Metal-organic charge transfer can produce biradicalstates and is mediated by conical intersectionsOksana Tishchenkoa,1, Ruifang Lia,b, and Donald G. Truhlara,1

aDepartment of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0431; and bDepartment of Chemistry,Nankai University, Tianjin 300071, People’s Republic of China

This article is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2009.

Edited by Nicholas J. Turro, Columbia University, New York, NY, and approved September 20, 2010 (received for review July 14, 2010)

The present paper illustrates key features of charge transfer be-tween calcium atoms and prototype conjugated hydrocarbons(ethylene, benzene, and coronene) as elucidated by electronicstructure calculations. One- and two-electron charge transfer iscontrolled by two sequential conical intersections. The two lowestelectronic states that undergo a conical intersection have closed-shell and open-shell dominant configurations correlating withthe 4s2 and 4s13d1 states of Ca, respectively. Unlike the neutral-ionic state crossing in, for example, hydrogen halides or alkalihalides, the path from separated reactants to the conical intersec-tion region is uphill and the charge-transferred state is a biradical.The lowest-energy adiabatic singlet state shows at least twominima along a single approach path of Ca to the π system: (i) avan der Waals complex with a doubly occupied highest molecularorbital, denoted ϕ2

1, and a small negative charge on Ca and (ii) anopen-shell singlet (biradical) at intermediate approach (Ca⋯C dis-tance ≈2.5–2.7 Å) with molecular orbital structure ϕ1ϕ2, where ϕ2

is an orbital showing significant charge transfer form Ca to theπ-system, leading to a one-electron multicentered bond. A thirdminimum (iii) at shorter distances along the same path correspond-ing to a closed-shell state with molecular orbital structure ϕ2

2has also been found; however, it does not necessarily representthe ground state at a given Ca⋯C distance in all three systems.The topography of the lowest adiabatic singlet potential energysurface is due to the one- and two-electron bonding patterns inCa-π complexes.

metal atom ∣ metastable state ∣ nature of metal-π binding ∣ one- andtwo-electron multicentered bonds ∣ triplet ground state

The interactions of metal atoms with alkenes, polyenes,aromatics, and graphene-based materials are important for

applications in catalysis (1, 2), molecular electronics (3–6), optoe-lectronic and sensing devices (7, 8), and hydrogen storage mate-rials (9–12). Charge transfer from the metal to the organic systemis one of the key features determining adsorption energies, reac-tivities, electronic structure, conductivities, and optical properties(4, 8). Most studies of charge transfer have focused on equili-brium geometries (4, 13–15), but the design of molecular devicesand the understanding of adsorption and reactivity also requireunderstanding the dependence of charge transfer on moleculargeometry (16). In this communication we show that charge trans-fer character can change rapidly and suddenly for small changesin geometry, and we explain this phenomenon in terms of conicalintersections (CIs). In addition we show that the charge transferstate can be a ferromagnetically coupled biradical, which is rele-vant to the possibility of control of magnetization by an electricfield in, for example, spintronics applications (17). A fundamen-tal understanding of the interfacial states of metal atoms inter-acting with conjugated π-systems is an essential element underly-ing rational molecular electronics design.

To model metal-π interactions, we first considered interactionsof a calcium atom with the coronene molecule, a small nonper-iodic model of graphene. Calculations using density functional

theory revealed the presence of two types of equilibrium struc-tures: one at a distance R between Ca and the closest carbonof about 3.8 Å with slightly negative charge on the calcium atom,and another state with a smaller R and with considerable shift ofelectrons from Ca to coronene; the latter is metastable. To get abetter understanding of the structure of potential energy surfacesthat govern the charge transfer process and of the nature ofelectronic states involved, we performed a detailed electronicstructure study using multiconfigurational wave functions forthe interaction of Ca with the smaller prototype molecules ben-zene and ethylene as well as coronene.

The closed-shell singlet potential energy surfaces of all threesystems have been found to possess similar features: an “outer”van der Waals minimum at a longer R and a metastable “inner”minimum, characterized by a different dominant electron config-uration, at a shorter R. Furthermore, at intermediate R the lowestsinglet state is open-shell, and the lowest-energy state appearsto be a triplet state. (Such states could be easily missed if onecalculated only closed-shell singlets, as is often done in computa-tional materials research.)

