of the large sensitivity of this complex to spin contamination (see above),4 VSSP contributions to Aiso are in fact comparably small (Figure 2, Table S13; SOMO contributions are absent due to symmetry reasons). This reflects the 3dxy character of the SOMO, which points between the four basal cyano ligands and thus interacts little with totally symmetrical linear combinations of either Mn-N or Mn-C-bonding character (the small VSSP contributions to Aiso are in fact distributed over about eight MOs with a1 symmetry, which even partly compensate each other). Instead, the SOMO polarizes in particular doubly occupied MOs with Mn≡N -bonding character, which cannot contribute directly to Aiso but cause the bulk of the spin contamination. Apart from strongly deteriorated Adip values (see above), the effects of spin contamination are thus mostly visible, more indirectly, in exaggerated negative CSSP contributions (and partly enhanced positive VSSP contributions as well): both 3s and 2s CSSP contributions are affected adversely by an overall erroneous 3d spin population. The more parameterized semi-local functionals (HCTH, -HCTH, M06-L, MN12-L, VSXC) exhibit comparable spin contamination (see above, Figure 1, Table S7) as GHs with moderate EXX admixture (MN15-L again stands out with larger spin contamination). This enhances the negative CSSP contributions and thus brings the overall Aiso results closer to the reference value, in spite of the also somewhat enhanced (positive) VSSP contributions. The completely unrealistic CSSP contributions for MN12-L and MN15-L are again notable and lead to overall too low Aiso in both cases. Disregarding such outliers from the set of Minnesota functionals and their deficiencies in describing CSSP, the GHs show the effects of increasing spin contamination with increasing EXX admixture in their enhanced negative CSSP contributions (Figure 2). Therefore GHs with larger EXX admixtures overshoot clearly towards too low Aiso values. The same holds for the middle-range HISS functional, while long-range corrected RSHs behave similarly as GHs with low EXX admixture. The vanishing VSSP contributions for the family of B97-related functionals are also notable. Finally, the LHs have similar CSSP contributions, but the two functionals with common LMF (Lh12ct-SsifPW92, Lh12ct-SsirPW92) feature slightly larger ones, and also essentially



vanishing VSSP contributions, leading to somewhat too negative Aiso compared to the reference value.

4.3 Further analysis of CSSP contributions as function of EXX admixture. Our earlier analyses had indicated,4,5 that an increase in EXX admixture in GHs for related semi-local ingredients of the functional enhances both the 3s and 2s CSSP contributions in absolute value. For the series BLYP (0% EXX admixture), B3LYP (20%), and BHLYP (50%), this is shown in Figure 3a for nine Mn complexes with a wide variety of oxidation and spin states (see Tables S15, S16, S17, S18 for numerical data, including further functionals). The 3s and 2s CSSP contributions to the spin density at the Mn nucleus are plotted as function of the net 3d spin population, reflecting how these core orbitals are polarized by the overall 3d -spin population (cf. Figure 3 in ref. 5). Both the negative slope of the 2s CSSP regression line and the positive one of the 3s CSSP regression line are enhanced by increasing EXX admixture. As the 2s contribution is larger in absolute value (see above), this enhances also the net (negative) CSSP contributions (cf., e.g., Figure 2 above). Additionally, the net 3d populations also grow with EXX admixture (as indicated by the data points moving to the right in Figure 3, left top panel, likely related to spin contamination), thereby further increasing the overall CSSP. Interestingly, matters are different in the -dependent meta-GGA-based, closely related series of functionals -HCTH (0% EXX admixture), -HCTH-hyb (15%), BMK (42%), Figure 3 (left middle panel): here the semi-local -HCTH meta-GGA functional gives the largest slopes (in fact much larger than both hybrids BMK and BHLYP!), whereas the largest EXX admixture of BMK provides the smallest slopes. While this different behavior compared to the GGAbased GHs in Figure 3 may be related to the -dependence, the highly parameterized nature of these meta-GGA-based functionals precludes a closer analysis. Interestingly, the net 3d spin populations also do not increase from -HCTH to -HCTH-hyb, but get largest with BMK, likely as a consequence of spin contamination with the latter functional. Due to these various different influences, the effect of EXX admixture on the net CSSP contributions is not clearcut, as sometimes BMK can still give the overall most negative CSSP values, sometimes HCTH (cf., e.g., Tables S10-S14).


