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PAPERNatalie Orms and Anna I. KrylovSinglet–triplet energy gaps and the degree of diradical character in binuclear copper molecular magnets characterized by spin-flip density functional theory

Volume 20 Number 19 21 May 2018 Pages 13095–13662

This journal is© the Owner Societies 2018 Phys. Chem. Chem. Phys., 2018, 20, 13127--13144 | 13127

Cite this:Phys.Chem.Chem.Phys.,

2018, 20, 13127

Singlet–triplet energy gaps and the degree ofdiradical character in binuclear copper molecularmagnets characterized by spin-flip densityfunctional theory†

Natalie Orms and Anna I. Krylov *

Molecular magnets, defined here as organic polyradicals, can be used as building blocks in the

fabrication of novel and structurally diverse magnetic light-weight materials. We present a theoretical

investigation of the lowest spin states of several binuclear copper diradicals. In contrast to previous

studies, we consider not only the energetics of the low-lying states (which are related to the exchange-

coupling parameter within the Heisenberg–Dirac–van-Vleck model), but also the character of the

diradical states themselves. We use natural orbitals, their occupations, and the number of effectively

unpaired electrons to quantify bonding patterns in these systems. We compare the performance of

spin-flip time-dependent density functional theory (SF-TDDFT) using various functionals and effective

core potentials against the wave function based approach, equation-of-motion spin-flip coupled-cluster

method with single and double substitutions (EOM-SF-CCSD). We find that SF-TDDFT paired with the

PBE50 and B5050LYP functionals performs comparably to EOM-SF-CCSD, with respect to both singlet–

triplet gaps and states’ characters. Visualization of frontier natural orbitals shows that the unpaired

electrons are localized on copper centers, in some cases exhibiting slight through-bond interaction via

copper d-orbitals and p-orbitals of neighboring ligand atoms. The analysis reveals considerable

interactions between the formally unpaired electrons in the antiferromagnetic diradicaloids, meaning

that they are poorly described by the Heisenberg–Dirac–van-Vleck model. Thus, for these systems the

experimentally derived exchange-coupling parameters are not directly comparable with the singlet–

triplet gaps. This explains systematic discrepancies between the computed singlet–triplet energy gaps

and the exchange-coupling parameters extracted from experiment.

I. Introduction

Molecular magnets (MMs) are organic polyradicals that have ahigh-spin ground state in the absence of an applied magneticfield.1,2 They can be used as building blocks to create novel,light-weight, and tunable magnetic materials, an alternative toconventional dense metallic or metal oxide magnets. MMs arealso of interest in the context of quantum computing andquantum information storage.3–5 Since the 1990’s much atten-tion has been given to the design and study of organometalliccomplexes containing d- and f-block elements—includingterbium, manganese, chromium, nickel, and copper—whoseelectronic structure (and consequent magnetic properties)

make them suitable candidates for use in magneticmaterials.6 Suitable candidates for magnetic applicationsshould have one or more unpaired, weakly interacting elec-trons. Here, we limit ourselves to systems containing only twounpaired electrons (diradicals).7–11 Electronic configurationsthat can be generated for two electrons in two orbitals areshown in Fig. 1.

The triplet state of a diradical can be represented by either asingle Slater determinant (high-spin Ms = 1 configuration (i)) orby a linear configuration of two Ms = 0 determinants with equalweights (configuration (ii)). Configurations (iii)–(v) give rise tothree singlet states. The energy gaps, relative state ordering,and relative weights of the Slater determinants (i.e., coefficients l)depend on the nature and energy separation of the respectivefrontier molecular orbitals (MOs) f1 and f2. The character of theMOs also determines the character of the wave function, e.g.,whether it is predominantly covalent (i.e., two electrons residingon different parts of a molecule) or ionic. Detailed analysis ofdifferent types of diradical electronic structure can be found in

Department of Chemistry, University of Southern California, Los Angeles, California

90089-0482, USA. E-mail: krylov@usc.edu

† Electronic supplementary information (ESI) available: Relevant Cartesian geo-metries, details of ECP, additional results for collinear kernels. See DOI: 10.1039/c7cp07356a

Received 31st October 2017,Accepted 22nd January 2018

DOI: 10.1039/c7cp07356a

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classic papers.7,8 In the context of MMs, the two states of interestare the lowest singlet and triplet states. In systems like binuclearcopper complexes, one expects these two states to have covalentwave functions in which the unpaired electrons are localized onthe two metal centers:

Cs,t(1,2) B [fCu1(1)fCu2(2) � fCu2(1)fCu1(2)]

� [a(1)b(2) 8 b(1)a(2)] (1)

with little-to-no contributions from ionic configurations

[fCu1(1)fCu1(2) + fCu2(1)fCu2(2)] � [a(1)b(2) � b(1)a(2)](2)

In these expressions, fCu1 and fCu2 denote orbitals localizedon the two copper centers, such as copper d-orbitals (perhapsincluding small contributions from the nearest ligand atoms).If the actual MOs hosting the unpaired electrons are deloca-lized and can be described as (nearly degenerate) bonding andantibonding combinations of fCu1 and fCu2 (case 1 in Fig. 1), asis the case in the MMs studied here, the triplet states aredescribed by configurations (i) and (ii) and have pure covalentcharacter, as Ct in eqn (1). The character of the lowest singletstate can vary, depending on the exact weights of configura-tions (iii) through (v). A purely covalent singlet wave function,Cs of eqn (1), corresponds to configuration (iv) with l = 1.A smaller value of l gives rise to the ionic configurations mixedinto the wave function. This happens when the interactionbetween the two centers stabilizes the bonding MO relative tothe antibonding one, either due to through-space or through-bond interactions. The ionic configurations can also appear inthe singlet wave functions due to mixing with configuration(iii), which, in contrast to (ii), has pure ionic character. Since allconfigurations, (iii)–(v), can contribute into the singlet state,the correct description of this state requires an electronicstructure method that treats (iii)–(v) on an equal footing. Inorder to describe relative energies of singlet and triplet states,the method should provide a balanced and unbiased descrip-tion of all four Ms = 0 configurations from Fig. 1.

From a theoretical perspective, the search for promisingMMs begins with first-principle calculations of the relevant

terms in the phenomenological spin Hamiltonian.2,12,13 Of allterms in the spin Hamiltonian, the most energetically signifi-cant one is the exchange-coupling interaction betweenunpaired, spatially separated electrons.14–16 The sign andmagnitude of electronic exchange-coupling between atomicspin centers determines whether a molecule will behave ferro-magnetically or antiferromagnetically upon exposure to anexternal magnetic field, and thus, whether the molecule issuitable for application in a magnetic material. In bimetallicdiradical complexes, like those considered here, the exchange-coupling constant equals the energy difference between thelowest singlet and triplet states.17 Thus, exchange-couplingterms of the spin Hamiltonian can be computed by ab initiomethods as singlet–triplet energy gaps. This approach can begeneralized to systems with more unpaired electrons andmultiple polyradical centers.18,19

The main challenge in applying this simple strategy is in themulticonfigurational character of the low-spin wave functions,which becomes evident by inspecting Fig. 1. The high-spinstates, such as Ms = 1 triplet (i), are well represented by a singleSlater determinant and, therefore, their energies can be reliablycomputed by standard single-reference methods, i.e., coupled-cluster theory or DFT. In contrast, wave functions of low-spinstates (ii)–(v) require at least two Slater determinants. Conse-quently, they are poorly described by single-reference methods.This imbalance in the description of high-spin and low-spinstates results in large errors in the computed singlet–triplet gaps.

