A self-consistent Hubbard U density-functional theory approachto the addition-elimination reactions of hydrocarbonson bare FeO+

Heather J. Kulika� and Nicola Marzarib�

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, USA

�Received 31 July 2008; accepted 29 August 2008; published online 7 October 2008�

We present a detailed analysis of the addition-elimination reaction pathways for the gas-phaseconversion of molecular hydrogen and methane on FeO+ to water and methanol, respectively, usingfirst-principles calculations. These two reactions represent paradigmatic, challenging test cases forelectronic structure approaches to transition-metal catalysis. We compare here density-functionalapproaches against state-of-the-art coupled-cluster and multireference quantum chemistryapproaches. The quantum chemical approaches are found to be in close agreement betweenthemselves as well as with the available experimental evidence. For the density-functionalcalculations, we employ a recently introduced ab initio, self-consistent Hubbard-like correction,coupled here with a generalized-gradient approximation �GGA� for the exchange-correlationfunctional. We find that our formulation provides a remarkable improvement in the description ofthe electronic structure, hybridization, and multiplet splittings for all calculated stationary pointsalong these reaction pathways. The Hubbard term, which is not a fitting parameter and, in principle,can augment any exchange-correlation functional, brings the density-functional theory results inclose agreement with the reference calculations. In particular, thermochemical errors as large as 1.4eV in the exit channels with the GGA functional are reduced by an order of magnitude, to less than0.1 eV on average; additionally, close agreement with the correlated-electron reference calculationsand experiments are achieved for intermediate spin splittings and structures, reaction exothermicity,and spin crossovers. The role that the Hubbard U term plays in improving both quantitative andqualitative descriptions of transition-metal chemistry is examined, and its strengths as well aspossible weaknesses are discussed in detail. © 2008 American Institute of Physics.�DOI: 10.1063/1.2987444�

I. INTRODUCTION

The gas-phase reactions of bare FeO+ cations with hy-drogen, methane, nitrogen gas, and other species have beenwidely studied as fundamental processes that can provideclues into the behavior of larger condensed-phase metal-oxosystems.1 These paradigmatic reactions have been studied bynumerous mass spectrometric techniques in detail,1–4 but ini-tial experimental results were surprisingly discordant withclassical transition-state theories.2 These results were pro-posed to be due to the crossing of multiple relevant spinsurfaces for the reactions,5,6 a feature now expected to berelevant in larger systems, which include the enzyme horse-radish peroxidase7 and model non-heme inorganic iron al-kane hydroxylation catalysts.8 The reaction of FeO+ with H2

has been under particularly extensive study for some timenow both theoretically5,6,9–13 and experimentally1,2,4,14 as themost fundamental example of the two-state reactivity para-digm. Additionally, the reaction of FeO+ with CH4 has beenstudied in detail, although primarily with density-functionaltheory �DFT�.15–18 Methane oxidation on FeO+ is particularly

relevant because of methane’s greater utility as a fuel uponconversion to methanol, as well as the relevance to enzy-matic systems such as methane monooxygenase.19

Reactions of FeO+ with CH4 and H2 are both known tobe exothermic by about 10 and 37 kcal/mol, respectively.However, these reactions are very inefficient: only about10% of all collisions lead to products in the case of methaneand 1% of all collisions lead to products for the hydrogencase. This inefficiency is particularly surprising because thesame-spin ground state reactant and product complexes, 6�+

FeO+ and 6D Fe+, make these reactions spin allowed.10 It hasalso been found that radical side products can form by dis-sociation of the reacting complex to FeOH+ and radical H orCH3. While the former products are weakly endothermic,CH3 radical formation is exothermic by roughly 3 kcal/mol.The decrease in observed rate constant with increasing en-ergy or temperature for both reactions is also in direct con-tradiction with the expected Arrhenius-like behavior for aspin-allowed reaction.5

Theoretical considerations pointed to crossings betweenthe sextet and quartet potential energy surfaces as the sourcefor this reaction inefficiency;5,6 it is nevertheless notable thatsubsequent studies,4–6,11,12 which employed density-functional techniques, have failed to agree with experimental

a�Electronic mail: hjkulik@mit.edu.b�Electronic mail: marzari@mit.edu.

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results or highly accurate quantum chemistry calculations.We will consider in detail here the reactions of FeO+ with H2

and CH4 as a challenging test for a Hubbard U density-functional approach, which we have recently introduced toimprove semilocal or hybrid density-functional descriptionsof transition-metal complexes.20 We specifically consider thegeneralized-gradient approximation �GGA� in DFT for theirefficiency and widespread use in the community;21 neverthe-less, the Hubbard correction introduced here could be ap-plied in conjunction with other exchange-correlation ap-proximations. In order to gauge the success of our approachon both of these reactions, we will compare DFT+U�GGA+U� results with both standard GGA results andhighly accurate quantum chemistry approaches.

A. The Hubbard U approach

We recently proposed augmenting semilocal exchange-correlation functionals with a Hubbard U term to improvethe description and understanding of transition-metal com-plexes with highly localized valence states.20 This approach,typically known as LDA+U or DFT+U, has been very suc-cessfully employed to improve the structural and energeticdescription of some strongly correlated solid-statematerials;22–24 we have recently shown that this method alsocorrects with great accuracy self-interaction errors inherentin the localized orbitals of transition-metal containing mol-ecules. Currently, however, the most widely used approachfor these molecular systems are the semiempirical densityfunctionals known as “hybrids,” which mix in a fraction ofHartree–Fock �HF� exchange-potential with LDA or GGAexchange-correlation functionals. While these hybrids, ex-emplified by the highly popular Becke-exchange basedB3LYP, have performed well on the first row, organic sys-tems for which they were parametrized, their extension totransition metals has yielded mixed results.6,25,26 In particu-lar, it has been shown that some transition-metal complexesare better suited to a reduced percentage of HF exchange�e.g., the Jacobsen–Katsuki catalysts25�, while others havebeen shown to be better described by an increased percent-age of HF exchange �e.g., heats of formation of smalltransition-metal halide compounds26�. Recent work has alsoexplored the usefulness of a Hubbard-like correction, al-though in a semiempirical way.27 Finally, higher order ap-proximations, which include the Laplacian of the density,have been considered by several groups to varying degreesof success.28–30 Our DFT+U approach is at variance withhybrid exchange-correlation functionals, for which the pa-rameters are fit to experimental data or higher order quantumchemistry calculations because the Hubbard U term is a fullyab initio property of the system that is determined directlyfrom the calculations at little cost.

In order to further motivate the use of the DFT+U ap-proach for single-site or few-site transition-metal complexes,we refer to an illustration suggested previously by Cococ-cioni and de Gironcoli.24 The proper energetic description ofan isolated atom in contact with a reservoir of electrons as afunction of its occupation number is a piecewise linear sta-tistical average of the integer occupation end points. On the

other hand, GGA or LDA incorrectly provide parabolic be-havior with a minimum often corresponding to fractionaloccupations.31 Additionally, the GGA or LDA estimates arequite close to the exact DFT energy for integer occupations,with larger divergences at fractional numbers �hence the util-ity of constrained density-functional approaches32,33�. Thedifference that remains between the exact and the GGA orLDA results is, thus, approximately piecewise parabolic andclose to zero at integer occupations.

