THE JOURNAL OF CHEMICAL PHYSICS 134, 064304 (2011)

Quantum chemical assessment of the binding energy of CuO+

Elixabete Rezabal,1 Jürgen Gauss,2 Jon M. Matxain,3 Robert Berger,4,5 MartinDiefenbach,1 and Max C. Holthausen1,a)1Institut für Anorganische und Analytische Chemie, Johann Wolfgang Goethe-Universität Frankfurt, D-60438Frankfurt am Main, Germany2Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany3Kimika Fakultatea, Euskal Herriko Unibertsitatea and Donostia International Physics Center (DIPC), P. K.1072, 20080 Donostia, Euskadi, Spain4Clemens-Schöpf Institut für Organische Chemie und Biochemie, Technische Universität Darmstadt, D-64287Darmstadt, Germany5Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, D-60438 Frankfurtam Main, Germany

(Received 2 November 2010; accepted 20 December 2010; published online 8 February 2011)

We present a detailed theoretical investigation on the dissociation energy of CuO+, carried out bymeans of coupled cluster theory, the multireference averaged coupled pair functional (MR-ACPF)approach, diffusion quantum Monte Carlo (DMC), and density functional theory (DFT). At the re-spective extrapolated basis set limits, most post-Hartree–Fock approaches agree within a narrowerror margin on a De value of 26.0 kcal mol!1 [coupled-cluster singles and doubles level augmentedby perturbative triples corrections, CCSD(T)], 25.8 kcal mol!1 (CCSDTQ via the high accuracyextrapolated ab initio thermochemistry protocol), and 25.6 kcal mol!1 (DMC), which is encourag-ing in view of the disaccording data published thus far. The configuration-interaction based MR-ACPF expansion, which includes single and double excitations only, gives a slightly lower value of24.1 kcal mol!1, indicating that large basis sets and triple excitation patterns are necessary ingredi-ents for a quantitative assessment. Our best estimate for D0 at the CCSD(T) level is 25.3 kcal mol!1,which is somewhat lower than the latest experimental value (D0 = 31.1 ± 2.8 kcal mol!1; reportedby the Armentrout group) [Int. J. Mass Spectrom. 182/183, 99 (1999)]. These highly correlated meth-ods are, however, computationally very demanding, and the results are therefore supplemented withthose of more affordable DFT calculations. If used in combination with moderately-sized basis sets,the M05 and M06 hybrid functionals turn out to be promising candidates for studies on much largersystems containing a [CuO]+ core. © 2011 American Institute of Physics. [doi:10.1063/1.3537797]

I. INTRODUCTION

Copper, along with iron, dominates the field of bi-ological oxygen chemistry and plays important roles inhomogeneous and heterogeneous catalysis.1 The impressivesubstrate specificity and selectivity of the oxidation reactionsmediated by copper-based metalloenzymes spurred vastresearch efforts in past decades, devoted to the understandingof structure–activity relationships associated with bindingand activation of dioxygen. Related bioinorganic researchhas led to a number of small synthetic copper complexes,which contain a [Cu2O2]2+ core as a key structural motifto mimic the biocatalytically active paragons. Experimentalstudies along these lines have impressively demonstratedthat dinuclear copper cores embedded in small polydentatedonor ligand frameworks can indeed mediate a selectivehydroxylation of aliphatic and aromatic C–H bonds.2 Inrecent years, the interplay between experiment and theory hasprovided detailed insight into the elementary steps underlyingthese oxygenation reactions.3–8

A substantial body of literature has canvassed the par-ticular chemical reactivities of various [Cu2O2]2+ core iso-

a)Author to whom correspondence should be addressed. Electronic mail:max.holthausen@chemie.uni-frankfurt.de.

mers, and the dominant coordination modes discussed inthe bioinorganic context are µ-!2:!2-peroxo, bis-µ-oxo, andtrans-µ-!1:!1-peroxo.9 It has been shown that the specificchemical reactivity of each of these cores is controlled bya variety of factors, including effects of the ligand environ-ment, solvent molecules, and counterions surrounding the ac-tive site. This situation leads to a highly intricate scenario forquantum chemistry already for small model systems,10, 11 ag-gravated by the substantial size of relevant bioinorganic sys-tems, let alone biological systems.

More recent research efforts focus on mononuclear[Cu–O2]+ complexes,12 and of particular current interest arerelated bioinorganic complexes that contain a mononuclear[CuO]+ core embodied in a donor ligand framework. Suchspecies have tentatively been suggested as the catalyticallyactive part of enzymes13–18 and synthetic systems.13, 19–24 Inrecent years several theoretical studies were performed toevaluate the reactivity of potential [CuO]+ intermediates,and subsequent experimental work inferred their role in theobserved reactivity.22, 25–27 This species, however, has es-caped experimental detection for decades, and only recentlyits existence has been proven in a combined theoretical andmass spectrometric study, which also demonstrated its highhydroxylation potential against C–H bonds.28

0021-9606/2011/134(6)/064304/13/$30.00 © 2011 American Institute of Physics134, 064304-1

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064304-2 Rezabal et al. J. Chem. Phys. 134, 064304 (2011)

Several decades before, the chemistry of bare, i.e., un-supported transition metal (TM)-oxo species has moved intothe focus of gas phase chemists, who studied these systems ingreat detail with highly elaborate matrix isolation and massspectrometric techniques.29–31 These techniques have enabledstudies on intrinsic reactivity patterns of isolated TM-oxocores under well defined conditions, i.e., without the com-plicating influence of ligand effects, counterions, and solventmolecules. In principle, quantum chemistry is ideally suitedto complement experimental gas phase studies, by providingreliable predictions of the thermodynamics, reaction barriers,and structures of many of these species by high level post-Hartree–Fock (post-HF) or density functional theory (DFT)methods.

The mass spectrometric evaluation of the thermochem-istry of bare MO+ species began in the 1960s,32, 33 and in-creasingly refined results have subsequently been obtainedemploying ever more sophisticated techniques. Of particularinterest in the present context is a series of studies performedon the binding energies of MO+, including the naked CuO+

ion.34, 35 Among all 3d TM-oxo species, CuO+ exhibits thelowest metal-oxo binding energy.36 This has been interpretedas a consequence of the fact that the formation of covalentbonds to Cu+, with its closed-shell 1S(3d10) ground state,involves admixture of the first excited 3D(3d 94s1) state tosome extent.37 Consequently, any chemical affinity to formthe Cu+–O bond would be compensated by the substantialenergetic costs of this promotion (64.8 kcal mol!1).38, 39 Ob-viously, the resulting inherent lability of the Cu+–O bond hasimplications for the unique oxygenation reactivity of relatedactive sites based on this building block.40 The dramatic scat-ter of experimental and theoretical gas phase data published asof today on the properties of CuO+, however, indicates an ir-ritating degree of uncertainty and impairs the relevance of anysuch interpretations to the bioinorganic chemistry of [CuO]+

species. To put the theoretical view on the parent diatomicion onto a solid basis, this paper aims at a rigorous quantumchemical assessment of CuO+ in its electronic ground statebased on systematically converged high-level ab initio results.

