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Strong electron correlations in cobalt valence tautomers M.X. LaBute, R.V. Kulkarni, R.G. Endres, D.L. Cox Department of Physics, University of California, Davis, CA 95616 (February 1, 2008) We haveexamined cobalt based valence tautomer molecules such as Co(SQ)2(phen) using density functional theory (DFT) and variational configuration interaction (VCI) approaches based upon a model Hamiltonian. Our DFT results extend earlier work by finding a reduced total energy gap (order 0.6 eV) between high temperature and low temperature states when we fully relax 2 the coordinates (relative to experimental ones). Furthermore we demonstrate that charge transfer 0 picturebaseduponformalvalenceargumentssucceedsqualitativelywhilefailing quantitativelydue 0 to strong covalency between the Co 3d orbitals and ligand p orbitals. With the VCI approach, 2 we argue that the high temperature, high spin phase is strongly mixed valent, with about 30% n admixture of Co(III) into the predominantly Co(II) ground state. We confirm this mixed valence a throughafittotheXANESspectra. Moreover,thestrongelectron correlations ofthemixedvalent J phase providean energy lowering of about 0.2-0.3 eV of thehigh temperaturephase relative tothe 6 low temperature one. Finally, we use the domain model to account for the extraordinarily large 1 entropy and enthalpy values associated with thetransition. 2 v I. INTRODUCTION • (1) On the theoretical side, density functional cal- 6 culations [11] have so far provided modest support 0 6 Transition metal complexes with redox active ligands to the above description, yielding stable high spin 0 havebeenthesubjectofextensivestudiesinrecentyears. states for the high temperature geometry, and sta- 1 Of particular interest are the valence tautomers based ble low spin states for the low temperature geome- 1 uponsemiquinoneandcatecholategroupsligatedtotran- try. However, more Co electron charge is found at 0 sitionmetalions[1,2]. These moleculeshaveintrinsicin- low T. / t terestascandidatesformechanicalandmagneticswitch- a • (2) For the Co(SQ) (phen) complex, x-ray absorp- m ing devices activated by temperature, pressure, or irra- 2 tion near edge spectra (XANES) spectra measured diation. They also present coordination environments - for the Co K-edge [10], there is additional spec- d and structural and magnetic conformation reminiscent, n tral weight at high energies (about 2-3 eV above at least, of the local properties in certain allosteric met- o the XANES peaks associated with transitions to alloproteins [3]. c 3d states) which has not yet been accounted for. : Inthispaper,wefocusonthecobaltbasedvalencetau- v i tomers of generic (high temperature) form Co(SQ)2Lp, • (3) The estimated entropy change ∆S at the tran- X where SQ represents a semiquinone ligand complex with sition inferred from fits to the effective moment is r formal charge -1 and spin 1/2, and Lp represents a neu- extraordinarilylarge,oforder 10-15R [5], whichis a tral redox passive counterligand such as phen, py2X difficult to account for from softened vibrational (X=O,Se,S,Te),tetramethylmethylenediamine(tmmda), modes at higher T. Concomitantly, a large en- ortetramethylethylenediamine(tmeda). Whencooledin thalpy of order 0.3 eV/molecule is required. Di- frozen organic solvents such as toluene, or in molecular rect measurements of ∆S [12] for molecular solid solidform(typicallyunderpressure),thesemoleculesun- forms give,on the other hand, a value of about 1.2 dergoa transition,inanarrowtemperatureregime(typ- R, consistent with the spin entropy change, which ically between 100 and 400K), from a high spin, high gives ∆H ≈0.03 eV/molecule. volume form, to a low spin, low volume form [4–9]. The momentchanges fromorder4-6µ athigh T to 2 µ at Motivated by these experimental and theoretical puz- B B low T, while the bond lengths of the inner coordination zles, we have set about to extend the description of the sphere typically decrease by about 10%, about 0.2˚A. cobalt valence tautomers driven in large measure by a Traditionally,itisassumedthatthereisachargetrans- desire to assess the role of strong electron correlations ferfromCototheSQligandsconcomitantwiththetran- in such systems. To this end, we have: (i) performed sition, with the enhanced ligand field splitting arising new, spin polarized, fully relaxeddensity functional the- fromthebondcontractionfavoringlowspinCo(III).This ory calculations. These calculations provide an energy pictureaccountsfortheobservedsusceptibilitydata,and lowering of the high T-low T splitting which partly re- is consistent with observed hyperfine splittings, optical ducesthedisagreementbetweenearlierDFTcalculations absorption. and experiment. A Mulliken population analysis reveals However, there remain some puzzles in this accepted 0.2fewer Co3delectronsforthehigh-T formwhencom- description of the valence tautomers, notably: paredtothe low-T form,inmarkedcontrastto the stan- 1 dard picture. We demonstrate, through an examination TheKohn-Shamequationsaresolvedwiththeexchange- of the projected densities of states, that the arguments correlation calculated using the generalized-gradient- baseduponformaloxidationstatesarequalitatively cor- corrections(GGA) approximation, in the fully ab initio rectwhenoneidentifiestheoverallmolecularorbitalsde- version