The multiconfigurational calculations of geometries and ener-gies presented here are multireference Moller–Plesset perturba-tion theory (MRMP2) (18, 19) calculations. Geometries andenergies were also found by M06-2X (20) density functional cal-culations. Single point coupled-cluster calculations [CCSD(T)](21) were also performed at M06-2X geometries. We will alsoshow partial sections of the potential energy surfaces calculatedby MS-CASPT2 (22). (Details of the electronic structure calcula-tions are given in Methods.)

These systems present the classic case of two-orbital configura-tion interaction with ϕ2

1, ϕ1ϕ2, and ϕ22 singlets and a ϕ1ϕ2 triplet.

In the notation that we use below for potential energy surfacesand stationary points, S0 denotes a state with the dominant con-figuration ϕ2

1, S1 and T1 denote the open-shell singlet and thecorresponding triplet, and S2 denotes a state with the dominantconfiguration ϕ2

2. We calculated these states with MRMP2, M06-2X, and CCSD(T) for ethylene and benzene and with M06-2X forcoronene. Note that the S0, S1, S2, and T1 labels apply to diabatic(23) states, and due to conical intersections (which becomeavoided crossings along most paths), a given adiabatic statecan have different labels at different geometries.

First we consider the amount of charge transfer as a function ofthe distance from Ca to the π-system. To ensure that the geome-tries considered are relevant to low-energy structures, we selectthe geometries making a path between the low-energy structures.Let R denote the distance from the Ca atom to the closest carbonatom in a molecular partner denoted P. Figs. 1 and 2 show the

Author contributions: O.T., R.L., and D.G.T. designed research, performed research,analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. E-mail: o_t@t1.chem.umn.edu or truhlar@umn.edu.

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ground singlet state energy of all three systems as a function of Ralong a path constructed as follows: At large R the path is thelinear synchronous path (LSP) (24, 25) from the global minimumS0 van der Waals complex, Ca⋯P, to the minimum-energy S1structure; and at smaller R, it is an LSP from the S1 minimumto an inner local minimum on the S2 surface, and this path is ex-tended linearly beyond the S2 minimum as well. Figs. 1 and 2 alsoshow the partial atomic charge q on Ca along these three paths.The jumps in q along a reaction path together with the multipleminima in potential energy are intriguing. Such jumps have alsobeen observed in chemical reactions involving metal atoms andsmall electronegative molecules (26–31), but have not been wellstudied in materials chemistry context. The charge transfermediated by conical intersections is particularly interesting inthe present case because of the negative electron affinity ofthe hydrocarbon molecules, because of the nonreactive natureof the potential energy surfaces, because the jumps are verysudden, and because of the technological importance of metal-π interactions. Although the acceptor of the transferred electronhas a negative electron affinity when isolated, it binds the electron

in the presence of the cation produced by the charge transfer, asin the NaH2 system (32).

The relative energies of optimized structures corresponding tokey minima are given in Table 1, and their molecular geometriesare shown in Figs. 3 and 4. In the structural labels, M denotes alocal minimum, SP a saddle point, and CI a conical intersection.The semiquantitative agreement between single-reference [M06-2X and CCSD(T)] and multireference (MRMP2) calculations inTable 1 is encouraging because previous work (33) showed pooragreement between these two kinds of calculations for Sc, V, andNi complexes with benzene. (Those systems also show a long-range global minimum and a short-range metastable state, some-times with a different spin; experiments (34, 35) are consistentwith observation of the long-range state.) Fig. 5, Upper and Low-er, show the energies of the four lowest electronic states of theCaEth and CaBen systems, respectively, as functions of the dis-tance R along the LSP that connects an inner S2-M and an outerS0-M minima. The shapes of the potentials along these paths aresimilar for the two systems. The ground electronic state, whichcorrelates adiabatically with the 4s2 state of the calcium atom,intersects the first excited singlet state twice: first at R of approxi-mately 3 Å, and then at a smaller R. At large R the first excitedsinglet state correlates adiabatically to the 4s13p1 state of Ca, andit is dominated by the 4s13d1 configuration of the Ca atom at geo-metries close to a conical intersection and at medium separationsof the reactants in a region before the earliest S0S1 conical inter-sections due to crossing of the P and D states as the reactantsapproach.