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Figure 3. Correlation of positive 3s and negative 2s CSSP contributions to the spin density at the Mn nucleus for nine Mn complexes (cf. Figure S2 for further plots, and Tables S15, S16, S17, S18 for numerical data) and 13 different functionals with the net 3d spin population obtained with the same functional. Symbols denote the different complexes while lines give a least-squares regression for all nine complexes, the slopes of which are tabulated.

Some differences between spin-channel and common LMFs in LHs are apparent from Figure 3 (right top panel): LHs with spin-channel t-LMF (Lh07t-SVWN and Lh14t-calPBE) provide a steeper 2s CSSP slope (comparable to BHLYP) than Lh12ct-SsirPW92 with common t-LMF (here the slope is comparable to B3LYP). Strikingly, the already relatively low slope for the 3s 29 ACS Paragon Plus Environment


CSSP for the spin-channel LMFs is further decreased for the common LMF (these differences hold for all common vs. spin-channel LMFs of this work, cf. Figure S1). This leads to the observed, overall more negative CSSP contributions for the latter, likely due to the abovementioned differential NDC cross contributions incorporated in a common LMF. The completely non-intuitive and likely unphysical behavior of the M05, M06, MN12-L, MN15-L, and MN15 functionals for CSSP is also reflected clearly in such plots (Figure 3; middle right and bottom left panels) and thereby shown to be a systematic feature: the semilocal M06-L and even more so MN15-L give very large slopes for both the 3s and 2s CSSP lines (cf. the similarities with the -HCTH meta-GGA), but a reasonable 3s/2s ratio between 0.42 and -0.67 (M06-2X exhibits the expected exaggerated slopes due to its large EXX admixture but also a reasonable ratio). The two hybrids M05 and M06 and the semi-local MN12-L give unusually small slopes for both lines, but most notably so for the 2s contributions (thereby giving very unusual 3s/2s ratios, see above). This behavior is more pronounced for M05 than for M06 and thus explains why the former functional gives even more erratic isotropic metal HFCs than the latter, but it gets extreme for MN12-L, where the 2s slope is very close to the baseline, but the 3s slope is negative. Similarly, in MN15 both slopes are negative, but here the 3s slope is still above the 2s one. Both slopes are relatively small, and therefore the total CSSP contributions seem relatively accurate, while the individual behaviors of the shells are unexpected and likely unphysical. It is important to note, that functionals with the acronym M06 share the overall underlying form but may exhibit very different values of the many semi-empirical parameters57 (the same holds for the M05-based series). This should be kept in mind, apart from the variations in the EXX admixture, when analyzing the widely varying behavior of the different functionals within a series for the present HFC evaluations. These results tie into an ongoing lively debate on the direction the development of exchangecorrelation functionals has taken in recent years. Some authors have argued that by focusing only on energy quantities in constructing highly parameterized functionals one has lost the right way, as electron densities may deteriorate.87 While one may in general argue about the importance of getting good electron densities in different regions of space (core or valence) and for different types of systems (atoms, molecules), it is clear that the present HFC evaluations clearly expose deficiencies in core-shell spin densities. The specific shortcomings of some of the Minnesota-family functionals are clearly linked to CSSP in various characteristic ways. 30 ACS Paragon Plus Environment

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Indeed, the previously noted density errors of such functionals also seem to pertain in particular to core rather than valence densities.87 This may reflect the fact that optimization of functionals for thermochemical and kinetic data does not take into account core-shell effects.