of Perdew, Burke and Ernzerhof [18]. We use rived from the nominal e symmetry about the Co site. Troullier-Martins norm-conserving pseudopotentials [19] g (ii) We have developed a model Hamiltonian for the ac- intheKleinman-Bylanderform[20]. ForCo,weusenon- tiveelectronicstateswhichwesolvebyavariationalwave linear partial-core corrections to account for exchange function calculation which includes strong electron cor- and correlation effects in the core region [21]. The ba- relationeffectsontheCosite(outsideDFT).Thiscalcu- sis set orbitals are obtained using the method of Sankey lationamountstoconfigurationinteractionwithaphysi- and Niklewski [22], generalized for multiple-ζ and polar- callymotivatedbasissetreductionwhichhasprovenuse- izationfunctions[23]. FortheCoatomthebasissetcon- fulforanalyzingsolidstatesystems. Wefindasignificant sistedofdouble-ζ 3dand4sorbitalswith4ppolarization further reduction in the high-T/low-T energy difference, orbitals. For C,N and O the basis set was double-ζ 2s with stronger electron correlation effects in the high-T and2porbitalswhereasforHitwasdouble-ζ 1sorbitals. phase(whichhas0.3lessCo3delectronsthanthelow-T The initial atomic coordinates are obtained using the phase). Inparticular,athigh-T the highspinCo(II)sig- experimentally determined high-temperature and low- nificantlyadmixeswithahighspinCo(III)stateinwhich temperature geometries for the phen complex. In both anSQ electronantiferromagneticallyscreens the Co mo- cases, following previous work [11], the tertiary butyl ment. Weshowthatthisagoodquantitativeaccountfor groups were removed and replaced with H atoms (at a the K-edge XANES data discussed above. (iii) We show distance ∼ 1˚A from the C atoms). The supercell ap- that a domain model [13,14]can resolve the discrepancy proximation was used and a cubic supercell of dimen- between inferred and direct measurements of ∆S,∆H. sion 38 Bohr was taken to ensure sufficient vacuum be- Specifically, we show that existing experimental data is tween neighboring tautomer molecules. The results ob- inconsistent with the assumption of a random mixture tained were checked for convergence by increasing su- of high-T and low-T forms of the tautomer and is best percell size to 45 Bohr. Conjugate-gradient relaxation explainedbyassumingthemoleculesformclustersofsize was performed on the system, with the experimental ge- ∼30-50. ometries providing the set of initial coordinates, to de- We stress that this description of the high-T phase of termine the minimum energy configurationwithin DFT. the valence tautomers makes them mixed valent in the The atoms were allowed to relax till the force on each sense used in the physics community to describe lan- atom was less than 0.04 eV/˚A. The electronic structure thanide compounds such as CeSn or SmS, in which was determined both for the initial geometry and the fi- 3 the ground state is a quantum superposition of states nalrelaxedgeometryandtheresultsarepresentedinthe withpredominantlytwotothreewelldefinedvalencesfor following. the rare earth ion. This mixed valence is distinct from the multi-site mixed valence of, e.g., the Creutz-Taube molecule. We distinguish this single site mixed valence B. Results from covalence, which is applicable to strong molecular orbital admixture at the single particle level. In this The ground-stateenergeticsobtainedbyus forthe ex- sense, a mixed valent single ion is intermediate between perimental geometries are in agreement with previous fully localized and fully covalent. Such molecular states ab initio calculations [11]. For the high-temperature ge- havealreadybeennoted,e.g.,inthecaseofcerocene[15], ometry, the electronic structure calculation yields high- whereformalvalenceargumentsgeneratetheexpectation spin (∼ 3/2) Co coupled ferromagnetically to spin 1/2 of diamagnetic, tetravalent Ce ions, while observations on each of the ligands. Furthermore we find that yield temperature independent paramagnetism. The Ce the low-spin configuration cannot be stabilized for the ion is mixed valent in this case. high-temperature geometry. In contrast, for the low- We discuss the DFT results in Sec. II, the many body temperature geometry, the low-spin (∼ 0) configuration theory in Sec. III, the domain model in Sec. IV, and is stabilized and the high-spin configuration is unstable. conclusions in Sec. V. In the low temperature geometry there is a net spin 1/2 for the complex which is located on the oxygen ligands. Furthermore, we find a net spin of 0.06 on the Co ion, II. ELECTRONIC STRUCTURE CALCULATIONS consistent with the observed hyperfine splitting of 30G or less [5], reduced by an order of magnitude compared A. Computational Method to, e.g., metallic Co. Thus, our results are in agreement with experimental data which show a transition from a We carry out the spin-polarized electronic structure high-spin state to a low-spin state as the temperature is calculations using the ab initio code SIESTA [16,17]. decreased. 2 Thereasonisofcourse,notcomplicated: theincreased high-spin geometry (7.1) to the low-spin geometry (7.3). 