II. ELECTRONIC STRUCTURE OF THE CuO+ CATION

Bond formation in CuO+ involves interactions of thecopper 3d and 4s orbitals with the oxygen 2s and 2p orbitals.As discussed in detail elsewhere,41, 42 hybridization of themetal 4s and 3d" orbitals effectively improves overlap andelectrostatic components of bond formation to the oxygenatom. The resulting (4s+3d)" and (4s!3d)" hybrid or-bitals form bonding (2" ; cf. Fig. 1) and antibonding (3" ")molecular orbitals (MOs) with the oxygen 2p" orbital,respectively. The 4" " orbital results from an antibondingcombination of the (4s!3d)" hybrid with the oxygen2p" orbital. The 1" valence orbital essentially consists ofthe 2s orbital of oxygen and is thus mainly nonbonding.Two degenerate sets of # -type orbitals (i.e., the bonding1#x and 1#y and the antibonding 2#"

x and 2#"y orbitals)

result from interactions between the corresponding 3d#

and 2p# orbitals of copper and oxygen, respectively. Thetwo metal 3d$ orbitals do not find an interacting orbital

FIG. 1. Qualitative molecular orbital (MO) scheme for the valence space ofCuO+ at r (CuO) = 1.756 Å (3%! ground state). MOs shown were obtainedby Hartree–Fock (HF) (left) and B3LYP (right) restricted open-shell calcu-lations (aug-cc-pVTZ(-PP) basis, isosurfaces plotted at 0.05 a!3/2

0 ; enumer-ation of orbitals includes only valence orbitals).

counterpart of $ symmetry at the oxygen atom and thusremain nonbonding (1$ set). Distributing the 16 valenceelectrons among these orbitals gives rise to a 3%! groundstate. The corresponding high-spin ground state is typicalfor the late 3d MO+ ions, and its electronic configuration(1" )2 (2" )2 (1#x )2 (1#y)2 (1$)4 (3" ")2 (2#"

x )1 (2#"y )1 (4" ")0

has also been identified for the isoelectronic neutral NiOcomplex.43 This configuration can be classified as a # -typeaxial biradical, and the close analogy to the 3%!

g high-spinground state of dioxygen has been inferred.44 However,because of the predominantly nonbonding nature of 1" ,and because both, the bonding 2" and the antibonding 3" "

orbitals are doubly occupied, the " -orbital space does notcontribute to bonding. The only bonding contribution arisesin the # -orbital space, i.e., the two bonding 1# orbitalsare each doubly occupied, while the two antibonding 2#"

orbitals are half filled. Thus, a formal bond order of 1results, and in its 3%! ground state CuO+ represents amolecule with a single # -bond without a " -bond. Severalelectronically excited states have been identified in formerab initio work, and the first excited 3& state, resulting froma (1" )2(2" )2(1# )4(1$)4(3" ")1(2#")3(4" ")0 configuration, iswell separated45, 46 from the electronic ground state, at leastin the Franck–Condon region. The two excited singlet states1' and 1%+ resulting from the (2#")2 configuration lie yetfurther above in energy.

III. PREVIOUS WORKS ON CuO+

Unfortunately, not much effort has yet been spent on thecharacterization of the spectroscopic properties of copper ox-ide ions. The first determination of the CuO+ binding energygoes back to gas phase bracketing experiments of Kappes

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064304-3 Binding energy of CuO+ J. Chem. Phys. 134, 064304 (2011)

TABLE I. CuO+ post-HF and DFT data available in literature.

D0 (kcal mol!1) re (Å) (e (cm-1) Ref.CCSD(T)a 20.5 1.835 369 49CCSD(T)b 23.4 50CCSD(T)c 23.1 50CCSD(T)d 19.4 51SA-CASSCFe 7.6 1.721 45MRSDCIe 17.6 1.678 802 45MRSDCI(+Q)e 22.0 1.682 45MRMPe 36.9 1.645 45CIPSIf 12.9g 2.132 258 52CASPT2c 26.5 50MRDCI-ACPFh 23.4i 1.794 600 46B3LYPa 32.1 1.790 526 49B3LYPj 29.3 1.811 487 36B3LYPk 31.2 1.809 494 51B3LYPe 34.2 1.765 45B3LYPl 33.1 1.775 50BP86a 48.0 1.749 625 49BLYPj 43.8 1.783 571 36BLYPe 49.5 1.745 45BPW91a 45.0 1.755 613 49BPW91j 40.1 1.777 553 36BOPa 45.7 1.753 45

aBasis set: Wachters + f for Cu and aug-cc-pVTZ for O.bQuadruple-) quality basis set for Cu, triple-) for O, and single point calculations onB3LYPl equilibrium structures.cQuadruple-) quality basis set, single point calculations on B3LYPl equilibrium struc-tures.dQuadruple-) quality basis set for Cu, aug-cc-pVTZ for O, and single point calculationson B3LYP k equilibrium structures.eDouble-) quality basis set for Cu and cc-pVTZ for O.fMixed triple and double-) quality basis set for Cu and double-) for O.g De value.hWachters+1s+2f basis set for Cu and triple-) quality basis set for O.iSee text.j6-311+G* basis set.kQuadruple-) quality basis set for Cu and triple–) quality basis set for O.lTZV(2P) basis set.

and Staley,47 who suggested 25 kcal mol!1 # D(Cu+–O)# 40 kcal mol!1. In 1990, the group of Armen-trout put forward the first direct determination of theCuO+ binding energy based on sophisticated guidedion beam mass spectrometric measurements (D0(Cu+–O)= 37.4 ± 3.5 kcal mol!1).34 The same group reinvestigatedthe system later and reported a corrected, lower value ofD0(Cu+–O) = 31.1 ± 2.8 kcal mol!1, which, to the best ofour knowledge, is the most reliable thermodynamic informa-tion on this species as of today.48

The first quantum chemical study relevant in this con-text was published by Hippe and Peyerimhoff.46 Exploringthe electronic and spectroscopic properties of neutral CuOby means of MRDCI-ACPF calculations, the study also re-ports re and (e of CuO+ (see Table I). Even though the dis-sociation energy was not explicitly mentioned, D0(Cu+–O)$ 23.4 kcal mol!1 can be estimated based on the computedadiabatic ionization energy (9.15 eV) of CuO, combined withthe computed De value for CuO (2.44 eV), the harmonic es-timate of the zero-point vibrational energy (0.04 eV) and theexperimental ionization energy of atomic copper (7.726 eV).When instead the experimental CuO dissociation energyD0 (2.75 eV) cited in Ref. 53 is employed, one arrives at an

estimate of D0 $ 30.5 kcal mol!1 from the data provided byHippe and Peyerimhoff (see Ref. 53).

The baton was passed to coupled cluster theory bySodupe et al.,51 who reported a D0 value of 19.4 kcal mol!1

as part of a study on the interaction of first row transitionmetal atoms with CO2. Their result represents the lower limitof binding energies reported in subsequent related coupledcluster studies by others,49, 50 all underestimating today’sbest experimental binding energy by at least 8 kcal mol!1.These data are compiled in Table I, together with results ob-tained applying multiconfigurational and density functionalapproaches. Here, also the multireference configurationinteraction (MRCI) approaches severely underestimatethe binding energy, whereas the results of Nakao et al.45

obtained by multireference perturbation theory (MRMP),overshoot experiment. The best agreement with experimentwas obtained by Delabie and Pierloot,50 who scantly mentiona preliminary—to the best of our knowledge so far yetunpublished—CASPT2 result of D0 = 26.5 kcal mol!1.Details of the calculations were not reported, however, whichimpedes any comprehensive discussion at this point.

Additional work based on DFT has been performed, andthe B3LYP hybrid functional has been most successful to re-produce experiment in cases, although the results for D0 rangebetween 29.3 and 37.6 kcal mol!1 depending on the basisset used.36, 45, 49–51, 54 Application of nonhybrid functionals re-sulted in systematically larger binding energies between 40.1and 49.5 kcal mol!1.36, 45, 49, 51

Clearly, the inconsistency of the molecular equilibriumdata compiled in Table I is quite unsatisfactory, and in particu-lar the scatter among the post-HF results is disturbing: Disso-ciation energies range from 12.9 to 36.9 kcal mol!1, harmonicvibrational wavenumbers are between 258 and 802 cm!1, andbond lengths span from 1.645 to 2.132 Å. The uncertaintyspreads especially among the multireference data, where boththe active spaces and the reference wavefunctions employedplay a decisive role. The current status thus highlights theneed for a rigorous evaluation of the molecular properties ofCuO+ in order to identify the sources of error and, at the sametime, to put forward a reliable reference value for this pivotalspecies to copper chemistry.