4.4 Systematic variation of EXX admixture for global and local hybrids. Our above comparisons of various functionals have shown that LHs with a common t-LMF do provide some improvement compared to GHs regarding enhanced CSSP without excessive VSSP and thus excessive spin contamination. We want to explore this possibility further by systematically varying the EXX admixture via the pre-factor for a) a GH, b) an LH with spinchannel t-LMF and c) an LH with common t-LMF. We keep all other aspects the same by mixing LSDA (Slater) and exact exchange and using the PW92 correlation functional (test calculations with short-range self-interaction-reduced PW92 correlation gave similar results and identical trends). The results for deviations from the reference values for spin densities at the nucleus and Adip, and for the expectation value of our four central manganese complexes are shown in Figure 4 (see Tables S19-S21 in SI for numerical data). The pre-factor (a for both t-LMFs, global EXX admixture a0 for the GH) is varied in steps of 0.1 from 0.0 to 1.0. For the GH this means that we go from pure Slater exchange with PW92 correlation to pure Hartree-Fock exchange with the same correlation functional. For example, a0 = 0.5 marks Becke’s original LSDA-based Half-and-Half functional,88 when used with PW92 correlation. For the LHs, the starting point is the same, whereas the end point marks an unscaled (spinchannel or common) t-LMF.32 Variation of EXX admixture for [Mn(CN)4]2-. We start again with our prototype CSSP case (see also results for [Cr(CO)4]+ in Table S19). We first note that spin contamination remains small even for pre-factors up to 1.0 (top right panel in Figure 4). Strikingly, the values for GH and for the LH with common t-LMF are closely parallel, with the GH giving somewhat larger values, as one might have expected. Both curves first increase slightly up to a pre-factor of about 0.3 and then decrease equally slowly. In contrast, the curve for the LH with spin-channel t-LMF continues to increase slowly but monotonically also after 0.3, up to a value that is about 0.018 above the nominal value for a sextet state. A striking result is found for the spin density at the Mn nucleus as function of the pre-factor (top left panel): the GH and common-LMF curves coincide almost perfectly, whereas the curve for the spin-channel LMF decreases more slowly! This indicates that a common t-LMF 31 ACS Paragon Plus Environment


reproduces essentially the CSSP of a GH that has as much EXX admixture as the maximum value of the LMF, suggesting that it is indeed the EXX admixture in regions close to the nucleus that controls the CSSP. Making the LMF depend on the spin-channel kinetic-energy densities (W and ) rather than on the total quantities (W and ) apparently not only misses some NDC cross terms (cf. eq. 6) but also seems to take away some of this CSSP, for reasons that are not fully clear at this point (plots of the LMFs show only rather subtle changes in the relevant core region which, however, do seem to affect CSSP substantially). We also note the essentially linear decrease of all three curves. Variation of EXX admixture for MnO. For this “intermediate VSSP case” the spin contamination of the LHs also remains small across the range of pre-factors (with the spinchannel LMF curve somewhat above the common LMF curve), while that of the GH increases much more rapidly (bottom right panel in Figure 4), albeit not as fast as for the strong VSSP/spin-contamination cases (see below). This is reflected also in the curves for Adip (middle bottom panel) where the GH produces a much larger slope. The behavior of the GH curve for Aiso as function of EXX admixture is very complicated (bottom left panel): after a slow decrease up to about 0.3 the curve turns slightly up until about 0.5, before dropping rapidly. The onset of significant spin contamination after around 0.4 certainly is important for this behavior, as spurious VSSP and the resulting spin contamination is known to deteriorate the 3d spin population4,5 (which is borne out by Adip). In contrast, the spin-channel LMF curve decreases slowly up to a pre-factor of about 0.5 before turning up equally slowly, generally remaining appreciably above the reference value. This shows that the spin-channel LMF is not able to generate sufficient CSSP even at large pre-factors. Finally, the curve for the common t-LMF decreases monotonically and passes through the reference value at an intermediate pre-factor of around 0.4-0.5. Variation of EXX admixture for MnO3 and [Mn(CN)4N]-. In spite of their rather different overall electronic structure (see above), these two “strong VSSP/spin-contamination” cases exhibit very similar behavior in Figure 4 and will thus be discussed together. As we had seen already further above, for a given functional, spin contamination for these two complexes behaves very similar, and this is confirmed by the curves in the right-hand panels. Clearly, the LH curves increase much more slowly than the GH curves, with the curve for the spin-channel t-LMF generally above that for the common t-LMF. Even at pre-factors of about 0.8, which is higher than thermochemically optimized values (see above), deviations from the nominal 32 ACS Paragon Plus Environment