Several strategies have been employed for describing theopen-shell states and exchange-coupling in MMs. Historically,the most popular are the broken symmetry (BS) methods20–22

and spin-restricted Kohn–Sham (REKS/ROKS) methods.22–24

Both approaches suffer from imbalance in their treatment—andsometimes, outright exclusion—of important configurationsdepicted in Fig. 1. For example, in BS approach all singlet andtriplet configurations are scrambled. While spin projectionallows one to formally separate singlet and triplet manifolds, itdoes not distinguish between open- and closed-shell singletstates (iii)–(v). Despite belonging to entirely different states(which may even have different spatial symmetry), these

Fig. 1 Wave functions of diradicals that are eigenfunctions of S2 (only configurations with positive spin projections are shown). Wave function (i)corresponds to the high-spin Ms = 1 triplet state. Wave functions (ii)–(v) correspond to the low-spin states: Ms = 0 singlets and triplets. Note that all Ms = 0configurations can be formally generated by a spin-flipping excitation of one electron from the high-spin Ms = 1 configuration. The character of the wavefunctions (i.e., covalent versus ionic) depends on the nature of molecular orbitals f1 and f2 and the value of l.

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configurations remain scrambled in spin-projected BS solutions.BS solutions are often contaminated by mixing with otherelectronic states, i.e., higher multiplets.25 Even more disturbingis the fact that spin-projectors are not uniquely defined and theresults are affected by specific choices of a projector.20,26–29

Furthermore, BS-DFT does not scale well with the number ofradical sites, as the number of BS solutions grows rapidly. Themost important concern is the quality of wave functions andspin-dependent properties extracted from BS-DFT. As pointedout by Neese,12 although broken-symmetry solutions representcharge density reasonably well, they result in erroneous spindensity. Consequently, the quality of broken-symmetry descrip-tion of properties that are determined by charge density and byspin density can be vastly different. Thus, despite being reason-ably successful in terms of predicting energy splittings betweenthe different components of multiplet spectra, BS-DFT approachdoes not offer a fully satisfactory solution to modeling MMs. Thedrawback of the REKS/ROKS scheme is that closed- and open-shell states are described by different sets of equations and thatthe effect of different dynamic correlation in the two manifoldsis not fully accounted for.30

A more balanced and robust alternative is the spin-flip(SF) family of methods.31–36 Within SF approach, a single-determinant high-spin triplet state is used as a reference fromwhich one computes the manifold of low-spin states arisingfrom configurations (ii)–(iv) in Fig. 1 by using spin-flippingexcitations. The performance of various variants of SF methodfor organic polyradicals has been extensively benchmarked,focusing on the energy gaps between the electronic states andtheir structures.9,34–42 Recently, we investigated the perfor-mance of SF methods in terms of characters of the underlyingwave functions.11 Specifically, we compared how differentfunctionals and wave function methods describe frontierorbitals, their occupations, and the number of effectivelyunpaired electrons. Towards this end, we employed density-based wave function analysis tools.43–45 By using naturalorbitals and their occupations, this analysis allows one tomap correlated multiconfigurational wave functions into asimple two-electrons-in-two-orbitals picture from Fig. 1.

The goal of this study is to investigate the performance of SFmethods for MM. In contrast to previously studied organicdi- and tri-radicals, typical singlet–triplet gaps in MM are muchsmaller, within several hundreds of wave numbers, which callsfor sub-chemical accuracy (1 kcal mol�1 = 350 cm�1). Addi-tional challenges arise due to the presence of heavy atoms,which might require using special basis sets and/or effectivecore potentials (ECPs). Although encouraging results of SFcalculations for several MMs with transition metals have beenreported before,18,19,25,46–48 no systematic studies of the effectof ECP (and/or basis sets) on multiplet splittings within theSF-TDDFT framework has been carried out. In the present work,we focus on a set of eight binuclear copper complexes whosemagnetic properties have been extensively studied.22,46,49–58

Given the above considerations and their large sizes, binuclearcopper diradicals represent a stringent test for the theory.We report the results of the EOM-SF-CCSD and SF-TDDFT

calculations of singlet–triplet gaps (which are equal toexchange-coupling within the assumptions of Heisenberg–Dirac–van-Vleck (HDvV) model14–16) and the bonding pattern(the shapes of frontier natural orbitals and the extent of inter-action between the formally unpaired electrons). We analyze theeffects of basis sets, ECPs, TDDFT kernel, the functional, andmolecular geometry.

The structure of the article is as follows: the next sectionprovides a brief theoretical overview of SF methods, the HDvVmodel14–16 for exchange-coupling, and density-based analysis.The following section outlines computational details. We thenillustrate the effects of the above mentioned variables oncomputed singlet–triplet gaps and wave function properties(frontier natural orbitals, their occupation, and the numberof effectively unpaired electrons) of the eight binuclear coppercompounds. In conclusion, we comment on the applicability ofthe HDvV model for this family of diradicals and the impor-tance of density-based analysis in the description of open-shellsystems.

II. Theoretical methodsA. Diradicals and spin-flip approach

As described above, the electronic states of diradicals,7,8,10,59

such as bicopper MMs, arise from two electrons distributed intwo nearly degenerate MOs. For same-spin electrons, there isonly one possible arrangement: Ms � 1. For the Ms = 0 states,more configurations can be generated, as illustrated in Fig. 1.All configurations depicted in Fig. 1 can contribute to diradicalspin-states of interest, which makes choosing an electronicstructure method that treats all states on an equal footing, asSF does, of the utmost importance. Because in the SF approachthere is no assumptions of the relative importance of config-urations from Fig. 1, these calculations allow one to unambigu-ously determine the state character, e.g., to quantify the extentof the interaction between the unpaired electrons and theweights of ionic configurations in the respective wave func-tions. Thus, SF methods are capable of distinguishing whethera given MM candidate falls within the regime of spatiallyseparated, weakly-interacting spin moments, as assumed withinthe HDvV model.

To describe diradicals within the SF framework, one beginswith the high-spin Ms = 1 triplet state, which is well representedby a single Slater determinant. The low-spin states (singlets andthe Ms = 0 triplets) appear as single excitations from the high-spin reference state. Thus, in the SF approach, a high-spin stateis used as the reference state and the target manifold of states isgenerated by applying a linear spin-flipping excitation operatorto the:60

Cs;tMs¼0 ¼ RMs¼�1C

tMs¼1: (3)

Using different approaches to describe correlation in thereference state gives rise to different SF methods.31–36,61 Ourprimary focus here is SF-TDDFT approach,34,35,62–64 which isvery attractive owing to its low computational cost. We compare

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the performance of several functionals chosen on the basis ofthe results of the previous studies.34,35 For the benchmarkingpurposes, we also report the results of EOM-SF-CCSD,33 a highlyaccurate wave-function method including advanced treatmentof dynamical correlation. A density-based wave function analysisallows quantitative comparison of the character of underlyingwave functions computed by different methods.