The middle minimum corresponds to a metastable biradical,which is found to have a similar bonding pattern in the Ca-ethy-lene, Ca-benzene, and Ca-coronene systems. In each case, a one-electron bond is formed between the Ca atom and the π-electronsystem, in which Ca and the hydrocarbon share one electron in anorbital ϕ2 formed by the interaction of the d-type orbital of Caand one of the π� orbitals of the hydrocarbon molecule. Becauseinteractions between two closed-shell species in the ground elec-tronic state are usually closed-shell singlets, the one-electronbonding in systems considered here is surprising. The origin ofthis unusual bonding type may be understood as follows: At largeintermolecular separation of the reactants, the Ca atom in itsground electronic state has a closed-shell 4s2 dominant configura-tion, whereas its 4s13d1 configuration corresponds to the secondexcited state (36). As the reactants approach, the 4s13d1 state ofCa is lowered in energy due to the interaction of the 3d orbitalwith a π� orbital of the hydrocarbon; therefore, this state under-

Fig. 1. Cuts through the S0, S1, and S2 states as functions of R for CaEth(Upper) and CaB (Lower) systems. The zero of energy corresponds to infinitelyseparated reactants. The path has C2 symmetry for CaEth and C2v symmetryfor CaBen. Dotted lines show the Hirshfeld charge on the Ca atom along theground electronic state singlet PES. Two sudden changes in the charge dis-tributions along this path occur at two conical intersections. All data shown inthis figure are obtained with M06-2X/def2-TZVPP (see Methods).

Fig. 2. Same as Fig. 1 except for CaCor system, using M06-2X/def2-SVP. Thepath has C2v symmetry.

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goes (avoided or actual, depending on the path) intersection withthe state correlating with the 4s13p1 state of Ca prior to the S0S1conical intersections, and finally it intersects the ground electro-nic state at S0S1-CI. At R smaller than its value at S0S1-CI, the4s13d1 configuration becomes the lowest-energy state; it corre-sponds to a biradical. The metastable biradical structure consti-tutes a middle minimum that is separated from reactants by an(avoided or actual) intersection of the two lowest singlet statesand is separated from the inner minimum by another (avoidedor actual) intersection of the S1 and S2 states. The intersectionsare actual (conical) intersections along the paths shown in Figs. 1and 2, but they would be (weakly) avoided for most nonsymmetricpaths. When conical intersections with biradical states have beenstudied in organic chemistry (37–57), they are often located muchhigher in energy relative to the ground electronic state and comeinto play only in photochemically induced processes, whereas thepresent intersections arising from the interaction of a π-electron

system with a metal atom are within 10–30 kcal∕mol from theisolated ground state reactants.

The biradical thus corresponds to a weakly quasibound com-plex of a hydrocarbon molecule with the Ca atom, in whichthe latter largely preserves its atomic nature with an 4s13d1

open-shell dominant configuration, bound to the hydrocarbonmolecule by a one-electron multicentered bond. A schematic re-presentation of the orbitals involved in the bonding is given inFig. 6. The CaEth complex is bihapto (η2) with the Ca atom an-chored to the two carbons by the three-centered one-electronbond formed from the dxy orbital of Ca and π� orbital of ethylene,and the CaBen complex is tetrahapto (η4) with the Ca atom an-chored to the carbons by the five-centered one-electron bondformed from the dxz orbital of Ca and one of the π� (e2u) orbitalsof benzene (see Fig. 6). The degeneracy of the two (e2u) orbitalsof benzene is lifted when the Ca atom approaches the benzenering, favoring the state with the highest orbital overlap.

At geometries close to the middle minimum, the lowest-energystate is a ϕ1ϕ2 triplet, whose energy is lower than the correspond-ing ϕ1ϕ2 singlet. This feature may be viewed as a consequence ofthe lower energy of the 4s3d 3D state of free Ca atom as com-pared to its 4s3d 1D state (36). The T1 triplet is close to theS1 singlet both geometrically and energetically. Spin changefor the lowest adiabatic state has also been observed in, e.g., in-teractions of transition metal atoms with benzene (13), in reac-tions of Mo with CH4 (58), and in more complex organometallicreactions (59, 60).