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value reach only about 0.5-0.6 for the spin-channel LMF and about 0.3-0.4 for the common LMF. This may be compared with the GH, which reaches deviations of about 1.0 already at EXX admixtures of 0.5. This shows that VSSP is clearly smaller for the LHs, due to lower EXX admixture in the bonding region (while we have seen above for the “pure CSSP cases” that CSSP is controlled by the EXX admixture closer to the Mn nucleus). And it shows that the NDC cross terms introduced by the common LMF help to reduce VSSP and spin contamination if all other aspects remain unchanged. Spin contamination is also reflected by the Adip curves (middle panels), which show a clear deterioration for the GH around 30% EXX admixture, whereas the LH curves turn upwards much later and much more slowly. For [Mn(CN)4N]-, all Adip curves first decrease slightly at low pre-factors before turning upwards. This reflects likely the effect of the strong nitride -donor ligand and the fact that moderate EXX admixtures help reduce delocalization errors that turn the Mn-N bond too covalent.89 The effects of spin contamination on Aiso for the GH are apparent for both complexes (left panels) in the very steep slope of the curve, where 30% EXX admixture leads already to an overshooting towards too negative values, while we know that larger EXX admixtures of 50% or more would be required to adequately treat CSSP4,5 (see above for the “pure” CSSP cases). The LH curves cross the reference values at pre-factors of around 0.5 ([Mn(CN)4N]-) or near 0.7 (MnO3), suggesting that we are closer to getting the right answer for the right reason compared to the GH, as spin contamination at these pre-factors is still moderate, while CSSP is expected to be described well. Notably, the Aiso curves are monotonous, having slightly negative curvature for the LHs but being almost linear for the GH. It is also interesting that the Aiso curves for the two LHs are almost on top of each other (except for the largest pre-factors with MnO3), in spite of the larger spin contamination and deviations for Adip with the spin-channel t-LMF. We note in passing that for MnO3 beyond 40% EXX admixture the GHs produce unstable wave functions (as diagnosed by the escf module in TURBOMOLE), and we have thus not included the data at higher pre-factors.



Figure 4. Influence of pre-factor (and thus of EXX admixture) on the performance of one GH and two LHs with LSDA (Slater) exchange and PW92 LSDA correlation. The x-axis marks the EXX admixture a0 for the GH and pre-factor a for a common or spin-channel t-LMF (eq. 7). Left panel: deviations of 𝛼−𝛽

spin densities at the metal nucleus (𝜌𝑁

) from experiment. Middle panel: deviations of dipolar HFCs

from experiment. Right panel: deviations of expectation value from the nominal value for the given spin state.


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4.5 Orbital analysis of GH vs LH pre-factor scan. To further unravel the origins of these dependencies on pre-factor for the GH and LHs, Figure 5 analyzes the CSSP, VSSP, and SOMO contributions for [Mn(CN)4]2- as CSSP case and MnO3 as VSSP/spin-contamination case, in the same way as we analyzed above the behavior for various functionals (cf. Figure 2). Numerical data are provided in Tables S22-S26 for the four central manganese complexes and for [Cr(CO)4]+.