B. The Heisenberg–Dirac–van-Vleck Hamiltonian forlocalized, weakly interacting electrons

The key assumption used to construct phenomenologicalspin Hamiltonians is that the unpaired electrons are spatiallyseparated and weakly interacting. The resulting magneticHamiltonian is:

HSPIN ¼X

A

SADASA þX

A

SAAA IA �X

AB

SAJSB (4)

where the SA are local electronic spin operators on nuclei A, theIA are nuclear spin moments, and D, A, and J represent the zero-field splitting, hyperfine, and exchange-coupling interaction,respectively.2,12,13 Magnetic parameters D, A, and J are extractedfrom experiment, typically by fitting electron paramagneticresonance, magnetic susceptibility, or other macroscopic prop-erties to a model equation.16,65–68 Solving this Hamiltonianallows one to predict magnetic behavior of the system in thepresence or absence of the magnetic field.

The parameters of eqn (4)—such as hyperfine couplings,zero-field splittings, exchange-couplings—can be extractedfrom experimental measurements or computed from correlatedmany-electron wave functions or DFT.2,12,13 All phenomeno-logical terms in eqn (4) can be computed using existing relativisticand non-relativistic electronic structure methods.21,22,46,47,58,69–75 Inthis work, we focus on the third, non-relativistic term, which givesrise to the HDvV Hamiltonian:14–16

HHDvV ¼ �X

AoB

JABSASB: (5)

The HDvV Hamiltonian describes weakly coupled localizedelectrons neglecting hyperfine couplings and zero-field split-tings. This phenomenological Hamiltonian can be mapped tothe exact many-body Hamiltonian using effective Hamiltoniantheory.13 By solving eqn (5), one can relate the energies ofdifferent spin states (singlets and triplets, in the case ofdiradicals) to the experimental magnetic parameter, J, knownas the exchange-coupling constant.17 Exchange-couplingsrepresent dressed, or effective interactions between thespins.13 The sign of J determines whether a molecular magnetbehaves ferromagnetically or antiferromagnetically in an externalmagnetic field.

Analytic solutions of eqn (5) are used to construct a formulaconnecting macroscopic magnetic susceptibility with exchange-couplings.76 This formula is used to extract the latter from thetemperature-dependent measurements of magnetic suscepti-bility. Thus, all experimental values of J derived from thesemeasurements equal to the true energy gaps only if the keyassumption of the HDvV model, that the spins are weakly

interacting and the contributions of the ionic configurationsinto the total wave function are small, hold true.

Eqn (5) assumes two (or more) weakly interacting, spatiallyseparated electronic spins localized on two (or more) atomiccenters, A and B. Obviously, the configurations arising frompaired (ionic) electronic configurations or locally excited(i.e., non-Hund) configurations are not admitted.77 The justifi-cation for this choice of the model Hamiltonian is that for theweakly interacting unpaired electrons, the ionic configurationsare much higher in energy than covalent triplet and singletconfigurations of eqn (1). Under these assumptions, the HDvVHamiltonian can be derived from the Hubbard model by usingquasidegenerate perturbation theory (see, for example, ref. 13).

What happens when ionic configurations are mixed withlarge weights into the singlet states in antiferromagneticspecies? In this case, the energy of the singlet diradical state isstabilized via configuration interaction, leading to an increasedsinglet–triplet gap. While the macroscopic properties, such astemperature dependence of magnetic susceptibility will reflectthis phenomenon, the application of the HDvV-based model tofit the data would produce J values that are too large, becauselarger values of J are necessary to describe the increased singlet–triplet gap in the absence of explicit ionic configurations in themodel. Thus, one should expect that for strongly AF species theexperimentally derived J values are systematically overestimated.

In the two-center systems, J is directly related to the energiesof spin states via the Lande interval rule.17 With an electronicstructure method that offers a balanced treatment of open andclosed-shell states, one can calculate J as an energy differencebetween the spin states:

E(S) � E(S � 1) = �JAB�S (6)

In diradicals S and S � 1 are triplet and singlet states: E(S) isthe energy of the lowest-energy triplet state and E(S � 1) is theenergy of the lowest singlet state. A positive J gives rise to aferromagnetic ground state, while negative J gives rise to anantiferromagnetic ground state. Note that a different form ofeqn (5) might be used,18,19 with a factor of 2 in front of thesum, giving rise to a different relationship between energygaps and exchange-coupling. Thus, for meaningful compar-isons between different studies, it is important to carefullycheck which form was used (and of course it is always safer toreport energy gaps between physical states, e.g., DEST in thecase of diradicals).

C. Quantifying radical character and bonding patterns bydensity-based analysis

Due to the complexity of many-body wave functions andarbitrariness in orbital choices, the interpretation of realisticwave functions (and often even of Kohn–Sham states) in termssimple two-electron-in-two orbitals picture (as in Fig. 1) is notstraightforward. In particular, in large open-shell systemscanonical Hartree–Fock of Kohn–Sham orbitals often providea poor representation of the frontier MOs even for relativelysimple high-spin states.11 Fig. 2 illustrates the difference

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between canonical Kohn–Sham orbitals and spin-density dif-ference for copper diradicals: while the excess spin densityexhibits an expected pattern (unpaired electrons are localizedon d-orbitals of copper), the two formally singly occupiedcanonical MOs are delocalized over the ligands. In this case,even assigning the state characters and verifying the correct-ness of the solution of the self-consistent field (SCF) procedureare problematic.

This problem is effectively addressed by using density-basedanalysis:43–45 natural orbitals and their occupations provide aclear and unambiguous picture of the essential electronicstructure of open-shell systems. Natural orbitals, which areeigen-functions of the one-particle density matrix, afford mostcompact representation of the wave function. The respectiveeigen-values can be interpreted as occupations. Thus, by com-puting natural orbitals for a given one-particle density matrixone can obtain frontier MOs representing, for example, theunpaired electrons in diradicals. The shapes of these orbitalsallow one to asses the extent of localization of the unpairedelectrons and their through-space and through bond interac-tions with each other and with other moieties. We note thatwhen two frontier orbitals are exactly singly occupied (such asin perfect diradicals), their choice is not unique and any linearcombination provides a legitimate representation (for example,in H2 with a completely broken bond, one can consider either apair of localized atomic-like orbitals or a delocalized bonding–antibonding pair).

By using natural occupations, one can define and computethe number of effectively unpaired electrons. In this work, weuse two indices, nu and nu,nl, proposed by Head-Gordon:43

nu ¼X

i

min �ni; 2� �nið Þ (7)

nu;nl ¼X

i

�ni2 2� �nið Þ2 (8)

In both equations, the sum runs over all natural orbitals andthe contributions of the doubly occupied and unoccupiedorbitals are exactly zero. While both formulas give identicalanswers for the limiting cases (such as a two-electron tripletstate), nu,nl delivers more physically meaningful and consistentresults for more complex wave functions. By comparing occu-pations of the frontier natural orbitals and nu,nl one can assesshow well a two-electrons-in-two-orbitals picture represents thereal wave function.