Table 1. Relative energies (kcal∕mol, with respect to the middle minimum for ethylene and benzene, respectively)

Structure PG* State ec† q‡ MRMP2 M06-2X CCSD(T)//M06-2X

EthyleneS2-M C2

1A b2 0.7 25.5 21.5 22.5S0S2-SP C2

1A a2 0.3 32.5 25.1S1-M C2

1B ðabÞS 0.3 3.45 0.87 1.62S1-Mip C2v

1A2 ða1a2ÞS 0.3 29.0 27.0 30.7T1-M C2

3B ðabÞT 0.3 0.0 0.0 0.0T1-Mip C2v

3A2 ða1a2ÞT 0.3 27.6 28.1 30.1S0-M C2v

1A1 a2 0.0 −11.9 −17.4S0-M C2v

1A1 a2 0.0 −11.7 −17.2BenzeneS2-M C2v

1A1 a22 0.6 18.9 25.2S1-M C2v

1A2 ða1a2ÞS 0.2 −0.25 0.68T1-M C2v

3A2 ða1a2ÞT 0.2 0.0 0.0T1-Mip C2v

3A2 ða1a2ÞT 0.3 46.1 43.9S0-M C2v

1A1 a21 0.0 −7.2 −15.4S0-Mip C2v

1A1 a21 0.0 −3.8 −13.5

All results in this table are calculated at the optimized molecular structures of the CaEth and CaBen systems.*Point group.†Dominant electron confuguration.‡Charge on Ca atom.

Fig. 3. Optimized structures of CaEth complex. Bond lengths are given atthe best electronic structure level available, namely, MRMP2∕CASSCFð2∕2Þ∕def2-TZVPP þ d for structures T1-M – S1-Minpl (inpl denotes in-plane), CASSCF(4/7)/def2-TZVPP for structure S0S1-CI, and M06-2X/def2-TZVPP for structuresS0-SP – S0-Minpl.

Fig. 4. Same as Fig. 3 except for CaBen, for which all geometries are fromM06-2X/def2-TZVPP calculations.

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Shapes of the bonding orbital along the LSPs in CaEth andCaBen complexes are shown in Fig. 5. The three-centeredone-electron bond in the CaEth complex leads to an elongationof the carbon–carbon distance by about 0.1 Å, and the five-centered one-electron bond in the CaBen complex leads to anelongation of the two opposite carbon–carbon bonds of the fourcarbons involved in the bonding by about the same amount. At

the minimum on the ~a3 B state of the CaEth complex, R is about2.5 Å. The four carbon atoms involved in the bonding in theCaBen complex are equally distant from the Ca atom by 2.64 Å,and the two remaining carbons are equally distant from the Caatom by 2.70 Å. The low-lying d orbitals of Ca have also beenfound (9) to play an important role in Ca binding to sp3 sites incovalent-bonded graphene (an all-graphitic 3D porous material).It is very important to understand the nature of the metal coor-dination because the coupling of a sudden charge transfer to amagnetic state provides an additional degree of freedom relevantto spintronics applications.

Structures S2-M of CaEth and CaBen correspond to localminima on the S2 state at small R. At these structures the portionof the valence wave function on Ca is dominated by the 3d2 (ϕ2

2)electron configuration at Ca atom, corresponding to a two-electron excitation from the 4s orbital of Ca to a hybrid 3d − π�orbital of the system. Because of a change from the biradical tothe closed-shell bonding pattern, these structures are character-ized by shorter R and longer carbon–carbon distances than thestructures at the middle minimum.

In addition to the three minima, a variety of other criticalpoints have been located on the potential energy surfaces of eachof the systems considered. Structure S0 -SP1 is the first order sad-dle point that connects S2-M with an outer van derWaals complexon the lowest-energy closed-shell adiabatic potential energy sur-face (PES). Structures T1-Mip and S1-Mip represent local minimaon the T1 and S1 potential surfaces analogous to the structuresT1-M and S1-M with the only difference being that the Ca atomin the former two cases is placed in the plane of the ethylenemolecule. These structures are found to be considerably higherin energy than their out-of-plane counterparts. An analogousT1-Mip structure has also been located on the lowest tripletpotential energy surface in the case of CaBen complex. Becauseof the rather high energy of this structure, no attempt was madeto locate a similar structure for an open-shell singlet state.