𝛼−𝛽

Figure 5. Individual contributions to 𝜌𝑁

(Mn) of [Mn(CN)4]2- (top) and MnO3 (bottom) as function of

pre-factor for one GH and two LHs (left panels), together with 3s/2s CSSP ratio (right panels). Open symbols in the left-hand panels provide individual 2s and 3s contributions to the overall core (CSSP) part. Above a pre-factor of 0.4, values for MnO3 with the GH are omitted, due to wave-function instability.

Starting with [Mn(CN)4]2-, we see that, as expected, the decrease of Aiso (via the spin density at the Mn nucleus) with increasing pre-factor is dominated by the more negative overall CSSP contributions (Figure 5, top). The latter decrease somewhat more quickly for the common tLMF, even though the individual contributions from 2s and 3s shells are overall smaller than in the other two cases. The reason is a peculiarly slow decrease of the 3s contribution (as also observed for MnO and [Cr(CO)4]+; Tables S23, S26), which in contrast to the other two 35 ACS Paragon Plus Environment


functionals adds to, rather than compensates, the trend of the 2s contributions. The decreases of the total spin density with pre-factor for the LH with common LMF and the GH are nevertheless closely similar, due to a slightly faster decrease of the VSSP contributions for the latter. Together with the relatively slow decrease of CSSP (partly due to an increasing 1s contribution, Figure 5), a slightly increasing VSSP contribution makes the LH with spinchannel LMF exhibit the overall slowest decrease of the total spin density at the nucleus. Therefore, even an increase of the pre-factor towards 1.0 does not allow this LH to reach the reference value, in contrast to the functional with common LMF. The variations of the CSSP contributions are also reflected in the 3s/2s ratio (top right panel), which decreases with prefactor for the GH, remains almost constant for the spin-channel LMF and increases for the common LMF.

Figure 6. Plot of the (unscaled) common and spin-channel t-LMF for [Mn(CN)4]2- (top), together with relevant Mn atomic orbital amplitudes (bottom); along (left) one Mn-C bond (Mn left, C right), or along (right) a vector pointing towards a triangular face of the complex (Mn atom left).


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These differences between common and spin-channel LMF may arise from subtle variations in spatial regions relevant for CSSP and/or VSSP. Figure 6 shows plots of the - and -parts of the spin-channel t-LMF, as well as the common t-LMF (both with a = 1.0), together with the relevant orbital amplitudes, either along an Mn-C bond or towards the center of a triangular face of the coordination tetrahedron. The plot reveals a significant relative shift between the and -components of the spin-channel t-LMF, with the common t-LMF in between. Notably, the largest shifts in the spin-channel LMF occur in the third peak of the LMF, in a region where both 3s and 3d orbitals have still significant amplitude (Figure 6), suggesting a strong influence on CSSP and particularly VSSP contributions. Notably, the EXX admixture in those regions is diminished in the -part of the spin-channel LMF and enhanced in the -part. It seems that in this way, the spin-channel LMF contributes to enhanced VSSP and thus spin contamination, compared to the common LMF. We may either view this as a deficiency of the spin-channel LMF, similar to previous observations for the SAOP model potential, which has been found to artificially enhance spin contamination in EPR-parameter calculations on open-shell transitionmetal complexes.11 Alternatively, the improved performance of the common LMF in this context may be seen as a differential correlation contribution from / cross terms (see above).23 For the VSSP/spin-contamination case MnO3 (Figure 5, bottom), the large positive SOMO contributions increase almost linearly with pre-factor, the negative VSSP and CSSP contributions become more negative (more slowly for the CSSP), resulting in an overall decreasing spin density at the Mn nucleus. As we had discussed further above for a more diverse set of functionals, here the sensitivity to spin contamination affects all three contributions. Consequently, all changes proceed more steeply with increasing pre-factor for the GH than for the LHs. The behavior of the latter two differs comparably less. At the pre-factor, where the total spin density at the nucleus crosses the reference value (below 0.3 for the GH, below 0.7 for both LHs), the various CSSP, VSSP, and SOMO contributions of the three functionals are not too different. However, the LH with common LMF exhibits the smallest VSSP and SOMO contributions (Table S24), reflecting the fact that at the given optimal pre-factor this functional also has the lowest spin contamination (0.957 for the common LMF, 1.154 for the spin-channel LMF, 1.107 for the GH; Table S20).



