Metal-organic charge transfer can produce biradical states and is mediated by conical intersections

Abstract

The present paper illustrates key features of charge transfer between calcium atoms and prototype conjugated hydrocarbons (ethylene, benzene, and coronene) as elucidated by electronic structure calculations. One- and two-electron charge transfer is controlled by two sequential conical intersections. The two lowest electronic states that undergo a conical intersection have closed-shell and open-shell dominant configurations correlating with the 4s² and 4s¹3d¹ states of Ca, respectively. Unlike the neutral-ionic state crossing in, for example, hydrogen halides or alkali halides, the path from separated reactants to the conical intersection region is uphill and the charge-transferred state is a biradical. The lowest-energy adiabatic singlet state shows at least two minima along a single approach path of Ca to the π system: (i) a van der Waals complex with a doubly occupied highest molecular orbital, denoted , and a small negative charge on Ca and (ii) an open-shell singlet (biradical) at intermediate approach (Ca⋯C distance ≈2.5–2.7 Å) with molecular orbital structure ϕ₁ϕ₂, where ϕ₂ is an orbital showing significant charge transfer form Ca to the π-system, leading to a one-electron multicentered bond. A third minimum (iii) at shorter distances along the same path corresponding to a closed-shell state with molecular orbital structure has also been found; however, it does not necessarily represent the ground state at a given Ca⋯C distance in all three systems. The topography of the lowest adiabatic singlet potential energy surface is due to the one- and two-electron bonding patterns in Ca-π complexes.

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), optoelectronic and sensing devices (7, 8), and hydrogen storage materials (9–12). Charge transfer from the metal to the organic system is one of the key features determining adsorption energies, reactivities, electronic structure, conductivities, and optical properties (4, 8). Most studies of charge transfer have focused on equilibrium geometries (4, 13–15), but the design of molecular devices and the understanding of adsorption and reactivity also require understanding the dependence of charge transfer on molecular geometry (16). In this communication we show that charge transfer character can change rapidly and suddenly for small changes in geometry, and we explain this phenomenon in terms of conical intersections (CIs). In addition we show that the charge transfer state can be a ferromagnetically coupled biradical, which is relevant to the possibility of control of magnetization by an electric field in, for example, spintronics applications (17). A fundamental understanding of the interfacial states of metal atoms interacting with conjugated π-systems is an essential element underlying rational molecular electronics design.

To model metal-π interactions, we first considered interactions of a calcium atom with the coronene molecule, a small nonperiodic model of graphene. Calculations using density functional theory revealed the presence of two types of equilibrium structures: one at a distance R between Ca and the closest carbon of about 3.8 Å with slightly negative charge on the calcium atom, and another state with a smaller R and with considerable shift of electrons from Ca to coronene; the latter is metastable. To get a better understanding of the structure of potential energy surfaces that govern the charge transfer process and of the nature of electronic states involved, we performed a detailed electronic structure study using multiconfigurational wave functions for the interaction of Ca with the smaller prototype molecules benzene and ethylene as well as coronene.

The closed-shell singlet potential energy surfaces of all three systems 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 configuration, at a shorter R. Furthermore, at intermediate R the lowest singlet state is open-shell, and the lowest-energy state appears to be a triplet state. (Such states could be easily missed if one calculated only closed-shell singlets, as is often done in computational materials research.)

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

These systems present the classic case of two-orbital configuration interaction with , ϕ₁ϕ₂, and singlets and a ϕ₁ϕ₂ triplet. In the notation that we use below for potential energy surfaces and stationary points, S₀ denotes a state with the dominant configuration , S₁ and T₁ denote the open-shell singlet and the corresponding triplet, and S₂ denotes a state with the dominant configuration . We calculated these states with MRMP2, M06-2X, and CCSD(T) for ethylene and benzene and with M06-2X for coronene. Note that the S₀, S₁, S₂, and T₁ labels apply to diabatic (23) states, and due to conical intersections (which become avoided crossings along most paths), a given adiabatic state can have different labels at different geometries.