3d-2p hybridization upon contraction increases the hy- This shouldbe contrastedwith the traditionalpicture in bridization induced ligand field splitting on the Co site, which we should expect the net d-charge to be lower by which disfavors Hund’s rule alignment of the spins (the 1 in the low-spin geometry due to charge transfer to the pointchargecontributionincreasesaswell). Theentropy ligands. cost is offset by the energy gain from increased ligand field energy. The ground-state energy difference between the two states without relaxation from the experimental geometry is found to be 1.3 eV, which is in good agreement with the previously obtained value of 1.2 eV. However this energy difference is too high to be consistent with the room-temperaturetransitionobservedinthe system. Experimentally the enthalpy change (∆H) between the two states is inferred from fits to the magnetic moment tobe∼0.3eV/molecule[5]whichismuchlowerthanthe theoretical value (as mentioned, the lone direct specific heat measurement suggests ∆H ≈ 0.03 eV/molecule). However, even this experimentally inferred value for the enthalpy corresponds to an entropy change of ∼ 14 R which is unphysically large. Thus we are faced with a discrepancybetweentheexperimentalresultsandtheen- ergetics obtained using DFT. This discrepancy can be partially resolved by not- ing that the experimentally determined ground-state FIG. 1. Partial densities of states (PDOS) for Co-valence geometry can differ slightly from the DFT ground- tautomers in the vicinity of the first ionization level. (a) Co state geometry. Thus to allow the system to relax 3d-projectedPDOSforthemajorityspininthehightemper- to the DFT ground-state configuration, we performed aturephase. (b)Co3d-projectedPDOSfortheminorityspin conjugate-gradient relaxation starting from the experi- inthehightemperaturephase. (c)Co3d-projectedPDOSfor mental geometries. Relaxing the geometries resulted in majority spin in the high temperature phase. There is neg- a substantial lowering of the energy difference between ligible difference in this case between majority and minority the high-spin and low-spin configurations to 0.6 eV, as spins. (d) PDOSjointly projected toSQ/CAT2p andCo 3d comparedto1.3eVforthe unrelaxedgeometries. Inthis levels in thelow temperaturephase. Notethat theformal eg levels, for example, are strongly covalently admixed between relaxation process, the bond distances between Co and theligands and theCo ion. ligands changes very little, and thus the relaxed geometry might be argued as a better comparison point for A resolution of this discrepancy is reached by analyz- experiment. In a previous study for the unrelaxed ex- ing the transition in terms of molecular orbital theory perimentalgeometries,Adams et al [5]obtaineda lower- and studying the projected density of states (PDOS) for ingoftheground-stateforthehigh-spinconfigurationby thetwogeometries. Intermsofmolecularorbitaltheory, performing a ‘broken-symmetry’ run. However, we have the transition can be summarized as follows. The d or- foundthatthebroken-symmetryconfigurationisnotsta- bitals on the metal are split into the t and e groups. bleanduponrelaxationtheground-stateconfigurationis 2g g Themetale levelsmixwiththeligandσorbitalstoform given by the ferromagneticalignment of the ligand spins g molecularorbitalswhicharebondingandantibondingin to the Co spin. However, the ground-state energy dif- character. The bonding levels are expected to be ligand ference obtained after relaxation is still too high to be dominated and the antibonding levels metal dominated. consistentwiththe lowtransitiontemperature anditin- The t levels don’t mix with the σ orbitals (ignoring, dicates that correlationeffects which are not included in 2g forsimplicity, the mixing with the ligandπ orbitals)and DFT play a significant role in determining the ground- the splitting between the t levels and the antibonding state energies. 2g molecularorbitals(dominatedbymetale levels)iswhat Performing the Mulliken population analysis for the g correspondstothecrystal-fieldsplitting. Inthehightem- relaxedgeometriesleadstoanotherdiscrepancywiththe perature geometry the lower crystal-field splitting stabi- traditionalinterpretationofvalencetautomerism. Inthis lizes the high-spinphase in whichthe antibonding e or- framework,thetransitionisdescribedasaspincrossover g bitals are occupied by two electrons and the t levels phenomenon associated with charge transfer from the 2g have1hole. Inthelowtemperaturegeometrythee lev- metal to the ligands. However the DFT results indi- g els are unoccupied and the t levels are filled and the cate that the net d-charge increases in going from the 2g 3 remainingelectronistransferredtotheligands. Thuswe consider the simplest model that possesses the salient havea crossoverfroma high-spinconfigurationto alow- features that are commonly accepted as essential for spin configuration associated with charge transfer from the combined metal-ligand charge transfer (CT)/spin the metal to the ligands. crossover of the tautomeric interconversion. We model A detailed analysis of the PDOS, shown in Fig. 1, for this with an Anderson Impurity Hamiltonian [25] , in both the high-spin and low-spin geometries reveals that which local electron correlation effects are treated ac- the above picture is indeed accurate, and yet it is also curately on the Co site, but neglected on the ‘metallic’ consistent with the DFT result that the net d-charge is carbon rings. greater in the low T phase. This is explained as fol- We have taken the structure of the Co(phen) complex lows : In