This picture is key to understanding the role of Hubbardfunctionals in complexes with a single transition metal,where intersite correlations are not relevant. We use here therotationally invariant “+U” functional24

EU��nmm�I� �� =

U

2 �I,�

Tr�nI��1 − nI��� , �1�

where nI� represents the occupation matrices of the localizedmanifold of states at site I with spin �, typically localized dor f states. In what follows, the occupation matrices are de-rived from projections of the molecular valence states ontothe atomic states of the isolated metal center, but any numberof approaches, such as maximally localized Wannier func-tions could be used.34 As written, the Hubbard U correctionis analogous to the parabolic corrections that must be madeto the energy functional in order to reproduce the true piece-wise linear behavior in the isolated limit we just described.In a molecular system, eliminating this curvature corre-sponds to improving the description of chemical hybridiza-tion by removing the tendency of d or f orbitals to over-delocalize from self-interaction errors. Moreover, thecurvature of the parabola �i.e., the strength of the Hubbard Ucorrection� is an intrinsic property of the system that caneasily be determined from a linear-response calculation, asmentioned below.

Note that there is a number of approximations in Eq. �1�,which we will not describe in detail �see Ref. 24�. First,construction of a DFT+U �here, GGA+U� functional neces-sitates the inclusion of a term, which corrects for doublecounting of the Hubbard term by contributions alreadypresent in the approximate exchange-correlation functional�here, GGA�. Second, in order to reach the simplified form ofEq. �1�, the higher order interactions of electrons of differentspins, traditionally identified by the term J, have been incor-porated into the term we use such that our U is effectivelyequal to the rigorously defined U↑↑−J↑↓. Our definition dis-tinguishes our results somewhat from those who employ a Jexplicitly, and this should be kept in mind when comparingresults of other LDA+U or GGA+U codes. Additionally, weprovide in the Appendix the results that show that pseudopo-tential oxidation state choice can have an effect on numericalvalues of the linear-response U while ultimately preserving aconsistent description of the system.

As shown by Cococcioni and de Gironcoli, the curvatureof the total energy with respect to occupation number can berecast, via Janak’s theorem, into a linear-response property

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U = +��I

KS

�nI−

��I

�nI= ��0

−1 − �−1�II, �2�

where �0 represents the bare, noninteracting reorganizationafter a potential shift � �which is in practice extracted fromthe first iteration of an SCF calculation�, while � representsthe self-consistent response to the same potential shift. Inpractice, a rigid potential shift, �, is applied on the localizedmanifold, and the resulting change in occupations is mea-sured. This linear-response approach is both computationallyaffordable and easy to employ for any structure for which awell-converged single point energy calculation can beobtained.

We have recently20 extended this approach to take intoconsideration the changes in the linear-response properties ofthe DFT+U ground state density with respect to the standardexchange-correlation functional �here, GGA�. In molecules,different electronic states may become more stable withGGA+U, or in the solid state, systems that are metallic withGGA can become insulators, leading to very different re-sponse properties. One might expect that consistency isachieved when the value of applied Uin yields a linear-response Uout that is equal to zero or alternatively, a fixedpoint in which Uout=Uin is reached. In fact, both assumptionsneglect the fact that there is already an inherent correlation inthe exchange-correlation functional that is augmented, andconsistency is achieved only when the added Hubbard corre-lation and the one intrinsic in the exchange-correlation func-tional are consistent. Simple algebra and the observation thatUout exhibits a linear behavior over a range of Uin lead to thedefinition of a self-consistent Uscf as

Uout = Uscf −Uin

m, �3�

where m is an effective degeneracy of the perturbed mani-fold. In practice, we obtain Uscf by extrapolating over a rangeof points �which follow the linear relationship� to Uin=0.From now on, we refer to the result of a standard-linearresponse calculation �obtained from Eq. �2�� as U0 and of theself-consistent extrapolated value as Uscf. Often, U0 and Uscf

are nearly identical but we have shown that Uscf occasionallyprovides a significant improvement, as in the case of the irondimmer.20 Lastly, although we focus on the use of the DFT+U functional for a single localized manifold, in some casesthe U terms on other manifolds can become relevant.20 In-clusion of the other manifolds simply requires a matrix for-mulation, as discussed in detail in Ref. 24, and we will dis-cuss explicitly when these manifolds �e.g., the 4s states inselect 3d64s1-based intermediates� play a non-negligiblerole.

A significant shortcoming of the Hubbard formulationthat should be highlighted is that total energies computed atdifferent values of U cannot be directly compared. In orderto construct comprehensive potential energy surfaces or re-action coordinate diagrams, we chose to use an average Uacross all relevant ionic structures and electronic states, areasonable approximation because of the proximity of indi-vidual values of Uscf to the global Uscf,av. In particular, sys-tems that have differing structure �e.g., equilibrium versus

stretched bonds� or coordination environments �e.g., bothnumber and character of interaction� often fortuitously ex-hibit reasonably similar values of linear-response U andtherefore their energetics may be easily compared at an av-eraged Uscf without bias. However, in some cases, energysplittings and barriers may be more accurately obtained byemploying a locally averaged U over only a small portion ofthe reaction coordinate, as we will later show. While webelieve that our approach has great promise for treatinglarge-scale systems with near-chemical accuracy, thanks toits negligible computational costs, we will first demonstrateits applicability to the well-known challenging test cases ofaddition-elimination reactions on FeO+ that are both repre-sentative of a broader class of catalytic processes.

II. METHODS

Calculations have been carried out primarily in twoforms: plane-wave, density-functional calculations, and lo-calized basis set, post-HF calculations, which we used for ahighly accurate reference to our DFT results. The plane-wavedensity-functional calculations were completed withthe QUANTUM-ESPRESSO package35 using thePerdew–Burke–Ernzerhof21 �PBE� generalized-gradient ap-proximation �GGA� both augmented with a linear-response,Hubbard-like U term, as previously outlined20,24 as well as inits standard form. The linear-response calculation of theHubbard U term is also implemented in this package. Ultra-soft pseudopotentials were employed with a plane-wave cut-off of 40 Ry for the wave function and 480 for the chargedensity for most structural relaxations to ensure accurateconvergence of forces as well as spin and symmetry statesplittings.36 Structural relaxations and nudged elastic band�NEB� optimizations37 were carried out at integer values ofthe Hubbard U to further assess the U dependence of struc-ture and energetics. In order to improve the resolution ofsteep reaction barriers, the climbing-image and variablespring methods were employed in conjunction with the NEBcalculations.38 For diatomic molecules, the harmonic fre-quencies and anharmonic contributions were obtained withhigh-order polynomial fits to the potential energy surface.

Post-HF approaches were employed to provide an accu-rate but computationally very expensive reference to com-pare the density-functional calculations against. The reactionwith hydrogen �nat=4� was studied primarily with GAUSS-IAN 03,39 and the reaction with methane �nat=7� was studiedwith MOLPRO.40 The primary single-reference method em-ployed was coupled cluster with singles, doubles, and pertur-bative triples �CCSD�T��. The T1 diagnostic was used as atool to identify structures with potentially strong multirefer-ence character or instability in the triples term, and selectmultireference configuration interaction �MRCI� and pertur-bation theory �CASPT2� calculation were employed to verifythe CCSD�T� results.41 All multireference calculations on ei-ther reaction were carried out using MOLPRO, which wasavailable to us on a 64 bit architecture. The same Pople-style6-311+ +G�3df ,3pd� basis set was used for all post-HF cal-culations to ensure consistency.