5. Conclusions and Outlook Hyperfine couplings (HFCs) are extremely subtle properties of individual nuclei in a molecule or solid, as they depend on the electron spin density at or around a specific nucleus and are influenced by a complicated interplay between contributions from the singly occupied orbital(s) of the systems plus, importantly, core- and valence-shell spin polarization effects (CSSP, VSSP). Transition-metal HFCs tend to be particularly demanding for a quantumchemical description, due to a variety of factors, including static correlation effects and spin contamination. Furthermore, in principle HFCs are also sensitive to relativistic effects. Here we have focused the quantum-chemical analysis on a number of 3d complexes studied before by coupled-cluster and DFT methods4,5,11,30 and examined a wide range of contemporary exchange-correlation functionals of which many had not been available at the time of the earlier studies. Our most important goal has been the critical evaluation of local hybrid functionals with position-dependent EXX admixture as a means to enhance CSSP while avoiding excessive VSSP and spin contamination. Indeed, having large EXX admixture in the core region but lower admixture in bonding regions does go some way towards improved functionals for HFC calculations, even though the local hybrids studied here still exhibit extremely simple oneparameter local mixing functions (LMFs) optimized exclusively for thermochemistry and reaction barriers. Significant improvements should certainly be possible by constructing more sophisticated functionals. This will be pursued in our continuing work. The present analyses have clearly exposed the advantage of using “common LMFs” compared to “spin-channel LMFs”, which will be important for the further development. The present evaluations have furthermore identified an encouraging performance of some contemporary -dependent meta-GGA functionals (-HCTH, VSXC, partly M06-L) in enhancing CSSP compared to simpler semi-local functionals. In contrast, two other recent metaGGA functionals from the Minnesota family, MN12-L and MN15-L, fail completely in describing CSSP, in characteristic but different ways. Global hybrids suffer from the well-known excessive VSSP and spin contamination at higher EXX admixtures, coming into play when the SOMO(s) are significantly metal-ligand antibonding. Apart from the EXX admixture, other aspects of the functional (-dependence, parameterization) of more recent global hybrids also affect the isotropic HFCs. Some global hybrids from the Minnesota family (M05, M06, MN15) exhibit additional shortcomings related 38 ACS Paragon Plus Environment

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to CSSP, again in characteristically different ways. While the first two of these functionals thus provided overall very poor isotropic HFCs, MN15 appears to benefit from some error compensation. A variety of long-range corrected hybrid functionals performed similarly as global hybrids with low EXX admixture, showing that the long-range corrections are not important for HFCs. The middle-range HISS functional performed similarly as global hybrids with about 35% EXX admixture, reflecting a corresponding amount of exact exchange in middle range. The present work has provided a basis upon which the construction of improved exchangecorrelation functionals will be pursued, aiming at good HFCs and other core-related properties (contact NMR shifts of paramagnetic systems, NMR spin-spin couplings, NMR shifts with spin-orbit contributions, Mössbauer isomer shifts, and possibly quantities from core-excitation or -ionization spectroscopies). HFCs are generally recommended as very sensitive probes of quantum-chemical methodologies, and they relate to the ongoing discussions of density- versus energy-based errors in DFT, showing that spectroscopic properties such as HFCs require clearly more in a functional than good overall energetics.

Acknowledgments. This work has been funded by DFG project KA1187/14-1. CS is grateful for a PhD scholarship of Studienstiftung des Deutschen Volkes.

Supporting Information. Tables and Figures with additional computational data. This material is available free of charge via the Internet at http://pubs.acs.org.



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