We recently applied this technique to prototypical di- andtriradicals.11 Here, we extend this approach11 to a set of 8binuclear copper diradicals. We quantify the degree of radicalcharacter using Head-Gordon’s indices,43 eqn (7) and (8), andvisualize natural frontier orbitals to characterize the localiza-tion and the interactions between the unpaired electrons.Visual inspection of natural orbitals, paired with the analysisof the respective natural occupations and the computed numberof effectively unpaired electrons, obtained from a variety of SF

Fig. 2 Spin difference densities, unrestricted singly occupied molecular and natural frontier orbitals (SOMOs and SONOs, respectively) of tripletCUAQAC02 (left) and CITLAT (right) at the B5050LYP/cc-pVTZ level of theory. CUAQAC02 has 202 electrons while CITLAT has 278, making their low-lying states and associated orbital surfaces numerous and complex. Reproduced with permission from ref. 11.

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methods, provides a robust way of validating the applicability ofthe HDvV model (and the Lande interval rule) for describing thisfamily of MM candidates.

III. Computational details

Fig. 3 and Fig. S1 (ESI†) show the experimental structures usedin calculations of singlet–triplet gaps of benchmark systems.Table 1 provides their associated experimental exchangecoupling constants. Counterions were removed from all struc-tures. The ferrocene group in the experimental PATFIA complexwas also removed (as in previous studies46,58), unless otherwisespecified.

Experimental geometries are used in all EOM-SF-CCSD andSF-TDDFT calculations, unless otherwise specified. To testpossible effects of uncertainty of the structure, we also con-sidered optimized structures of the high-spin triplet states ofBISDOW, PATFIA (without the ferrocene group), and CITLAT.These optimizations were performed using the oB97X-D78

functional and all-electron cc-pVTZ basis. The results foroptimized structures are presented in ESI.†

In the tables below, we report singlet–triplet gaps, DEST,defined as:

DEST = ES � ET. (9)

within the HDvV model, DEST = J, by virtue of eqn (6).We compared the following DFT functionals in the

SF-TDDFT calculations of DEST: LDA (with Slater exchange andVWN correlation), several members of the Becke-exchange/LYPcorrelation family: BLYP,79,80 B3LYP,81 and B5050LYP(50% Hartree–Fock + 8% Slater + 42% Becke for exchangeand 19% VWN + 81% LYP for correlation).35 Of the P86correlation (with Becke exchange) and PW91 families, we chosethe BP86, B3P86,82,83 PW91, and B3PW91 functionals.84–87

From the PBE family, we selected the PBE, PBE0 (75% PBEand 25% Hartree–Fock exchange, 100% PBE correlation),PBE50 (50% PBE and 50% Hartree–Fock exchange and 100%PBE correlation), and oPBEh (80% PBE, 20% short-rangeHartree–Fock exchange and 100% long-range Hartree–Fockexchange, PBE correlation) functionals.88,89 Of the Minnesotafamily of functionals, we chose M06 (hybrid with 27% Hartree–Fock exchange),90 M06-L (meta-GGA),90 GAM (GGA),91 MN15-L(meta-GGA),92 and MN15 (hybrid with 44% Hartree–Fockexchange and MN15 correlation).93 All of these functionals(with the exception of GAM, MN15-L and MN15, which arerecent additions to the Minnesota family) have been extensivelybenchmarked with SF-TDDFT for organic polyradicals, whereinhybrid functionals such as PBE50, B5050LYP, and PBE0 wereshown to produce relative state energies approaching chemicalaccuracy.34,35 In this paper, we present a similar error analysisof collinear and non-collinear SF-TDDFT with the abovefunctionals in calculating energy gaps of binuclear coppercomplexes.

We also present EOM-SF-CCSD/cc-pVDZ energy splittings forselected complexes. The core electrons were frozen in all EOMcalculations. For all SF-TDDFT and EOM-SF-CCSD calculations,we report only electronic energy separations between the singletand Ms = 0 triplet states (DEST).60 Because of the similarity of the

Table 1 Experimental exchange-coupling constants for eight binuclearcopper diradicals shown in Fig. 3

Complex J (cm�1) Ref.

BISDOW �382 50CUAQAC02 �286 51CAVXUS �19 52 and 53PATFIA �11 54Cu2Cl6

2� 0, �40 55XAMBUI 2 56YAFZOU 111 57CITLAT 113 49

Fig. 3 Eight binuclear copper complexes included in this study. Structures are denoted with their Cambridge Structural Database names.107,108

Complexes 1, 3, 4, 6, and 7 have a charge of 2+; complex 2 is neutral; complex 5 has a charge of 2�; and complex 8 has a charge of 1+. The respectiveLewis structures are shown in Fig. S1 in ESI.†

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electronic structure of the singlet and triplet states in the caseof very weakly interacting electrons, we expect that both thestructures and vibrational frequencies of the two states arevery close.

We performed density and wave function analysis using thelibwfa module44 contained in the Q-Chem electronic structurepackage.94,95 We analyzed singlet and triplet states of all eightbenchmark complexes at the PBE50/cc-pVDZ level of theory,and six of the eight complexes (BISDOW, CUAQAC02, Cu2Cl6

�,XAMBUI, YAFZOU, and CITLAT) at the EOM-SF-CCSD/cc-pVDZlevel of theory. PBE50, B97, and LDA functionals were used withthe non-collinear SF-TDDFT kernel35,96,97 in the analysis ofCUAQAC02. Collinear SF-TDDFT with the B5050LYP functional,which was recommended in the original SF-TDDFT paper,34

was also used in the density analysis of CUAQAC02. We alsopresent frontier natural orbitals of the full experimental struc-ture of the PATFIA complex, which includes a ferrocene group,at the collinear SF-TDDFT/cc-pVDZ level with the B5050LYPfunctional. We compare these results to those obtained fromthe simplified structure without the ferrocene group.

We compared the performance of Dunning’s cc-pVDZ andcc-pVTZ all-electron basis sets for all eight complexes andseveral functionals. Table S1 in the ESI† provides an atom-by-atom description of the effective core potentials (ECPs) andmatching basis functions used in our study. Included areLANL2DZ (a non-relativistic ECP calibrated with Hartree–Focktotal energies),98 SRSC (a relativistic ECP fitted with all electron-eigenvalues and charges),99 and CRENBL (a relativistic ECPfitted with Hartree–Fock valence orbital energies).100,101 Alsoincluded is the ECP10MDF pseudopotential,102,103 a relativisticECP designed to reproduce valence energy spectra. ECP10MDFhas been shown to perform well in the EOM-CCSD analysis ofsmall copper compounds.104,105

We performed all calculations with the Q-Chem electronicstructure package.94,95 Molecular orbitals were rendered usingIQmol106 and natural orbitals were rendered using Jmol.