IV. METHODS

Post-HF calculations have been performed with the MOL-PRO program package,55 the CFOUR program package,56

and the MRCC program of Kállay57, 58 interfaced to CFOUR

and MOLPRO. The coupled cluster (CC) method used withMOLPRO was always the CC singles and doubles approx-imation augmented by a perturbative treatment of tripleexcitations [CCSD(T)].59, 60 For open-shell systems, the spin-unrestricted ansatz was employed based on a restrictedopen-shell Hartree–Fock (ROHF) reference determinant[ROHF-CCSD(T)].60 Equilibrium structures and harmonicvibrational frequencies were obtained at this level employ-ing numerical techniques to evaluate gradient and Hessianelements. The hierarchy of Dunning’s correlation-consistentcc-pVXZ and aug-cc-pVXZ basis sets has been used forthe oxygen atom61 and the corresponding cc-pVXZ-PP and

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064304-4 Rezabal et al. J. Chem. Phys. 134, 064304 (2011)

aug-cc-pVXZ-PP basis sets62 were used for copper in combi-nation with the relativistic pseudopotential of Figgen et al.63

The 3d electrons of Cu+ and the 2s and 2p electrons of Owere correlated. Additional calculations were performed withthe cc-pwCVXZ(-PP) basis set series to evaluate valence plus(outer) core correlation effects. All electrons not implicitlyincluded in the pseudopotential have been correlated in thesecases.

Higher levels in the coupled cluster hierarchy,namely coupled cluster up to (iterative) triple excita-tions (CCSDT)64–67 and quadruple excitations (CCSDTQ)58

were employed as implemented in the MRCC program,which was invoked via the interface of the MOLPRO programpackage for calculations of atomic excitation energies. Onlyvalence electrons were correlated when basis sets of thecc-pVXZ(-PP) and aug-cc-pVXZ(-PP) series were used forO(Cu). The MRCC program was also employed in combina-tion with the CFOUR program package for calculations withinthe high accuracy extrapolated ab initio thermochemistry(HEAT) protocol68–70 for the atomic excitation and molecularatomization energies as outlined below.

A variant of the HEAT protocol68–70 was employedto obtain high-level benchmark data for the CuO+ dis-sociation energy within an all-electron framework thataccounts for relativistic effects in a perturbative scalar-relativistic framework. Calculations were carried out withthe all-electron basis sets from Dunning’s cc-pVXZ and cc-pwCVXZ hierarchies.61, 71, 72 The HF complete basis set esti-mates (ECBS

SCF ) were obtained via

ESCF,X = ECBSSCF + A e!BX, (1)

based on calculations with X = 3, 4, 5. The extrapolatedCCSD(T) energy ('ECBS

CCSD(T)) was obtained via Eq. (2) (seebelow) based on calculations with X = 4, 5. The 'ECCSDT

increment, which accounts for the effect of an iterative in-stead of a perturbative treatment of triple excitations, wascomputed with the cc-pVTZ basis set. Higher level correla-tion effects ('EHLC) were estimated at the CCSDTQ levelusing the cc-pVDZ basis at O and a dzp basis73 at Cu. Scalar-relativistic corrections 'Erel were computed via second or-der direct perturbation theory (DPT2)74 using an uncontractedcc-pwCVQZ basis set (cc-pwCVQZ-unc, note that the addi-tional steep s and p functions in the cc-pwCVQZ set needto be skipped in order to avoid linear dependencies in thebasis). Spin-orbit coupling contributions and diagonal Born–Oppenheimer corrections were neglected.

Multireference configuration interaction (MRCI) typecalculations were carried out at the size extensive averagedcoupled pair functional (MR-ACPF) level employing sin-gle and double excitations in conjunction with the aug-cc-pVXZ(-PP) basis sets (X = 3, 4, and 5). The active spacein the preceding complete active space SCF (CASSCF) cal-culations comprised the 3d 4s 4d (Cu) and 2p (O) orbitalsresulting in approx. 1.25 % 106 configuration state functions(CSFs). The remaining six occupied (semi)core 3s 3p (Cu)and 1s 2s (O) orbitals were optimized for each state but werekept doubly occupied. This choice of orbitals in the activespace sums to 14 electrons in 14 orbitals and yields a well-balanced set, which covers valence correlation of the " /" "

and # /#" space as well as radial correlation of the d-electrons(involving the 3d/4d orbitals) of copper; the latter has beenidentified as the most important contribution to correlationenergy in late first-row transition metal atoms75 (viz. “3ddouble shell effect”).76, 77 To arrive at a still manageablenumber of CSFs in the subsequent MR-ACPF treatment, atruncated reference wavefunction was used by selection ofconfigurations based on their coefficients in the CASSCFexpansion. Note, however, that this truncation/selectionscheme forces the CASSCF to use CSFs as the N -electron ba-sis instead of the generally more efficient Slater determinants,which, in turn, considerably increases the computationaleffort in the CASSCF part—in this particular case by a fac-tor of about 10–20. Restrictions to the reference wavefunc-tion directly influence the results, which manifests, for exam-ple, in slight discontinuities in the potential energy curve. Inorder to achieve a balanced N -electron basis at a still fea-sible computational effort, we chose a fairly low threshold,such that in the MR-ACPF calculations all reference config-urations with a coefficient larger than 0.005 were considered(truncated reference CI). Common selection schemes employselection thresholds of approx. 0.01–0.1. As in the CASSCFpart, no excitations were allowed from the six previously de-fined (semi)core orbitals. The 2s (O) orbital, was, however, in-cluded in the correlation treatment. This results in approx. 100reference orbital occupation patterns (configurations) givingrise to about 1200 reference CSFs and, depending on the sizeof the basis set, in 2–6 % 107 internally contracted configu-rations. Equilibrium bond lengths, harmonic, and anharmonicvibrational frequencies were obtained from polynomial least-squares fits through single-point scans around the minimumstructure.

The fixed node78, 79 diffusion quantum Monte Carlo(DMC) calculations were performed using the CASINO

program80 at r (Cu–O) = 1.756 Å (CCSD(T)/aug-cc-pV&Z(-PP) value), and for infinitely separated O and Cu+. Forthe DMC calculations, the trial wavefunctions used in thiswork are written as a product of a Slater determinant anda recently developed Jastrow factor,81 which is the sumof homogeneous, isotropic electron–electron terms, isotropicelectron–nucleus terms centered on the nuclei, and isotropicelectron–electron–nucleus terms, also centered on the nu-clei. Two methods have been used to generate the deter-minantal part of the wavefunction, namely, unrestricted HFand B3LYP in combination with the TZVP basis of Schäfer,Huber, and Ahlrichs73, 82 for O and Cu. The basis set on cop-per was supplemented with a diffuse s function (with an expo-nent 0.33 times that of the most diffuse s function on the orig-inal set), two sets of p functions optimized by Wachters83 forthe excited states, one set of diffuse pure angular momentumd functions (optimized by Hay),84 and three sets of uncon-tracted pure angular momentum f functions, including bothtight and diffuse exponents, as recommended by Ragavachariand Trucks.85 We denote this basis set as TZVP+G(3df,2p).Relativistic effects were accounted for by adding the scalarrelativistic corrections 'Erel from DPT2 calculations (seeabove) to the results of the all-electron DMC calculations. Fora detailed description of the DMC approach the reader is re-ferred to Refs. 86–89.

