Systematic study of ﬁrst-row transition-metal diatomic ...hjkgrp.mit.edu/sites/default/files/pub_reprints/10... · the ionic transition-metal ﬂuorides. The ionic nature of the

Systematic study of first-row transition-metal diatomic molecules:A self-consistent DFT+U approach

Heather J. Kulika� and Nicola MarzariDepartment of Materials Science and Engineering, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA

�Received 3 July 2010; accepted 24 August 2010; published online 17 September 2010�

We present a systematic first-principles study of the equilibrium bond lengths, harmonicfrequencies, dissociation energies, ground state symmetries, and spin state splittings of 22 diatomicmolecules comprised of a first-row 3d transition-metal and a main-group element �H, C, N, O, or F�.Diatomic molecules are building blocks of the key molecular bonding motifs in biological andinorganic catalytic systems, but, at the same time, their small size permits a thorough study by eventhe most computationally expensive quantum chemistry approaches. The results of severaldensity-functional theory �DFT� approaches including hybrid, generalized-gradient, andgeneralized-gradient augmented with Hubbard U exchange-correlation functionals are presented.We compare these efficiently calculated DFT results with the highly accurate but computationallyexpensive post-Hartree–Fock approaches multireference configuration interaction �MRCI� andcoupled cluster �CCSD�T�� as well as experimental values, where available. We show that byemploying a Hubbard U approach, we systematically reduce average errors in state splittings anddissociation energies by a factor of 3. We are also able to reassign the ground state of four moleculesimproperly identified by hybrid or generalized-gradient approaches and provide correct assignmentof all ground state symmetries as compared against experimental assignment and MRCI reference.By providing accuracy comparable to more expensive quantum chemistry approaches with therobust scaling of the generalized-gradient approximation, our DFT+U approach permits the studyof very large scale systems with vastly improved results. © 2010 American Institute of Physics.�doi:10.1063/1.3489110�

I. INTRODUCTION

Diatomic molecules comprised of a single transition-metal atom and an organic ligand atom represent the simplestbuilding block of much larger, catalytically relevanttransition-metal complexes. The highly unsaturated characterof diatomic molecules makes them challenging to study withelectronic structure approaches. The most expensive tech-niques may be used to study these molecules, up to andincluding full configuration interaction, but the sheer numberof possible states that are close in energy in the case of anopen-shell diatomic molecule can make it difficult to ascer-tain the ground state spin or symmetry.1 Of the 22 moleculeswe consider in this work, several molecules have provenchallenging to characterize consistently using both theoreti-cal and experimental approaches, including prototypical hy-dride FeH2,3 and the oxides VO4,5 and NiO.6,7 Close study ofdiatomic molecules can reveal information about related re-active intermediates, as is in the case of high-valent, high-spin metal-oxo species which are active in both the methaneto methanol conversion8 and in halogenation at an enzymeactive site.9

We consider here the challenging but relevant cases oftransition-metal hydrides, carbides, nitrides, oxides, andfluorides. Expansive studies of transition-metal diatomicmolecules have been completed in recent years with a variety

of hybrid functionals and functionals that improve upon thegeneralized-gradient approximation �GGA� with inclusion ofhigher order terms �meta-GGAs�.10–14 The hybrid meta-GGAs in particular12–15 have successfully described proper-ties of the ground states and, in some cases, excited states ofsmall molecules on which they have been tested. Neverthe-less, this is the first completed study of exchange-correlationfunctionals augmented with a “+U” term �referred to here asDFT+U� on a broad class of transition-metal containingmolecules. The DFT+U method includes an approximationthat corrects a standard exchange-correlation functional toreduce self-interaction error. It is based on a Hubbard modelapproach16,17 for treating strongly correlated systems andhas, thus, been widely used in the solid state physicscommunity.18 We recently showed, however, that the sameapproach may be applied to relevant molecular catalyticcomplexes ranging from small molecules1,8 to models ofmetalloenzymes.9

A simplified, rotationally invariant DFT+U approach1,19

adds a Hubbard term that is included in each self-consistentiteration of a DFT calculation and follows the form

EU =U

2 �I,�

Tr�nI��1 − nI�� , �1�

where nI� is the occupation matrix of the relevant localizedmanifold �e.g., 3d electrons� at site I with spin �. This func-tional form, which is tied to the exact correction needed fora�Electronic mail: [email protected].

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simple exchange-correlation functionals in the limit of anatomic system,19,20 penalizes fractional occupations and ap-proaches zero as n approaches 0 or 1.

Occupations of localized atomic levels in a molecule thatenter into our EU expression may typically be determinedfrom �1� projections onto orbitals �e.g., atomic orbitals andWannier functions�, �2� population analysis methods, or �3�integration of the angular-momentum-decomposed chargedensity within a certain distance of the atom.19 Generally, allof these forms lead to the same expression for the occupationmatrix elements,

nmm�I� = �

k,vfkv

� ��kv� �Pmm�

I ��kv� , �2�

where �kv� is the valence electronic wavefunction corre-

sponding to a given molecular state with spin � and fkv� is the

occupation of the molecular level. The Pmm�I are generalized

projection operators that satisfy a number of rules19 includ-ing that orthogonality of projectors for a single site is main-tained. In this work, we project onto the atomic orbitals thatare derived during pseudopotential generation with projec-tors of the form

Pmm�I = ��m

I ��m�I � , �3�

where ��mI is a valence atomic orbital with angular momen-

tum �lm �choice of the localized manifold determines l�.Projections onto atomic orbitals are particularly useful be-cause they are general and permit comparison across manydifferent coordination environments and structures. How-ever, other formulations in which localized molecular orbit-als �e.g., Wannier functions� are used as the basis for theprojections may be useful when only a small variation incoordination environment is considered.

In molecules, this “+U” augmentation of the functionalimproves upon the overhybridization that occurs in bonds asa result of the self-interaction error present in most function-als. As an added benefit, it has been shown19 that the Uvariable in the EU functional may be calculated directly fromlinear-response

�IJ =�2E

��I�J=

�nI

��J, �4�

where �IJ is the linear-response function obtained from ap-plying an arbitrary shift � to the potential on the site J thatresults in a reorganization of the occupations, n on site I. Asimilar expression may be obtained for the noninteractingcase, as a linear shift in the potential can still result in arehybridization that must be removed from our overall ex-pression to determine U. The value of Hubbard U for a sys-tem is then obtained as

U = �0−1 − �−1, �5�

where � is simply a number if we are only interested in asingle manifold and site or it is a matrix in the case of mul-tiple sites or manifolds. Overall, since U is directly calcu-lated as a system-dependent property and not treated as aparameter, it provides an improvement over other functionalforms that rely on fitting multiple parameters to achieve the

best results. In the following, we highlight the utility of thisDFT+U approach on different classes of small molecules.

II. METHODS

Plane-wave density-functional calculations were com-pleted with the QUANTUM ESPRESSO package21 using thePerdew–Burke–Ernzerhof22 GGA. We also augmented thisstandard GGA functional with a self-consistent, linear-response, Hubbard U term, as previously outlined.1,19 Ultra-soft pseudopotentials were used with a plane-wave cutoff of30 Ry for the wavefunction and 300 Ry for the charge den-sity to ensure basis-set-dependent convergence of forces andspin state splittings.23 Previous works have demonstrated thatultrasoft pseudopotentials perform well when comparedagainst all-electron results for both first-row elements24 andtransition-metals25 including the challenging cases of Fe, Co,and Ni. We have also recently shown that some propertiesare sensitive to the charge state in which the pseudopotentialwas generated, but that any variation between results fromdifferent pseudopotentials is actually decreased with inclu-sion of a Hubbard term.8 The harmonic frequencies wereobtained from a linear fit to the first derivative of the energyat equilibrium.

Post-Hartree–Fock �HF� approaches were employed toprovide an accurate but computationally expensive referencefor the density-functional calculations. Both single-referenceand multireference methods were employed using differentcodes, but the same Pople-style 6-311++G�3df,3pd� basisset was used for all post-HF calculations to ensure consis-tency. The single-reference method, coupled cluster withsingles, doubles, and perturbative triples �CCSD�T��, wascarried out as implemented in GAUSSIAN,26 and the T1 diag-nostic was used as a tool to identify structures with poten-tially strong multireference character or instability in thetriples term.27 All multireference configuration interaction�MRCI� calculations were carried out using MOLPRO.28 Forthis work we primarily compare against experimental valuesand provide the single-reference and multireference results toprovide both a comparison of the agreement between experi-ment and expensive but, in some cases, more accuratepost-HF approaches. The post-HF approaches also can beused to determine whether discrepancies between density-functional theory results and experiment are due to differ-ences in the states identified computationally and experimen-tally. For a more detailed consideration of the multireferencecharacter of transition-metal complexes and how that influ-ences DFT+U results, see the supporting information pro-vided along with Ref. 8. Hybrid density-functional calcula-tions �B3LYP� were also employed using GAUSSIAN as acomparison to the plane-wave DFT results.

III. RESULTS

In order to examine bonding motifs we expect to be thebuilding blocks of catalytic mechanisms, we consider prima-rily the bonds transition-metals form with first-row atomligands. The hydrides are useful for understanding the rolethat 4s electrons play in transition-metals, as the majority ofbonding in these systems occurs via M 4s-H 1s interactions.