the high T phase there is an occupancy of the as the physical basis for our model. We represent the o- e −σantibondingmolecularorbitalsof2electronsinthe quinone ligands by their benzene ring skeletons and the g spin-up channel. In the low T phase one of these elec- oxygen atoms which are nearest neighbor to the metal trons gets transferred to the Co t level and the other site,andneglectoctahedralsymmetrybreakingattheCo 2g is transferred to the ligand SQ level. However this does site. The model system therefore consists of the cobalt nottranslateintoaloweringofchargeontheCoatomby atom, the 3d-levels split by the cubic field arising from one. There is significant covalency between the metal-e the local six-fold coordination of the oxygen atoms, the g levels and the ligand-σ orbitals. So when 2 spin-up elec- nitrogen atoms of the (N-N) complex, and the aromatic tronsaretransferredfromthee −σantibondingorbitals rings of the semiquinones. g thatcorrespondstoalossofjust1.4fortheCoatomand We restrict our model to the electronically active π not2. Ontheotherhandbecausethereisgreateroverlap orbitalsofthesemiquinoneligandsandthe3dt ande 2g g in the low-Tphase there is greatermetalcontributionto orbitals of the Co ion. the occupied e −σ bonding orbitals (which are ligand The LCAO Hamiltonian using second-quantizednota- g dominated). The difference between the metal contribution for the simple model we have just described is tionstothee −σ bondingorbitalis∼0.6(higherinthe g low T phase). So the net metal e occupancy in going H =− t c† c (1) g 0 iαjβ iασ jβσ from high T to low T changes by −1.4+0.6 = −0.8 i.e iXα,jβ σ by less than 1 electron and not 2 as the formal picture indicates. Thisismorethancompensatedbytheincrease where c† and c are, respectively the fermion cre- iασ jβσ in the occupancy of the t2g level by 1 and hence we see ation and annihilation operators of the aforementioned that the net charge increases in the low T phase. orbital set where i and j are site indices, α and β are or- Thus we have seen that the DFT results support the bital symmetry-adapted labels, and σ refers to the spin traditionalqualitativepictureforvalencetautomerismat degeneracy. Thet areon-siteenergiesforiα=jβ and iαjβ the same time clarifying that there is no effective charge areelectron-hoppingintegralsforiα6=jβ.Theon-site3d transferinthesystem. Howeverthereisstillasignificant energies are discrepancy between the calculated and observed energy differences between the two states. In the next section ǫ =ǫ +∆ǫ (2) dγσ dσ LF we will explore the role of correlation effects in reducing this discrepancy. where |hiασ |∆V |dγσi|2 ∆ǫ =− (3) LF ǫ −ǫ (iασ) III. VARIATIONAL MANY BODY THEORY Xiασ dσ p −0.4∆ if γ =x2 −y2,3z2 −r2 + 0 Inthis section,we showthatelectron-electroninterac- (cid:26)0.6∆0 if γ =xy,yz,xz tioneffectscanprovideafurtherreductionof∆H beyond DFT, and that they provide an explanation for hereto- where ∆V is the potential barrier that an electron hop- foreunexplainedspectralintensityinthehigh-TXANES pingfromtheCoatomtooneoftheligandNorOatoms spectra. Moreover, they confirm that like cerocene, the musttunnelthrough,iisthesitesumoverthe4oxygens high-T phaseofthe valence tautomersis mixedvalentin and 2 nitrogen and α refers to the three 2p-orbitals on the sense usedwithin the physicscommunity to describe those sites. γ sums over the degeneracy of irreps of Oh SmS, CeSn3 and other intermetallic compounds. and ǫdσ is the bare d-electron energy while ǫp(iασ) are the ligand atom on-site energies. ∆ is the electrostatic 0 contributionto the ligandfield(LF). ǫ then playsthe dγσ A. Model Hamiltonian role of the on-site 3d-orbital energy in the model. We include electron-electron interactions only for the In order to investigate the possible role for correla- Co 3delectrons. These are certainly the mostsignificant tion effects in the cobalt valence tautomers, we wish to owing to the more localized character of the states, and 4 experience with transition metal oxide solids show this Hamiltonian of Eqns.(1) and (4) within this basis of to be a good starting assumption. Including also the multi-electron wavefunctions. These determinants are spin-orbit coupling, we thus add to H built from the valence orbital set of our model. 0 For the ‘impurity’ model described above, the varia- Hd =Ud nγσnγ′σ′ +JH ~sγσ·~sγ′σ′ tional wavefunction method (VCI) has been shown to γσX6=γ′σ′ γX>γ′ successfully describe transition metal or rare earth ions σ>σ′ embedded both in metals andin metal oxides[28,29]. In Nd thepresentmolecularcontextweregarditasaphysically + ξ(~r )~l ·~s (4) i i i motivated basis set reduction: we have ‘divided’ to con- Xi=1 querbyemphasizingfirstthestrongestsourceofelectron- where U is the direct Coulomb integral and J < 0 is electron correlations, expanding in the hybridization d H the ferromagnetic Hunds’ rule exchange coupling. The about the atomic limit for the Co ion. This said, the occupancy operator is defined as n = d† d where approach yields results which are non-perturbative in V, γσ γσ γσ d† createsanelectroninthe3d-orbitalγ withspinσ. ~l and systematically controlled by two handles: (i) the γσ i spin+orbital degeneracy (the lowest order results within istheorbitalangularmomentumoftheithd-electronand a restricted Hilbert space become exact for large degen- ~s istheone-electronspin. ξ(~r )isthefree-ionparameter. i i eracy), with particle-hole excitations suppressed by in- For parameters, we make use of the tabulated on-site verse powers of the degeneracy relative to the starting energies and also the distance parameterizations of the state, and(ii) the presenceof excitationgapswhichsup- hopping integrals of Harrison [26] . The hopping matrix press contributions from particle-hole excitations about elements are written as linear combinations of these integrals using Slater-Koster theory [27]. U and J are the leading order configurations. d H The composition of the valence electron ground state treated as model parameters. We are currently working will indicate the significance of correlations. The repul- to constrain these parameters from information gained sive nature ofthe Coulombinteractionwilltend to drive through ab-initio methods. However, we may obtain d-electrons off the metal site. This should manifest as reasonable estimates by fitting the results of the model coherent quantum mechanical tunneling between states calculation to X-ray absorption data. This places constraints on the charge transfer gap and also U -U , of nd electrons and states belonging to the nd-1 configu- dc d ration (valence fluctuations). If such effects are insignif- which is the difference between the Coulomb integrals icant, we should expect the type of single-determinant describing the core hole-3d electron and 3d-3d interac- ground state that is treated very well within Hartree- tions, respectively. Fock theory. The high and low temperature expressions of the Theconstructionofthemany-bodybasisisbrieflyout- charge transfer gap are given by lined. ThestatesareSlaterdeterminantsthatwillconsist 3 of tensor products of cobalt states and ligand states, i.e. ∆ht ≃ǫ −0.4∆ − J +6U −ǫ CT d 0 2 H d L | ΓβSM ;L i where Γ is the crystal point-group irrep, S α and (5) β is its degeneracy, S is the total spin, M is the spin S ∆lCtT≃ǫd+0.6∆0+6Ud−ǫL+1.74eV degeneracy. Lα is the ligand state label which is used only if the αth orbital is occupied. Fig. 2 depicts the where ǫ refers to the energy of an electron localized states we consider in the variational ansatz. We restrict L in a ligand orbital and 1.74 eV refers to the hybridiza- ourHilbertspacetothetwolowlyingstatescontaining6 tion contribution to the ligand field splitting in the low- or7Co3delectrons;strongCoulombrepulsionlegislates temperature geometry. against other configurations. Thus, the Hamiltonian is essentially that of a single- Magnetic susceptibility and x-ray absorption data, as impurityAndersonModel,withtheCoionprovidingthe well as first-principles calculations, have suggested the localizedstatesandtheSQligandsplayingtheroleofthe relevance of three closely- lying multiplets for the 3d- ‘metal’. Suchaheuristicmappingtothismodelallowsus electrons: the 4T of the high-spin (h.s.) 3d7 (S = 3), 1 2 to pursue a method of solution that we discuss in some the 2E of low-spin (l.s.) 3d7 (S = 1), and the 1A of 2 1 detail in the next subsection. l.s.-3d6 (S = 0). In the spin sector, we consider only the stretched state of maximal M within the S manifold. S These would be expected to be the lowest-lying due to B. Variational Wave Function the Hund’s rule energy. Wealsoincludeanintermediatespinsectorofstates(S We proceed to solve for the ground state by perform- = 1, 3d6(3T and 3T ) which couple to the h.s. and l.s. 1 2 ingavariationalconfigurationinteraction(VCI)withina states through the spin-orbit interaction and to l.s.-3d7 restrictedbasissetconsistingofthelowestenergysingle- by CT. Slater determinants. This amounts to diagonalizing the We assume that the SQ/CAT eigenstates of H are p 5 adequately treated within the framework of Hu¨ckel theory. This provides the standard approximate method of dealing with π∗ electrons. We employ these methods to find the unoccupied states of the two o-quinone ligands and also their decomposition onto the constituent atomic orbital basis. Once these states have been obtained, Eqns.(1) and (4) may be diagonalizedwithin the truncated basis of tensor product states. # 1 − 3 # 4 − 6 L L FIG. 3. Quantum weights of single determinants in # 13 # 7 − 12 Co(phen) ground state wavefunction. The vertical axis mea- L L sures | αi |2 where αi are the variational coefficients. The model parameters correspond to ∆ht = -0.05 eV and ∆lt CT CT = 0.42 eV. # 14 − 15 The mostsurprisingaspectofthese results is the reso- L nancestateinthehigh-Tphasefeaturinglargeadmixture with the h.s.-3d6 state. There is a concomitant lowering of the energy such that the high-T phase is stabilized FIG. 2. Single-Slater determinants which comprise the with respect to the low-T phase by ∼0.4 eV. This en- many-body basis. States (1-3) are 4T1 of h.s.-3d7, (4-6) be- tanglement of states would not be resolved by a mean- longto5T2 ofh.s.-3d6,(7-12)are3T1,3T2 of(S=1)3d6,(13) is1A1 which isl.s.-3d6,and(14-15) is2Eofl.s.-3d7. Lrepre- field theory like Hartree-Fockor DFT which both utilize a single-determinant groundstate composed of effective sentsthedoubly-degeneratespin-downLUMOofthequinone single-particle molecular orbitals. ligands. An estimate should be made of the contribution of thesecorrelationstothe high-Tgroundstateenergy. We C. Results canseparateout the contributionto the resonantenergy lowering due to covalency by finding the energy differ- The Hamiltonian is diagonalized in both the high-T ence between single determinant h.s.