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064304-5 Binding energy of CuO+ J. Chem. Phys. 134, 064304 (2011)

DFT calculations were performed employing the func-tionals PBE,90 BP86,91, 92 BLYP,93, 94 TPSS,95 TPSSh,95

B3LYP,92, 93, 96, 97, BHandHLYP, M05,98 and M05-2X,99 asimplemented in the GAUSSIAN03 program.100 Additionalcalculations have been performed with the GAUSSIAN09version101 of the program employing the M06-L,102 M06,and M06-2X functionals103 and the recently devised B2PLYPdouble hybrid functional.104

Post-HF energies obtained with the (aug-)cc-pVXZ(-PP) basis sets (X = 4, 5) have been extrapolated tothe basis set limit (denoted CBS) using the ansatz of Halkieret al.105, 106 for the correlation energy (see Eq. (2), X being thecardinal number) and Jensen’s107 exponential extrapolationscheme modified by Martin108 for HF and DFT energies [seeEq. (3)]; here L denotes the highest angular momentum quan-tum number of the corresponding basis set. The latter was alsoemployed within the MR-ACPF approach to extrapolate theCASSCF reference energies to the basis set limit, because thebasis set convergence of the uncorrelated CASSCF energies(lacking dynamical correlation) is expected to be comparableto that of HF energies (see, e.g., Refs. 109 and 110). To avoidinconsistencies in the extrapolation procedure, which mightarise from irregularities in the truncated reference potential(see also related figure in the supplemental material111), thereference energies of the full CASSCF expansion were used.The (dynamic) correlation energies corresponding to thedifference between those CASSCF reference energies andthe MR-ACPF total energies were extrapolated according toEq. (2):

EX = ECBScorr + A X!3. (2)

EL = ECBSSCF ! A (L + 1) e!9

'L . (3)

Following previous successful work,108, 112–114 an empiricalmixed exponential/Gaussian scheme suggested by Petersonet al.112 [see Eq. (4)] was used for corresponding X = 3, 4,and 5 extrapolations of bond lengths and harmonic vibrationalfrequencies,

PX = PCBS + A e!(X!1) + B e!(X!1)2. (4)

Additional DFT calculations were carried out with basissets of limited size [LANL2DZ/D95V,115, 116 6-31+G(d,p),117

SVP,118 TZVP,73, 82 and def2-TZVPP (Ref. 119)] for Cu andO to assess the performance of production level methodscommonly applied in reactivity studies on extended cop-per/oxygen complexes.

V. RESULTS

A. 3D ! 1S Atomic excitation energies

As already alluded to above, the formation of cova-lent bonds to Cu+ involves hybridization of the metal 4sand 3d" orbitals, which corresponds to an admixture of the3D(3d 94s1) excited state to some extent. Because computa-tion of the dissociation energy of CuO+ only involves thelow-spin 1S atomic ground state in the dissociation limit,whereas the molecular bonding mechanism partially involves

TABLE II. Coupled cluster results for the atomic 3d94s1 ( 3d10 excitationenergy, 'ESD, as a function of the basis set quality.

'ESD(kcal mol!1)

X 2 3 4 5 CBS '

CCSD cc-pVXZ-PP 56.53 57.41 58.33 58.70 59.12aug-cc-pVXZ-PP 59.25 57.54 58.47 58.75 59.08 !0.04cc-pwCVXZ-PP 54.32 56.80 58.50 59.28 60.14 1.02best estimatea 60.10

CCSD(T) cc-pVXZ-PP 59.09 60.72 62.28 62.86 63.51aug-cc-pVXZ-PP 62.49 61.43 62.62 62.98 63.39 !0.12cc-pwCVXZ-PP 57.19 60.18 62.33 63.32 64.39 0.88best estimateb 64.27

CCSDT cc-pVXZ-PP 58.64 60.16 61.56 62.06 62.62aug-cc-pVXZ-PP 62.04 60.74 61.83 62.15 62.52 !0.10cc-pwCVXZ-PP 56.76 59.69 61.72 62.64 63.63 1.01best estimatec 63.53

CCSDTQ cc-pVXZ-PP 58.78 60.25 61.66 (62.76)aug-cc-pVXZ-PP 62.13 60.85cc-pwCVXZ-PP 56.91best estimated 63.64

HEAT 65.40Expe 64.76f

aCCSD/CBS + 'aug + 'cv.bCCSD(T)/CBS + 'aug + 'cv.cCCSDT/CBS + 'aug + 'cv.dCCSDT/best + 'CCSDTQ,QZ + ''CCSDTQ,aug,TZ + ''CCSDTQ,cv,DZ, with the ''

corrections being determined as the difference between the CCSDTQ aug/cv incrementfor a given X -tuple zeta basis set and the corresponding CCSDT aug/cv increment.eRef. 38 and 39.fWeighted avg., calculated as

!"JmaxJmin

(2J + 1) E J

#/(2L + 1)(2S + 1).

the 3d94s1 configuration, errors in the computed 3D ( 1Satomic excitation energies do not cancel out and might bepresent to some extent also in the molecular calculation. It ishence obvious that an accurate description of the 3D(3d 94s1)( 1S(d10) atomic state splitting is of fundamental importancefor any quantum chemical method to properly describe thebonding in CuO+. Bauschlicher et al.41 suggested in fact thatthe error in the atomic separations can be taken as a measureof the accuracy of the molecular calculations within the post-HF framework.

CCSD, CCSD(T), CCSDT, and CCSDTQ results for theatomic 3d94s1( 3d10 excitation energy of the Cu+ ion as afunction of the basis set quality are listed in Table II. Increaseof the basis set size leads in all cases to convergence to the ex-trapolated basis set limit from below. This may to some extentbe rationalized as a consequence of the significantly differentsizes of 3d and 4s orbitals in the Cu+ ion: a ratio of )r4s*/)r3d *= 2.82 results from numerical Hartree–Fock calculations,41

and the average interelectronic distance is much smaller inthe 3d10 configuration than in the 3d 94s1 configuration. Con-sequently, insufficient recovery of electron correlation energywill destabilize the energy of the 1S ground state relative to theenergy of the 3D excited state, resulting in an underestimationof the state energy separation. While smooth convergence tothe basis set limit is generally observed for the three basis setseries used here, augmentation of the basis with diffuse func-tions leads to faster convergence, as already noted by Petersonand coworkers; this was interpreted as a consequence of sd

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064304-6 Rezabal et al. J. Chem. Phys. 134, 064304 (2011)

correlation effects which profit from addition of diffuse f-typefunctions in the aug-cc-pVXZ-PP basis set series.62 The onlyirregularity in Table II is observed for the augmented double-) basis, which provides a larger value of the excitation energythan the augmented triple-) basis.

Not unexpectedly, augmentation of the basis by diffusefunctions has only a minute influence on the extrapolated ba-sis set limits for CCSD and CCSD(T) in Cu+. Inclusion of3sp core-valence correlation effects, however, increases thecomputed excitation energy by 1.02 and 0.88 kcal mol!1 atthe CCSD and CCSD(T) levels of theory, respectively. Usingthe effects of diffuse functions ('aug) and core-valence corre-lation ('cv) obtained at the respective basis set limits as ad-ditive corrections to the extrapolated CCSD(T)/cc-pV&Z-PPresult leads to an excitation energy of 64.28 kcal mol!1. Thisbest estimate agrees well with the J -averaged experimentalvalue of 64.76 kcal mol!1.39 The remaining difference is ofthe same order of magnitude as spin-orbit coupling effects(Figgen et al. suggest a SO correction of 0.25 kcal mol!1),63

which are not considered in the present calculations.Inclusion of higher level electron correlation effects

within CCSDT and CCSDTQ leads to excitation energiesdeviating somewhat more from experiment than the cor-responding CCSD(T) values. Comparison of CCSD andCCSD(T)/CCSDT results illustrates the importance of per-turbative or iterative triple excitations in line with the qual-itative discussion of the correlation problem stated above:the best estimate for the CCSD basis set limit underesti-mates the experimental value by more than 4 kcal mol!1.The estimate based upon the application of the HEAT pro-

tocol for the excitation energy gives 65.40 kcal mol!1 andthus overshoots the experimental excitation energy by a sim-ilar order of magnitude as the best estimate for CCSDTQdoes. The difference between the formally equivalent CCS-DTQ and HEAT results mainly stems from the use of basissets of different quality and use of a pseudopotential versusan all-electron approach. With respect to the pseudopoten-tial employed for copper we note that the inherent error be-tween the numerical multiconfiguration Dirac-HF-Coulomball-electron (MCDHF) reference value and the correspondingtwo-component pseudopotential result for the 3D ( 1S exci-tation amounts to !0.6 kcal mol!1.120