114103-2 H. J. Kulik and N. Marzari J. Chem. Phys. 133, 114103 �2010�





These systems also require the introduction of a Hubbard Uterm on the 4s manifold, termed U4s, which we discuss indetail. Transition-metal oxides are most representative of keycovalent bonds in reaction intermediates: that is, high-spinmetal-oxo bonds participate in a number of key catalyticcycles. Transition-metal carbides and nitrides have beenstudied much less extensively than the oxides. Finally, wecontrast the carbide, nitride, oxide, and hydride cases withthe ionic transition-metal fluorides. The ionic nature of thefluorides diminishes the role and necessity of a U term, sinceinteger electron donation rather than fractional electron shar-ing predominates in ionic bonds. Lastly, the size and relativesimplicity of diatomic molecules make them particularlyideal to study nuances of the GGA+U method. The differ-ences in isoelectronic compounds and numerical stability ofthe GGA+U method are considered.

A. Transition-metal hydrides: The role of U4s

The electronic structure of transition-metal hydrideshelps to highlight the broader applicability of a DFT+U ap-proach to other valence manifolds aside from 3d states. Thedominant mode of bonding in the metal hydrides �M and H,respectively� is M�4s�-H�1s� �-bonding interactions. Theelectron density from the 3d manifold plays a more indirectrole in bonding through limited M�3dz2�-H�1s� �-bondinginteractions as well as the significant 4s-3dz2 hybridizationpresent in �4s orbitals. All other 3d density is highly local-ized on the metal, and the corresponding states closely re-semble the isolated atomic density. It follows, therefore, thatthe U3d of all of these systems is quite low, ranging fromzero to less than 3 eV. In turn, a U4s should ensure accuratestructural and energetic descriptions of the electronic statesin question. We consider several prototypical cases: early-�CaH, ScH�, mid-�CrH, MnH, FeH�, and late-�CuH, ZnH�transition-metals in order to illustrate key trends in the hy-drides.

Commensurate with observations of low levels of�-bonding from 3d-1s hybridization and nearly empty or,conversely, nearly filled 3d levels, the early and latetransition-metal hydrides share in common low values of U3d

�between 0 and 1.5 eV�. The overall trends for the value ofU3d show a maximum at the midrow transition-metal hy-drides, compared to the empty or filled 3d transition-metalhydrides �see Fig. 1�. The value of U3d is most critical for thehalf-filled hydrides; thus, it is important to note that thelinear-response approach we use to assign values of U is keyhere both for selecting the appropriate projection manifoldsand determining whether a Hubbard U term is necessary fordescriptions in bonding. The values of the U4s exhibit a trendsimilar to that of the U3d values. The lowest U4s,0.6–0.7 eV, is observed at the extrema: CaH and ZnH.Moving toward the center of the periodic table, values of U4s

increase from 1.0 �CuH� to over 2.0 eV �ScH, CrH, MnH�, inseveral cases exceeding the value of U3d for that system.Overall, the values of linear-response U on the 3d and 4smanifolds confirm the intuitive chemical picture that 4sbonding interactions are important, particularly in the

mid-row cases where there is a competition between 4s- and3d-derived orbital occupation. Here, GGA+U may provekey in describing the splitting of the lowest electronic statesof the approximately half-filled 3d manifold cases, CrH,MnH, and FeH, which can be very sensitive to the methodemployed.2,3,13,29–32 The lowest-lying electronic state for theearly and late transition-metal hydrides is well separated en-ergetically and so most methods, including GGA, can cor-rectly characterize these states.

The structural properties, including equilibrium bondlength, frequency, and dissociation energy, are calculated forthe first-row, transition-metal hydrides �Table I�. In additionto the ground state, these properties were also calculated incases where a second spin’s lowest electronic state residedwithin 2 eV of the ground state. For each electronic state, thestructural and vibrational properties were calculated at theGGA and GGA+U levels of theory and compared againstavailable experimental values, where available �many ofwhich are summarized in Ref. 40�. In addition to the stan-dard approach of applying only a U3d, the combined effect ofthe linear-response U3d and U4s were considered �referred tohereafter as GGA+U3d/4s�.

For the hydrides with nearly empty 3d occupations,CaH, and nearly fully, ZnH, the small values of linear-response U correspond to minimal changes in structural andenergetic properties when comparing GGA and GGA+U ap-proaches. In the case of X 2�+ CaH, there are only threevalence electrons, which result in a � bond between Ca�4s�and H�1s� and a nonbonding 4s orbital on Ca.41 In the caseof the X 2�+ ZnH, the bonding also is dominated by 4s in-teractions with a closed 3d10 manifold and a �2��,1 bondingscheme.

The calculated equilibrium bond lengths for these twocases are lengthened by under 0.01 Å upon addition of eithera U3d or U3d/4s term. The harmonic frequencies of these ex-trema also vary by less than 50 cm−1 with addition of the Uterm, and the reduction in frequency is not necessarily asignificant improvement over GGA. Dissociation energiesfor these states also varies by only 0.10 eV with a U3d/4s

0.4

1.8

1.3

1.5

2.8

1.3

0.0

0.7

2.2

2.0 2.0

1.3

1.0

0.6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

CaH ScH CrH MnH FeH CuH ZnH

U3d

U4s

FIG. 1. Values of Hubbard U �in eV� for several transition-metal hydridesordered with respect to their placement in the periodic table. The values ofU3d �blue� and U4s �red� are maximal for the midrow transition-metal hy-drides and decrease for nearly full shell cases, such as ZnH, or nearly emptycases, such as CaH.

114103-3 DFT+U for transition-metal diatomic molecules J. Chem. Phys. 133, 114103 �2010�





approach, and, in this case, the inclusion of a U generallyimproves properties as a result of the GGA tendency to over-estimate dissociation energies.

Results for CuH are similar to those for ZnH, thoughwith one less 4s electron available for bonding. The X 1�+

state has a valence electron configuration with a full 3d10

manifold and bonding occurring primarily through a doublyoccupied �2�4s�, while the lowest triplet state, a 3�+ CuHinstead has a �� occupation. Structural �Table I� and ener-getic �Table II� properties for both CuH states are largelyimproved by GGA+U3d/4s. Results for the early and latetransition-metal hydrides confirm the trend implied by the

TABLE I. Structural properties of transition-metal hydrides. Bond lengths �angstrom�, frequencies �cm−1�, and dissociation energies �eV� are compared againstexperimental values for CaH �Ref. 33�, ScH �Ref. 34�, CrH �Refs. 35 and 36�, MnH �Ref. 37�, FeH �Ref. 37�, CuH �Ref. 38�, and ZnH �Ref. 39� �ground stateproperties from Ref. 40�.

B3LYP GGA GGA+U3d GGA+U3d/4s CCSD�T� MRCI Expt.a

CaH re 1.985 1.972 1.977 1.978 2.020 2.042 2.0002�+ �e 1292 1295 1291 1287 1296 1362 1298

De 2.00 1.95 1.93 1.83 1.70 1.74 1.78ScH re 1.750 1.760 1.772 1.783 1.787 1.787 1.7751�+ �e 1621 1603 1591 1596 1601 1589 1547

De 3.40 2.59 2.15 2.04 2.10 2.06 2.06ScH re 1.844 1.844 1.847 1.877 1.900 1.908 ¯

3 �e 1460 1481 1446 1453 1467 1437 ¯

De 2.14 2.33 2.16 2.07 2.08 1.81 ¯

CrH re 1.632 1.600 1.626 1.620 1.630 1.742 1.6724�+ �e 1737 1709 1706 1741 1727 1636 ¯

De 1.99 1.96 1.90 1.88 1.91 1.98 ¯

CrH re 1.662 1.650 1.664 1.668 1.652 1.678 1.6566�+ �e 1632 1657 1611 1586 1531 1581 ¯

De 2.38 2.16 2.07 1.98 2.05 1.92 2.03MnH re 1.632 1.592 1.600 1.617 1.691 1.662 1.6055�+ �e 1737 1729 1695 1692 1509 1565 1720

De 1.99 1.84 1.75 1.66 1.71 1.59 ¯

MnH re 1.735 1.714 1.726 1.725 1.740 1.744 1.7317�+ �e 1507 1546 1530 1527 1552 1580 1548

De 1.70 1.75 1.62 1.38 1.40 1.25 1.35FeH re 1.558 1.560 1.564 1.587 1.570 1.583 1.5894 �e 1734 1916 1853 1824 1829 1644 1827

De 1.81 2.13 2.00 1.78 1.65 1.78 1.70FeH re 1.682 1.670 1.681 1.701 1.685 1.690 1.7706 �e 1581 1626 1565 1539 1577 1648 ¯

De 2.31 1.81 1.73 1.51 1.58 1.51 ¯

CuH re 1.484 1.487 1.495 1.500 1.486 1.460 1.4631�+ �e 1854 1973 1887 1924 1884 2229 1941

De 2.72 2.79 2.68 2.64 2.58 2.65 2.63CuH re 1.585 1.590 1.598 1.599 1.625 1.617 1.5673�+ �e 1485 1502 1477 1481 1603 1716 ¯

De 0.30 0.49 0.47 0.47 0.73 1.06 ¯

ZnH re 1.622 1.605 1.605 1.609 1.605 1.586 1.5942�+ �e 1512 1543 1543 1503 1527 1767 1603

De 1.00 1.05 1.05 0.95 0.98 1.04 1.01

aReference 40.