-3d7 and the high-T and low-T geometries using the optimized coordinates ground state of our calculation. This will be the energy from the fully relaxed DFT runs. Fig. 3 is a histogram omitted by DFT calculations. For completeness, we in- depicting how the quantum weight is distributed among clude the possibility ofthe localelectrontunneling to all the determinants of our variational ansatz ground state excited states of the o-quinone ligands. This leads to in both the high andlow temperature phases. The high- an extra energy of ∼0.2 eV that needs to be subtracted T phase suggests a definite non-trivial role for correla- from the enthalpy difference between high- and low-T. tions with a mixed valent state of 69% h.s.-3d7, 28% This further reduces the 0.6 eV obtained from our re- h.s.-3d6 with an electron delocalized over the catechol laxed DFT calculations. The small magnitude of the ligand, and the remainder (∼3%) in l.s.-3d7. The single- gap ∆CT seems to be indicative of the system’s prox- determinant description works quite well in the low-T imity to a charge-transfer instability within our model. phase with 98.5% of the weight residing in the l.s.-3d6 Thisalsojustifiesthehighdegreeofinter-configurational state, and (∼1.5%) residing in the intermediate spin 3d6 admixture in the high-T phase. ∆hCtT is nearly zero , state. The small mixing between low-spin and higher- slightly stabilized on the nd = 7 side. ∆lCtT is larger and spinwithinasingleconfigurationisdue tothespin-orbit has changed signs, so nd = 6 is stable. It will be seen coupling. Thereforeourlow-Tphaseisexactlyconsistent inthe next sectionthatnear-edgeX-rayabsorptiondata with previous results of both ab initio calculations and mandates some constraints as to where the high-T and experiment. low-T phases must reside relative to the instability. 6 D. XANES calculation where ω is the incident photon energy, Eα′ and |α′i are the eigenstates of the Hamiltonian H′ which is the same We find that data taken in X-ray absorption studies system described by H in the presence of the core-hole near the cobaltK-edge [10]corroboratesthe existence of perturbation. E0 and |Ψ0i are the unperturbed ground the h.s.-3d6/3d7 mixed valence in the high-T state. We state energy and wavefunction of our model calculation. first discuss features of the experimental spectra. The Tˆ is the transition operator on-set of the 1s-4p absorption edge is ∼7712.2 eV. The Tˆ =W d† s (9) region of immediate interest to us is the low-intensity γσ σ Xγσ pre-edge interpreted as being composed of quadrupolar- allowed 1s-3d transitions. (Actually, the Co ions lack where we have neglected the orbital angular momentum inversion symmetry, though if restricted to just nearest dependenceinW,thesingle-electron1s-3dtransitionma- neighbors there is an approximate mirror symmetry for- trix element. bidding mixing between 4p and 3d states on site. The We now discuss our results. Fig. 4(a) shows the spec- symmetry breaking by the SQ rings will yield an admix- tra directly resulting from the model we have developed ture of 4p and 3d which is likely to dominate the direct inthe previoussections. The experimentalXANES data quadrupole matrix elements.) Data taken here provides hasbeensuperimposedoverourresults. Fig4(b)depicts a means to probe the nominal valence of the cobalt d- the result of a single-determinant l.s.-3d7 low-T ground orbitals. AsthetemperatureisreducedfromT ,three stateforthepurposesofcomparison;sincethetautomers amb mainstructuresbecomeapparent. Thelowestenergyfea- areontheedgeofthechargetransferinstability,itmight tureisashoulderat7708eVthatlosesspectralweightas be possible by suitable selection of the redox passive lig- the temperature is reduced. In the high-T phase, there andtoaccessthis low-spinCo(II) state. We havenotes- is an intense peak at 7709.6 eV (which shifts up in en- timated the broadening generatedby electronic coupling ergyby∼0.3eVatlowertemperatures)thatgains∼56% to vibrationaldegrees of freedom and other contributing in height. The third structure is a broad shoulder at factors to finite lifetime effects. This limits our compari- ∼7712.3 eV that all but completely disappears at low son to peak positions and relative weights. temperatures. While the first two structures have been We first discuss the high-T result. Fig 4(a) features assignedthe1s-3dt and1s-3de transitions,thislatter four main temperature dependent peaks that may be di- 2g g feature has yet to be identified. rectlyrelatedtotheXASdata. Wefirstnotethatourcal- We have used our model as the basis to calculate a culations reproduce the relative peak heights quite well. theoretical XANES. It has been shown that the core The peak at 7708.03 eV matches the shoulder found in electrons in transition metal and rare earth compounds that region and is the result of a down-spin 1s electron do not efficiently screen the 1s-core hole left behind af- transferring into the t2g. The shoulder feature has been ter the photoemission process. Thus the energies of previously assigned this transition. The peak at 7709.6 the 3d-valence orbitals are significantly renormalized by eVistheconsequenceofthetransferofadown-spinelec- Coulomb interactionwith the core-hole. Treated pertur- tron into an eg level, also consistent with previous as- batively this amountsto adown-shiftin energyforthese signment. The energy distance between these two struc- orbitals. It has also been shown that in order to obtain tures(∼1.60eV)isapproximatelythevalueoftheligand- good agreement with experimental data, a term must field splitting which we would expect on the basis of the be added to the Hamiltonian that accounts for this core assigned transitions. There are two very closely-spaced hole-valence electron interaction [30], peaks at 7710.79and 7710.8eV. These aretransitions of down-spin electrons into the t shell of 3d7 final states. 