We conclude at this point that the atomic excitation en-ergy problem constitutes a non-negligible source of error forcoupled cluster theory (4 kcal mol!1 at CCSD, about 1 to2 kcal mol!1 for CCSDT and CCSDTQ treatments) as part ofthe problem to determine an accurate dissociation energy forCuO+. Large basis sets are equally indispensable for a solidassessment of this property. The best estimate for CCSD(T)shows nearly quantitative agreement with experiment and isused as a reference value further below.

B. Molecular equilibrium properties

Table III lists CCSD(T) and CCSD results for dissoci-ation energies, bond lengths, and harmonic vibrational fre-quencies as a function of the basis set size. Commencingour discussion with computed dissociation energies, we notethat improvement of the basis set quality from double-) toquintuple-) quality leads to a systematic increase of the com-

TABLE III. CCSD(T) and CCSD results for CuO+: dissociation energies De and D0, equilibrium bond length re and harmonic vibrational wavenumber (eas a function of the basis set quality.

X 2 3 4 5 CBS '

CCSD(T) De(kcal mol!1) cc-pVXZ(-PP) 23.65 23.82 25.11 25.92 27.11aug-cc-pVXZ(-PP) 23.72 25.68 26.54 26.65 26.86 !0.25cc-pwCVXZ(-PP) 23.84 24.68 25.36 25.75 26.25 !0.86best estimatea 26.00

re(Å) cc-pVXZ(-PP) 1.839 1.780 1.766 1.759 1.755aug-cc-pVXZ(-PP) 1.812 1.768 1.759 1.757 1.757 0.002cc-pwCVXZ(-PP) 1.836 1.770 1.766 1.765 1.764 0.009best estimatea 1.766

(e(cm!1) cc-pVXZ(-PP) 357.3 453.8 483.0 494.6 501.3aug-cc-pVXZ(-PP) 402.7 482.7 495.5 503.0 507.4 6.1cc-pwCVXZ(-PP) 451.1 474.2 485.0 487.1 488.3 !13.0best estimatea 494.4

D0(kcal mol!1) best estimateb 25.29

CCSD De(kcal mol!1) cc-pVXZ(-PP) 20.49 19.25 20.00 20.55 20.84aug-cc-pVXZ(-PP) 19.99 20.66 21.15 21.16 20.48 !0.36cc-pwCVXZ(-PP) 20.63 19.86 20.29 20.57 20.95 0.11best estimatec 20.59

re(Å) cc-pVXZ(-PP) 1.939 1.873 1.873 1.846 1.830aug-cc-pVXZ(-PP) 1.901 1.851 1.841 1.841 1.841 0.011cc-pwCVXZ(-PP) 1.935 1.854 1.849 1.847 1.845 0.015best estimatec 1.856

aCCSD(T)/CBS + 'aug + 'cv.b De CCSD(T)/CBS ! ZPE ; ZPE = 1

2 (e = 0.707 kcal mol!1.cCCSD/CBS + 'aug + 'cv.

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064304-7 Binding energy of CuO+ J. Chem. Phys. 134, 064304 (2011)

puted De for all three basis set series employed for bothmethods. As already observed for the computed atomic statesplittings, addition of diffuse functions in the aug-cc-pVXZ(-PP) series leads to improved convergence to the basisset limit, but the extrapolated dissociation energies agree withthe cc-pV&Z(-PP) results within 0.25 and 0.40 kcal mol!1

at the CCSD(T) and CCSD levels of theory, respectively. In-clusion of core-valence correlation reduces De at the basis setlimit by !0.86 kcal mol!1 for CCSD(T) calculations, whereasa minute increase by 0.1 kcal mol!1 is obtained at the CCSDlevel. Adding 'aug and 'cv to the extrapolated CCSD(T)/cc-pV&Z(-PP) result we arrive at De = 26.0 kcal mol!1 as ourbest estimate. Systematically lower dissociation energies areobtained at the CCSD level for all basis sets employed, anda significantly lower best estimate of De = 20.6 kcal mol!1

results at this level of theory, underestimating the correspond-ing CCSD(T) result by 5.4 kcal mol!1. To qualitatively as-sess the influence of limitations of the atomic basis sets usedin our calculations we evaluated the basis set superpositionerror (BSSE) at the CCSD(T) level by means of the coun-terpoise procedure of Boys and Bernardi122 (see correspond-ing table in the supplemental material111). While a significantBSSE of about 4 kcal mol!1 is obtained for the double-)quality cc-pVDZ(-PP) and cc-pwCVDZ(-PP) basis sets, theaddition of diffuse functions in the aug-cc-pVDZ(-PP) basisset reduces this error to 1.8 kcal mol!1. Increasing the ba-sis set size leads to a systematic reduction of the BSSE, eachincrement of the basis set cardinal number effectively halv-ing the computed BSSE at least. A residual BSSE of about0.2 kcal mol!1 remains employing basis sets of quintuple-)quality. Note that in general, the counterpoise procedure over-estimates the BSSE,123, 124 because according to Pauli’s exclu-sion principle, in the molecular dimer only the space of theunoccupied orbitals of the partner fragment is available to thecorresponding monomer, while the counterpoise correction tothe monomer employs the full basis set of its partner.

To assess the importance of higher level electron correla-tion effects, these were accounted for within the HEAT pro-tocol. As shown in Table IV, contributions to the dissociationenergy from iterative triples and iterative quadruples are ofsimilar magnitude but opposite sign. The resulting cancella-tion renders the computationally much cheaper CCSD(T) ap-proach particularly valuable. Note that at the HF/cc-pV&Zlevel, the CuO+ molecule is unbound at the equilibrium bondlength of the HEAT protocol. Relativistic corrections obtainedat the DPT2 level are a factor of 3 larger than the CCSDT and

TABLE IV. HEAT calculation contributions to the De of CuO+.

De (kcal mol!1) re (Å)ECBS

HF $ HF/cc-pV&Z !5.51'ECBS

CCSD(T) $ ' CCSD(T)/cc-pwCV&Z 26.08'ECCSDT $ ' CCSDT/cc-pVTZ 1.61'EHLC $ ' CCSDTQ/dzp !1.43'Erel $ DPT2 CCSD(T)/cc-pwCVQZ-unc 5.05Sum of contributions 25.80 1.759a

aObtained at the HEAT level as described in Ref. 121 with the relativistic correctionconsidered as an additive shift in the equilibrium distance, re = re(HEAT, nonrelativis-tic) + 'r (DPT2, CCSD(T)/cc-pwCVQZ-unc).

CCSDTQ contributions and thus of special importance for anaccurate estimate of the dissociation energy. The final HEATresult for the De value of CuO+ amounts to 25.8 kcal mol!1,in excellent agreement with the CCSD(T) estimate obtainedwithin the pseudopotential framework.