TABLE II. Energy splitting �in eV� of the two lowest spin states of several hydrides. In the cases of CrH �Refs.35 and 36�, MnH �Ref. 37�, FeH �Ref. 37�, and CuH �Ref. 38�, experimental values are available and comparedagainst the B3LYP, GGA, GGA+U, CCSD�T�, and MRCI results. Negative values indicate an incorrect groundstate.

B3LYP GGA GGA+U3d GGA+U3d/4s CCSD�T� MRCI Expt.

ScH 1�+→ 3 0.12 0.13 0.08 0.12 0.12 0.25 ¯

CrH 6�+→ 4�+ 1.18 1.65 1.52 1.40 1.37 1.10 1.39MnH 7�+→ 5�+ 0.12 0.01 0.13 0.21 0.22 0.30 0.21FeH 4→ 6 0.50 0.57 0.45 0.26 0.27 0.27 0.24CuH 1�+→ 3�+ 2.41 2.24 2.31 2.43 2.45 2.48 2.42






values of linear-response U: a GGA description of the prop-erties of these systems is largely sufficient, but a GGA+Uapproach does not significantly undermine the proper de-scription of bonding already present in GGA.

While seemingly similar to the cases of CaH, CuH, andZnH, ScH, with roughly one 3d electron available for bond-ing, does introduce complexity not seen previously in theearly or late transition-metal hydrides. The ground state ofScH is experimentally known to be X 1�+,with the valence electron configuration of�2�Sc 4s /3d-H 1s��2�Sc 4s ,nonbonding�.34,40,42 Earliertheoretical predictions identified a 3 ground state43,44 thatpreferentially occupied ��4s��3d�. While no experimentalresults are available for the a 3 state, higher level resultsare able to confirm the X 1�+ ground state of ScH. The re-sults from GGA suggest a 3 ground state, but GGA+U3d/4s predicts the correct ground state and obtains im-proved agreement with both CCSD�T� and MRCI �Table II�.Structurally, the GGA+U3d/4s results are in improved agree-ment with CCSD�T�, MRCI, and experiment for the X 1�+

state. In particular, the dissociation energy is improved, re-ducing an overestimate by GGA of over 0.5 eV to very goodagreement within 0.02 eV. The equilibrium bond length isoverelongated, but it still provides an improved agreementwith experiment over GGA. The equilibrium frequency forGGA+U3d/4s agrees very well with CCSD�T� and MRCI,which also predict a harmonic frequency roughly 50 cm−1

larger than that obtained experimentally. While experimentalstructural properties are not available for the a 3 case, theGGA+U bond length, harmonic frequency, and dissociationenergy are all in improved agreement with availableCCSD�T� and MRCI calculations over GGA.

We next consider the most challenging cases, the mid-row transition-metal hydrides, CrH, MnH, and FeH. Formost of the midrow hydrides, the splitting of the two lowestspin states is quite small because they correspond to ferro-magnetic and antiferromagnetic coupling of the hydrogen 1sspin with the lowest energy spin and configuration of theisolated transition-metal atom. Manganese hydride is an ex-cellent example; the hydrogen 1s density can couple to theMn atom density to produce either a high-spin septet state,7�+, or, alternatively, a low-spin 5�+ state. Both early andmore recent density-functional calculations have shown apreference for the low-spin 5�+ state;13,29 however, the ex-perimental ground state of MnH has been determined to be7�+.37,45,46 An isolated Mn atom has a 6S5/2 ground statecorresponding to 3d54s2 valence occupation. The major elec-tronic difference between the two molecular states is that 1sdensity in the high-spin state, 7�+, populates exclusively spinup 4s-derived orbitals, including bonding ��4s� and antibond-ing ��4s

� �. The quintet preferentially occupies exclusivelybonding �4s molecular orbitals in both the up and down spinchannels �see Fig. 2�.

Traditional views of molecular bonding would predictthe 5�+ state to be lower in energy, since formally bondingorbitals are commonly occupied before the associated anti-bonding orbitals. However, there are several physical expla-nations for why the 7�+ state is in fact the ground state. Thedifference in bond lengths, re, from the quintet to the septet

is only 0.12 Å �from 1.62 to 1.73 Å for GGA+U3d/4s�. Whilewe expect the bond order to be reduced nominally by one ingoing from 5�+ to 7�+, the �4s

� state is only weakly antibond-ing and possesses between nonbonding and antibondingcharacter, a feature previously observed in photoelectronspectra.47 It follows, also, that the placement of �4s

� densityon the Mn site minimizes overlap with the other 3d statesalmost as much as the �4s orbital does via 4s-1s hybridiza-tion, but in this case it achieves it by occupying a diffuseorbital away from the molecular bond �see Fig. 2�. Finally,the presence of H 1s density in the minority spin channel for5�+ induces charge transfer from Mn 3d states into the spindown �3d orbital where the density had been weak in thecase of the septet. This reduces the net magnetic moment ofthe 3d states further, increasing the overall energetic cost offorming the 5�+ state with respect to 7�+.

A GGA description finds the 5�+ and 7�+ states to benearly degenerate. The B3LYP hybrid functional does worsethan GGA by stabilizing the wrong state, 5�+ by 0.12 eVwith respect to the correct septet ground state. However,upon inclusion of a U3d/4s term to GGA, we recover a split-ting of 0.21 eV, in excellent agreement with both CCSD�T�and experiment.37 The structural properties, on the otherhand, change relatively little with the 5�+ and 7�+ statesexhibiting similar bond elongation �0.01–0.03 Å� for GGA+U3d/4s over GGA and a reduction in harmonic frequency byabout 20–40 cm−1. Lastly, the most significant change, re-sulting from the difference in relative energy at the minimabetween GGA and GGA+U, the septet and quintet dissocia-tion energies decrease from 1.75 to 1.38 eV and from 1.84 to1.69 eV, respectively. While the combined GGA+U3d/4s ap-proach modulates structural properties in these hydrides onlyvery weakly, the relative spin state ordering and properground state is only determined with a U term on both 3dand 4s states.

Iron hydride �FeH� presents another challenging case fortheoretical approaches, and a relatively large amount isknown experimentally about its low-lying electronicstates.37,48–50 The ground state is experimentally known to beX 4 with a 3d7 derived valence configuration of �2�2�3 anda 4s derived configuration that doubly occupies a bonding �orbital.37 The valence configuration of the excited a 6 stateconsists instead of 3d6 derived ��2�3 orbitals and a greater3d-4s mixing in the 4s2 derived �2��. Early theoretical ap-proaches predicted a quartet ground state,2 while more re-cently, a variety of post-Hartree–Fock approaches were em-

�4s

�3d

�*4s��

��

��

��

��

��

��

��

FIG. 2. The valence orbitals which participate in bonding for the 7�+ and5�+ states of MnH �left� as well as qualitative occupations an ordering of themolecular orbitals in both of these electronic states �right�.






ployed to achieve a quartet-sextet splitting consistent withexperiment.3,30 Both our GGA and B3LYP results overesti-mate the quartet-sextet splitting, while the GGA+U3d/4s re-sults �0.26 eV� are in very good agreement with the 0.24 eVexperimental value. The GGA description of 4 and 6

structural properties is slightly overbinding with respect toexperimental values-the bond length is underestimated by0.03 Å in the quartet case and 0.10 Å in the sextet case.Harmonic frequencies and dissociation energies are alsooverestimated for the GGA description of 4 by 100 cm−1

and 0.40 eV, respectively, but GGA+U3d/4s produces agree-ment within 3 cm−1 and 0.08 eV for the same properties.Overall, this GGA+U result improves greatly over the ap-parent excessive delocalization of the 3d manifold in bothGGA and B3LYP.

In contrast with MnH and FeH, the X 6�+ ground stateof chromium hydride �CrH� is well separated from otherlow-lying states �Table II�. Both X 6�+ and the excited a 4�+

state have a valence electron configuration corresponding to�2��2�2 where the bonding � orbitals are derived fromM�4s�-H�1s� hybridization and �� is a roughly nonbondingorbital of 4s and 3dz2 character localized on the metal. The6�+ state is adiabatically connected to the ground state, 7S,of the chromium atom, while the 4�+ state is derived from anexcited state of the atom, 5S, which is excited by nearly 1 eVwith respect the ground state.31 As a result, GGA structuraland energetic descriptions are in improved agreement withexperiment over the other midrow TM hydrides. The GGAbond length for the sextet ground state is within 0.006 Å ofexperiment, but the GGA+U3d/4s result overelongates thebond slightly. Nevertheless, the slight overestimate of De byGGA is improved upon with GGA+U3d/4s for this state, andGGA+U3d/4s also improves the structural description of thequartet. The 6�+→ 4�+ splitting is slightly underestimated byB3LYP and overestimated by GGA with respect to experi-ment and CCSD�T� �Table II�. The GGA result is qualita-tively correct, but GGA+U3d/4s achieves quantitative agree-ment, yielding a splitting within 0.01 eV of experiment.