IV. Results and discussionA. The comparison of singlet–triplet gaps computed bydifferent methods

We begin by comparing computed singlet–triplet gaps againstexperimentally derived exchange-coupling constants. Mean absoluteerrors (MAEs) for each DFT functional and EOM-SF-CCSD arepresented in Fig. 4. Tabulated mean errors (ME), mean absoluteerrors (DMAE), and standard deviations of the error (DSTD) areprovided in the ESI† for the non-collinear kernel, the PBE0,PBE50, and B5050LYP functionals, and various all-electron basissets and ECPs.

Hybrid functionals—in particular, LRC-oPBEh, PBE0,PBE50, and B5050LYP—outperform LDA and GGA functionalsand approach the accuracy of EOM-SF-CCSD. As will be shownin subsequent sections, errors against the experiment are non-uniform among antiferromagnetic (AF) and ferromagnetic (F)complexes and the functionals and kernels that yield theclosest agreement with experiment and EOM-SF-CCSD varydepending on the sign of J (i.e., whether the complex exhibitsa singlet or triplet ground state). Although the lowest MAE isobserved for the MN15 functional, energy differences betweenthe high-spin triplet reference and the Ms = 0 component of thetriplet state are often large (on the order of 2 eV, in contrastwith typical values of 0.1–0.2 observed for other functionals,see, for example, Table S2 in ESI†) and spin-contamination oftarget states high, ranging between 0.01 and 0.33, whereasfor other functionals such as B5050LYP and PBE50, spin-contamination is typically within 0.01–0.02. M06 shows largererrors relative to MN15. Compared to B5050LYP, M06 MAE isalmost three times larger, likely because of a too-small fractionof the exact exchange (27%).

The non-collinear kernel outperforms the collinearSF-TDDFT kernel for all functionals, with the exception ofhybrids PBE50 and B5050LYP, where the collinear kernel showsmodest improvement. For GGA functionals, use of the collinear

Fig. 4 Mean absolute error (MAE) in the singlet–triplet gap for the eight copper benchmark systems. Error relative to experimental values of theexchange-coupling constant, J, are presented for 18 density functionals and EOM-SF-CCSD. The all-electron cc-pVDZ basis set was used for all atoms.

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kernel is tantamount to Koopmans’ estimate of the gaps usingKohn–Sham orbital energies, as the spin-flipped configurationsare not coupled in the absence of the Hartree–Fock exchange.34

Consequently, energy errors are high for these functionals,relative to results obtained with the non-collinear kernel, andspin contamination of target states significant (see ESI†).

The scatter plot in Fig. 5 illustrates the singlet–triplet gaps ofall eight complexes computed by several hybrid functionals andEOM-SF-CCSD. An idealized line showing perfect agreementbetween experimental values of J and theoretical singlet–tripletgaps is also shown. All five selected functionals capture thequalitative trend and reproduce the sign of the exchange-coupling. The errors are relatively small for complexes withnear-zero experimental exchange-coupling and for complexeswith a triplet ground state. We observe better agreement withexperiment when hybrid functionals with 50% HF exchange areapplied to molecules with a triplet ground state. For mostfunctionals, the discrepancy between theoretical and experi-mental values are quite large for the two complexes with singletground states, BISDOW and CUAQAC02. Table 2, in which

EOM-SF-CCSD/cc-pVDZ results for exchange-coupling are pre-sented, shows the same trend. Again, we note good agreementwith experiment for complexes that exhibit near-zero or ferro-magnetic coupling, while substantial discrepancy is observed foranti-ferromagnetic complexes. It will be shown in Section 4.6that this discrepancy arises not from electronic structure methodfailure, but the limitations of the HDvV model in the case of thesinglet ground states of more strongly AF complexes.

B. SF-TDDFT: the effect of ECP and basis sets

Tables 3–6 provide singlet–triplet gaps calculated with fourhybrid functionals, and seven ECP/basis set combinations.For the PBE0 and oPBEh functionals, results obtained withthe collinear kernel are tabulated in the ESI.† The use ofsmaller basis sets and ECPs affords computational savings,which is attractive for large systems, however, careful bench-marking is required to determine the effect of additionalapproximations on the computed quantities. We find thatregardless of functional, there is little difference in quality ofresults when a triple-zeta, rather than double-zeta, basis set isused, justifying the choice of a smaller DZ-quality basis forcalculations of exchange-coupling in binuclear copper diradicals.Use of an ECP (regardless of ECP choice) appears to yield slightlybetter agreement with experiment for AF complexes, while little-to-no effect is observed for F complexes.

C. SF-TDDFT: collinear versus non-collinear kernel

Fig. 4, Tables 3–6 present error analysis and values of thesinglet–triplet gap computed with the collinear SF-TDDFTkernel and using the geometries from crystal structures. Fig. 6compares the singlet–triplet gap calculated for the BISDOW (AF),PATFIA (weakly AF), and CITLAT (F) complexes using the collinearand non-collinear SF-TDDFT kernels. The ECP10MDF/cc-pVDZbasis is used, where the ECP is applied only to copper atoms.Errors are non-uniform with respect to both functional and kernel

Table 2 EOM-SF-CCSD/cc-pVDZ singlet–triplet gaps (computed usingthe X-ray structures) and experimental exchange-coupling constants,in cm�1

Complex EOM-CCSD DEST hS2irefa hS2ita hS2isa Exp. J

BISDOW �208b 2.115 2.013 0.016 �382CUAQAC02 �180c 2.006 2.000 0.005 �286Cu2Cl6

2� �16c 2.009 1.997 0.009 0, �40XAMBUI 0.0 2.006 2.003 0.001 2YAFZOU 86c 2.022 — — 111CITLAT 59d 2.008 1.980 0.028 113

a hS2iref denotes hS2i of the Ms = 1 reference determinant. hS2it, and hS2isdenote the Ms = 0 target triplet and singlet states, respectively. b Choleskydecomposition with a tolerance of 1 � 10�2. Virtual orbitals frozen above4.5 hartree (total of 14). c Cholesky decomposition with a toleranceof 1 � 10�3. d EOM-SF-MP2 approximation. Virtual orbitals above 4hartree (total of 20).

Fig. 5 Theoretical singlet–triplet gaps computed with the PBE0, o-PBEh, B3LYP, and B5050LYP functionals (y-axis) versus experimentally derived valuesof the exchange-coupling parameter (x-axis). The inset in the bottom right zooms into the region spanning the range of �50 to 10 cm�1. Dasheddiagonal line marks perfect agreement between theoretical and experimental values. The all-electron cc-pVDZ basis was applied to all atoms. Thecollinear SF-TDDFT kernel was used with the B5050LYP functional, and non-collinear SF-TDDFT kernel with all other functionals.

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choice, a result that is consistent with findings for other com-plexes not included in Fig. 6.