In order to construct the CCSD�T� hydrogen on FeO+

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reaction coordinate, we estimated equilibrium CCSD�T� ge-ometries for each intermediate. The low-dimensionality ofthe potential energy surface permitted calculation of a finemesh of single point energies about 0.01–0.02 Å apart anddeduction of a minimum energy configuration. These geom-etries are also useful to determine errors in geometries in theGGA and GGA+U methods. For transition states �TSs�, weused GGA+U or GGA geometries to calculate single pointCCSD�T� energies since discretizing the PES at the TSs wastoo computationally expensive.42

III. RESULTS

We consider first the value of U calculated for each in-termediate and transition state for the reactions with bothhydrogen and methane, which helps us identify the broadsimilarities of the two cases �see Table I�. As mentioned pre-viously, in order to obtain a global reaction coordinate, wemust use energies calculated with a single value of HubbardU, which we choose as an average, Uscf,av, over all states.Since total energies at differing values of U are not compa-rable, it is important to pay attention to large deviation inUscf at any stationary point from the global average. Theoverall Uscf,av we obtain for the H2 reaction is 4.93 eV, andthe value for the CH4 reaction is very similar at 5.09 eV. TheUscf is on average larger than the value calculated from GGA�U0� for most geometries. We find the smallest discrepancybetween the self-consistent and non-self-consistent values ofU for reaction intermediates, while the differences for TSsare as large as 1.5 eV, as is in the case of 6TS-2CH4

. For mostsingle points in these two reactions, the individual Uscf val-ues are within 1 eV of the Uscf,av. Based upon typical cases�e.g., Int-3 splittings in Fig. 5 and sextet TS-1 barriers inTable I�, employing GGA+U with a value of Uscf,av, which is1 eV too small or too large for the appropriate structure,

likely yields energy differences in the range of 0.01–0.1 eV,and this uncertainty can, in many cases, be decreased byexamining splittings with a locally averaged value of Uscf.

Trends in the values of Uscf are very consistent for thetwo reactions’ quartets and sextets �see Table I�. While theexact numbers between the two reactions are not identical,the similarities highlight the fact that coordination and elec-tronic configuration contribute most strongly to the calcu-lated U value. Overall, we observe a larger Uscf in the rangeof 5.5–6.5 at the entrance channel, where the iron center isweakly three- or twofold coordinated in the Int-1 structure�see Fig. 1�. At the first reactant barrier, Uscf decreases mono-tonically for all TS-1 and Int-2 structures, and this change isconcomitant with a reduction in FeO+ bond order and coor-dination.

At the second reaction barrier, the quartet and sextet sur-faces exhibit different linear-response properties. This behav-ior of Uscf is strongly affected by differences in the chemicalcoupling of the quartet and sextet Fe+ centers with the or-ganic ions in the two spin surfaces. The 3d64s1-like sextet’stwo minority spin electrons are equally divided between thed and s manifolds, and this results in an increased promi-nence of the Uscf,4s for 6TS-2H2

and an overall lower Uscf,3d

around 3–4 eV for both reactions. While the extension toinclude 4s states, and thus a U4s, is straightforward, it playsa role exclusively in the class of weakly bound moleculeswhere the 4s-3d hybridization dominates and 3d- and4s-derived molecular states are close in energy. The sextetTS, especially in the methane case, is indeed quite weaklybound, and we will later discuss this in greater detail. The TS

FIG. 1. Comparison of the first intermediate along sextet and quartet spinsurfaces with GGA and GGA+U for reactions of FeO+ with CH4 �top� andH2 �bottom�.

FIG. 2. Comparison of the second and third intermediates along sextet andquartet spin surfaces with GGA and GGA+U for reactions of FeO+ withCH4 �top� and H2 �bottom�.

TABLE I. U0 and Uscf for intermediates �Int� and TSs along the H2 and CH4

reaction coordinates. Quartet surface values in upper part of the table, sextetin the lower part.

H2 CH4

U0 Uscf U0 Uscf

QuartetFeO+ 5.84 5.50 5.84 5.50Int-1 5.89 6.31 6.21 6.23TS-1 5.34 5.74 5.97 6.05Int-2 4.87 4.95 4.67 5.10TS-2 4.52 5.73 4.68 5.47Int-3 1.81 1.89 2.15 2.33Avg 4.71 5.02 4.92 5.11

SextetFeO+ 5.37 5.48 5.37 5.48Int-1 6.46 6.41 6.07 6.17TS-1 4.47 4.84 4.59 4.59Int-2 4.18 4.17 3.93 4.37TS-2 3.27 3.17 2.72 4.35Int-3 4.79 5.00 5.32 5.47Avg 4.76 4.84 4.67 5.07

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along the quartet surface, which has a 3d7 character, is moretightly bound, and the Uscf of 4TS-2 is close to Uscf,av. AllInt-3 structures at the exit channel closely resemble a weaklybound Fe+ ion with the product water or methanol �see Fig.2�. Consequently, the Fe+ charge density is increasinglyatomiclike, and the values of the Hubbard U are significantlylower at this point in the reaction surface for the quartet ataround 2 eV �the sextet displays a Uscf considerably closer tothe average�.

We highlight that the Uscf from linear response for eachstationary point of a given configuration and electronic sym-metry is calculated separately. We also consider the energeticevolution of each state individually and thus take into ac-count any changes in state energetics ordering with increas-ing values of Hubbard U. Since these two reactions involveunsaturated metal centers for which the number and charac-ter of metal-ligand bonds change across various spin states,they provide a far more challenging probe of the robustnessof our method compared to more typical problems, whichinvolve saturated metals or fewer changes in metal-ligandbonding. From the trends we have thus far observed in Uscf

along the reaction coordinate, we anticipate the GGA+Uscf,av energetic description to be optimal for the first halfof the reaction, while it should be slightly poorer for thesecond half of the reaction �particularly at 6TS-2�. Evenwhen an accurate description of the global reaction coordi-nate may be difficult to achieve, a locally averaged Uscf at akey stationary point can greatly improve the structural andenergetic estimate.20

A. The low-lying states of the FeO+ reactant

We previously demonstrated20 that the structure and en-ergetics of low-lying states of FeO+ are described withgreatly improved accuracy with GGA+U over standardGGA by comparing against high level quantum chemistrymethods �CCSD�T��. For both GGA and GGA+U, the FeO+

ground state is a d6s1 6�+ with full spin up 3d and 4s mani-folds, as well as fractional occupation of spin-down � and�dz2 orbitals. Elongation of the FeO+ bond from 1.62 Å

�GGA� to 1.66 Å �GGA+U=5 eV� is concomitant with en-hancement of spin up 3d density in �� states and subsequentreduction of the partial 3d occupation of spin-down � orbit-als. This change in hybridization also reduced the harmonicfrequency by about 150 cm−1 in improved agreement withCCSD�T� results.20 Several low-lying quartet FeO+ states areclose in energy, including 4�, 4�, and 4�. While 4� and 4�are nearly degenerate in energy with GGA, GGA+U stabi-lizes 4� preferentially by nearly 0.75 eV over the otherstates, in agreement with ours and other CCSD�T�results.12,20,43 The equilibrium bond length of 4� increasessignificantly from 1.58 �GGA� to 1.75 Å �GGA+U� and theharmonic frequency decreases by from 1038 to 612 cm−1

due to an even more pronounced increase in �� occupation inthis case than in the sextet.

For both the quartet and the sextet, two competing fac-tors exist—a physically relevant hybridization of the Fe 3delectrons with O 2p electrons to form the bonding � orbitals

as well as the spurious tendency of the 3d electrons todelocalize—the latter of which is corrected and greatly re-duced by the Hubbard U term.

It is of particular note that the GGA+U sextet-quartetsplitting �0.54 eV� is in remarkable agreement with theCCSD�T� value �within three hundredths of an eV� and fre-quencies for the sextet and the quartet are within 25 and21 cm−1 of their CCSD�T� values, respectively, compared toa discrepancy as large as 400 cm−1 for GGA.20 Obtainingthe correct multiplet splittings for FeO+ low-lying states iscrucial to ensure that the entire reaction coordinate is welldescribed.