Theoretical splittings of the transition-metal hydrides arecompared to experimental results, where available, in TableII. All methods perform well for the singlet-triplet splitting inCuH, which has a full 3d10 manifold in both states. Thepartially occupied cases, CrH, MnH, and FeH, are more chal-lenging. The hybrid functional, B3LYP, fails to correctly as-sign the ground state of MnH and produces errors around 0.2eV for FeH and CrH. The largest errors exhibited by GGAare in the inversion of the singlet and triplet state ordering inScH and in the near-degeneracy predicted for MnH. Overall,a GGA+U approach with a U4s as well as a U3d vastly im-proves the prediction of state ordering and splitting-from er-rors as large as 0.20 eV for GGA to errors on average around0.01 eV for GGA+U. Structural properties are also improvedoverall with GGA+U3d/4s �Table I�. GGA errors, primarilymarked by overestimates in the harmonic frequency and dis-sociation energy and underestimates of bond lengths are re-duced by GGA+U in most cases.

B. Transition-metal carbides and nitrides

The transition-metal carbides and transition-metal ni-trides are covalent species that are relatively poorly studiedcompared to the other molecules considered in this work.Carbides and nitrides are also quite challenging to study be-cause of the open-shell, closely spaced levels of the isolatedcarbon and nitrogen atoms. We consider three cases, FeC,CoC, and CrN as examples of this class of diatomic mol-ecules and first provide the calculated values of Hubbard Uin Table III.

The low-lying states of iron carbide have been well-characterized both experimentally51–53,58 andcomputationally.56,58 The ground state of FeC, 3, has a va-lence configuration best characterized as �2�4�3��4s�1. Thelowest-lying singlet state, 1, differs from the ground stateonly by the spin-coupling of a � orbital. Both GGA andB3LYP underestimate the triplet-singlet splitting by 0.08 and0.17 eV, respectively. GGA+U instead slightly overestimatesthe splitting �0.50 eV� but in overall improved agreementwith the experimental value of 0.43 eV �Table IV�. Structuraldata are known experimentally for both states and GGAoverbinds in both cases. The GGA results underestimate re

by about 0.02 Å and overestimate both the harmonic fre-quency and dissociation energy by around 100 cm−1 and 0.6eV, respectively �Table V�. GGA+U reverses these trends,though too severely, and, in turn, it overestimates the bondlength by about 0.04–0.06 Å and underestimates both theharmonic frequency and dissociation energy by at least100 cm−1 and 0.2 eV, respectively.

Several experimental works have discussed the proper-ties of the 2�+ ground state of CoC as well as a 2 state thatlies within 0.08 eV.54,59,60 While no excited quartet or sextetstates have been isolated, theoretical results57 indicate 4 asa likely candidate for the lowest-lying quartet state. The 2�+

state is described as �2�4�4��4s�1 and the 4 state differs by

TABLE III. Linear-response values of Hubbard U �in eV� for the transitioncarbides, FeC and CoC and nitrides, CrN in their low spin �L.S.� and high-spin �H.S.� states. The standard result, U0, is shown, as well as the self-consistently obtained Uscf for each state and an average value for both states,Uscf,av.

L.S. U0 Uscf H.S. U0 Uscf Uscf,av

FeC 1 4.75 4.38 3 4.01 4.17 4.27CoC 2�+ 5.05 4.85 4 4.92 4.71 4.78CrN 2�− 4.26 4.28 4�− 3.85 3.87 4.07

TABLE IV. Energy splitting �in eV� of the two lowest spin states of severalcarbides and nitrides. In the cases of FeC �Ref. 53� and CrN �Ref. 55�,experimental values are available and compared against the B3LYP, GGA,and MRCI results �Refs. 56 and 57�.

B3LYP GGA GGA+U MRCIa Expt.

FeC 3→ 1 0.26 0.35 0.50 0.42 0.43CoC 2�+→ 4 0.20 0.64 0.80 0.82 ¯

CrN 4�−→ 2�− 0.56 0.83 1.17 0.72 ¯

aLiterature MRCI results for FeC �Ref. 56� and CoC �Ref. 57� are presentedhere. Since literature MRCI results were unavailable for CrN, our own cal-culations are provided.






occupying a majority spin �� orbital instead of a minorityspin � orbital. The doublet-quartet splitting obtained fromB3LYP �0.20 eV� and GGA �0.64 eV� are both too smallwith respect to MRCI results �0.82 eV�.57 Inclusion of aHubbard term improves agreement of the 2�+→ 4 splitting�0.80 eV� to within 0.02 eV of the MRCI value. GGA+Ualso provides an improved bond length �1.528 Å� and har-monic frequency �981 cm−1� over GGA when comparedagainst both MRCI and experiment. These results suggestthat the overbinding exhibited by GGA in the case of CoC isproperly treated with a Hubbard term and does not produceexcessive underbinding, as was the case in FeC. The fuller3d manifold in Co with respect to Fe may have prevented theunderpopulation of bonding orbitals that apparently contrib-utes to excessive bond elongation in 3 FeC.

Chromium nitride is known experimentally55,61,62 tohave a 4�− ground state with a �2�4�2��4s�1 valence con-figuration. Experimental results indicate the lowest-lyingdoublet state to be 2�−, which has a configuration that differsonly by the spin of the valence ��4s� orbital, at around 1 eVabove the quartet state, but this determination relies on par-allels with the isoelectronic VO molecule. The GGA+U de-scriptions of 4�− and 2�− states are in reasonable agreementwith MRCI results �Table V�.

Overall, GGA+U does not significantly improve thestructural descriptions of the few examples provided here�FeC, CoC, and CrN� over GGA. The small number of mol-ecules considered as well as the limited experimental datacan inflate the apparent average errors �see Table XIV�. Po-tential sources of errors in GGA+U descriptions may be aresult of the fact that here the 2p manifold of carbon ornitrogen is distinctly partially filled. The interplay betweenthe occupation of the 2p states and 3d on the metal could betreated through inclusion of a U term on the 2p states, per-

haps in the formalism of the recently introduced LDA+U+V approach.63 Additionally, these molecules may belong toa class of systems in which the linear-response U dependsstrongly on internuclear separation. Ongoing work also in-corporates such variations in order to combat the tendency ofstandard GGA+U to overelongate some bonds.

C. Transition-metal oxides

We consider the low-lying states of eight transition-metal oxides spanning from CaO to ZnO. The oxides arefundamental because high-valent, high-spin M–O speciesplay a key role in many reactions �e.g., the addition-elimination formation of methanol from methane1,8 and al-kane halogenation in the active site of SyrB2�.9,73 Addition-ally, the transition-metal oxides often have closely spacedlevels of differing spin and symmetry that challenge the pre-dictive abilities of many computational approaches. Theshort metal-oxo bonds in these molecules are characterizedby covalent interactions between M�3d� and O�2p� electronsto form � and � molecular orbitals. The GGA+U approachcan play a distinct role in tuning the molecular bonds of thelow-lying states of early-�CaO, TiO, VO�, mid-�MnO, FeO�,and late-�CoO, NiO, ZnO� transition-metal oxides. The val-ues of the linear-response U0 and the self-consistent form,Uscf, for all of the relevant transition-metal oxides have beencalculated �see Table VI�. For the 3d0 configuration of CaO,a zero U0 on the 3d manifold is obtained. In this case, weobtain a small U on the 4s1 states which do participate inbonding.64 Similarly, the 3d10 ZnO molecule exhibits nomeasurable U on the 3d states. Intermediate transition-metaloxides exhibit values of U between 3 and 5.5 eV with thehighest values of U being for the early- to midrow oxidesTiO �5.6 eV� and VO �4.6 eV�.

TABLE V. Structural properties of transition-metal carbides �FeC �Refs. 51–53� and CoC �Ref. 54�� and nitrides �CrN� �Ref. 55� at several levels of theory.Bond lengths �angstrom�, frequencies �cm−1�, and dissociation energies �eV� are compared against experimental values, where available �obtained from Ref.40 if not otherwise noted�. Literature MRCI results are presented here, where available �Refs. 56 and 57�.


FeC re 1.604 1.578 1.659 1.605 1.5913 �e 608 901 607 810 867

De 3.59 4.57 3.68 3.41 3.90FeC re 1.566 1.549 1.601 1.585 1.5741 �e 741 1050 835 882 938

De 4.02 4.55 4.43 4.66 ¯

CoC re 1.509 1.513 1.528 1.548 1.5612�+ �e 917 1092 981 965 934

De 3.43 4.24 3.53 3.49 ¯

CoC re 1.616 1.602 1.620 1.641 ¯

4 �e 821 933 847 863 ¯

De 3.28 3.60 2.73 2.69 ¯

CrN re 1.547 1.554 1.675 1.572 1.5604�− �e 861 980 683 939 1050

De 3.53 4.17 3.93 3.21 4.20CrN re 1.513 1.527 1.559 1.555 ¯

2�− �e 1149 1110 892 1090 ¯

De 4.66 4.80 2.73 2.18 ¯

aLiterature MRCI results for FeC �Ref. 56� and CoC �Ref. 57� are presented here. Since literature MRCI results were unavailable for CrN, our owncalculations are provided.