For BISDOW, the most anti-ferromagnetic complex in ourstudy, the modest Hartree–Fock exchange of the PBE0 and LRC-oPBEh functionals paired with the non-collinear kernelappears to yield results in closest agreement with the experi-mental values of J (but in disagreement with EOM-SF-CCSD).However, even with the PBE0 and LRC-oPBEh functionals, thediscrepancy is always larger than for ferromagnetic complexes.Exchange-coupling between copper centers in CITLAT, themost ferromagnetic complex in our study, is best captured byfunctionals with a higher percentage of Hartree–Fock exchange,such as in PBE50 and B5050LYP. The sign of the weakly AFcomplex, PATFIA, is captured by both the PBE0 and LRC-oPBEhfunctionals, regardless of kernel choice, while the magnitude isbetter represented by the PBE50 or B5050LYP functionals andthe non-collinear kernel.

D. BS-DFT versus SF-DFT

Two studies compared the performance of BS-DFT and SF-DFTfor the binuclear copper MMs.25,46 Ziegler and co-workersfocused on the performance of SF constricted variationalDFT, SF-CV-DFT; they reported excellent performance ofSF-CV-DFT with the B5050LYP functional and extensively dis-cussed limitations of the BS-DFT and REKS/ROKS approaches.

Table 4 Singlet–triplet gaps (cm�1) for complexes 1–8 calculated usingSF-TDDFT with the PBE0 method and the non-collinear TDDFT kernel

Basis

Complex

1 2 3 4 5 6 7 8

cc-pVDZ �523 �354 �27 �35 �23 2 201 232cc-pVTZ �506 �362 �26 �33 35 1 197 227ECP10MDF/cc-pVDZ �486 �326 �27 �40 23 �1 186 213ECP10MDF/cc-pVTZ �456 �326 �25 �40 40 1 84 204LANL2DZ �615 �603 �21 39 0 0 250 267SRSC �471 �449 �19 21 �10 1 191 208CRENBL �474 �459 �19 31 �75 0 202 223

Exp. J �382 �286 �19 �11 (0, �40) �2 111 113

Table 5 Singlet–triplet gaps (cm�1) for complexes 1–8 calculated using SF-TDDFT with the PBE50 method

Basis

Complex

1 2 3 4 5 6 7 8

NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb

cc-pVDZ �173 �219 �121 �154 �9 �11 0 60 55 �3 0 0 93 77 120 78cc-pVTZ �158 �211 �119 �153 �7 �11 1 �57 63 7 1 �1 90 60 118 77ECP10MDF/cc-pVDZ �158 �213 �115 �148 8 11 15 �60 �52 �3 �1 1 88 57 112 75ECP10MDF/cc-pVTZ �148 �199 �111 �143 �6 �11 �8 �56 61 15 0 0 84 56 110 74LANL2DZ �203 �262 �208 �269 �9 �10 15 �60 �18 �85 0 0 141 80 114 86SRSC �160 �208 �156 �198 �7 �9 �15 �46 7 — 0 0 79 60 99 73CRENBL �161 �211 �158 �201 �7 �15 �15 �50 �35 �90 0 0 92 60 99 74

Exp. J �382 �286 �19 �11 0, �40 �2 111 113

a Non-collinear TDDFT kernel. b Collinear TDDFT kernel.

Table 3 Singlet–triplet gaps (cm�1) for complexes 1–8 calculated usingSF-TDDFT with the LRC-oPBEh method and the non-collinear TDDFTkernel

Basis

Complex

1 2 3 4 5 6 7 8

cc-pVDZ �444 �319 �24 �35 39 0 198 231ECP10MDF/cc-pVDZ �416 �296 �24 �44 40 0 184 213LANL2DZ �510 �532 �20 38 �2 �1 244 263

Exp. J �382 �286 �19 �11 (0, �40) �2 111 113

Table 6 Singlet–triplet gaps (cm�1) for complexes 1–8 calculated using SF-TDDFT with the B5050LYP method

Basis

Complex

1 2 3 4 5 6 7 8

NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb NCa Cb

cc-pVDZ �155 �212 �114 �149 �8 �11 2 �57 45 �7 �1 0 84 61 109 77cc-pVTZ �150 �205 �115 �149 �6 �10 2 �56 53 2 �1 0 82 60 108 77ECP10MDF/cc-pVDZ �142 �205 �103 �140 7 11 9 �57 44 0 1 1 75 56 98 74ECP10MDF/cc-pVTZ �133 �191 �99 �136 �2 �10 �8 �56 52 12 1 0 73 53 92 73LANL2DZ �204 �255 �211 �261 �7 �10 7 �31 �27 �83 0 �1 97 79 110 85SRSC �162 �199 �158 �192 �7 �9 22 �38 �7 — 0 0 72 58 90 72CRENBL �165 �205 �162 �199 �9 �9 6 �42 �48 �95 �2 0 79 60 94 73

Exp. J �382 �286 �19 �11 0, �40 �2 111 113

a Non-collinear TDDFT kernel. b Collinear TDDFT kernel.

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They reported BS-DFT results obtained with severalfunctionals, illustrating strong dependence of the exchange-coupling values on the functional. The errors of BS-DFTwith B5050LYP functional were comparable to those ofSF-CV-DFT.

Truhlar and co-workers46 carried out a detailed benchmarkstudy in which they compared collinear SF-TDDFT with BS-DFT(and with REKS) using several functionals with varying amountof Hartree–Fock exchange (X): B3LYP (X = 20), M06 (X = 27),B3LYP40 (X = 40), B1LYP40 (X = 40), B1PW40 (X = 40), BMK

Fig. 6 Singlet–triplet gaps for complexes 1–8 calculated using the PBE0 (A), LRC-oPBEh (B), PBE50 (C), and B5050LYP (D) density functionals. Theperformance of the non-collinear and collinear TDDFT kernels is compared in (C) and (D). The ECP10MDF pseudopotential and matching cc-pVDZ-PPbasis set were applied to Cu atoms. The cc-pVDZ basis was used for all other atoms.

Table 7 Singlet–triplet gaps (cm�1) for optimized geometries of complexes 1, 4 and 8, compared with gaps obtained using the experimentallydetermined molecular geometries

Method Basis

Complex 1 Complex 4 Complex 8

X-Ray Optimized X-Ray Optimized X-Ray Optimized

PBE0 cc-pVDZ �523 �524 �35 �57 109 DRSa

ECP10MDF/ccpvdz �486 �459 �40 �61 213 233LANL2DZ �615 �571 39 �12 267 313

LRC-oPBEh cc-pVDZ �444 �418 �35 �59 231 259ECP10MDF/ccpvdz �416 �394 �44 �64 213 233LANL2DZ �510 �473 38 �15 263 306

PBE50 cc-pVDZ �173 �161 0 �24 120 113ECP10MDF/ccpvdz �158 �148 15 �31 112 104LANL2DZ �203 �180 15 �16 114 134

B5050LYP cc-pVDZ �155 �152 2 �25 109 102ECP10MDF/ccpvdz �142 �133 9 �32 98 90LANL2DZ �204 �165 7 �19 110 115

Exp. J �382 �11 113

a SCF calculation converged to a wrong reference state.