Computed equilibrium bond lengths re show regular con-vergence with the basis set size. Complete basis set limitswere obtained via Eq. (4). A significant shortening of re is ob-served when going from double-) to triple-) basis sets, whilefurther improvements to quintuple-) quality lead to rapid con-vergence to the respective basis set limits with little variance.As for the computed dissociation energies, extrapolation tothe cc-pV&Z(-PP) and aug-cc-pV&Z(-PP) basis set limitsleads to nearly identical results (1.755 Å and 1.757 Å, re-spectively). Inclusion of core-valence correlation effects leadsto a slightly larger re of 1.764 Å at the basis set limit. As-suming incremental behavior of the effect of diffuse func-tions and core-valence correlation, we obtain a bond lengthof re = 1.766 Å, which we suggest as our best estimate forthis property within the pseudopotential method. This valueagrees well with the HEAT result of re = 1.759 Å, which hasbeen obtained by correcting the corresponding nonrelativisticvalue (as described in Ref. 121) by a relativistic shift obtainedat the DPT2 level in CCSD(T)/cc-pwCVQZ-unc calculations.Systematically larger bond lengths are obtained from struc-ture optimizations performed at the CCSD level of theory. Theindividual results by and large follow the same trends as ob-served for CCSD(T) optimized structures upon basis set im-provement, but a notably longer bond length of re = 1.856 Åis obtained as our best estimate at this level of theory.

The trends described above also hold for the computedharmonic vibrational frequencies, which systematically con-verge to the basis set limits from below. Inclusion of diffusefunctions increases the harmonic vibrational wavenumber by6.1 cm!1 and core-valence correlation effects lead to a de-crease by !13.0 cm!1 at the respective basis set limits. Ourresulting best estimate is (e = 494.4 cm!1. Based on thisvalue we obtain a vibrational zero-point energy correction of0.7 kcal mol!1, and we suggest D0 = 25.3 kcal mol!1 asour best estimate for the bond dissociation energy obtainedat the extrapolated CCSD(T)/aug-cc-pwCV&Z(-PP) basis setlimit. Anharmonic contributions to D0 are of the order of0.01 kcal mol!1 (see Table V) and are thus insignificant.

TABLE V. CBS extrapolation of MR-ACPF/aug-cc-pVXZ(-PP) data for theatomic 3D ( 1S excitation energy of Cu+, 'ESD, and for the molecularproperties of CuO+, as a function of the basis set quality.

X 3 4 5 CBS'ESD(kcal mol!1) 61.52 62.65 62.98 63.32re(Å) 1.761 1.752 1.751 1.750(e(cm!1) 412.6 432.8 433.0 433.0(exe(cm!1) !15.2 7.9 10.8 12.4De(kcal mol!1) 23.50 24.09 24.11 24.13D0(kcal mol!1)a 22.90 23.48 23.49 23.52

aCalculated including the first anharmonicity constant as D0 = De ! 12 (e + 1

4 (exe,where the harmonic and anharmonic contributions to D0 amount to 1

2 (e

= 0.62 kcal mol!1 and 14 (exe = 0.01 kcal mol!1.

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064304-8 Rezabal et al. J. Chem. Phys. 134, 064304 (2011)

FIG. 2. Schematic valence-MO diagram of the active space of CuO+ at re= 1.763 Å, obtained from CASSCF (left) and MR-ACPF (right) calculations(aug-cc-pVTZ(-PP) basis, isosurfaces plotted at 0.05 a!3/2

0 ). The orbitals areordered according to the CASSCF eigenvalues, and the respective occupationnumbers are given.

C. Multireference averaged coupled pair functional(MR-ACPF) calculations

The scatter of the hitherto published data implies a size-able degree of uncertainty, particularly among the multirefer-ence results (see Table I). Although in transition metal con-taining molecules near-degeneracy effects can play a role,and, in certain cases, only multiconfiguration treatments mayprovide a realistic picture of such complexes, an appre-ciable proportion of multireference character is not to beexpected here, considering the analogy to 3%!

g O2.44 Inour coupled cluster calculations, we accordingly obtain T1-diagnostics125 ranging from 0.043 [cc-pVDZ(-PP)] to 0.055[cc-pV5Z(-PP)], which imply a more or less tempered case inthis regard. In puzzling contrast, however, the ground statewavefunction of CuO+ was reported to exhibit remarkablemultireference character in the work of Nakao et al.45

To assess whether the single reference approach usedin coupled cluster theory is still appropriate, MRCI-typecalculations at the size extensive averaged coupled pairfunctional (MR-ACPF) level were performed. Next to sizeextensivity, a well-balanced active space including fullvalence correlation as well as a suitable reference wave-function are crucial parameters for an adequate descriptionof the electronic structure and of the binding energy. Wethus included the "/" " and #/#" valence space as wellas the 3d and 4d orbitals in the active space, the latter torecover radial correlation of the almost fully occupied 3dorbitals on the metal atom (see Fig. 2). The MR-ACPF pro-cedure used in our work is based on a truncated CAS ref-erence (see Computational Details). The squared norm ofthe selected configurations in these MR-ACPF expansions is

0.997, and the coefficient C0 of the reference wavefunctionamounts to 0.959, which demonstrates that the chosen ap-proach is well-adjusted. In the full (nontruncated) CASSCFwavefunction of the 3%! ground state, the leading root repre-senting the (1" )2 (2" )2 (1# )4 (1$)4 (3" ")2 (2#")2 (4" ")0 con-figuration has a coefficient of 0.970, while the next lowerroots have coefficients of less than 0.06 (see supplemen-tal material111); these roots involve 2" + 4" " [essentially2p" (O) + 4s (Cu)] and 1# + 2#" [essentially 3d# (Cu)+ 2p# (O)] excitations. This picture also holds for the MR-ACPF wavefunction, where the reference coefficient of theleading root amounts to 0.930 and that of the next lower rootsto less than 0.062. Thus, use of a single reference approachrepresenting the leading root appears well justified for a de-scription of the 3%! ground state wavefunction.

We note that Nakao et al.45 found in their MR-CISD cal-culations rather different configuration weights of 0.77 for theleading root and 0.41 for the next lower roots, which cor-respond to a degenerate 1' state, and thus suggest a strongmultireference character of the 3%! ground state wavefunc-tion. This can, however, most likely be attributed to the in-herently multiconfigurational nature of the reference wave-function, which was averaged over the manifold of the lowesttwelve or so triplet and singlet states, and is thus not opti-mized to the actual ground state. More importantly, the activespace chosen in this work is substantially smaller, because itonly includes the 2s 2p (O) and 3d 4s (Cu) orbitals, which arealready singly or doubly occupied in the 3D(Cu+) asymptote(see Sec. II). This is prone to induce deficiencies especiallyin the # and $ space of the molecular state of the referencewavefunction, an indicator of which is also the non-negligiblepopulation of the 3#" and 2$ orbitals in our calculationswith an enhanced active space (Fig. 2). In our state-specificCASSCF calculation with a substantially more flexible activespace, this alarming multiconfigurational effect accordinglyvanishes.

The computed atomic 3D ( 1S excitation energiesof Cu+ (entry 'ESD in Table V) compare favorably wellwith the extrapolated CCSD(T)/aug-cc-pV&Z(-PP) values(Table II). The data for X = 5 are identical, while those ofX = 3 and 4 converge faster to the CBS limit at the MR-ACPF/aug-cc-pV&Z(-PP) level of theory as compared tothe corresponding CCSD(T) values, which causes a slightlylower CBS entry of 63.32 kcal mol!1. Although the MR-ACPF approach includes single and double excitations only,the results mirror the CCSD(T) rather than the CCSD data.Similarly, the molecular properties calculated at the MR-ACPF/aug-cc-pVX&Z(-PP) level of theory are closer to theCCSD(T) results. At 1.750 Å, the equilibrium bond lengthreflects the extrapolated CCSD(T)/aug-cc-pV&Z(-PP) resultof 1.756 Å fairly closely (cf. re = 1.841 Å at the extrapo-lated CCSD/aug-cc-pV&Z(-PP) limit). The dissociation en-ergy amounts to 24.1 kcal mol!1 at the extrapolated MR-ACPF/aug-cc-pV&Z(-PP) level of theory, which falls in be-tween the CCSD(T) and CCSD aug-cc-pV&Z(-PP) extrap-olated results. This lower De of 24.1 kcal mol!1 as com-pared to the CCSD(T) aug-cc-pV&Z(-PP) extrapolated value(26.9 kcal mol!1) is also consistent with the lower CBS valuefor (e in Table V (433 cm!1 vs. 507 cm!1 in Table III).