First we consider the amount of charge transfer as a function of the distance from Ca to the π-system. To ensure that the geometries considered are relevant to low-energy structures, we select the geometries making a path between the low-energy structures. Let R denote the distance from the Ca atom to the closest carbon atom in a molecular partner denoted P. Figs. 1 and 2 show the ground singlet state energy of all three systems as a function of R along a path constructed as follows: At large R the path is the linear synchronous path (LSP) (24, 25) from the global minimum S₀ van der Waals complex, Ca⋯P, to the minimum-energy S₁ structure; and at smaller R, it is an LSP from the S₁ minimum to an inner local minimum on the S₂ surface, and this path is extended linearly beyond the S₂ minimum as well. Figs. 1 and 2 also show the partial atomic charge q on Ca along these three paths. The jumps in q along a reaction path together with the multiple minima in potential energy are intriguing. Such jumps have also been observed in chemical reactions involving metal atoms and small electronegative molecules (26–31), but have not been well studied in materials chemistry context. The charge transfer mediated by conical intersections is particularly interesting in the present case because of the negative electron affinity of the hydrocarbon molecules, because of the nonreactive nature of the potential energy surfaces, because the jumps are very sudden, and because of the technological importance of metal-π interactions. Although the acceptor of the transferred electron has a negative electron affinity when isolated, it binds the electron in the presence of the cation produced by the charge transfer, as in the NaH₂ system (32).

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 infinitely separated reactants. The path has C₂ symmetry for CaEth and C_2v symmetry for CaBen. Dotted lines show the Hirshfeld charge on the Ca atom along the ground electronic state singlet PES. Two sudden changes in the charge distributions along this path occur at two conical intersections. All data shown in this 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. The path has C_2v symmetry.

The relative energies of optimized structures corresponding to key minima are given in Table 1, and their molecular geometries are shown in Figs. 3 and 4. In the structural labels, M denotes a local 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 in Table 1 is encouraging because previous work (33) showed poor agreement between these two kinds of calculations for Sc, V, and Ni complexes with benzene. (Those systems also show a long-range global minimum and a short-range metastable state, sometimes with a different spin; experiments (34, 35) are consistent with observation of the long-range state.) Fig. 5, Upper and Lower, show the energies of the four lowest electronic states of the CaEth and CaBen systems, respectively, as functions of the distance R along the LSP that connects an inner S₂-M and an outer S₀-M minima. The shapes of the potentials along these paths are similar for the two systems. The ground electronic state, which correlates adiabatically with the 4s² state of the calcium atom, intersects the first excited singlet state twice: first at R of approximately 3 Å, and then at a smaller R. At large R the first excited singlet state correlates adiabatically to the 4s¹3p¹ state of Ca, and it is dominated by the 4s¹3d¹ configuration of the Ca atom at geometries close to a conical intersection and at medium separations of the reactants in a region before the earliest S₀S₁ conical intersections due to crossing of the P and D states as the reactants approach.

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
Ethylene
S2-M	C2	1A	b2	0.7	25.5	21.5	22.5
S0S2-SP	C2	1A	a2	0.3		32.5	25.1
S1-M	C2	1B	(ab)S	0.3	3.45	0.87	1.62
S1-Mip	C2v		(a1a2)S	0.3	29.0	27.0	30.7
T1-M	C2	3B	(ab)T	0.3	0.0	0.0	0.0
T1-Mip	C2v		(a1a2)T	0.3	27.6	28.1	30.1
S0-M	C2v		a2	0.0		−11.9	−17.4
S0-M	C2v		a2	0.0		−11.7	−17.2
Benzene
S2-M	C2v			0.6		18.9	25.2
S1-M	C2v		(a1a2)S	0.2		−0.25	0.68
T1-M	C2v		(a1a2)T	0.2		0.0	0.0
T1-Mip	C2v		(a1a2)T	0.3		46.1	43.9
S0-M	C2v			0.0		−7.2	−15.4
S0-Mip	C2v			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 at the best electronic structure level available, namely, MRMP2/CASSCF(2/2)/def2-TZVPP + d for structures T₁-M – S₁-M_inpl (inpl denotes in-plane), CASSCF(4/7)/def2-TZVPP for structure S₀S₁-CI, and M06-2X/def2-TZVPP for structures S₀-SP – S₀-M_inpl.