B. The first reaction intermediate

Free hydrogen or methane complexes weakly with FeO+

at the metal site to form the first intermediate �Int-1� of thereaction, which has not been isolated experimentally1 �seeFig. 1�. Along the sextet surface �both with GGA and GGA+U�, the electronic structure of the FeO+ state is only weaklyperturbed by the bound CH4 or H2, as is evidenced by thefact that the Fe-O bond lengthens by only 0.01 Å with re-spect to its bare FeO+ value, and the preferred binding sitefor the reactant is parallel to the Fe–O bond. The H–H bondlength, 0.79 Å, is lengthened by only 0.01 Å with respect tothe isolated molecule. In 6Int-1CH4

, the methane tetrahedronflattens slightly, with the three hydrogens closest to the C–Febond exhibiting a dihedral of 135° instead of the idealizedvalue of 120°. While the effect is subtle, GGA+U does infact tune the relative energy of this intermediate with respectto the reactants.

In contrast to the sextet, the 4Int-1 exhibits a more dra-matic response structurally and energetically to the value ofU, and thus demonstrates the utility of a Hubbard U ap-proach. While the binding of hydrogen or methane to theFeO cation occurs parallel to the metal-oxo bond in the sex-tet case, the ligand may instead preferentially bind in a man-ner perpendicular to the bond. Early density-functional cal-culations predicted that this nonplanar geometry for the4Int-1H2

was the preferred structure.5,6 However, we showedthat this stabilization by GGA was not preserved in theGGA+U result for the reaction with hydrogen. Our GGA+U results reversed the preference of GGA and stabilized theplanar structure for U above 3 eV and by nearly 0.5 eV at aU of 5 eV, with respect to the lowest-lying planar geometryelectronic state �see Fig. 1�. In both planar and nonplanarorientations, the primary bonding occurs via bonding overlapbetween H 1s density and the FeO+ bonding � molecularorbitals �3dz2 +2pz�. In the nonplanar geometry, additionalbinding of the complex occurs via overlap of the two hydro-gens’ 1s density with the xy atomiclike orbitals of the metalcenter. The GGA preference of the nonplanar hydrogen bind-ing site by nearly 0.5 eV may be viewed as an unphysicalartifact of self-interaction error similar to the erroneous en-ergetic ordering of binding sites of small organic moleculeson metal surfaces.44 Importantly, our CCSD�T� calculationsconfirm the GGA+U result that the planar geometry is moreenergetically favorable.20 The nonplanar geometry is not a

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stable sextet intermediate because the favorable interactionoccurs via the minority spin channel, which is not present inthe sextet case.

It follows that GGA shows a similar preference for thenonplanar geometry for methane reaction 4Int-1 complexes�see Fig. 1�. This effect is reduced in the methane case forseveral reasons: the density of only one out of four hydro-gens is available for overlap with the localized orbitals of Fe,it is not possible to position the hydrogens or carbon to over-lap with states, and, moreover, C–Fe interactions are un-able to contribute to formation of a bent structure. Indeed,while a nonplanar GGA structure is preferred for the4Int-1CH4

by about 0.11 eV, GGA+U stabilizes the planarstructure for all values of U greater than 1 eV and overall byabout 0.5 eV at the globally averaged U of 5 eV. MRCIresults for the methane reaction agree with this state order-ing, and our 4Int-1 GGA+U results demonstrate the dramaticimprovement in both structural and energetic descriptionbrought about by addition of a Hubbard U term.

C. The first reaction barrier

The first catalytic step in the reaction of FeO+ with meth-ane or hydrogen is the abstraction of a single hydrogen fromthe reactants by oxygen. The TS associated with this step forall reactants and spin surfaces is four centered and all bondsto the soon to be abstracted hydrogen are highly stretched.Along the sextet surface, GGAs have been shown to properlyreproduce the characteristic experimentally steep barrier5,6,20

�see Table II�. This large activation barrier for both sextetreaction surfaces stems from the short and tight bond of the6�+ FeO+ molecule, which is stretched by 0.1 Å to a valueof 1.72 Å in the GGA TS. The H2 reactant bond is stretchedto 1.05 Å, and a new, weak O–H interaction at 1.34 Å isformed. This transition-state structure changes very littlefrom GGA to GGA+U, although with a slightly longer Fe–Obond and shorter H–H bond, and the barrier height estimatesare similar for GGA �1.01 eV� and GGA+U �0.97 eV�. Themethane reaction has a similarly high activation barrier alongthe sextet surface, which changes relatively little from GGA�1.35 eV� to GGA+U �1.29 eV�. Structurally, the Fe–O bondis stretched to the same extent as in the hydrogen reactionand the C–H bond is stretched by 0.3 Å from its equilibriumvalue of 1.10 Å, and changes in structure as a result of aug-menting a +U term are negligible. While the forward barriersfor both reactions do not change significantly with the value

of U, the change in the back-reaction barriers �or the step’sexothermicity� for both the hydrogen and methane case isquite large.

In contrast to the sextet surfaces, the quartet reactionbarrier demonstrates considerable sensitivity to the value ofthe Hubbard U, as we would have predicted after the studyof 4Int-1. The NEB reaction path we study uses the planar4Int-1 as an end point, although it is a metastable minimumfor GGA and low to intermediate U values. The path alsopasses through a nonplanar GGA 4Int-1-like structure beforereaching the stretched transition state, which is the globalminimum for GGA but disappears as a stable intermediatefor intermediate values of GGA+U. We previously showed20

that a GGA barrier for the hydrogen reaction is steep, atabout 0.4 eV with respect to the nonplanar Int-1, and thequartet TS-1 lies above the reactant energies, in contrast toexperimental evidence1 �see Fig. 3�. With GGA+U, a shal-low reaction surface is recovered with an activation energyof 0.20 eV, and the TS lies 0.16 eV below the ground statereactants. The reduction in the barrier estimate is likely de-rived from the weakening of the Fe–O interaction in the TSfrom 1.61 to 1.72 Å, thereby permitting increased bondingwith the reactant complex.

As with the hydrogen case, the GGA 4Int-1CH4nonplanar

geometry appears in the GGA reaction coordinate as a localminimum stabilized by 0.1 eV. The TS also resembles itsanalogous sextet TS-1 with the methyl group sitting nearlyperpendicular to the Fe–O bond and the leaving hydrogenshared equally between the carbon and oxygen.

The local minimum from the nonplanar structure disap-pears in the GGA+U reaction coordinate, and the overallreaction barrier increases to nearly 0.65 eV for U=5 eV as aresult of the reduced relative energy of the planar 4Int-1 endpoint �see Fig. 3�. If we choose instead as a reference pointeither the nonplanar geometry or the crossing point of thequartet and sextet coordinates, the barrier estimate is reducedeven further to 0.32 or 0.25 eV, respectively. Additionally,this GGA+U transition state also lies below the reactant en-ergy by 0.4 eV, compared to 0.3 eV for GGA. The quartetsurface for both reactions demonstrates considerable sensi-

TABLE II. Forward and back-reaction barriers as a function of U for thefirst sextet barrier of the reactions of FeO+ with methane and hydrogen ineV.

CH4 H2

U �Efwd �Eback �Efwd �Eback

0 1.35 1.88 1.01 1.601 1.35 1.88 1.01 1.682 1.35 1.89 1.00 1.773 1.34 1.92 0.99 1.854 1.33 1.96 0.98 1.945 1.29 2.00 0.97 2.02

0 1 2 3 4 5Hubbard U (eV)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

ActivationEnergy(eV)

�Efwd,CH

4

�Efwd’,CH

4

�Eback,CH

4

�Efwd,H

2

�Eback,H

2

FIG. 3. �Color online� Forward and back-reaction barriers as a function of Ufrom U=0 �GGA� to U=5 �the global Uscf,av� for the first quartet barrier ofthe reactions of FeO+ with methane and hydrogen in eV. For the methanereaction, �Efwd� refers to the forward reaction barrier calculated from thenonplanar geometry.

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tivity to the value of U applied, which must be close to theUscf in order to accurately reproduce experimental and accu-rate theoretical estimations.

