Spin states in polynuclear clusters: The [Fe₂O₂] core of the methane monooxygenase active site

Spin Contamination in Unrestricted Kohn–Sham Determinants

When a Slater determinant is constructed, the total spin S can be controlled in two ways. First, a total 〈_z〉 = M_S value of M_S = S can be adjusted in the initial guess determinant by choosing the appropriate number of α and β electrons, N^α and N^β, because any Slater determinant |Ψ〉 is an eigenfunction of the operator with the eigenvalue ,

(1)

For unrestricted Slater determinants, the total spin expectation value 〈²〉 is in general larger than the desired value of M_S(M_S + 1). Second, the 〈²〉 expectation value can be forced to be equal to M_S(M_S + 1) during the SCF iterations by applying the Lagrangian multiplier technique.36 This, however, leads in general to determinants with a higher energy than an unrestricted MO optimization without this constraint would yield.36 It can also lead to convergence problems in geometry optimizations. Slater determinants in Hartree–Fock theory should be eigenfunctions of the total spin operator, which commutes with the (spin-independent) nonrelativistic many-electron Hamiltonian. However, the Hamiltonian used in unrestricted KS-DFT, which depends on the spin density, describes a system of noninteracting fermions with same electronic density as the true electronic system. Because the KS surrogate system can exactly be described by a single Slater determinant (the potential energy terms are local), one may not request that Slater determinants used in KS-DFT should be eigenfunctions of ². Pople, Gill, and Handy have advocated that unrestricted KS determinants should not be spin eigenfunctions, because they are no wave functions, that is, they are not the wave functions of fully interacting electronic systems.37 In a similar spirit, Perdew, Savin, and Burke argued that there is no obvious reason why the KS noninteracting wave function should always have the same symmetry as the true ground-state wave function.38 There appears to be no reason why the disadvantages of the spin-constrained MO optimization should be accepted in KS-DFT. Without this constraint, unrestricted KS determinants may (and must) exhibit spin contamination.

Spin contamination denotes the difference between the 〈²〉 expectation value resulting from a quantum chemical calculation and the S(S + 1) eigenvalue, which is usually equal to M_S(M_S + 1) for the targeted S = M_S determinants. The actual 〈²〉 expectation value of a Slater determinant depends on the overlap integrals O$$ = 〈ψ$$|ψ$$〉 of the spatial parts |ψ$$〉 and |ψ$$〉 of the α and β spin orbitals |ϕ$$〉 and |ϕ$$〉 (see, e.g., ref.39),

(2)

as can be easily shown (N^α is assumed to be equal to or larger than N^β). In the case of a closed-shell determinant, the number of α electrons N^α equals the number of β electrons N^β, and the orthonormal set of spatial parts of the α orbitals is equal to the orthonormal set of spatial parts of the β orbitals. Therefore, each spatial part of an α spin orbital |ϕ$$〉 overlaps completely with the spatial part of the β spin orbital |ϕ$$〉, but not with with the spatial part of any other β orbital, 〈ψ$$|ψ$$〉 = O$$ = δ_ij. This reduces the double sum in eq. (2) to a summation over N^β with individual terms all equal to 1. Therefore, the sum equals N^β and the total spin expectation value indicates a pure spin state. A high-spin determinant in the strict sense, which consists of α orbitals only, is not spin-contaminated either, because N^β is equal to zero and the overlap matrix O^βα of α and β orbitals has no entries at all.

A high-spin determinant in the sense commonly used when talking about transition metal complexes and clusters, however, has as many excess α orbitals N^α – N^β as there are valence electrons on the metal centers of the molecule it describes. Note that this is a qualitative picture taylored for the single-determinant wave functions that allows one to clearly identify those molecular orbitals with characteristic atomic d orbital contributions from the metal centers. Experience shows that such a determinant is in most cases not spin-contaminated. This can be explained by taking an idealized unrestricted determinant as an example, which may consist of “closed-shell” orbitals, indicating pairs of α and β spin orbitals with equal spatial parts, and where only the valence electrons of the metal centers are occupying orbitals whose spatial part does not (or only little) overlap with the spatial part of any orbital with the other spin. These orbitals are termed “magnetic orbitals” (for a rigorous definition of magnetic orbitals and how they can be obtained from the canonic orbitals by unitary transformations, see, e.g., refs.40-42). For the hs determinant, all valence electrons on the metal centers have the same spin (by convention: α). The sum in eq. (2) then consists of N^β contributions of one from the (normalized) spatially fully overlapping pairs of α and β orbitals, and N^α – N^β contributions of zero from the magnetic α orbitals that do not overlap spatially with any β orbital. Note that the spatial parts of all β orbitals are equal to one of the other N^βα orbitals and the spatial parts of the α orbitals are by construction orthonormal among themselves. Thus, the sum is equal to N^β, and the 〈²〉 expectation value is equal to M_S(M_S + 1).