2g δH =−U (1−s†s ) d† d (6) Goingup inin energybyanamountequalto the ligand- dc σ σ γσ γσ Xσ Xγ field,wefindaclusteroflow-intensitypeaksintheinter- val 7712.32eV - 7712.42eV that correspondto transfers so our Hamiltonian in Eqns.(1) and (4) becomes H′ intothe e levelsofthe 3d7 finalstates. Theselatterfea- g tures can be identified with the spectral weight centered H′ =H +δH (7) about 7712.3 eV. This assignment is consistent with the disappearance of these peaks at low-T. To our knowl- where s† is the 1s-core electron creation operator, U σ dc edge, this is the first attempt to interpret these higher- is the 1s-3d Coulomb integral. The expressionwe use to energysatellitestructures. Thisfeatureisthusuniqueto calculate the X-ray absorption coefficient I for the 1s-3d the type of highly admixed 3d7/3d6 ground-state we are transition is the following which is similar to a weighted proposing for high-T. It can now be seen that the dis- many-body density of states tance between the peaks at ∼7709.6 eV and the cluster centeredabout7712.4eV(andalsothe distancebetween I(ω)= |hα′ |Tˆ |Ψ0i|2 δ(ω−Eα′ +E0) (8) the peak at 7708.03 and the peaks at ∼7710.8 eV) of Xα′ ∼2.5 eV provides a constraint for the value of U -U . dc d 7 metal-ligandCTandmayberelevantforsomespeciesof tautomerwhichonlyexhibitsspin-crossoveronthemetal site. Whilethecentralpeakat7709.9eVagreeswellwith thedata,thereisthefeatureofaugmentedweightatlow energies(∼7708.05eV)atlow-Tratherthanthesuppres- sion seen in experiment. Since the absorptiondata seem to lack the twin peak structure (resulting from Hund’s rule exchange splitting of up- and down- spin electron transitionsto the singly-occupiede shell), this seems to g be anuntenable possibilityforthe low-Tgroundstate,at least with current choices for redox passive ligands. IV. DOMAIN MODEL As noted in the introduction, another problem with the traditional interpretation of valence tautomerism is the extraordinarily large entropy change (10-15 R) associated with the transition. However, it should be noted that this entropy change is inferred only indirectly from measurements of the magnetic susceptibility and opti- FIG.4. (a)ModelXANEScalculationshownwithexperi- cal absorption. On the the other hand, measurements mentalresultsfor∆ht =-0.05eV,∆lt =0.416eV,E = ofthe specific heatas a function of temperature done by CT CT shift -1.05 eV, and U -U = 2.36 eV. Experimental curves show Abakumovetal[12],whichconstituteamoredirectmea- dc d change in XANES with temperature. The solid line corre- surementoftheentropy,giveamuchlowerentropyvalue sponds to the high-T phase and the (— ··· —) line is the (∼ 1 R). This discrepancy leads us to examine the set of low-T phase. (b) Low-T XANES calculation resulting from assumptions used to infer the entropy change from the l.s.-3d7 groundstate. ∆lt = -0.004 eV, E = 0.737 eV, CT shift susceptibilitydata. Thesecanbe summarizedasfollows. and U -U = 2.43 eV dc d At any given temperature T, the system is taken to be a two component mixture of the high-spin and low- We have introduced an energy shift between the high- spin forms of the molecule. The fraction of atoms in and low-T spectra, E . The pragmatic intent is to fit shift the high-spinform(f)is determinedthermodynamically the data. The underlying physical reasoning is twofold. by minimizing the molar Gibbs free energy (G) which is We have used identical model parameters for both the given by high-Tandlow-Tcalculations,allowingonlythedistance dependenceofthehoppingintegralstochange. Wewould G=fG +(1−f)G −RT[fln(f)+(1−f)ln(1−f)] hs ls expect some renormalization of some of these parame- (10) ters due to effects like screening. More significantly, we haveomittedsometermsfromoursimplemodelHamilto- wherethelasttermistheentropyofmixingforarandom nianthat would be quite different in the two geometries, two-componentmixture. Minimizing the above equation e.g. cohesionenergies,off-siteCoulombinteractions,etc. with respect to f then gives us the functional form for Presumably, these would all have to be incorporated in thevariationoff withT whichusedtoinfertheenthalpy such a shift. The low-T spectra in Fig 4(a) consists of 4 (∆H) and entropy (∆S) change from the experimental main peak structures concentratedin the region7709.66 data. - 7710.8 eV. These are all associated with transferring a However the above derivation makes the assumption 1s-electron into the empty e shell of the l.s.-3d6 state. g that the system can be described as a random two- The heights and location of these agree quite well with component mixture. If there is clustering present in the the single broadened peak of experiment. The weights system, then the above equations have to be modified. have not been altered relative to the high-T phase, giv- The simplest generalization which accounts for this is ing very good consistency with experiment. the domain model [13,14]. In this model, clustering is Fig 4(b) depicts a low-T XANES resulting from a l.s.