D. The second reaction intermediate

Upon abstraction of a hydrogen Habs from the reactants,the hydrogen in turn binds to oxygen to form the stable Int-2,and the Fe–O bond lengthens further. This low symmetrystructure possesses only a single low-lying electronic stateand the energetic splitting of the sextet and quartet structuresis close in energy ��E� 50 meV� for both reactions. Thesextet state is the GGA and GGA+U ground state for themethane case, while GGA+U preferentially stabilizes thehydrogen reaction’s 4Int-2 over the sextet, reversing theGGA ordering. The sextet structures have a planar configu-ration with the ligands at obtuse angles relative to the line ofthe Fe–O bond; the quartet structures instead prefer a ligand-Fe–O angle of about 90° and an O–H bond directed out ofthe plane. For both reactions and spin surfaces, the Fe–Obond is elongated with respect to the TS-1 value. In 6Int-2,the Fe–O bond is about 1.74 Å for GGA and lengthens fur-ther to about 1.77 Å for GGA+U. The total elongation forthe GGA+U 4Int-2 is 0.08 Å with respect to its 1.69 ÅGGA value. The second reaction step, of which Int-2 is thestarting point, is the concerted translation of the ligand �H orCH3� at the iron site to a midway point atop the FeO bondconcomitant with a rotation of the already abstracted hydro-gen. This geometric rearrangement culminates in the forma-tion of the water or methanol leaving group and correspondsto a dramatic change in coordination environment for the Fecenter that goes from having two strong ligands to one strongand one weak ligand, finally to a single very weak ligand.These coordination changes provide a stringent test forwhether GGA+U can treat each of them with equalaccuracy.

E. The second reaction barrier

Previous results6,20 have shown the quartet barrier to liebelow the sextet barrier, and the geometric structure plays asignificant role in the relative barrier heights. In the 4Int-2H2structure, a ligand-FeO angle of 90° requires less translationto position the ligand directly over the Fe–O bond. The GGAreaction barrier for the hydrogen reaction is about 0.42 eV,but it is significantly reduced by GGA+U �=5 eV� to 0.13eV �see Fig. 4�. The Fe–O bond elongates further from thelength in 4Int-2 to 1.80 Å in the GGA+U 4TS-2H2

. Thetransferring hydrogen is shared nearly equally between Feand O, with bond lengths of 1.60 and 1.44 Å, respectively.The GGA+U reaction barrier for the hydrogen reaction isconsistent with the shallow, barrierless quartet surface pre-dicted by experimental evidence and our own highly accu-rate theoretical results.2,6,20

For the methane reaction, the second quartet GGA bar-rier is very steep at nearly 1.4 eV, while with GGA+U theactivation energy is halved to about 0.7 eV �see Table IV�.The GGA 4TS-2CH4

possesses rather long Fe–C and C–Obond lengths, suggestive of a weakly bound methyl radicaland FeOH+ complex. However, as U increases above 2 eV,

the Fe–C bond shortens, and there is an overall increase inthe CH3–FeOH+ binding for the TS. The GGA+U resultsclearly favor a bound TS, while the GGA results remainambiguous. The relative energy to dissociated radical prod-ucts is even more key in interpreting the 6TS-2 results, as wewill later show. While the GGA+U quartet surface barrier islarger than its hydrogen counterpart, 4TS-2CH4

remains be-low the reactant energies and requires less rearrangementthan the sextet surface does. Generally, at the exit channel weobserve a shallower quartet surface with GGA+U than wedid with GGA, and we also observe an increase in the back-reaction barrier consistent with increased exothermicity ofthe reaction.

The second barrier along the sextet surface is highly sen-sitive to the value of the Hubbard U but in divergent waysfor the hydrogen and methane reactions. Along the hydrogenreaction coordinate, the GGA sextet surface is relativelysteep with an activation energy of 1.29 eV. The GGA+Uactivation barrier is markedly decreased from GGA to 0.82eV with the Uscf,av of 5 eV �see Table III�. We have previ-ously shown that this value is an underestimate, and the de-crease in activation energy with the value of U for the hy-drogen reaction is due to two factors. First, the Uscf

calculated from the 3d states is low at about 3.27, and sec-ond, the 4s electrons play an increasingly important role inbonding in this region of the sextet reaction coordinate. TheGGA geometry of 6TS-2, however, is not dissimilar to the4TS-2 structure with an Fe–H bond length of 1.50 Å andstretched O–H bond of 1.65 Å, which elongates further by

0 1 2 3 4 5Hubbard U (eV)

0.0

0.2

0.4

0.6

0.8

1.01.2

1.4

1.6

1.8

2.02.2

2.4

2.6

ActivationEnergy(eV)

�Efwd,CH

4

�Eback,CH

4

�Efwd,H

2

�Eback,H

2

FIG. 4. �Color online� Forward and back-reaction barriers as a function ofincreasing U from 0 eV �GGA� to 5 eV �the global Uscf,av� for the secondquartet barrier of both reactions with methane and hydrogen.

TABLE III. Forward and back-reaction barriers as a function of U for thesecond sextet barrier of both reactions with methane and hydrogen.

CH4 H2

U �Efwd �Eback �Efwd �Eback

0 1.87 1.26 1.22 2.011 1.68 1.39 1.19 2.232 1.51 1.52 1.10 2.413 1.36 1.69 1.00 2.644 1.18 1.79 0.90 2.865 1.03 1.90 0.82 3.01

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about 0.03 Å for GGA+U. The different reaction barrierbehaviors of the two spin surfaces originates from both elec-tronic structure differences �as evidenced by the differingvalues of linear-response U� and the differing degrees of dis-placement from the Int-2 geometry required to reach the TS.

Specifically, the sextet reaction coordinate is generallyderived from Fe+ atomic states, which are 3d64s1 in configu-ration �unlike the quartet 3d7�. The role of the 4s becomesmost pronounced for weaker, long-distance binding interac-tions associated with the stretched bonds of the TS. By ex-tending the linear-response calculation to a matrix formula-tion that takes into account the response of the 4s and 3dsimultaneously, we obtain a Uscf,4s of 4 eV and an increasedUscf,3d of 4 eV as well. While a matrix formulation was al-ready present in the original derivation,24 its application tostudying both 3d and 4s electrons simultaneously has notbeen as crucial for solid-state applications as it is here inmolecules with strong 3d-4s hybridization as well as closespacing of the 3d and 4s levels. By applying a Hubbardcorrection on both the 4s and 3d electrons �4 eV for both, asmentioned before and as obtained from linear response�, weobtain an improved estimate of the barrier height of 1.16 eV,as compared to the 0.82 eV value obtained from GGA+U3d=5 eV. The back-reaction barrier remains unchangedat 3.01 eV. Both forward and back-reaction barrier estimatesare in improved agreement with the CCSD�T� values of 1.1and 3.0 eV, respectively. Alternatively, a locally averagedUscf of the 3d states alone, 3.6 eV, provides an improvedbarrier estimate of about 0.96 eV. For cases where the role ofthe 3d states are reduced, a low Uscf can provide a usefulreminder to consider also binding interactions due to the 4s,which are otherwise overshadowed by the 3d states for com-plexes with short bond lengths and strong metal-ligand 3dhybridization.