Fig. 4.

Same as Fig. 3 except for CaBen, for which all geometries are from M06-2X/def2-TZVPP calculations.

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-electron multicentered bond between Ca and the hydrocarbon) in these systems as functions of the same distance. Arrows indicate approximate position of the optimized structures given in Table 1. The zero of energy in each case is at the outer minimum.

The middle minimum corresponds to a metastable biradical, which is found to have a similar bonding pattern in the Ca-ethylene, Ca-benzene, and Ca-coronene systems. In each case, a one-electron bond is formed between the Ca atom and the π-electron system, in which Ca and the hydrocarbon share one electron in an orbital ϕ₂ formed by the interaction of the d-type orbital of Ca and one of the π^∗ orbitals of the hydrocarbon molecule. Because interactions between two closed-shell species in the ground electronic state are usually closed-shell singlets, the one-electron bonding in systems considered here is surprising. The origin of this unusual bonding type may be understood as follows: At large intermolecular separation of the reactants, the Ca atom in its ground electronic state has a closed-shell 4s² dominant configuration, whereas its 4s¹3d¹ configuration corresponds to the second excited state (36). As the reactants approach, the 4s¹3d¹ state of Ca is lowered in energy due to the interaction of the 3d orbital with a π^∗ orbital of the hydrocarbon; therefore, this state undergoes (avoided or actual, depending on the path) intersection with the state correlating with the 4s¹3p¹ state of Ca prior to the S₀S₁ conical intersections, and finally it intersects the ground electronic state at S₀S₁-CI. At R smaller than its value at S₀S₁-CI, the 4s¹3d¹ configuration becomes the lowest-energy state; it corresponds to a biradical. The metastable biradical structure constitutes a middle minimum that is separated from reactants by an (avoided or actual) intersection of the two lowest singlet states and is separated from the inner minimum by another (avoided or actual) intersection of the S₁ and S₂ states. The intersections are actual (conical) intersections along the paths shown in Figs. 1 and 2, but they would be (weakly) avoided for most nonsymmetric paths. When conical intersections with biradical states have been studied in organic chemistry (37–57), they are often located much higher in energy relative to the ground electronic state and come into play only in photochemically induced processes, whereas the present intersections arising from the interaction of a π-electron system with a metal atom are within 10–30 kcal/mol from the isolated ground state reactants.

The biradical thus corresponds to a weakly quasibound complex of a hydrocarbon molecule with the Ca atom, in which the latter largely preserves its atomic nature with an 4s¹3d¹ open-shell dominant configuration, bound to the hydrocarbon molecule by a one-electron multicentered bond. A schematic representation of the orbitals involved in the bonding is given in Fig. 6. The CaEth complex is bihapto (η²) with the Ca atom anchored to the two carbons by the three-centered one-electron bond formed from the d_xy orbital of Ca and π^∗ orbital of ethylene, and the CaBen complex is tetrahapto (η⁴) with the Ca atom anchored to the carbons by the five-centered one-electron bond formed from the d_xz orbital of Ca and one of the π^∗ (e_2u) orbitals of benzene (see Fig. 6). The degeneracy of the two (e_2u) orbitals of benzene is lifted when the Ca atom approaches the benzene ring, favoring the state with the highest orbital overlap.

Fig. 6.

Schematic representation of the key molecular orbitals (one of the Ca atom, and another on the hydrocaron molecule) involved in bond making for CaEth (Left) and CaBen (Right) complexes.

At geometries close to the middle minimum, the lowest-energy state is a ϕ₁ϕ₂ triplet, whose energy is lower than the corresponding ϕ₁ϕ₂ singlet. This feature may be viewed as a consequence of the lower energy of the 4s3d ³D state of free Ca atom as compared to its 4s3d ¹D state (36). The T₁ triplet is close to the S₁ singlet both geometrically and energetically. Spin change for the lowest adiabatic state has also been observed in, e.g., interactions of transition metal atoms with benzene (13), in reactions of Mo with CH₄ (58), and in more complex organometallic reactions (59, 60).