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064304-9 Binding energy of CuO+ J. Chem. Phys. 134, 064304 (2011)

TABLE VI. De of Cu+–O calculated by DMC at re = 1.756 Åa.

De(kcal mol!1)DMC (HF) 19.32b ± 0.03DMC (B3LYP) 25.63b ± 0.03

aCCSD(T)/aug-cc-pV&Z(-PP) value.bIncluding 'Erel(DPT2) = 5.0 kcal mol!1.

Including zero-point vibrational contributions, we obtain D0

= 23.5 kcal mol!1. Generally, according to Table V, the re-sults are more or less resemblant from X = 4 upward and im-ply convergence at the MR-ACPF/aug-cc-pVQZ(-PP) level.Pursuant to the above, both, the CCSD(T) approach and theMR-ACPF ansatz give descriptions of similar quality. That isto say, the static correlation energy from the multireferenceCI-type approach with single and double excitations is recov-ered to some extent by the perturbative triple excitations inthe single reference CCSD(T) numbers. The CCSD(T) ap-proach, however, gains 2–3 kcal mol!1 in binding energywith respect to the MR-ACPF ansatz. Obviously, triple excita-tions are essential for a more complete recovery of correlationenergy.

D. Diffusion quantum Monte Carlo (DMC) calculations

DMC has proven to be a very accurate tool for the studyof energetic parameters in TM–O species126–129 depending onthe nodal surface of a given Slater determinant, with the po-tential of performing as accurately as highly electron correlat-ing post-HF methods.130 In this work DMC calculations areperformed based on HF and DFT (B3LYP) wavefunctions,and the results are compared to the CC values.

The DMC results in Table VI are calculated with a time-step of 0.005 Eh/¯. Scalar relativistic corrections are takenfrom DPT2 calculations and are included as an additive in-crement (see computational details). A large discrepancy be-tween both levels is noted, which arises from the differencesbetween Hartree–Fock molecular orbitals (HF-MO) and theKohn–Sham MOs (KS-MO, see Fig. 1). The HF-MOs showhighly localized orbitals forming the Cu+–O bond, while theKS-MO set shows a similar contribution of the Cu and theO, in particular in the # and #" sets. Obviously, the latterprovides a better description of the covalent bond betweenthe Cu and O, as documented in the literature for the isoelec-tronic NiO,43, 131 and other TM oxides.127 As already pointedout, the accuracy of the DMC calculation highly relies onthe correct description of the nodal surface of the wavefunc-tion. The highly localized HF # orbitals thus yield a low De.A proper description of orbital nodal structures by KS-MOson the other hand leads to De = 25.63 kcal mol!1, in ex-cellent agreement with HEAT and CCSD(T) data (0.2 and0.5 kcal mol!1 deviation, respectively) presented above.

E. DFT calculations

Carefully calibrated DFT constitutes a powerful toolfor studies on TM species at low computational cost. It is,however, well known that the performance of DFT strongly

TABLE VII. CBS extrapolated HF and DFT/aug-cc-pV&Z(-PP) first exci-tation energy of the Cu+ cation, 'ESD, together with dissociation energies,D0, the Cu–O bond length, re, and the harmonic vibrational wavenumbers,(e.

'ESD D0 re(Å) (e(cm!1)(kcal mol!1) (kcal mol!1)

HF 14.1 10.1 2.201 223PBE 56.3 54.4 1.723 657BP86 56.1 54.2 1.722 661BLYP 52.1 53.1 1.736 643B3LYP 49.1 36.8 1.750 588BHandHLYP 40.1 20.9 1.878 337TPSS 56.4 48.6 1.731 647TPSSh 54.9 40.4 1.740 618M05 80.0 25.2 1.814 485M05-2X 63.3 21.5 1.918 354M06-L 52.8 46.8 1.727 623M06 64.1 30.9 1.767 543M06-2X 59.0 21.7 1.963 318B2PLYP 61.4 32.3 1.759 563Exp. 64.8a 31.1 ± 2.8b

Best est.c 64.3 25.3 1.766 494

aRefs. 38 and 39.bRef. 48.cPresent work (CCSD(T)/CBS).

depends on the nature of systems being studied and usuallyshows large variations with the exchange-correlation func-tional and the particular basis set chosen. This is also clearlyseen in Table I. Consequently, the use of DFT requires a care-ful assessment of the influence of both, the density functional(DF) and the basis set for each new class of chemical systemsunder study. In this section we thus present an assessment ofsome promising DFs, which would be typical choices in re-activity studies on copper systems. We first discuss the per-formance of a variety of functionals at the respective basis setlimits, thereby evaluating the intrinsic functional performancefree of basis set effects (see Table VII; HF results are also in-cluded for comparison). For the sake of brevity, only CBSvalues obtained from the augmented basis sets will be dis-cussed here. The full set of data is provided in the supplemen-tal material.111 In addition, we combined selected functionalswith basis sets of limited size to identify a computationallyaffordable, yet reasonably accurate level of DFT, which couldeasily be applied in studies of much larger related complexesin the bioinorganic framework.

Commencing our discussion with the computed3D ( 1S atomic excitation we note a variation between40.1 and 80.0 kcal mol!1 (cf. Table VII), which illustratesfundamental problems of DFT to properly describe atomicexcitation energies.132 Pure functionals like BP86, PBE, andTPSS underestimate the experimental excitation energy byabout 8 kcal mol!1, while the BLYP and M06-L functionalsfurther increase this error by 4 kcal mol!1. It has already beendocumented132 that the additional Fock exchange contribu-tion in hybrid functionals leads to a further underestimationof the reference value due to the well-known bias of HFtheory towards high-spin over low-spin states. This can beclearly observed with the B3LYP and BHandHLYP function-als employing 20% and 50% of Fock exchange, respectively.

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064304-10 Rezabal et al. J. Chem. Phys. 134, 064304 (2011)

Here, the respective deviations from experiment amount to15.7 and 24.7 kcal mol!1, artificially stabilizing the excitedstate almost linearly with the percentage of Fock exchangeadmixed to the parent BLYP GGA. A similar effect of theFock exchange contribution is observed in the performance ofthe Minnesota DF series. The excitation energy decreases by16.7 kcal mol!1 from M05 to M05-2X (28% and 56% Fockexchange, respectively) and by 5.1 kcal mol!1 from M06to M06-2X (27% and 54% Fock exchange, respectively);an irregularity to this trend occurs, however, for the nonhy-brid functional M06-L, where 'ESD is lowest in the M06series. TPSSh drops only by 1.5 kcal mol!1 as compared toTPSS, due to the low percentage of Fock exchange (10%).B2PLYP, bearing 53% Fock exchange and 27% perturbativecorrelation, shows a fair performance but obviously, itsparametrization leaves room for improvement. Overall,B2PLYP, M05-2X, and M06 perform quite satisfactorily,deviating only by !3.4, !1.5, and !0.7 kcal mol!1 from theexperimental excitation energy.