CaO and ZnO are examples of oxides that are not ad-equately treated with a traditional U on 3d states alone. Thelowest-lying singlet and triplet states of CaO have been iden-tified experimentally.64,74 The 1�+ ground state is character-ized by a valence occupation of �2�4, where the orbitals arepredominantly derived from O 2s and 2p atomic orbitals.The lowest triplet state, 3�, is described as �2�3��4s�1 andis closely related to the A 1� state. The experimental 1�+

→ 3� splitting is 1.02 eV �Table VII�. The GGA singlet-triplet splitting is 0.3 eV too large, while inclusion of a cal-culated U4s=1.6 eV �the U3d is zero; see Table VI� bringsthe GGA+U result into vastly improved agreement within0.01 eV �Tables VI and VII�. The effect of this U4s on struc-ture is subtle: GGA+U4s bonds are longer than GGA forboth 1�+ and 3� by about 0.02 Å and harmonic frequenciesare reduced by about 30–40 cm−1 �Table VIII�. The GGA+U structural properties are generally in improved agree-ment with experiment over GGA, but re for the 3� state isstill underestimated. In the case of ZnO, the calculated valueof U is zero and we therefore only have GGA results �TableVI�. ZnO has been experimentally observed in a 1�+ groundstate69–71 and in the low-lying 3� state.71 The highest occu-pied valence orbitals in the 1�+ state are in the configuration��4s�2��,4, while the 3� state differs by population of a ma-jority spin �� over a minority spin �� orbital. Previous com-putational results75 indicated that B3LYP inverts the singlet-triplet splitting, and our B3LYP results confirm this �TableVII�. On the other hand, GGA correctly predicts the orderingbut slightly overestimates the 1�+→ 3� splitting �0.51 eV

versus 0.31 eV for experiment�. Structurally, GGA bondlengths and harmonic frequencies are within 0.02 Å and30 cm−1 of experiment. The hybridization of 3d electronsplays a less prominent role in both CaO and ZnO, and, there-fore, properties of these molecules are similarly treated bydiffering exchange-correlation functionals.

The early- to midrow transition-metal oxides TiO andVO have the largest values of Uscf for all the oxides consid-ered at 5.6 and 4.6 eV, respectively. The ground state 3 TiOhas been characterized including the triplet state’s interactionwith several excited singlet states.72,76 The valence electronconfiguration of X 3 is �2�4��4s�1�1 and the lowest singletstate, a 1, differ only in the spin-coupling of the highest��4s� and � molecular orbitals. While B3LYP underestimatesthe triplet-singlet splitting �0.24 eV� with respect to experi-ment �0.43 eV�, GGA and GGA+U values of 0.40 and 0.41eV, respectively, are in improved agreement. Structurally,GGA yields bond lengths and frequencies in good agreementwith experiment and MRCI, and GGA+U preserves thesefeatures, save for slightly underestimating the �e of 3. TheHubbard term most noticeably improves dissociation energyestimates with respect to experiment and MRCI compari-sons. Numerous studies have been completed on the lowest-lying quartet and doublet states of VO.4,5,77–80 The groundstate of VO is 4�− with an electron configuration of�2�4��4s�1�2. Results from anion photoelectronspectroscopy4 suggest the lowest doublet state to be 2�−,which differs from the quartet configuration only in the spin-coupling of ��4s�. More recent experimental results focus

TABLE VI. Linear-response values of Hubbard U �in eV� for the transition-metal oxides CaO, TiO, VO, MnO,FeO, CoO, NiO, and ZnO in their L.S. and H.S. states. The standard result, U0, is shown, as well as theself-consistently obtained Uscf for each state and an average value for both states, Uscf,av. For CaO, the U listedis obtained on the 4s manifold, as the 3d state occupation is minimal, and this is denoted in italics in the table.


CaO 1�+ 1.25 1.51 3� 1.45 1.62 1.57TiO 1 5.01 5.12 3 5.72 6.03 5.58VO 2�− 4.67 4.82 4�− 4.39 4.46 4.64MnO 4� 3.32 3.53 6�+ 3.39 3.40 3.47FeO 7�+ 3.14 3.15 5 2.94 3.01 3.08CoO 4 3.71 3.73 6 3.65 3.68 3.71NiO 1� 3.89 4.06 3�− 4.23 4.34 4.20ZnO 1�+ 0.00 0.00 3� 0.00 0.00 0.00

TABLE VII. Energy splitting �in eV� of the two lowest spin states of several oxides. In the cases of CaO �Ref.64�, TiO �Ref. 72�, VO �Ref. 4�, MnO �Ref. 66�, FeO �Ref. 68�, NiO �Ref. 7�, and ZnO �Ref. 71�, experimentalvalues are available and compared against the B3LYP, GGA, GGA+U, CCSD�T�, and MRCI results. Negativevalues indicate an incorrect ground state.

B3LYP GGA GGA+U CCSD�T� MRCI Expt.

CaO 1�+→ 3� 1.00 1.32 1.01 0.92 0.93 1.02TiO 3→ 1 0.24 0.40 0.41 0.55 0.46 0.43VO 4�−→ 2�− 0.49 0.79 0.64 0.69 0.68 0.70MnO 6�+→ 4� 1.22 1.49 1.04 1.14 1.15 1.08FeO 5→ 7�+ 0.98 0.76 0.24 0.15 0.13 0.14CoO 4→ 6 0.86 0.82 0.85 0.74 0.80 ¯

NiO 3�−→ 1� 0.40 0.78 1.24 0.84 1.08 1.26ZnO 1�+→ 3� 0.06 0.51 0.51 0.21 0.25 0.31






instead on a low-lying 2 state,5 but published MRCIresults67 support the 2�− assignment. As with TiO, B3LYPunderestimates the quartet-doublet splitting �0.49 eV� withrespect to experiment �0.70 eV�. In this case, the GGA 4�−

→ 2�− splitting is a slight overestimate �0.79 eV�, whichGGA+U improves upon �0.64 eV�, despite a slight overcor-rection. The dissociation energy of the X 4�− state is im-proved by GGA+U over GGA. In both TiO and VO, GGA+U improves energetic descriptions while qualitatively pre-serving structural descriptions already well-described byGGA.

The next two transition-metal oxides, MnO and FeO,have lower Uscf,av values around 3 eV �Table VI�. The 6�+

ground state of MnO and low-lying quartet states have beenstudied experimentally.66,81 The X 6�+ state is best describedas �2�4�2��4s�1��,2 and the lowest-lying quartet state, 4�,favors population of a minority spin ��4s�� molecular orbitalover a majority spin ��. The structural description of the 6�+

state in GGA+U �re=1.674 Å, �e=817 cm−1 , De

=3.91 eV� is in improved agreement over B3LYP and GGA,both of which overbind. The harmonic frequency of 4�,which is roughly estimated from photoelectronspectroscopy,66 is much higher when calculated in all meth-ods than in experiment. Since agreement between GGA+Uand both CCSD�T� and MRCI is excellent for this state, it islikely that the experimental value should be refined. Thesextet-quartet splitting is 1.08 eV experimentally, and bothB3LYP �1.22 eV� and GGA �1.49 eV� produce slight over-estimates. Inclusion of a Hubbard term improves the agree-

ment of the 6�+→ 4� splitting to within 0.04 eV, andGGA+U overall provides the best structural and energeticdescription. The ground state and low-lying excited states ofFeO have been studied extensively, and, in 1977, the groundstate of FeO was reassigned from 5�+ to 5.82 Twenty yearslater, the lowest-lying septet state was determined to be 7�+

by anion photoelectron spectroscopy.68,83 Other experimentshave been carried out that yielded information about low-lying triplet states that are within 1 eV of the quintet groundstate.84 The X 5 state has an electron configuration of�2�4�2��4s�2��,2, and the a 7�+ state, as with the excitedstate of MnO, populates a minority spin ��4s�� instead of amajority spin �. The 5→ 7�+ splitting has been experimen-tally measured to be only 0.14 eV,68 but B3LYP and GGAfind the splitting to be much larger, 0.98 and 0.76 eV, respec-tively. Inclusion of the Hubbard term greatly improves thequintet-septet splitting to within 0.1 eV of the experimentalvalue. The bond lengths and harmonic frequencies of both5 and 7�+ from GGA are already in reasonable agreementwith experiment but are further improved by GGA+U. MnOand FeO are challenging but compelling examples where theDFT+U approach improves structural and energetic resultsfrom other exchange-correlation functionals.

Finally, the high 3d occupancy cobalt and nickel oxides,whose electronic states remain challenging to assign, areconsidered. Several experiments on CoO suggest the groundstate is 4,36,85 while previous density-functional calcula-tions instead predicted a 4�− state.86 More recent computa-tional results, however, provide further support for a 4

TABLE VIII. Structural properties of early transition-metal oxides at several levels of theory. Bond lengths �angstrom�, frequencies �cm−1�, and dissociationenergies �eV� are compared against experimental values for CaO �Ref. 64�, TiO �Ref. 65�, VO �Ref. 4�, and MnO �Ref. 66�, where available �ground stateproperties obtained from Ref. 40 if not otherwise noted�. MRCI literature results, which are available for VO �Ref. 67�, are compared here along with our owncalculated values for the remaining molecules.