Shapes of the bonding orbital along the LSPs in CaEth and CaBen complexes are shown in Fig. 5. The three-centered one-electron bond in the CaEth complex leads to an elongation of the carbon–carbon distance by about 0.1 Å, and the five-centered one-electron bond in the CaBen complex leads to an elongation of the two opposite carbon–carbon bonds of the four carbons involved in the bonding by about the same amount. At the minimum on the B state of the CaEth complex, R is about 2.5 Å. The four carbon atoms involved in the bonding in the CaBen complex are equally distant from the Ca atom by 2.64 Å, and the two remaining carbons are equally distant from the Ca atom by 2.70 Å. The low-lying d orbitals of Ca have also been found (9) to play an important role in Ca binding to sp³ sites in covalent-bonded graphene (an all-graphitic 3D porous material). It is very important to understand the nature of the metal coordination because the coupling of a sudden charge transfer to a magnetic state provides an additional degree of freedom relevant to spintronics applications.

Structures S₂-M of CaEth and CaBen correspond to local minima on the S₂ state at small R. At these structures the portion of the valence wave function on Ca is dominated by the 3d² () 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 to the closed-shell bonding pattern, these structures are characterized by shorter R and longer carbon–carbon distances than the structures at the middle minimum.

In addition to the three minima, a variety of other critical points have been located on the potential energy surfaces of each of the systems considered. Structure S₀ -SP1 is the first order saddle point that connects S₂-M with an outer van der Waals complex on the lowest-energy closed-shell adiabatic potential energy surface (PES). Structures T₁-M_ip and S₁-M_ip represent local minima on the T₁ and S₁ potential surfaces analogous to the structures T₁-M and S₁-M with the only difference being that the Ca atom in the former two cases is placed in the plane of the ethylene molecule. These structures are found to be considerably higher in energy than their out-of-plane counterparts. An analogous T₁-M_ip structure has also been located on the lowest triplet potential energy surface in the case of CaBen complex. Because of the rather high energy of this structure, no attempt was made to locate a similar structure for an open-shell singlet state.

Structure S₀S₁-CI is a minimum-energy point on the S₀S₁ conical intersection of the two lowest singlet states. This structure, as optimized by complete active space self-consistent (CASSCF)(2/2)/def2-TZVPP, exceeds the energy of the E-S₁-M inner minimum of the CaEth complex by 13.9 kcal/mol. Calculations including dynamical electron correlation are expected to decrease 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 well studied (68). The effect of a CI on a thermally activated reaction depends on the energetic accessibility of the dividing surface and of the seam of intersection. Examples of thermally activated reactions where the CI seam is (or can be) accessed under usual conditions involve chemiluminescence, energy transfer, reactions involving metal atoms (29, 32), and hydrogen atom abstractions from phenolic antioxidants (69). The CIs in systems considered here are particularly interesting because of their relevance in nonreactive processes, such as attachment of metal atoms to aromatic systems, which is of utmost importance in organometallic chemistry and materials chemistry.

We now return to the coronene–calcium system, which will be abbreviated CaCor. The coronene-Ca cation has been recently studied by photodissociation (70), but we know of no experiment on the neutral. As in the cases of the smaller prototype systems, three kinds of equilibrium structures with dominant electron configurations , (ϕ₁ϕ₂), and exist that correspond to the outer, middle, and the inner minima, respectively (optimized molecular structures are shown in Fig. 7, and their energies are given in Table 2). Structure S₀-M_cr (cr denotes central ring) () corresponds to the van der Waals minimum (no change in the electron configurations of the fragments relative to the separated fragments; R ≈ 3.8 Å), structures T₁-M_cr [(ϕ₁ϕ₂)T], and S₁-M_cr [(ϕ₁ϕ₂)S] correspond to the middle minima (one-electron bond between Ca and the coronene molecule; R ≈ 2.7 Å) analogously to structures T₁-M and S₁-M for the CaEth and the CaBen systems, and structures S₂-M_cr () and S₂-M_or (or denotes outer ring) () correspond to the inner minima (two-electron bond between Ca and the coronene molecule; R ≈ 2.4–2.5 Å) analogously to the structures S₂-M in the two smaller systems. The notable distinctions between the CaCor and the CaBen systems are the following: (i) there are more than one attachment site for the metal atom in the former and (ii) due to the lower energy of the first several unoccupied molecular orbitals in the former system, several excited states at medium R become lower in energy as compared to the CaBen system.