- accounted for by considering domains with n number of 3d7 groundstate. This regime should be investigated molecules per domain (n is like a mean-field parameter sincethismanifoldofstatesliesincloseenergeticproxim- corresponding to the average size of the domain, with itytothel.s.-3d6singletstate. Asshown,thislow-Tstate n=1 correspondingto the randomtwo-componentmix- can be accessed with only very slight tuning of the pa- ture). In this case the entropy of mixing is now given rameters. Suchaninterconversionwouldnecessarilylack by 8 S =−R/n [fln(f)+(1−f)ln(1−f)] (11) tautomer literature [12]. We have also shown that DFT mix provideslittle evidence for net chargetransfer offthe Co Correspondingly the variation of f with temperature is ion to the SQ ligands (indeed, the 3d weight increases modestly in the low-T phase!). On the other hand, a f =1/[1+en∆RH(1/T−1/Tc)] (12) carefulexaminationofthePDOSshowsthatthequalita- tivepicture emergingfromthe formalvalence arguments The key point is that the values inferred from exper- is stillcorrect: the orbitalswith the largestadmixtureof iment have used the above form with n = 1 giving un- 3d−e symmetry are depopulated in going through the physicallylargevaluesfor∆H and∆S. Howeverifthere g transition. is clustering (n>1) then the inferred value of ∆H (and Next,weshowedthatavariationalconfigurationinter- correspondingly ∆S) will be lowered by a factor of n action approach could produce a further lowering of the leading to more reasonable values. This leads us to the high-Tenthalpyrelativetothelow-Tentropybyproduc- question: Howcanweinferwhattheappropriatevalueof inga30%admixtureofCo(II)andCo(III)configuration. n is? According to the domain model the answer can be Atthelowestvariationalorder,weestimatean0.2-0.3eV obtained from specific heat measurements which should lowering of energy due to electron correlation effects be- indicate apeak inthe specific heat(C )centeredaround p yondDFT,whichbringsuswithinanorderofmagnitude thetransitiontemperatureT . The‘jump’inthespecific c ofthedirectlyobservedenthalpy. Thishigh-Tstateisin- heat is given by C (T ) − 1(C (T ) + C (T )) where p c 2 ls c hs c trinsically “CI”: no unitary transformation in the single C (T )(C (T )) is obtained by linear extrapolation at ls c hs c particle orbital space can remove the multiconfiguration T =T ofthelow(high)temperaturespecificheatvalues. c character. It is, in fact, a mixed valent state associated Thenumberofmoleculesperdomainnisthenrelatedto with the Co ion and the more extended and weakly cor- this jump by related states of the SQ rings. We have also shown that 4 R T2 1 thishigh-Tstate,whenproperaccountingismadeforthe n= (∆H)c2[Cp(Tc)− 2(Cls(Tc)+Chs(Tc))] (13) core-hole interaction of the 3d electrons, can explain the previously observed spectral weight shift well above the Itshouldbenotedthatthespecificheatmeasurements dominant 1s-3dpeak for high temperatures. The weight willprovideadirectmeasurementofof∆S (andthereby isascribedtotransitionstotheadmixedhighspinCo(III) ∆H)andwithinthedomainmodelthiscanbeusedtoes- configuration. timaten. Forthetautomers,onlyonesetofexperiments Finally, we have proposed a domain model to account done so far have measured the specific heat. From the for the large ∆S,∆H values inferred from susceptibil- data given by Abakumov et al for the bpy complex, we ity and optical absorption data (as compared to direct make the following estimates : The ‘jump’ in C corre- measurement). In this view, the transition corresponds p spondsto∼150J/molKwhichgivesachangeinentropy to a broadened first order phase transition of molecular of∆S ∼1.2R. Usingthesevaluesandtheequationforn clusters of order 20-60. In the dilute solvent, it might in the domain model we estimate n≈50. Thus the data be possible to observe such clustering with elastic light for the specific heat shows evidence for clustering in the scatteringexperiments. Althoughtheheatcapacitymea- tautomers and is inconsistent with the original assump- surements in the interesting temperature range are dif- tion of a random two-component mixture. The latter ficult, we certainly would encourage further experiments assumption leads to unphysically high inferred values of on tautomers. ∆H and ∆S and we propose that measurements of the We hope to apply this combined DFT and VCI ap- specific heat can be used to resolve this discrepancy and proach to other tautomer systems and to analyses of to self-consistently check for the applicability of the do- thecloselyrelatedphenomenaobservedinthe prosthetic main model to the system. complexes of metalloproteins and metalloenzymes. Acknowledgements. We would like to thank P. Or- dejón, E. Artacho, D. Sańchez-Portal and J. M. Soler V. CONCLUSIONS for providing us with their ab initio code SIESTA. We acknowledge useful discussions with R.R.P. Singh, A. In conclusion, we have extended the theory of cobalt Shreve, and R. Weht. The work at Davis was supported valence tautomers to resolve some of the puzzles sur- bytheU.S.DepartmentofEnergy,OfficeofBasicEnergy rounding various data. 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