The methane 6TS-2 barrier varies strongly with HubbardU by decreasing drastically from 1.87 eV for GGA to 1.03eV for GGA+U �see Table III�. Unlike 6TS-2H2

, this TSexhibits a larger Uscf,3d of 4.35 relatively close to the Uscf,av

value of 5 eV, while the linear-response U4s is nearly zero,highlighting how the formalism permits detection of whichmanifolds are most relevant. The GGA 6TS-2CH4

structure,like its quartet analog, resembles a methyl radical looselybound to an FeOH+ fragment. The GGA coordination dis-tances of 2.83 Å for Fe–C and 2.33 Å for C–O suggest thatvery diffuse interactions are responsible for binding. Thelong Fe–C bond length in the GGA structure corresponds toweak bonding via overlap of the Fe 4s density with theC 2p, which is also weakly antibonding with respect to theC–O bonding. The loosely bound GGA structure points tosignificant competition between the elimination step and dis-sociation of the TS to radical side products, which are in factboth endothermic by 0.4 eV and around 0.05 eV, respec-tively. The similarity of the GGA TS-26 to a free methylradical and FeOH+ moiety is further substantiated by com-paring the TS geometries with isolated products. The C–Hbond lengths and H–C–H angles are nearly identical to theisolated methyl radical �1.085 Å and 119°�, and there is avery slight distortion from the plane of CH3 with a dihedralof 170°. The nearly dissociated nature of the 6TS-2CH4

GGA

structure is much greater than in the quartet case as a resultof the spins of the isolated radical fragments—quintetFeOH+ and doublet CH3—which couple with greater bond-ing interaction if the spins are opposed, as in the quartet, thanif the majority spins match, as is the case in the sextet.

Bond lengths for the more tightly bound GGA�U�=5 eV� TS are shortened to 2.49 Å for Fe–C and 2.26 Åfor C–O. The GGA+U structural differences are largely dueto an increased population of three-center bonds involvingthe Fe 3dxz, O 2p, and C 2p orbitals in lieu of weaker in-teractions between Fe 4s and C 2p. In order to demonstratein greater detail how the Hubbard U term tunes the TS-26

binding strength, we measure the changes in Fe–O bondlength ��RFeO� as well as the difference between the TS-26

CH3 dihedral and the planar value in the isolated methylradical ���CH3

�. With increasing values of U up to 2 eV, weat first see a decrease in the Fe–O bond and slight decrease inthe dihedral �see Table IV�. However, for U=3 eV, the dif-ference in the Fe–O bond length increases as charge transferout of the Fe–O bond and into the CH3 moiety increasesconcomitant with a dihedral change from an increased polar-ization of the Fe–O bond. These changes in electronic struc-ture are evidenced by a 5% increase in the Fe–O bond lengthand 10% difference in the dihedral in the U=5 eV structure.The TS is also stabilized by over 0.6 eV when compared tothe dissociated products and a now weakly exothermic sideproduct. An accurate estimate of the relative height of thisbarrier both to reactants and to the radical side products is aparticularly sensitive test for any ab initio method; GGA+U is in very good agreement with experimental findingsand the MRCI reference calculation, while GGA fails to pro-vide a physically reasonable description of the reaction.

F. The third reaction intermediate

The exit channel Int-3 resembles an isolated Fe+ ionweakly ligated by the leaving oxygen of either the water ormethanol �see Fig. 2�. The weak metal-ligand interactionproduces many closely spaced low-lying electronic statesdiffering only by their minority spin orbital occupations,which presents the challenge to resolve ordering of statesseparated by a few hundredths of an eV. Based upon the�E6→4=0.23 eV splitting of isolated Fe+, the sextet Int-3should be most stable, but common density functionals erro-neously predict the quartet instead to be most stable.5,6,20

Since the metal-ligand binding is equally weak in both reac-

TABLE IV. 6TS-2CH4structural and energetic changes with U. The �E are

in eV and the structures are percent difference from the bond length ordihedral for the isolated moiety at the same value of U as in the transitionstate.

U �ERct-TS26 �EFeOH+-TS26 �RFeO �%� ��CH3�%�

0 0.02 0.36 3.16 5.751 −0.22 0.48 2.88 5.692 −0.46 0.56 2.61 5.243 −0.67 0.61 4.83 9.104 −0.92 0.68 4.95 9.645 �1.07 0.78 5.13 10.02

134314-8 H. J. Kulik and N. Marzari J. Chem. Phys. 129, 134314 �2008�

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tions, the Int-3 structures of both reactions should be studiedtogether. In the sextet case, a single 3d electron resides in thespin-down manifold, and the � and orbitals retain theirdegeneracy. For all 6Int-3 structures, occupation of a or-bital, which minimizes overlap with any of the water mol-ecules is preferred. The occupation of a � orbital yields anelectronic state roughly 0.1 eV higher in energy. Populationof the � orbital is prohibitive, as it corresponds to the maxi-mal overlap with the remainder of the electron density, and itresides roughly 0.5 eV above the lowest state. None of theserelative symmetry orderings change significantly with in-creasing values of U, although the geometry does change.The two lowest GGA sextet Int-3H2

states have a long Fe–Obond length of about 2.06 Å, which lengthens to 2.14 Å forGGA+U. For the methane case, the GGA equilibrium Fe–Obond length is slightly shorter, about 2.03 Å, but it lengthensthe same amount to about 2.11 Å with GGA+U.

For the quartet Int-3 structures, GGA Fe–O bonds areabout 1.93 Å for the lowest Int-3H2

states and lengthen toabout 2.03 Å for GGA+U �=5 eV�. The methanol complexis similar with a GGA bond length of 1.90 Å extending to1.99 Å for GGA+U. With quartet states, a greater numberof permutations of the occupied orbitals is possible, and thedegeneracy present in the sextet Int-3 � and orbitals is nowbroken. Population of the �dxz

orbital yields states lower inenergy than those occupying �dyz

because �dyzforms a

weakly repulsive antibonding orbital with stray 2pz densityfrom the oxygen. The two lowest states favor instead � or-bital occupation in combination with either a dx2−y2 or �dxz

orbital. The symmetries that are derived from alternative per-mutations of the two 3d electrons reside roughly 0.1 to 0.4eV above these two states. For GGA, the lowest state occu-pies the orbital and is favorable by about 0.04 eV, but at aU of around 3 eV, the two lowest states reverse their orderingand instead, the � occupation is preferred by about 0.09 eV�at U=5 eV� �Fig. 5�.

While GGA erroneously predicts the Int-3 ground stateto be a quartet, with increasing U the two lowest sextet statesbecome lower in energy than the lowest quartets. This is animprovement upon GGA, which found the quartet to belower in energy by 0.1 eV and, thus, does not predict spin

inversion at the exit channel. At a U of 5 eV, we find thesextet-quartet ordering reversed and a state splitting of about0.21 eV. This value may be an overestimate compared to ourown and other CCSD�T� calculations,12 which place the sex-tet to quartet splitting closer to 0.12 eV. Because the splittingis very sensitive to the application of the Hubbard U, it isadvantageous to obtain the splitting at the true U of thissystem. Each state has an individual Uscf, which should becalculated separately and then averaged locally over all4Int-3 and 6Int-3 states, rather than using the globally aver-aged Uscf, which is used to construct a global reaction coor-dinate. By employing an average Uscf of 3.5 eV from the twolowest electronic states, the GGA+U splitting is found to be0.12 eV, in improved agreement with CCSD�T� results. Thelowest sextet and quartet GGA+U electronic states also ex-hibit consistent symmetry with those obtained fromCCSD�T�. Interestingly, as we show in Fig. 5, the state split-tings and their dependence on Hubbard U is nearly indepen-dent of ligand identity, thus showing that the GGA errors andGGA+U corrections are both systematic in nature. Thisanalysis suggests also suggests that Int-3CH4

sextet-quartet

0 1 2 3 4 5Hubbard U (eV)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Rel

ativ

eE

nerg

y(e

V)

4B1: dz2, dxz4A2: dz2,dx2-y2

6B1: dxz6A2: dxy

FIG. 5. �Color online� Comparison of the energy of the two lowest-lyingsextet Int-3 states �blue� and quartet Int-3 states �red� for both methanol�solid lines� and water �dashed� ligands.