Due to the lack of experimental spectroscopic parametersfor CuO+, the CCSD(T)/CBS data will be taken as the refer-ence for comparison in the following. Excellent agreementin bond lengths is obtained for M06, B2PLYP, and B3LYP,which all deviate by less than 0.02 Å. Those DFs with largeramounts of Fock exchange, i.e., BHandHLYP, M06-2X andM05-2X, overestimate re by more than 0.1 Å, while the purefunctionals slightly underestimate re. Accordingly, the har-monic vibrational wavenumber is overestimated by the purefunctionals by more than 100 cm!1, and the inclusion of Fockexchange tends to lower this value, reaching the best agree-ment for B3LYP, M05, M06, and B2PLYP. Functionals withlarger portions of Fock exchange underestimate the referenceharmonic vibrational wavenumber by around 150 cm!1.

The computed dissociation energies show an enormousspread of 40 kcal mol!1, similar to that seen for the excita-tion energies (see Table VII), with a comparable influence ofthe components of the DFs. In this case, the pure function-als PBE, BP86, BLYP, TPSS, and M06-L largely overesti-mate the CCSD(T)/CBS reference of D0 = 25.3 kcal mol!1,by about 25 kcal mol!1. As expected from the pure HF re-sults (cf. Table VII), the inclusion of Fock exchange low-ers the binding energy, leading therefore to more accuratebut still too large values for TPSSh, B2PLYP, B3LYP, M05,and M06, and to a slight underestimation of dissociation en-ergies for BHandHLYP, M05-2X, and M06-2X. M05 per-forms best with a D0 almost identical to our best estimate(!0.1 kcal mol!1 deviation)—however, as noted above, thecomputed atomic excitation energy largely overestimates theexperimental value.

The quite satisfactory dissociation energies obtained byBHandHLYP, M05-2X, and M06-2X are accompanied by un-acceptable deviations for bond lengths and harmonic vibra-tional wavenumbers (see above). Instead, those DFs bearinga moderate amount of Fock exchange contribution show areasonable and balanced overall performance; consequently,B3LYP, M05, M06, and B2PLYP were additionally used incombination with popular basis sets of limited size, namelyLANL2DZ, 6-31G+(d,p), SVP, TZVP, and def2-TZVPP, ig-noring relativity.

TABLE VIII. Dissociation energies D0 together with the Cu–O bond lengthre and the harmonic vibrational wavenumbers (e for selected DF/basis setcombinations.

D0(kcal mol!1) re(Å) (e(cm!1)B3LYP SVP 32.7 1.835 472

TZVP 29.6 1.818 4826-31+G(d,p) 32.5 1.785 534LANL2DZ 24.9 1.902 405def2-TZVPP 30.4 1.804 497

B2PLYP SVP 24.0 1.863 381TZVP 22.6 1.819 4206-31+G(d,p) 26.2 1.779 494LANL2DZ 18.0 1.982 303def2-TZVPP 24.6 1.793 457

M05 SVP 25.5 1.884 438TZVP 21.0 1.875 3946-31+G(d,p) 24.4 1.837 448LANL2DZ 16.8 2.028 245def2-TZVPP 21.6 1.860 406

M06 SVP 27.6 1.858 413TZVP 25.5 1.843 4326-31+G(d,p) 24.3 1.816 462LANL2DZ 21.4 1.940 355def2-TZVPP 26.3 1.827 446

Overall, computed bond lengths and harmonic vibra-tional frequencies are in reasonable agreement with those cal-culated at the CCSD(T) level (see Table VIII), with the excep-tion of LANL2DZ data. The dissociation energy, instead, ismore sensitive to the choice of the DF and basis set. The vari-ation of the latter leads to differences of up to 4 kcal mol!1

for each of the functionals, variation of the functionals leadsto variations in the range of 8 kcal mol!1. The use of basis setsof limited size leads in cases to better agreement with the ref-erence data compared to the corresponding DFT/CBS results.Obviously, error compensation is in effect.

In summary, for those levels of DFT which accu-rately reproduce the spectroscopic parameters within 0.1 Åand 50 cm!1 from the reference values, use of the small6-31+G(d,p) basis set represents a viable compromise be-tween computational effort and accuracy: the B2PLYPdouble-hybrid as well as the M05 and M06 functionals repro-duce the CCSD(T) best estimate of D0 within 1 kcal mol!1.The B3LYP/6-31+G(d,p) level of DFT, in turn, overshootsthis target value of 25.3 kcal mol!1 by some 7 kcal mol!1.

VI. CONCLUSIONS

In the present contribution, a detailed study on the dis-sociation energy and spectroscopic parameters of CuO+ hasbeen carried out by means of several highly correlated post-HF methods. At the respective extrapolated basis set limits,most post-HF approaches agree within a narrow error marginon a De value of 26.0 kcal mol!1 [CCSD(T)], 25.8 kcal mol!1

(CCSDTQ via the HEAT protocol), and 25.6 kcal mol!1

(DMC), which is encouraging in view of the disaccording

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064304-11 Binding energy of CuO+ J. Chem. Phys. 134, 064304 (2011)

data published thus far. The CISD-based MR-ACPF expan-sion gives a slightly lower value of 24.1 kcal mol!1, indicat-ing that large basis sets and triple excitation patterns are nec-essary ingredients for a quantitative assessment. The singlereference schemes limited to single and double excitations donot provide a satisfactory representation of the Cu–O bond,which further documents that the perturbative (T) term is es-sential to obtain accurate results. All post-HF methods showconvergence with the basis set size beyond quadruple-) qual-ity. A sufficient recovery of correlation energy further relieson the correct representation of the # orbital space, whichforms the covalent bond in the 3%! ground state of CuO+.If these orbitals are too strongly localized, as happens inpure HF wavefunctions, the bond energy obtained will be toolow.

With an adequately balanced configuration space, the re-sults from size extensive multireference CI expansions nicelyvalidate the CCSD(T) data in terms of the description of boththe ground state reference wavefunction as well as the atomicand molecular properties. A sufficiently flexible active spacein the MRCI procedure is, however, essential for a coherentdescription of the electronic ground state. All in all, our mostaccurate calculations result in a dissociation energy somewhatlower than the experimental value, and we suggest that thetrue dissociation energy is close to the lower boundary of theexperimental value.

The assessment of various DFT flavors at the CBS limitshows that only inclusion of (a moderate amount of) Fockexchange provides reasonably accurate results. The best per-formance for the binding energy was found for the M05 hy-brid, which, however, shows large deviations for the atomicexcitation energy. Reasonable accuracy, in combination withthe limited-sized 6-31+G(d,p) basis set, is obtained with theM05, M06, and B2PLYP functionals; the B3LYP functionalperforms well for re and (e, but overestimates the binding en-ergy by more than 7 kcal mol!1. Although clearly further cal-ibration studies are indicated for related systems, these find-ings show that computationally affordable, yet sufficientlyreliable methods for extended copper-oxo systems are athand.

ACKNOWLEDGMENTS

The work in Frankfurt was supported by the DeutscheForschungsgemeinschaft (SFB/TR49), the Beilstein InstitutFrankfurt/Main, and the City Solar AG Bad Kreuznach. Thework in Mainz was supported by the Deutsche Forschungs-gemeinschaft and the Fonds der Chemischen Industrie. JMMwould like to thank the Spanish Ministry of Science andInnovation for funding through a Ramon y Cajal position(RYC2008-03216) and the SGI/IZO-SGIker UPV/EHU (sup-ported by Fondo Social Europeo and MCyT) for generousallocation of computational resources. RB acknowledges theVolkwagen Foundation for financial support. Computer timeand excellent service was provided by the Center for Scien-tific Computing (CSC) Frankfurt and the Hochschulrechen-zentrum (HRZ/HHLR) Darmstadt. We are grateful to Prof.Michael Dolg (Köln) for discussions and for providing us

with four-component MCDHF data as used in the develop-ment of the pseudopotential for copper.

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