Fig. 7.

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

Table 2.

Relative energies at the optimized molecular structures of the coronene + Ca system^*

Structure	PG	State	ec	q	M06-2X
S2-Mcr	C2v			0.7	42.8
S2-Mor	Cs		a′′	0.7	18.6
S1-Mor	Cs		(a′a′′)S	0.3	−0.2
S1-Mcr	C2v		(a1b1)S	0.4	6.1
T1-Mor	Cs		(a′a′′)T	0.3	0.0
T1-Mcr	C2v		(a1b2)T	0.4	7.5
T1-Mcr	C2v		(a1b1)T	0.4	7.5
T1-Mcr	C2v		(a1a2)T	0.3	19.0
S0-Mcr	C2v			0.0	−3.3

The energies in this table are in kcal/mol, with respect to the inner minimum, ).

^*See footnotes to Table 1.

Structures in which Ca is bound to an outer ring (S₁-M_or, T₁-M_or, and S₂-M_or) are found to be energetically lower than structures in which Ca is bound to the central ring (i.e., than the corresponding structures S₁-M_cr, T₁-M_cr, and S₂-M_cr). This feature may be rationalized by considering that the geometry distortion accompanying the metal-π bond formation is energetically more favorable for an outer ring than for the central ring. Mobility of the metal atom between different rings in both singlet and triplet manifolds represents an interesting question inviting a further study. The electronic state at the T₁-M_cr minimum-energy geometry is found to be (nearly) degenerate (with the and components in C_2v symmetry). Furthermore, the excited state has an energy minimum only ≈12 kcal/mol above the ground in the vicinity of the T₁-M_cr structure. On the contrary, the excited states above the T₁-M minima in the case of CaBen system are appreciably higher in energy. Fig. 8, Lower, depicts the eight lowest electronic states of the CaCor system as a function of R for the perpendicular approach of Ca toward the center of the coronene molecule. Because of the large size of this system, the results are given only at the state-averaged (SA)-CASSCF level. For comparison, Fig. 8, Upper and Middle, show the SA-CASSCF results for the two smaller prototype systems. The similarity in shapes of the adiabatic PESs corresponding to the S₀, S₁, T₁, and S₂ states in all three cases implies that the basic features described above must be valid for interactions of Ca with a variety of aromatic molecules.

Fig. 8.

Cuts through the lowest adiabatic potential energy surfaces for CaEth (Upper) and CaBen (Middle) and CaCor (Lower) systems calculated by SA-CASSCF as functions of R. In the latter case, additional states are shown because of their low energies.

In summary, we find that approach of a Ca atom to an extended π-system involves three kinds of minima in the PES: a van der Waals minimum (i), a biradical formed by partial single charge transfer (ii), and a closed-shell singlet formed by partial double charge transfer (iii). The shift in charge in moving from i to ii or from ii to iii occurs suddenly (over a small range of geometry) because of two sequential (along a path) conical intersections. In the entrance channel, the and ϕ₁ϕ₂ states intersect in a region where the energy of the d orbital of the metal atom is lowered by interaction with the π-type lowest unoccupied molecular orbital of a molecule. The recognition that a sudden redistribution of charge may occur for a small change in geometry provides a computational design handle for molecular electronic switching devices if a conical intersection can be engineered to occur near a stable or metastable state. In the middle range the lowest-energy state overall is a metastable triplet so that the charge transfer may be coupled to magnetic changes. The singlet-triplet splitting is small. The metastability and the magnetic moment of the lowest-energy charge-transferred state at the intermediate approach distance are surprising and also invite exploitation.