Int-1 Int-2TS-1-2.0

-1.5

-1.0

-0.5

0.0

0.5

Int-1 Int-2TS-1-2.0

-1.5

-1.0

-0.5

0.0

0.5

Rel

ativ

eE

nerg

y(e

V)

GGA+UGGA

FIG. 6. �Color online� The TS-1 NEB minimum energy paths �blue� forGGA �left� and GGA+U �right� are compared against CCSD�T� �black�reference energies for both quartet �dashed� and sextet �solid� surfaces of thereaction of FeO+ with H2. Here, U is the globally averaged Uscf,av of 5 eV.The results of the different methods were aligned only once at 6Int-1.

Int-2 Int-3TS-2

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

Int-2 Int-3TS-2

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

Rel

ativ

eE

nerg

y(e

V)

GGA+UGGA

FIG. 7. �Color online� The TS-2 NEB minimum energy paths for GGA �left�and GGA+U �right� �both in blue� are compared against CCSD�T� �black�reference energies for both quartet �dashed� and sextet �solid� surfaces of thereaction of FeO+ with H2. The results of different methods were alignedonly once at 6Int-1. The GGA+U results were determined at the globallyaveraged Uscf,av of 5 eV, excluding the points calculated at locally averagedU3d=4 eV and U4s=4 eV �shown in green�.

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splittings for GGA+U with a locally averaged U is appro-priate at about 0.13 eV, reduced from the value of 0.22 eVusing the global Uscf,av.

IV. DISCUSSION

We have shown that for the addition-elimination of hy-drogen on FeO+, GGA+U reaction coordinate provides asubstantial improvement over GGA when compared againstexperiment or highly accurate CCSD�T� reference calcula-tions. The GGA reaction coordinate �see Figs. 6 and 7� is indirect contradiction with available experimental evidenceand theory because it �1� underestimates the exothermicityby over 1 eV, �2� the quartet surface is too steep to rational-ize a “shallow, barrierless” reaction,2,6 �3� spin splittings atintermediates show large errors, and �4� there is no spincrossover at the exit channel.45 The majority of these errorsin the GGA reaction coordinate are localized to the secondreaction barrier �see Fig. 7�, with the exception of the errorsin the quartet Int-1 structures and barriers in the first barriersurface �see Fig. 6�. While the first GGA sextet TS barrierlargely agrees with CCSD�T� results, the second GGA bar-rier exhibits much greater errors for the exothermicity andstate ordering at Int-3. In contrast, the GGA+U �=5 eV�reaction coordinate �see Figs. 6 and 7� correctly estimatesand predicts the reaction exothermicity, the shallow but ex-cited quartet surface, barrier heights, and spin crossover atthe exit channel �see Fig. 7�. Systematic improvements onGGA+Uscf,av results on the largest discrepancies �e.g., at thehydrogen 6TS-2 barrier� were achieved by including explic-itly the 4s manifold, highly relevant for isolated transition-metal complexes, or by locally averaging the value of Uscf.Additionally, the portions of the reaction coordinate that areproperly estimated by GGA in the first TS barrier �see Fig.6�, namely, the sextet barrier steepness and structures, arepreserved in GGA+U, while the errors of the GGA approachin the second TS barrier �see Fig. 7� are greatly improvedupon.

In order to further measure the quantitative accuracy ofthe GGA+U hydrogen on FeO+ reaction coordinate, we es-timated equilibrium CCSD�T� geometries for each interme-diate. The low dimensionality of the potential energy surfacepermitted calculation of a fine mesh of single point energiesabout 0.01–0.02 Å apart and deduction of a minimum en-ergy configuration. These geometries are also useful to de-termine errors in geometries in the GGA and GGA+U meth-ods. For TSs, we used GGA+U or GGA geometries tocalculate single point CCSD�T� energies since discretizingthe PES at the TSs was too computationally expensive.42

To further measure the quantitative accuracy of theGGA+U reaction coordinate, we average the error of theGGA+U and GGA splittings for each intermediate with re-spect to the CCSD�T� splittings for that intermediate. Wealso compare here to B3LYP results that we obtained buthave not discussed in detail because they exhibited the sameinaccuracies, which were previously discussed inliterature.5,6 The average error for the five multiplet splittingsalong the reaction coordinate is greatly reduced to 0.04 eVwith GGA+U from GGA, which exhibits an average error of

0.20 eV. The commonly employed B3LYP functional per-forms poorly with an average error of 0.30 eV in splittings.Not only does GGA+U improve errors by fivefold overGGA but using the CCSD�T� geometries we found from in-terpolating a potential energy scan, we also observe thatGGA+U on average produces improved geometries. Meanerrors for GGA+U geometries are reduced from 4.3 pm �forGGA� to 2.2 pm, and are slightly worse than B3LYP geom-etries with an average error of 1.3 pm. Although the B3LYPfunctional shows a slight improvement in geometries, thesevenfold improvement in spin splittings GGA+U providesis much more key. The improvement of the B3LYP geom-etries over GGA and GGA+U could also be attributed todifferences between the bonds that hydrogen forms in all-electron, localized basis set methods and plane-wave pseudo-potential methods.

The GGA description of FeO+ on methane demonstrateseven greater discordance with theory and known experimen-tal results than the hydrogen reaction. In fact, using GGA wepredict both the reactions to create methanol and to formradical side products to be endothermic by 0.2 and 0.4 eV,respectively �see Fig. 8�. Additionally, the sextet TS-2 barrierresides slightly above the reactant energies. The remainingGGA discrepancies are comparable to those for the reactionwith hydrogen and include spin splittings at intermediates,

_

___

_

__

_

_

_

__

__

_

___

_

__

_

__

___

_ __

_

FeO+/CH4 1 2 3 Fe+/CH3OHTS1 TS2

-3.0 -3.0

-2.0 -2.0

-1.0 -1.0

0.0 0.0

1.0 1.0

RelativeEnergy(eV)

FeOH++CH3

FIG. 9. �Color online� Comparison of GGA+U=5 eV �blue� and MRCI�black� reaction coordinates for quartet �dashed� and sextet �solid� surfacesof the reaction of FeO+ with CH4.

_

_

_

_

_

_

_

_

__

__ _

_

__

___

_

__

_

__

___

_ __

_

FeO+/CH4 1 2 3 Fe+/CH3OHTS1 TS2

-3.0 -3.0

-2.0 -2.0

-1.0 -1.0

0.0 0.0

1.0 1.0

RelativeEnergy(eV)

FeOH++CH3

FIG. 8. �Color online� Comparison of GGA �blue� and MRCI �black� reac-tion coordinates for quartet �dashed� and sextet �solid� surfaces of the reac-tion of FeO+ with CH4.

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barrier estimates, and the absence of a spin crossover at theexit channel. When we instead compare our GGA+U resultsto highly accurate MRCI calculations, we observe quantita-tive agreement between the two methods �see Fig. 9�. TheGGA+U quartet surface is shallower but not barrierless, inagreement with MRCI results. We should also note that ex-perimentally a large kinetic isotope effect is observed,1 sug-gesting that quantum-mechanical tunneling through classicalbarriers may need also to be considered. We see an improvedestimate of exothermicity of both main and side reactionsand now also preserve spin crossover at the exit channel. Thedescription of the quartet Int-1 geometry is also improved. Acomparison of the accuracy of GGA+U geometries is be-yond the scope of this study due to the higher dimensionalityof the PES over that for the four atom reaction.

We can further examine the accuracy of GGA+U intreating the energetics of these reactions by decomposing theexothermicity for all spins and reaction steps, as summarizedin Table V. While GGA produces errors as large as 1.4 eVfor the methane reaction and 1.1 eV for the hydrogen reac-tion, GGA+U is in very good agreement with accurate quan-tum chemistry. The largest errors, such as the 0.32 eV under-estimate at the sextet TS-2H2

barrier can be systematicallyimproved by extensions that either include a U4s or a locallyaveraged Uscf.