B3LYP GGA GGA+U CCSD�T� MRCI Expt.a

CaO re 1.814 1.816 1.829 1.848 1.836 1.8221�+ �e 788 770 745 772 786 732

De 6.00 5.15 4.97 5.92 5.32 �4.80CaO re 2.078 1.939 1.961 1.991 2.030 2.0803� �e 554 591 558 563 558 556

De 3.28 3.25 2.95 5.02 3.03 3.08TiO re 1.585 1.613 1.626 1.618 1.601 1.6191 �e 1090 1029 1007 929 1043 1018

De 4.80 5.92 4.78 5.11 4.98 ¯

TiO re 1.612 1.617 1.647 1.626 1.628 1.6203 �e 1029 1040 961 1026 1014 1009

De 7.09 7.35 6.93 7.07 6.86 6.98VO re 1.567 1.588 1.603 1.559 1.577 ¯

2�− �e 1063 1100 1057 1132 1059 1090De 5.25 5.96 4.98 5.02 5.94 ¯

VO re 1.579 1.598 1.628 1.582 1.584 1.5894�− �e 1045 1053 951 1010 1041 1011

De 6.60 6.75 6.49 6.45 6.62 6.50MnO re 1.604 1.605 1.619 1.609 1.614 ¯

4� �e 945 882 863 886 890 660De 3.33 4.62 3.25 2.97 2.94 ¯

MnO re 1.633 1.628 1.674 1.651 1.678 1.6466�+ �e 900 904 817 817 805 840

De 4.73 5.46 3.91 3.83 3.87 3.88

aReference 40.






ground state with an electron configuration of�2�4�3��4s�2��,2.87 These same theoretical results suggestthat a 6 state is the next lowest state of an excited spin witha configuration that differs by unoccupying a minority spin��4s� orbital and instead occupying a majority spin ��4s��

orbital. The quartet-sextet splitting has not been measuredexperimentally, but all methodologies employed in this workpredict a splitting within 0.1 eV of 0.8 eV �Table VII�. Nev-ertheless, GGA and B3LYP both significantly overbind thestructure of X 4 CoO, while GGA+U achieves very goodagreement for the bond length �1.634 Å� within 0.005 Å ofexperiment and good agreement of the harmonic frequencyand dissociation energies, within 50 cm−1 and 0.06 eV, re-spectively �Table IX�. Structural properties for the 6�+ stateare not available but agreement is achieved here betweenGGA+U and MRCI. The ground state of NiO, 3�−, has beenstudied experimentally,88 and the lowest singlet state is be-lieved to be 1� at 1.26 eV above the triplet state7 despiteother studies which have suggested that the lowest singlet isinstead 1 only 0.98 eV above.6 Ramond et al.7 instead sug-gested that the state which had been assigned as 1 was anexcited triplet state. The electron configuration of X 3�− isbest described as �2�4�4��4s�2��,2��4s��,2 and the 1� statepopulates a minority spin �� orbital instead of a majorityspin ��4s��. Both B3LYP and GGA underestimate the triplet-singlet splitting by 0.8 and 0.4 eV, respectively, while addi-tion of a “+U” term improves agreement. Structural proper-ties for the GGA+U 3�− are in good agreement: theharmonic frequency and dissociation energy are within3 cm−1 and 0.16 eV of experiment, respectively. While the

bond lengths from both GGA �1.641 Å� and GGA+U �1.653Å� are overestimates with respect to experiment, they areboth in good agreement with the CCSD�T� result. In the caseof the excited 1� state, only the bond length has been ex-perimentally determined, but GGA+U also provides goodagreement with this known value. Despite the fact that CoOand NiO have both historically been challenging to charac-terize, GGA+U describes the low-lying states of both ofthese molecules accurately.

Structural properties were calculated for the earlytransition-metal oxides �CaO, TiO, VO, and MnO� �TableIX� and the late transition-metal oxides �FeO, CoO, NiO, andZnO� �Table IX�. For the oxides, the largest difference be-tween GGA and GGA+U is immediately observed to be thedissociation energies. Using a GGA approach, the minima ofthe potential energy curves of transition-metal oxides areoverstabilized. The 3d electrons are strongly delocalized inthe � and � molecular bonds compared to the isolatedatomic 3d states, and this makes the GGA energy at theminimum artificially lower than it should be. In turn, thiserror manifests itself most typically in overestimates of dis-sociation energies by as much as 1 eV. By comparison, ef-fects on the equilibrium bond length are much more subtle.The largest differences between GGA and GGA+U equilib-rium bond lengths are around 0.03 Å, with most beingaround 0.01–0.02 Å. In most cases, the GGA number is aslight underestimate of the experimental or CCSD�T�/MRCIbond lengths, but the GGA+U value in several cases may bean overestimate. This overelongation is due to some inherentpenalty on all bonding interactions, but this effect is system-

TABLE IX. Structural properties of late transition-metal oxides at several levels of theory. Bond lengths �angstrom�, frequencies �cm−1�, and dissociationenergies �eV� are compared against experimental values for FeO �Ref. 68�, CoO �Ref. 36�, NiO �Ref. 7�, and ZnO �Refs. 69–71�, where available �ground stateproperties obtained from Ref. 40 if not otherwise noted�.

B3LYP GGA GGA+U CCSD�T� MRCI Expt.

FeO re 1.611 1.604 1.623 1.618 1.608 1.6165 �e 903 948 895 901 876 880

De 5.34 5.38 4.35 4.16 4.36 4.22FeO re 1.573 1.677 1.685 1.591 1.692 ¯

7�+ �e 988 873 847 949 873 887De 4.03 4.01 3.03 3.12 3.21 ¯

CoO re 1.591 1.613 1.634 1.643 1.636 1.6294 �e 1001 824 800 825 879 853

De 4.14 5.10 4.05 3.87 3.90 3.99CoO re 1.658 1.652 1.667 1.637 1.668 ¯

6 �e 814 764 722 749 790 ¯

De 1.54 3.84 3.53 3.61 3.57 ¯

NiO re 1.758 1.642 1.657 1.670 1.721 1.6621� �e 613 838 752 668 752 ¯

De 2.55 2.81 2.75 2.75 2.73 ¯

NiO re 1.626 1.641 1.653 1.649 1.690 1.6273�− �e 884 861 841 818 832 838

De 3.73 4.85 4.08 4.01 3.81 3.92ZnO re 1.711 1.698 1.698 1.702 1.720 1.7241�+ �e 767 764 764 763 751 770

De 3.75 3.96 3.96 3.89 3.86 ¯

ZnO re 1.884 1.866 1.866 1.871 1.845 1.8503� �e 537 510 510 523 529 540

De 1.35 1.46 1.46 1.42 1.27 ¯






atic and thus may be anticipated when interpreting results.The experimental fundamental frequencies can only approxi-mately be compared to the harmonic frequencies, �e, wecalculated. However, GGA in general overestimates the �e ofmost of the molecules while the value from GGA+U is aslight underestimate for the same reasons the bond is slightlyoverelongated. In all cases, the GGA+U energetic splittingsare in excellent agreement with experiment, especially com-pared to the other exchange-correlation functionals.

D. Trends from isoelectronic 6�+ states

In order to isolate the effects of electron affinity, metal-ligand hybridization, and net molecular charge on the rolethat GGA+U plays, we consider an isoelectronic series ofmolecules which have the same spin and symmetry associ-ated with the 6�+ term symbol. We have studied five mol-ecules including two charged species, FeO+ and CrO−, twohighly covalent molecules, FeN and MnO, and the ionic CrF.Calculations of the Hubbard U for each of these speciesshows that the charged, covalent FeO+ exhibits a larger Uscf

of 5.5 eV compared to its closest neutral comparison, FeN at4.38 eV �Table X�. By comparison, when the increase incharge facilitates an increase in relative ionicity, as in CrO−,the Uscf, in this case 2.85 eV, decreases with respect to theclosest neutral comparison, MnO at 3.41 eV but is still largerthan the neutral ionic CrF at 2.04 eV. Overall, these resultsconfirm that more covalent species exhibit much higher val-ues of U than their ionic counterparts. The results are alsomanifested by the equilibrium bond lengths and harmonicfrequencies of the species. For FeN, FeO+, and MnO, theequilibrium bond lengths are all around 1.65 Å and the har-monic frequencies reside around 780–850 cm−1. The moreionic CrO− and CrF, on the other hand, exhibit longer bondlengths around 1.8 Å and lower harmonic frequencies around600 cm−1. The effect of a Uscf term is greatest on the morecovalent group of 6�+ states although the overall effects onbond length and frequency are modest. The greatest differ-ence with inclusion of a Uscf term is the dissociation energy,which decreases by as much as 1.5 eV in the case of 6�+