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(X = 42), MPW1K (X = 42.8), B3LYP54 (X = 54), M06-2X (X = 54),M06-HF (X = 100). Of this set, B3LYP54 is very similar toB5050LYP. Their test set comprised 12 binuclear copperMMs, including all 8 MMs investigated here. This study

reported that SF-TDDFT approach performs systematically bet-ter than the spin-projected weighted-average BS strategy,although the optimal percentage of Hartree–Fock exchangewas found to be slightly different for the two approaches (about40% for SF-TDDFT and 35% for spin-projected weighted-average BS-DFT). In agreement with our findings, they pointedout that the SF-TDDFT exchange-couplings show a much stron-ger dependence on the percentage of Hartree–Fock exchangethan on the other details of the exchange functionals or thenature of the correlation functionals.

E. The effect of geometry on the computed gaps

To investigate the possible effect of geometry on the computedsinglet–triplet gaps, we carried out geometry optimizations forthree of the eight benchmark complexes: BISDOW, PATFIA, andCITLAT. The resulting structures are shown in Fig. S2 in ESI.† Themost important structural parameter is the distance between theradical sites (Cu atoms). We found that the internuclear Cu

Table 8 NC-PBE50/cc-pVDZ energy splittings (cm�1) and wave functionproperties of lowest singlet and Ms = 0 triplet states of all eight benchmarkcomplexes

Molecule DEST

nu nu,nl hS2i

Singlet Triplet Singlet Triplet Singlet Triplet

BISDOW �173 1.81 2.01 1.96 2.00 0.01 2.02CUAQAC02 �121 1.84 2.01 1.97 2.00 0.01 2.01CAVXUS �9 1.96 2.01 2.00 2.00 0.07 1.97PATFIA 0 1.86 1.98 1.98 2.00 0.36 1.68Cu2Cl6

2� 55 1.85 2.02 1.97 2.00 0.02 2.03XAMBUI 0 2.00 2.01 2.00 2.00 0.01 2.02YAFZOU 93 1.97 2.01 2.00 2.00 0.01 2.02CITLAT 120 1.96 2.01 2.00 2.00 0.01 2.02

Fig. 7 Frontier natural orbitals of lowest singlet and Ms = 0 triplet states of copper diradicals at the PBE50/cc-pVDZ level of theory. With the exception ofYAFZOU, for other systems only a orbitals of the triplet states are shown, since spatial extent of relevant paired a and b natural orbitals—and theappearance of frontier natural orbitals associated with the lowest singlet and triplet states—does not differ. %n = |na + nb|, with |na � nb| provided inparentheses. %ns and %nt correspond to %n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively.

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distance of the optimized triplet geometries of BISDOW andCITLAT changed very little, with an increase of 0.03 Å and adecrease of 0.02 Å, respectively. We observed a larger change inthe optimized PATFIA complex (0.19 Å), possibly due to theabsence of the ferrocene group in the simplified structure, a topicthat is examined more thoroughly in the next section.

Table 7 summarizes exchange-coupling constants computedfor the two types of geometries, four hybrid functionals andthree ECP/basis combinations. Overall, we find that singlet–triplet gaps computed at a given level of theory changes verylittle with subtle changes in nuclear geometry, indicating thatpossible uncertainties in the experimentally derived geometriescannot account for the discrepancies between the computed andexperimental exchange-couplings. Only for PATFIA, we observe aslightly better agreement between theory and experiment whenusing the optimized structure. Overall, these results justify usingcrystal structures for exchange-couplings calculations.

F. Wavefunction analysis

Energies alone are not sufficient to assess the performance ofelectronic structure methods.11 To gain further insight into

performance of SF-TDDFT, we now investigate the characters ofthe underlying electronic states and compare relevant quanti-ties (such as the number of effectively unpaired electrons)computed by SF-TDDFT and EOM-SF-CCSD. Table 8 presentsdensity-based analysis of the lowest singlet and triplet SF-TDDFT states of all eight complexes at the PBE50/cc-pVDZ levelof theory. Fig. 7 shows the associated frontier natural orbitals.

As expected, for all triplet states the number of effectivelyunpaired electrons as given by nu,nl is exactly 2 (nu gives slightlylarger values, similarly to our previous study or organicdiradicals11). We observe that for the BISDOW and PATFIAcomplexes—for which the recommended SF-TDDFT methodsand EOM-SF-CCSD consistently disagree with experiment—thenumber of unpaired electrons (i.e., values of nu and nu,nl) in theground-state singlets are less than 2, indicative of weakerdiradical character and mixing in of the ionic configurations.These effects are outside of the domain of the HDvV Hamilto-nian. The weakly AF complexes Cu2Cl6

2� and PATFIA, for whichtop-performing SF-TDDFT methods disagree with experimentin both magnitude and sign, also exhibit weaker diradicalcharacter. CAVXUS (a weakly AF complex), XAMBUI (a complexthat exhibits zero exchange-coupling between radical centers),and the ferromagnetic YAFZOU and CITLAT complexes all havesinglet states with the nu,nl values at the ideal diradical value of2. These are also the four complexes for which top-performingDFT functionals and EOM-SF-CCSD agree well with the experi-mental exchange-coupling values.

Spin-contamination appears to be minor: for most com-plexes the deviation from the exact hS2i values are 0.01–0.03.In CAVCUS, the deviation is slightly larger (0.07 for the singletstate). The largest spin-contamination is observed in PATFIAwhere the hS2i of the singlet state is 0.36. Spin-balance is also

Table 9 SF-TDDFT/cc-pVTZ and EOM-SF-CCSD/cc-pVDZ energy split-tings (cm�1) and wave function properties of lowest singlet and Ms = 0triplet spin states of CUAQAC02

Method DEST

nu nu,nl hS2i

Singlet Triplet Singlet Triplet Singlet Triplet

NC-PBE50 �123 1.84 2.01 1.97 2.00 0.01 2.01COL-B5050LYP �148 1.81 2.01 1.96 2.00 0.01 2.01NC-LDA �1420 0.57 2.00 0.47 2.00 0.01 2.01NC-B97 71 2.75 2.82 2.59 2.69 1.07 1.77EOM-SF-CCSD �171 1.79 1.91 1.96 2.00 0.00 2.00

Fig. 8 Frontier a natural orbitals of lowest singlet and Ms = 0 triplet states of CUAQAC02 at the SF-TDDFT/cc-pVTZ and EOM-SF-CCSD/cc-pVDZ levelsof theory. Natural orbital surfaces obtained with three different density functionals are compared. The collinear TDDFT kernel was used with theB5050LYP functional. %n = |na + nb|, with |na � nb| provided in parentheses. %ns and %nt correspond to %n values obtained from the occupancies of the singletand triplet natural orbitals, respectively.

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evidenced by the natural orbitals occupations: |na � nb| valuesof exactly zero for BISDOW, CUAQAC02, Cu2Cl6

2�, XAMBUI,and CITLAT.

The frontier natural orbitals depicted in Fig. 7 are all of thedxy or dyz type, exhibiting weak interaction with p-orbitals ofnearest ligand atoms. Mixing of open- and closed-shell config-urations is symmetry-allowed for CAVXUS, PATFIA, and YAF-ZOU. When spin-traced occupation numbers are exactly equal,both localized and delocalized frontier orbitals provide a validrepresentation of the density. Thus, the apparent localization ofthe frontier orbitals in CAVXUS, PATFIA, and the Ms = 0 tripletstate of CITLAT does not result from the spatial or spinsymmetry breaking. We note that populations associated withfrontier natural orbitals are consistent with the computednumber of the unpaired electrons, meaning that electronicstructure of these compounds maps well into two-electrons-in-two-orbitals problem.