Structure S0S1-CI is a minimum-energy point on the S0S1 con-ical intersection of the two lowest singlet states. This structure, asoptimized by complete active space self-consistent (CASSCF)(2/2)/def2-TZVPP, exceeds the energy of the E-S1-M innerminimum of the CaEth complex by 13.9 kcal∕mol. Calculationsincluding dynamical electron correlation are expected to de-crease this energy difference. Although the importance of CIs(61–65) in photochemical processes is now widely recognized(66, 67), their role in thermally activated processes is not wellstudied (68). The effect of a CI on a thermally activated reactiondepends on the energetic accessibility of the dividing surface andof the seam of intersection. Examples of thermally activated re-actions where the CI seam is (or can be) accessed under usualconditions involve chemiluminescence, energy transfer, reactionsinvolving metal atoms (29, 32), and hydrogen atom abstractionsfrom phenolic antioxidants (69). The CIs in systems consideredhere are particularly interesting because of their relevance innonreactive processes, such as attachment of metal atoms to aro-matic systems, which is of utmost importance in organometallicchemistry and materials chemistry.

We now return to the coronene–calcium system, which will beabbreviated CaCor. The coronene-Ca cation has been recentlystudied by photodissociation (70), but we know of no experimenton the neutral. As in the cases of the smaller prototype systems,three kinds of equilibrium structures with dominant electronconfigurations ϕ2

1, (ϕ1ϕ2), and ϕ22 exist that correspond to the out-

er, middle, and the inner minima, respectively (optimized mole-cular structures are shown in Fig. 7, and their energies are givenin Table 2). Structure S0-Mcr (cr denotes central ring) (ϕ2

1) cor-responds to the van der Waals minimum (no change in the elec-tron configurations of the fragments relative to the separatedfragments; R ≈ 3.8 Å), structures T1-Mcr [ðϕ1ϕ2ÞT], and S1-Mcr[ðϕ1ϕ2ÞS] correspond to the middle minima (one-electron bond

Fig. 5. Cuts through four lowest potential energy surfaces for CaEth (Upper)and CaBen (Lower) systems calculated by MS-CASPT2 as functions of R.Also shown are the shapes of ϕ2 (the orbital that forms the one-electronmulticentered bond between Ca and the hydrocarbon) in these systems asfunctions of the same distance. Arrows indicate approximate position ofthe optimized structures given in Table 1. The zero of energy in each caseis at the outer minimum.

Fig. 6. Schematic representation of the keymolecular orbitals (one of the Caatom, and another on the hydrocaron molecule) involved in bondmaking forCaEth (Left) and CaBen (Right) complexes.

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between Ca and the coronene molecule; R ≈ 2.7 Å) analogouslyto structures T1-M and S1-M for the CaEth and the CaBen sys-tems, and structures S2-Mcr (ϕ2

2) and S2-Mor (or denotes outerring) (ϕ2

2) correspond to the inner minima (two-electron bondbetween Ca and the coronene molecule; R ≈ 2.4–2.5 Å) analo-gously to the structures S2-M in the two smaller systems. The no-table distinctions between the CaCor and the CaBen systems arethe following: (i) there are more than one attachment site for themetal atom in the former and (ii) due to the lower energy of thefirst several unoccupied molecular orbitals in the former system,several excited states at medium R become lower in energy ascompared to the CaBen system.

Structures in which Ca is bound to an outer ring (S1-Mor,T1-Mor, and S2-Mor) are found to be energetically lower thanstructures in which Ca is bound to the central ring (i.e., thanthe corresponding structures S1-Mcr, T1-Mcr, and S2-Mcr). Thisfeature may be rationalized by considering that the geometry dis-tortion accompanying the metal-π bond formation is energeticallymore favorable for an outer ring than for the central ring. Mo-bility of the metal atom between different rings in both singletand triplet manifolds represents an interesting question invitinga further study. The electronic state at the T1-Mcr minimum-energy geometry is found to be (nearly) degenerate (with the3B1 and