MnO.More clues to the differences between these isoelectronic

states lie in the differences between the corresponding frac-tional eigenvalues for each state in the occupation matrix�see Table X�. All states have roughly integer occupationscorresponding to spin up manifold �2�2, but the relative oc-cupations of a partially occupied spin up � and spin down �

and �2 vary widely among the isoelectronic series. The morecovalent FeN, FeO+, and MnO exhibit large spin up � occu-pations corresponding to the filling of both � and �� orbitals.The Uscf increases the total occupation of spin up � further,while the spin-down bonding � occupations decrease. Thesechanges are concomitant with the small increase in bondlength and more significant decreases in dissociation energyobserved for these molecules. The � occupations amount toabout 0.5e− in each orbital, roughly the contribution of anatomic 3d orbital to a bonding � molecular orbital. The moreionic CrF and CrO−, on the other hand, exhibit much loweroccupations in the aforementioned orbitals due to enhancedcharge transfer into the ligand. The spin up � occupation isreduced to about half and the 3d density which was in ��

orbitals is instead transferred to the F or O ligands. The othermolecular orbitals in the minority spin are similarly reducedfrom the covalent counterparts. Note that the U only mod-estly alters the values of the occupations from their GGAvalues by enhancing majority spin antibonding orbitals andslightly decreasing the occupation of minority spin bondingstates. These results also demonstrate that the occupationmatrix can be a valuable tool for predicting the role andanticipated magnitude of a U term prior to a linear-responsecalculation. The ability to tie chemical intuition to values ofU is key, particularly in cases where numerical errors fromsmall linear-response values can make it difficult to deter-mine the appropriate value of U for the system �see Appen-dix�.

E. Transition-metal fluorides: The Hubbard U inpartially ionic systems

Several transition-metal fluorides ranging from the earlytransition-metals with low 3d occupation �ScF� to midrow�CrF, FeF� and late transition-metals �CuF� are presentedhere. The fluorides are unique because the molecular bond ispredominantly ionic. In our approach, the Hubbard termmodulates hybridization between the metal center’s 3d statesand the states of the ligand, which is small for the fluorides.The linear-response values of U0 and the self-consistentlyobtained Uscf are calculated for the two lowest spin states ofScF, CrF, FeF, and CuF �Table XI�. The values of U for allthe fluorides considered is similar, between 1.75 and 2.75 eVon average, with no discernible trend related to the occu-pancy of the 3d manifold, save for a slightly lower U forCuF. Such small values of U suggest that a GGA approachmay already be sufficiently accurate to describe these sys-

TABLE X. Occupations of both � and � orbitals from eigenvalues of the occupation matrix for each isoelec-tronic 6�+ state computed at the GGA and GGA+Uscf levels. The arrows indicate whether the orbital belongsto majority �↑� or minority �↓� spin, and the Uscf of each molecule is also listed.

GGA

Uscf

GGA+Uscf

�↑ �↓ �↓ �↑ �↓ �↓

FeN 0.87 0.48 0.66 4.38 0.91 0.33 0.73FeO+ 0.94 0.41 0.55 5.50 0.98 0.24 0.64MnO 0.72 0.25 0.24 3.41 0.81 0.20 0.17CrF 0.61 0.12 0.07 2.85 0.57 0.15 0.09CrO− 0.27 0.16 0.07 2.04 0.27 0.14 0.06






tems. The effects that the modest values of U have on theelectronic and structural properties of these fluorides shouldbe limited compared to carbides, nitrides, and oxides. Nev-ertheless, we calculate structural properties of two states ofeach of these fluorides with both GGA and a GGA+Uscf,av

approach in order to ensure that GGA+U preserves anyproperties already properly described in GGA �Table XII�.

We first consider the early transition-metal fluoride ScF,which has a 1�+ ground state.89 A large number of singletand triplet electronic states have been identified and well-characterized experimentally for scandium fluoride.89,90,97–102

The 1�+ ground state has a valence electron configuration �2

where there is some 4s-3dz2 mixing in the contribution fromSc. The first lowest triplet state, 3, has instead a �� valenceconfiguration. The 1�+→ 3 splitting �Table XIII� is overes-timated slightly by GGA, and B3LYP incorrectly assigns the3 state as the ground state. Such a result is consistent withthe picture that � orbitals are overstabilized with respect to

more localized � orbitals as a result of self-interaction inGGA. The B3LYP hybrid functional likely overcorrects forthe self-interaction errors in this case. Structurally, the GGAbond length and harmonic frequency for 1�+ of 1.797 Å and745 cm−1 are consistent with the experimental values of1.788 Å and 736 cm−1, while GGA+U slightly overesti-mates the bond �1.811 Å� and underestimates the harmonicfrequency �722 cm−1�. Nevertheless, GGA+U agreementwith experiment is still quite good and GGA+U is actuallyin improved agreement with MRCI results that similarly un-derestimate bond strength with respect to experiment. For the3 state, the trends between GGA and GGA+U are repeated,with the GGA bond length �1.842 Å� and harmonic fre-quency �685 cm−1� increased by 0.03 Å and decreased by

TABLE XI. Linear-response values of Hubbard U �in eV� for the transition-metal fluorides ScF, CrF, FeF, and CuF in their L.S. and H.S. states. Thestandard result, U0, is shown, as well as the self-consistently obtained Uscf

for each state and an average value for both states, Uscf,av.


ScF 1�+ 2.64 2.70 3 2.52 2.60 2.65CrF 4�+ 0.93 1.94 6�+ 1.83 2.04 1.99FeF 4 2.34 2.60 6 1.54 1.80 2.20CuF 1�+ 1.00 1.85 3�+ 0.84 1.63 1.74

TABLE XII. Comparison of experimental structural properties, including bond lengths �angstrom�, frequencies �cm−1�, and dissociation energies �eV�, of early�ScF� �Refs. 89 and 90�, mid �CrF, FeF� �Ref. 91�, and late �CuF� �Refs. 92 and 93� transition-metal fluorides against calculated B3LYP, GGA, and GGA+U values as well as literature MRCI results �Refs. 94–96�. Ground state experimental properties obtained from Ref. 40 if not otherwise noted.


ScF re 1.795 1.797 1.811 1.824 1.7881�+ �e 709 745 722 733 736

De 6.73 6.86 6.02 5.95 6.21ScF re 1.855 1.842 1.873 1.899 1.8673 �e 644 685 639 629 649

De 5.49 5.61 5.12 5.70 ¯

CrF re 1.783 1.774 1.781 1.801 ¯

4�+ �e 647 680 670 671 ¯

De 3.95 3.71 3.61 4.58 ¯

CrF re 1.800 1.794 1.816 1.800 1.7846�+ �e 636 667 631 655 664

De 4.36 4.82 4.41 4.72 4.61FeF re 1.759 1.742 1.762 1.776 1.7394 �e 660 657 642 663 684

De 4.45 4.47 4.40 4.08 ¯

FeF re 1.790 1.785 1.801 1.801 1.7806 �e 651 635 625 650 663

De 4.30 4.41 4.35 4.74 4.64CuF re 1.772 1.782 1.794 1.752 1.7451�+ �e 600 596 578 611 623

De 4.47 4.71 4.41 4.12 4.46CuF re 1.762 1.800 1.798 1.743 1.7383�+ �e 625 596 606 680 674

De 2.15 2.22 2.20 2.84 ¯

aLiterature MRCI results for ScF �Ref. 94�, CrF and FeF �Ref. 95�, and CuF �Ref. 96� are presented here.

TABLE XIII. Energy splitting �in eV� of the two lowest spin states ofseveral fluorides. In the cases of ScF �Ref. 89�, FeF �Ref. 91�, and CuF�Refs. 92 and 93� experimental values are available and compared againstthe B3LYP, GGA, GGA+U, and literature MRCI results �Refs. 94–96�.Negative values indicate an incorrect ground state.


ScF 1�+→ 3 0.16 0.40 0.32 0.25 0.24CrF 6�+→ 4�+ 1.43 1.47 1.73 1.05 ¯

FeF 6→ 4 0.05 0.15 0.39 0.66 0.62CuF 1�+→ 3�+ 1.86 1.60 1.68 1.75 1.81

aResults for ScF �Ref. 94�, CrF �Ref. 95�, and FeF and CuF �Ref. 96� areobtained from literature MRCI results. For CuF MRCI splitting, the resultsfrom CCSD�T� and MRCI+Q from Ref. 96 are cited because they are invastly improved agreement with the experimental result over MRCI alone.






46 cm−1, respectively, in GGA+U. In this case, the GGA+U properties are again in improved agreement with MRCIand also now with experiment. While the singlet-triplet or-dering is correctly predicted by GGA �Table XIII�, GGA+U decreases the magnitude of the splitting from 0.40 to0.32 eV, in improved agreement with the experimental valueof 0.24 eV.