The dyz bonding/anti-bonding pair of frontier orbitalsobserved in CUAQAC02 is unique among the other copperdiradicals considered in our study. The dd* natural orbital

orientation suggests a level of electron–electron interactionthat might preclude the use of the HDvV Hamiltonian formodeling exchange-coupling, and lead to the observed imper-fect diradical character of the singlet ground state.

Using the interesting case of CUAQAC02, Table 9 and Fig. 8illustrate the agreement between the SF-TTDFT/PBE50, SF-TDDFT/B5050LYP, and EOM-SF-CCSD, not just for the relativeenergetics of the states of interest, but their associated densi-ties. While LDA and B97 fail to accurately describe stateenergetics and the degree of diradical character associated withthe singlet (and, in the case of B97, the triplet) states, theshapes of the frontier natural orbitals predicted by LDA areconsistent with EOM-SF-CCSD, PBE50 and B5050LYP. Themain difference between LDA and other methods is in naturaloccupations (and, consequently, the number of effectivelyunpaired electrons): LDA’s singlet state has much strongerclosed-shell character. Shown in the ESI† (Fig. S3), B97 singlyoccupied natural orbitals exhibit a ss-type interaction withequatorial nearest-neighbor oxygen atoms, meaning that thelowest energy triplet state returned by this functional is alto-gether inconsistent with the other four methods. Using stan-dard troubleshooting tools, we were not able to find an SCFsolution with this functional that corresponds to the samebonding pattern as predicted by other functionals. Using awrong reference state leads to large errors in the singlet tripletgaps and large spin-contamination (Table 9). This case illus-trates the utility of using natural orbitals for detecting proble-matic situations with SCF convergence (as one can see fromFig. 2, the inspection of canonical orbitals is not very instructivefor these highly non-Koopmans systems).

Table 10 B5050LYP/cc-pVDZ SF-TDDFT energy splittings (cm�1) andwave function properties of lowest singlet and Ms = 0 triplet spin statesof simplified PATFIA and the full experimental structure, PATFIA +Fe(C5H5)2. The collinear TDDFT kernel was applied

Molecule DEST

nu nu,nl hS2i

Singlet Triplet Singlet Triplet Singlet Triplet

PATFIA �57 1.80 2.01 1.95 2.00 0.02 2.02PATFIA + Fe(C5H5)2 �60 1.80 2.01 1.96 2.00 0.05 1.98

Fig. 9 Frontier a natural orbitals of lowest singlet and Ms = 0 triplet states of simplified PATFIA and the full experimental structure, PATFIA + Fe(C5H5)2.Natural orbitals were obtained at the B5050LYP/cc-pVDZ level of theory with the collinear TDDFT kernel applied. %n = |na + nb|, with |na � nb| provided inparentheses. %ns and %nt correspond to %n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively.

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Finally, we consider the case of PATFIA when the fullexperimental structure that includes the ferrocene group isused in the SF-TDDFT calculation. Results at the B5050LYP/cc-pVDZ level of theory (with the collinear TDDFT kernel) aresummarized for simplified PATFIA and PATFIA + Fe(C5H5)2 inTable 10 and Fig. 9. The observed localization of unpairedelectron density on the copper atoms in PATFIA + (C5H5)2, andthe consistency in the computed values of the singlet–tripletgap, nu and nu,nl, suggest that the ferrocene group indeed doesnot interact with unpaired electrons on the copper centers, andthat the ubiquitous but thus-far unsubstantiated use of thesimplified structure in theoretical calculations of exchange-coupling is valid.

V. Conclusions

We report a benchmark study of eight binuclear copper dir-adicals, focusing on the singlet–triplet energy gaps (which arerelated to exchange-coupling constants within the HDvVmodel) and the associated wave functions. We carefully com-pare SF-TDDFT methods with selected functionals against ahigh-level wave function based method, EOM-SF-CCSD. Inagreement with previous calibration studies of organic di- andtriradicals,11,34,35 we find that hybrid functionals outperformLDA and GGA-type functionals. The non-collinear kernel vastlyimproves the quality of SF-TDDFT results for GGA and LDAfunctionals, with modest improvement for most hybrids. Per-formance of DZ and TZ-quality Dunning’s basis sets are com-parable, while the use of various ECPs has little effect on thecomputed value of DEST, suggesting that computational savingscan be achieved without sacrificing quality of results whenextending the approaches outlined in this work to largerbinuclear (or trinuclear) copper complexes.

Small variations in nuclear geometries—specifically, a slightincrease or decrease in the internuclear separation between theCu atoms that host the unpaired electrons—does not impactthe computed value of DEST, validating the use of experimen-tally determined geometries (or slightly modified symmetrizedgeometries) in calculations of exchange-coupling. Furthermore,the presence of the ferrocene group that is often excluded inab initio studies of exchange-coupling in the PATFIA complexdid not alter the value of DEST or the underlying density,justifying the use of the simplified structure.

We employ density-based analysis to gain insight into theinteraction between the unpaired electrons in copper diradicalsand, in so doing, discover that only five of the eight binuclearcopper complexes exhibit perfect or near-perfect diradicalcharacter in their lowest singlet and triplet states. This findingsuggests that while the HDvV Hamiltonian might be an appro-priate model for exchange-coupling in CAVXUS, PATFIA,Cu2Cl6

2�, XAMBUI, and CITLAT, quality of results should beexpected to suffer for complexes like BISDOW and CUAQAC02,wherein ionic configurations are mixed into the singlet groundstates. For these species, the experimentally derived values of Jare expected to be systematically overestimated. Thus, the

observed discrepancies between theoretical and experimentalvalues for AF molecules arise not due to the limitations oftheoretical methods, but due to the limitations of the HDvVmodel. Spin-contamination appears to be minor in this class ofcompounds.

We have illustrated that the PBE50 or B5050LYP functionals(with a modest choice of basis set and ECP) can be paired withthe SF-TDDFT approach to paint a robust picture of theenergetics of MM candidates, with a level of accuracy compar-able to EOM-SF-CCSD. When combined with the density-basedanalysis of the relevant spin states, one can determine whetheror not the unpaired electrons in a given MM obey the under-lying physics of the HDvV Hamiltonian. Screening of MMcandidates based solely on degree of radical character observedin the low-lying spin states is made possible. Recent work byMayhall and Head-Gordon extends the formalism of SF tocomputation of exchange-coupling constants in complexes withgreater than two radical sites and/or unpaired electrons.18,19

Conflicts of interest

The authors declare the following competing financialinterest(s): A. I. K. is a member of the Board of Directors anda part-owner of Q-Chem, Inc.

Acknowledgements

This work is supported by the Department of Energy throughthe DE-FG02-05ER15685 grant (A. I. K.). A. I. K. thanks Prof.Joachim Sauer for stimulating discussions and his suggestionsregarding the effects of ECPs and basis sets.

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