3B2 components in C2v symmetry). Furthermore, the ex-cited 3A2 state has an energy minimum only ≈12 kcal∕mol abovethe ground ~a3B1 in the vicinity of the T1-Mcr structure. On thecontrary, the excited states above the T1-M minima in the caseof CaBen system are appreciably higher in energy. Fig. 8, Lower,depicts the eight lowest electronic states of the CaCor system as afunction of R for the perpendicular approach of Ca toward thecenter of the coronene molecule. Because of the large size of thissystem, the results are given only at the state-averaged (SA)-CASSCF level. For comparison, Fig. 8, Upper and Middle, show

Fig. 7. Optimized (M06-2X/def2-TZVPP) structures of the CaCor system.

Fig. 8. Cuts through the lowest adiabatic potential energy surfaces forCaEth (Upper) and CaBen (Middle) and CaCor (Lower) systems calculatedby SA-CASSCF as functions of R. In the latter case, additional states are shownbecause of their low energies.

Table 2. Relative energies at the optimized molecular structuresof the coronene + Ca system*

Structure PG State ec q M06-2X

S2-Mcr C2v1A1 b2

2 0.7 42.8S2-Mor Cs

1A0 a00 0.7 18.6S1-Mor Cs

1A00 ða0a00ÞS 0.3 −0.2S1-Mcr C2v

1B1 ða1b1ÞS 0.4 6.1T1-Mor Cs

3A00 ða0a00ÞT 0.3 0.0T1-Mcr C2v

3B2 ða1b2ÞT 0.4 7.5T1-Mcr C2v

3B1 ða1b1ÞT 0.4 7.5T1-Mcr C2v

3A2 ða1a2ÞT 0.3 19.0S0-Mcr C2v

1A1 a21 0.0 −3.3

The energies in this table are in kcal∕mol, with respect to the innerminimum, ~a3A00).*See footnotes to Table 1.

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the SA-CASSCF results for the two smaller prototype systems.The similarity in shapes of the adiabatic PESs correspondingto the S0, S1, T1, and S2 states in all three cases implies thatthe basic features described above must be valid for interactionsof Ca with a variety of aromatic molecules.

In summary, we find that approach of a Ca atom to an ex-tended π-system involves three kinds of minima in the PES: avan der Waals minimum (i), a biradical formed by partial singlecharge transfer (ii), and a closed-shell singlet formed by partialdouble charge transfer (iii). The shift in charge in moving fromi to ii or from ii to iii occurs suddenly (over a small range of geo-metry) because of two sequential (along a path) conical intersec-tions. In the entrance channel, the ϕ2

1 and ϕ1ϕ2 states intersect ina region where the energy of the d orbital of the metal atom islowered by interaction with the π-type lowest unoccupied mole-cular orbital of a molecule. The recognition that a sudden redis-tribution of charge may occur for a small change in geometryprovides a computational design handle for molecular electronicswitching devices if a conical intersection can be engineered tooccur near a stable or metastable state. In the middle rangethe lowest-energy state overall is a metastable triplet so thatthe charge transfer may be coupled to magnetic changes. Thesinglet-triplet splitting is small. The metastability and the mag-netic moment of the lowest-energy charge-transferred state atthe intermediate approach distance are surprising and also inviteexploitation.

Organized charge transfer is a key element of photoactive re-ceptors and electronic devices for sensing, signal conversion, andmemory. The transfer of a significant amount of charge when asystem parameter is varied a small amount provides a control me-chanism for charge transfer. The biradical character of the initialcharge transfer state provides a mechanism for a magnetic statechange to be coupled to the motion of the metal atom. The para-meter here is a geometrical coordinate, but in a device sensitivityto a geometrical coordinate could be replaced by sensitivity to aphysical parameter such as pressure, external field, or substrate(71). The high sensitivity could then be manifested in a large hy-perpolarizability or sensitive control of charge or magnetizationstate. The present results show that it is important to consider notjust equilibrium structures but also nearby features of the poten-tial energy surfaces that indicate accessible mechanisms for pro-cesses that change the charge distribution or magnetic state.