The midrow transition-metal fluorides, CrF and FeF, arecharacterized by having similar valence electron configura-tions in high-spin and low-spin states that differ only by thespin-coupling of the relevant electrons. Chromium fluoridehas a 6�+ ground state that has been carefully characterizedexperimentally.103–105 No quartet states of CrF have thus farbeen characterized, but highly accurate MRCI results95 pre-dict a 4�+ state as the lowest quartet state of CrF. In bothcases, the electronic configuration is �1�2�2 with the differ-ence in states derived from the spin-coupling of the molecu-lar orbitals. Both structural and energetic differences be-tween GGA and GGA+U for this approach are small andthey are in very good agreement with MRCI �Ref. 95� andexperiment.40 The ground state spin sextet states of ironfluoride106–108 and the quartet states91 have been character-ized experimentally. The X 6 and a 4 states of FeF bothhave a valence electron configuration of ��2�3. GGA cor-rectly predicts the sextet ground state, but it predicts a sextet-quartet splitting of only 0.15 eV compared to the experimen-tal value of 0.62 eV. The hybrid functional B3LYPunderestimates the sextet-quartet splitting even further andGGA+U achieves the best density-functional result with apredicted splitting of around 0.40 eV. The structure of FeF�Table XII� is described similarly by GGA and GGA+U.

Copper fluoride has a 1�+ ground state with a valenceelectron configuration of �2�4��,4�4, and the lowest tripletstate, 3�+, is best described as ��4��,4�4.92,93 The �→�� promotion is prohibitive, and B3LYP, GGA, andGGA+U all correctly predict the magnitude of the singlet-triplet splitting within 0.2 eV of the 1.81 eV experimentalvalue �Table XIII�, while providing reasonable structuralagreement with experiment �Table XII�.

Structural properties were calculated for the transition-metal fluorides, ScF, CrF, FeF, and CuF, and are presentedalong with available experimental results in Table XII.40,89–93

Since the values of Uscf are small, overall structural changesare also much smaller compared to the oxides. The elonga-tion of the bonds in GGA+U is around 0.01–0.02 Å with thelargest being about 0.03 Å. The harmonic frequencies forGGA+U are decreased with respect to the GGA values by25 cm−1 on average. The GGA+U bond lengths and har-monic frequencies only sometimes improve upon GGA withrespect to experimental values and this suggests that forsome structural properties of the fluorides a GGA approach ispreferable. The dissociation energies, however, particularlyfor 1�+ ScF and 1�+ CuF, are reduced by 0.3–0.8 eV inimproved agreement with experimental numbers. Overall,the Hubbard term plays a strong role in partitioning the oc-cupying bonding and antibonding states in covalent mol-ecules, but it plays a less significant role in ionic moleculeswhere this feature is also less chemically relevant. The 1�+

ScF state has partially covalent character and it shows the

most significant improvement with GGA+U. The energeticsplittings obtained with GGA+U are, in fact, significantlyimproved with respect to both GGA and B3LYP �Table XIII�.The GGA results are already significantly better than B3LYP,which misidentifies the ground state of ScF as 3 and pre-dicts the 4 and 6 states of FeF to be nearly degenerate.While GGA correctly predicts the ground state for all of thefluorides, the GGA+U agreement with experiment is an im-provement in all fluoride cases. Overall, GGA is typicallysufficient for describing fluorides structurally, but in order toproperly assign states and provide their relative energetics, aGGA+U approach proves beneficial.

IV. 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. Of the 22 diatomic molecules studied here, averageerrors in the state splittings of diatomic oxides and hydridesare reduced from 0.24 and 0.30 eV, respectively, to only 0.01and 0.06 eV, with inclusion of a “+U” term �Table XIV�. Wealso successfully assign the nature of the ground state spinand symmetry of these molecules with a GGA+U approach.Importantly, the Hubbard U term is not used as a fittingparameter but it is a true linear-response property of thetransition-metal complex which may in principle augmentany exchange-correlation density-functional. The practicallimitations to this approach stem from the complexity of thesystems which we wish to study: evolution of the coordina-tion environment along a global reaction coordinate may re-

TABLE XIV. Mean absolute errors of different exchange-correlation func-tionals �B3LYP, GGA, and GGA+U� as compared against available experi-mental data and averaged over hydrides, carbides/nitrides, oxides, and fluo-rides. The E refers to the splitting between the ground state and the loweststate of the second-lowest spin manifold. Numbers in boldface indicate thefunctional that yields the minimum error for each property within a class ofmolecules.

B3LYP GGA GGA+U GGA+U3d/4s

Hydridesre�pm� 2.8 3.1 2.5 2.4�e�cm−1� 51.1 31.4 29.3 28.6De�eV� 0.22 0.17 0.09 0.03E�eV� 0.20 0.24 0.13 0.01

Carbides and nitridesre�pm� 2.2 2.3 6.1�e�cm−1� 165.5 93.5 194.3De�eV� 0.50 0.35 0.26E�eV� 0.17 0.08 0.07

Oxidesre�pm� 2.2 2.4 2.6�e�cm−1� 45.8 30.8 25.9De�eV� 0.49 0.70 0.09E�eV� 0.38 0.30 0.06

Fluoridesre�pm� 1.7 2.2 3.1�e�cm−1� 24.0 29.7 35.7De�eV� 0.28 0.34 0.18E�eV� 0.34 0.28 0.15






sult in local deviations from a globally averaged Hubbard U,bring to the forefront the role of 4s contributions, or evenrequire consideration of a U on the 2p states of the ligand.

Overall, the DFT+U results �here, GGA+U� have beenshown to provide for systematic improvement over GGA forall systems considered thus far. The inexpensive linear-response U calculation also acts as a probe for the relativeutility of the DFT+U approach for a given transition-metalcomplex; that is, if the Hubbard U calculated is small ornearly zero, a standard description may be sufficient. Webelieve that this work provides an excellent reference for thepotential that DFT+U can also have in describing larger-scale systems with high accuracy.

ACKNOWLEDGMENTS

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

APPENDIX: NUMERICAL STABILITY OF THE LINEAR-RESPONSE CALCULATION

In addition to calculating properties of several types oftransition-metal containing diatomic molecules in order tolearn more about the role the U plays on the chemical bond,these molecules also provide excellent test cases for under-standing the practical limits of our approach. The technique

used to determine the linear-response U for each electronicstate potentially suffers from numerical instability. The cal-culated value of U serves as a probe to the chemical natureof the molecule and should reflect what is known about thebonding interactions in the molecule. In our approach, wecalculate a response function, �, of the occupations as afunction of a potential shift, �, which is inverted to obtainthe U of our system. Typically, when the manifold of interestcontains nearly integer occupations, the response functionsbecome small, nearly zero. Since U is proportional to �−1,the result diverges in these cases. An example of this prob-lem is best manifested by the ground state of ZnH, 2�+,which has nearly a 3d10 configuration. The occupations aretheir lowest at 9.97 under significant compression of thebond but rise quickly to 9.99 at the equilibrium and asymp-totically approach 10.00 as the molecule is dissociated �seeFig. 3�. Because the occupation matrix on the 3d manifold ofZnH is qualitatively as well as nearly quantitatively 3d10, themanifold should intuitively yield very little response.

The bare and converged response functions of the 3dmanifold of ZnH were measured and are shown in Fig. 4.Convergence criteria on total energy and diagonalization aswell as higher cutoffs for the plane-wave basis set can in-crease the accuracy of the projections calculated from thedensity. However, regardless of tight convergence criteria,there will always be a small amount of scatter on the datawhich should in particular affect the bare response functionbecause this property is calculated from the first self-consistent step, the initial diagonalization of the density uponresponse to the potential shift, �. The response functions areof the order of −0.002 e− /eV for bond lengths near equilib-rium. A best fit polynomial to both the bare and convergedalso is nearly identical for most bond lengths. However,when the two curves and data are inverted, the reciprocal ofthe response functions are very large, of the order of 4–5000.The small numerical noise in the two response curves givesrise therefore to a large value of U of around 11 eV.

The U calculated from the data points as well as thedifference of the two best fit polynomials is shown in Fig. 5.The scatter of the data from the best fit curve demonstratessome of the error bar obtained in inverting these small num-bers. In fact, the fact that the calculated U changes sign sorapidly with varying bond length hints strongly at rounding

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Zn-H Distance (Å)

9.96

9.97

9.98

9.99

10.00

Occu

pati

on

s

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Zn-H Distance (Å)

0.00

0.25

0.50

0.75

1.00

En

erg

y(e

V)

FIG. 3. The binding energy curve �blue circles� and the bond length depen-dent 3d projection occupations �red squares� for 2�+ ZnH.

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Zn-H Distance (Å)

-5e-03

-4e-03

-3e-03

-2e-03

-1e-03

0e+00

dn

/dα

(ev

-1)

Converged

Bare

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Zn-H Distance (Å)

0e+00

1e+03

2e+03

3e+03

4e+03

5e+03

6e+03

7e+03

8e+03

χ-1(e

V)

Converged

Bare

FIG. 4. The bare �red circles� and converged �black squares� response functions, �, for the 2�+ ZnH state over a range of bond lengths �left� as well as theinverse of the response functions, �−1 �right�. The best fit polynomial curves are also shown as solid lines.






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1.2 1.4 1.6 1.8 2

Zn-H Distance (Å)-20

-15

-10

-5

0

5

10

15

20

Hu

bb

ard

U(e

V)

FIG. 5. The value of U calculated from linear-response at various bondlengths for ZnH from the inverse linear-response functions is shown in blackcircles. Additionally, the difference of the two best fit lines of the linear-response functions is shown as a black curve.




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