Organized charge transfer is a key element of photoactive receptors and electronic devices for sensing, signal conversion, and memory. The transfer of a significant amount of charge when a system parameter is varied a small amount provides a control mechanism for charge transfer. The biradical character of the initial charge transfer state provides a mechanism for a magnetic state change to be coupled to the motion of the metal atom. The parameter here is a geometrical coordinate, but in a device sensitivity to a geometrical coordinate could be replaced by sensitivity to a physical parameter such as pressure, external field, or substrate (71). The high sensitivity could then be manifested in a large hyperpolarizability or sensitive control of charge or magnetization state. The present results show that it is important to consider not just equilibrium structures but also nearby features of the potential energy surfaces that indicate accessible mechanisms for processes that change the charge distribution or magnetic state.

Methods

The MRMP2 results in Table 1 are based on CASSCF (72, 73) reference wave functions with two electrons in two active orbitals, ϕ₁ and ϕ₂; this active space is denoted here as (2/{ϕ₁,ϕ₂}) or (2/{4s,3d}). The (2/{ϕ₁,ϕ₂}) active space is the minimal active space required to get a qualitatively correct description of the shapes of the lowest potential energy surfaces and the three (middle, M-S₁ and M-T₁, and inner, M-S₂) mechanistically relevant energy minima. 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 of the hydrocarbon at medium and small R, and it is a pure 3d orbital of Ca at larger R; the latter orbital is defined (among the set of 5 3d orbitals) as the one that has the largest overlap with a π^∗ lowest unoccupied molecular orbital of the molecule. In MRMP2 calculations all orbitals were correlated; i.e., no core orbitals were frozen. Molecular geometries were fully optimized by MRMP2 with numerical gradients.

Partial atomic charges in Tables 1 and 2 are Hirshfeld charges (74) obtained by density functional theory calculations with the M06-2X density functional. (The Hirshfeld charges are about 25% smaller than charges obtained 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 multireference perturbation theory (MS-CASPT2) based on SA-CASSCF reference wave functions with all four states weighted 25%. For the CaEth system, the active space used in the reference SA-CASSCF wave function involves four active electrons in seven active orbitals. This active space is larger than the active space used in geometry optimizations by MRMP2; in addition to the two above-mentioned orbitals (4s and 3d orbitals of Ca), it also includes the π, π^∗ orbitals of ethylene, and the 3p_x, 3p_y, and 3p_z orbitals of Ca; this active space is denoted (4/4) or (4/{4s,3d,3p_x,3p_y,3p_z,π,π^∗}). For CaBen system, the SA-CASSCF wave function is based on the (2/{ϕ₁,ϕ₂}) active space as described above.

The potential energy curves shown in Fig. 8 are obtained by SA-CASSCF calculations. In the case of the CaCor system, the SA-CASSCF wave function is constructed using the (2/4) active space, where the four active orbitals include the lowest-energy orbital in each of the four symmetries, in order to include additional electronic states. In this case, seven states (three lowest triplets 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-CASSCF potential curves for CaEth and CaBen systems shown in Fig. 8 are obtained using the same active spaces as in the correlated MS-CASPT2 calculations. The CASSCF calculations include only nondynamical electron correlation, whereas the other methods include both nondynamical and dynamical correlation.

The one-electron basis set used in coupled cluster, including all single and double excitations and a quasiperturbative treatment of connected triple excitations [CCSD(T)], MS-CASPT2, and most M06-2X and SA-CASSCF calculations is def2-TZVPP (75), and in the MRMP2 calculations it is def2-TZVPP plus one s and one p diffuse functions on carbon atoms with the exponents 0.04402 (76) and 0.03569 (76), respectively. The exceptions are the M06-2X calculations for CaCor shown in Fig. 2 and the SA-CASSCF calculations for the same system shown in Fig. 7; these are performed with the def2-SVP (75) basis set. MRMP2 calculations are performed with gamess (77), M06-2X and CCSD(T) calculations are performed with gaussian (78), and SA-CASSCF and MS-CASPT2 calculations are performed with molpro (79).