MethodsThe MRMP2 results in Table 1 are based on CASSCF (72, 73) reference wavefunctions with two electrons in two active orbitals, ϕ1 and ϕ2; this activespace is denoted here as ð2∕fϕ1;ϕ2gÞ or ð2∕f4s;3dgÞ. The ð2∕fϕ1;ϕ2gÞ activespace is the minimal active space required to get a qualitatively correct de-scription of the shapes of the lowest potential energy surfaces and the three(middle, M-S1 and M-T1, and inner, M-S2) mechanistically relevant energyminima. In terms of atomic orbitals, ϕ1 is mainly composed of the 4s orbital

on Ca, whereas ϕ2 is a combination of the 3d orbital of Ca and a π� orbital ofthe hydrocarbon at medium and small R, and it is a pure 3d orbital of Ca atlarger R; the latter orbital is defined (among the set of 5 3d orbitals) as theone that has the largest overlap with a π� lowest unoccupied molecularorbital of the molecule. In MRMP2 calculations all orbitals were correlated;i.e., no core orbitals were frozen. Molecular geometries were fully optimizedby MRMP2 with numerical gradients.

Partial atomic charges in Tables 1 and 2 are Hirshfeld charges (74) ob-tained by density functional theory calculations with the M06-2X densityfunctional. (The Hirshfeld charges are about 25% smaller than chargesobtained by fitting the electrostatic potential.)

M06-2X density functional calculations are spin-unrestricted (i.e., spin-polarized) for the biradical states and spin-restricted for S0 and S2.

Cuts through the PESs shown in Fig. 5 are obtained by multistate multi-reference perturbation theory (MS-CASPT2) based on SA-CASSCF referencewave functions with all four states weighted 25%. For the CaEth system,the active space used in the reference SA-CASSCF wave function involves fouractive electrons in seven active orbitals. This active space is larger than theactive space used in geometry optimizations by MRMP2; in addition to thetwo above-mentioned orbitals (4s and 3d orbitals of Ca), it also includes the π,π� orbitals of ethylene, and the 3px , 3py , and 3pz orbitals of Ca; this activespace is denoted (4∕4) or ð4∕f4s;3d;3px;3py ;3pz;π;π�gÞ. For CaBen system,the SA-CASSCF wave function is based on the ð2∕fϕ1;ϕ2gÞ active space asdescribed above.

The potential energy curves shown in Fig. 8 are obtained by SA-CASSCFcalculations. In the case of the CaCor system, the SA-CASSCF wave function isconstructed using the (2∕4) active space, where the four active orbitals in-clude the lowest-energy orbital in each of the four symmetries, in orderto include additional electronic states. In this case, seven states (three lowesttriplets and four lowest singlets) are weighted equally in the state average.For the CaBen system, these additional states are energetically higher and,for that reason, the state average includes only four states. The SA-CASSCFpotential curves for CaEth and CaBen systems shown in Fig. 8 are obtainedusing the same active spaces as in the correlated MS-CASPT2 calculations. TheCASSCF calculations include only nondynamical electron correlation, whereasthe other methods include both nondynamical and dynamical correlation.

The one-electron basis set used in coupled cluster, including all single anddouble excitations and a quasiperturbative treatment of connected triple ex-citations [CCSD(T)], MS-CASPT2, and most M06-2X and SA-CASSCF calcula-tions is def2-TZVPP (75), and in the MRMP2 calculations it is def2-TZVPPplus one s and one p diffuse functions on carbon atoms with the exponents0.04402 (76) and 0.03569 (76), respectively. The exceptions are the M06-2Xcalculations for CaCor shown in Fig. 2 and the SA-CASSCF calculations forthe same system shown in Fig. 7; these are performed with the def2-SVP(75) basis set. MRMP2 calculations are performed with GAMESS (77), M06-2Xand CCSD(T) calculations are performed with GAUSSIAN (78), and SA-CASSCFand MS-CASPT2 calculations are performed with MOLPRO (79).

ACKNOWLEDGMENTS. The authors are grateful to Doreen Leopold for helpfuldiscussion. This work was supported in part by the National Science Founda-tion under Grant CHE09-56776, by a Molecular Science Computing FacilityComputational Grand Challenge grant at the Environmental MolecularScience Laboratory of Pacific Northwest National Laboratory (computertime), and by the Minnesota Supercomputing Institute (computer time).

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