V. CONCLUSIONS

In conclusion, we have demonstrated that a novel Hub-bard U approach can greatly ameliorate the shortcomings ofcommonly employed functionals for transition-metal chem-istry. Using the popular PBE-GGA approximation, we findthat activation barriers, stationary point spin splittings, andreaction energies exhibit errors as large a 1.4 eV for both themethane and the hydrogen systems discussed. By augment-ing our GGA exchange-correlation functional with a +Uterm, we reduced these errors by an order of magnitude to,on average, 0.1 eV with respect to the best available quantumchemical methods �such as CCSD�T� and MRCI� as well asexperimental values. Importantly, the Hubbard U term is notused as a fitting parameter but it is a true linear-responseproperty of the transition-metal complex, which may, in prin-ciple, augment any exchange-correlation density functional.The practical limitations to this approach stem from the com-plexity of the systems that we wish to study: Evolution of thecoordination environment along a global reaction coordinatemay result in local deviations from a globally averaged Hub-bard U or bring to the forefront the role of 4s contributions.Overall, the DFT+U results �here, GGA+U� have been

shown to provide for systematic improvement over GGA forall systems considered thus far by treating, for the first timewithin a DFT framework, the energetics of differing spinsurfaces and electronic states with the same level of accu-racy. The inexpensive linear-response U calculation also actsas a probe for the relative utility of the DFT+U approach fora given transition-metal complex, that is, if the Hubbard Ucalculated is small or nearly zero, a standard description maybe sufficient. We believe that this work paves the way forstudying a myriad of large-scale systems, which contain tran-sition metals with both accuracy and efficiency.

ACKNOWLEDGMENTS

This work was supported by a NSF graduate fellowshipand ARO-MURI DAAD-19-03-1-0169. Computational fa-cilities were provided through the NSF Grant No. DMR-0414849 and the PNNL Grant No. EMSL-UP-9597.

APPENDIX: PSEUDOPOTENTIAL DEPENDENCE

We consider here the effect that different details of theatomic projections as a consequence of pseudopotentialchoice can have on both the determination of the Hubbard Uand its effect on physical properties. It is important to notethat the value of U obtained for the same structure but withdiffering pseudopotentials may easily differ by as much as2–3 eV, particularly if the pseudopotentials were generated indifferent oxidation states. Although this value might seemhigh, it reflects the changes that atomic orbitals undergo as afunction of the total atomic charge. In addition, as the valueof U corresponds to how much correlation is missing fromthe standard LDA or GGA, it follows that the poorer the

TABLE VI. Comparison for several pseudopotentials of structural proper-ties including equilibrium bond length �re�Å�� and harmonic frequency��e�cm−1�� and state splittings �in eV� of the 6�+ and 4� states of FeO+. TheGGA+U results are obtained at the respective Uscf of each pseudopotential.

Fe2+�d5.5s0.5� Fe0.5+�d6.5s1� Fe�d6s2�

GGA �U=0 eV�

6�+ re�Å�, �e�cm−1� 1.59, 902 1.63, 906 1.62, 9014� re�Å�, �e�cm−1� 1.52, 1033 1.57, 1040 1.56, 1038�E6→4 �eV� 0.76 0.82 0.84

GGA+Uscf �eV� 7.0 4.9 5.5

6�+ re�Å�, �e�cm−1� 1.65, 745 1.66, 751 1.66, 7494� re�Å�, �e�cm−1� 1.76, 606 1.75, 613 1.75, 612�E6→4 �eV� 0.50 0.52 0.54

TABLE V. Net energy required for reaction steps �in eV� for GGA, GGA+U �=5 eV�, and post-HF methods,where the post-HF method for the H2 reaction is CCSD�T� and the CH4 reaction is MRCI.

TS-14 TS-16 TS-24 TS-26

Method CH4 H2 CH4 H2 CH4 H2 CH4 H2

GGA −1.14 −1.04 −0.54 −0.59 0.46 −0.92 0.61 −0.79GGA+U −1.07 −1.42 −0.71 −1.05 −0.84 −2.02 −0.87 −2.19Post-HF −0.99 −1.35 −0.75 −0.96 −0.87 −1.82 −0.80 −1.87

134314-11 Addition-elimination reactions of hydrocarbons on bare FeO+ J. Chem. Phys. 129, 134314 �2008�

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match between the oxidation states of the pseudopotentialand the physical system, the higher the value of the linear-response U will be. For reference, we provide here compari-son of FeO+ sextet and quartet state properties calculatedusing pseudopotentials generated in various oxidation states.The Fe2+ �d5.5s0.5�, Fe0.5+ �d6.5s1.0�, and Fe �d6s2, usedthroughout the remainder of this paper� pseudopotentials hadvalues of Uscf of 7.0, 4.9, and 5.5 eV, respectively. As theelectronic configuration of FeO+ states is either d6s1 or d7s0,it is reasonable that the second pseudopotential produces thelowest value of U.

In comparing LDA+U or GGA+U calculations of dif-fering pseudopotentials, total energies and, subsequently,highest occupied molecular orbital-lowest unoccupied mo-lecular orbital or band gap values for different values of Uare not likely to be comparable. However, the properties de-rived from each spin and symmetry’s ground state density,including frequencies, bond lengths, and state splittings, maybe compared. The GGA results for each of the three casesshow significant differences for some structural propertiesand energetic splittings �see Table VI�. While the harmonicfrequencies for 4� and 6�+ of 1040 and 900 cm−1 are iden-tical for all pseudopotentials within the accuracy of the fit-ting procedure, more significant differences are observed inthe bond lengths. The most ionic pseudopotential �d5.5s0.5�yields the shortest GGA value with 4� and 6�+ at 1.52 and1.59 Å, respectively, while the other two pseudopotentialsyield bond lengths within 0.01 Å of each other. The GGAspin splittings are quite close at 0.82 and 0.84 eV for theFe0.5+ and Fe pseudopotentials, respectively, while the moreionic Fe2+ yields 0.76 eV. Upon augmentation of a Hubbardterm using the Uscf appropriate to each respective pseudopo-tential, the bond lengths of all three pseudopotentials arebrought into agreement with each other within 0.01 Å, theharmonic frequencies agree within 6–7 cm−1, and the spinsplittings for all three pseudopotentials agree within 0.04 eV.Overall, an even smaller spread in values is observed forGGA+U than for GGA �see Table VI�. Even a poor pseudo-potential choice will remedy itself so long as the propervalue of U, Uscf, is used. These observations demonstratethat perfect pseudopotential selection is not mandatory, butthe best pseudopotential choice is one which clearly reflectsthe oxidation states and charges of the physical system ofinterest.

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QUANTUM-ESPRESSO is a community project for high-quality quantum-simulation software, based on DFT, and coordinated by Paolo Giannozzi.See http://www.quantum-espresso.org and http://www.pwscf.org.

36 State symmetry assignments were made based upon the absolute totalangular momentum, , of the density and the overall symmetry of thecomponent densities, �, which remain as “good” quantum numbers in ourdensity-functional formalism. Additionally, the preserved spin quantum

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MOLPRO, a package of ab initio programs designed by H. J. Werner, P. J.Knowles, R. Lindh et al., Version 2006.1, see http://www.molpro.net,Cardiff UK, 2006.

41 See EPAPS Document No. E-JCPSA6-129-009837 for multireferenceand single-reference quantum chemistry details including method com-parisons regarding multireference character, as well as HubbardU-dependent geometries of intermediates and TSs at integer values of U.For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.

42 We note that energetic differences at the CCSD�T� level of the differentTS geometries were small and did not noticeably affect barrier heightestimates.

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45 The reaction coordinates of different methods have been aligned at the6Int−1. The curves associated with GGA and GGA+U reaction coordi-nates are splines of points from the minimum energy path as determinedby NEB calculations.

134314-12 H. J. Kulik and N. Marzari J. Chem. Phys. 129, 134314 �2008�

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