Such a qualitative picture might be close to reality, if either the unpaired electrons are occupying molecular orbitals with predominantly d atomic orbital character while the the core orbitals of the metals and the orbitals of the ligands keep their closed-shell character, or if the MOs can be transformed into a form close to this situation. This is confirmed by the fact that most hs (in the common, not in the strict sense) determinants are indeed not spin-contaminated.

Regarding the BS determinant, where the number of α and β electrons is the same and the total spin is equal to zero, while the local spins are not (the (nonzero) number of α electrons located at one metal atom is equal to the number of β electrons located at the other one), the sum in eq. (2) is lower than N^β. In the extreme case the sum is equal to N^β minus the number of unpaired α or β electrons. This is the number of “closed-shell” pairs of orbitals, whose overlap integrals each contribute a value of one to the sum, whereas the contributions from the overlap of the magnetic orbitals are much smaller than one. Therefore, spin contamination in unrestricted determinants is introduced by a lack of spatial overlap of pairs of α and β orbitals.

Heisenberg Model of Electron Spin Coupling

The energy difference of the ferromagnetic and the antiferromagnetic state of a system with two coupled magnetic centers A and B is usually dominated by the exchange contribution of the unpaired electrons to the energies. Dirac, Heisenberg, and van Vleck have pointed out that such a system can be described phenomenologically by a vector coupling model,43-46 in which the Heisenberg Hamiltonian is represented by47

(3)

Note that the form of this operator is typical for the quantum mechanical description of two coupled angular momenta. This model does not describe any electrons, but only two interacting local spin vectors. The orientation of the two vectors is allowed to be either parallel (ferromagnetic coupling) or antiparallel (antiferromagnetic coupling). The local spins, S_A and S_B, are traditionally assigned by chemical intuition. Ŝ_A · Ŝ_B has also been considered as a quantum chemical operator that can act upon electronic wave functions. The expectation value of 〈Ŝ_A · Ŝ_B〉 can be calculated in a rigorous way using the local projector technique proposed by Clark and Davidson31 (see below).

The coupling constant J_AB is determined experimentally such that the corresponding Heisenberg Hamiltonian is consistent with the measured temperature dependence of the magnetic susceptibility. It could as well be determined by calculating the energies of the ferromagnetically and the antiferromagnetically coupled state. Noodleman has proposed a scheme that allows the calculation of J_AB based on the energy difference between a BS determinant that describes the antiferromagnetically coupled state (usually with S = 0 if the two local spins on A and B are equal) and a determinant that describes the ferromagnetically coupled state (usually the hs state if no intermediate local spins are considered).35 BS and hs states are related by an inversion of the spins of all unpaired electrons on one center and thus have the same local S_A and S_B quantum numbers, but M$$ or M$$ with inverted sign. Noodleman's original expression has been used and extended by other authors. A comparison of different ways of calculating Heisenberg coupling constants with BS approaches for systems containing two magnetic centers was given by Soda et al.,19 who also addressed the strong dependence of the coupling constant on the parameters that determine the exchange–correlation potential in the density functional. Heisenberg coupling constants can be calculated according to Noodleman's expression35 as

(4)

or with an expression propsed by Noodleman35 and employed by Bencini et al.48 and Ruiz et al.49 as

(5)

where “BS” has been replaced by “0” to indicate that this expression only holds strictly for spin eigenfunctions with S = 0. Yamaguchi suggests to use19

(6)

and with Clark and Davidson's local spin expectation values,31, 50 we obtain

(7)

According to Soda and Yamaguchi,19 Noodleman's expression [eq. (4)] holds in the limit of weakly interacting and the expression of Bencini et al.48 and Ruiz et al.49 [eq. (5)] in the limit of strongly interacting magnetic centers (which is in the extremest case equal to a closed-shell molecule). Their own expression [eq. (6)] reduces to the appropriate limit, depending on the interaction strengh. This can be explained by considering that the spin contamination of a high-spin determinant is in general low,

(8)

and by employing eq. (2) for estimating the 〈²〉 expectation value of the M_S = 0 determinant. If all pairs of α and β orbitals overlap completely, the determinant is a closed-shell determinant, so ^BS〈²〉 is equal to zero, and ^hs〈²〉 – ^BS〈²〉 is equal to S_hs(S_hs + 1), the denominator in eq. (5). If there are magnetic orbitals that do not overlap at all (BS determinant), the sum in eq. (2) is a sum of Nβ – 2S_A overlap integrals equal to one (the sum of unpaired electrons on A or B, and thus the number of magnetic orbitals of α or β spin, is equal to 2S_A). Then the sum is equal to N^β – 2S_A and 〈²〉 equals 2S_A = S_hs, and ^hs〈²〉 – ^BS〈²〉 equals S_hs(S_hs + 1) – S_hs = S$$, which is the denominator of eq. (4). Furthermore, Clark and Davidson's expression also reduces to Yamaguchi's expression in case that both the high-spin and the BS determinant are eigenfunctions of the local spin operators $$ and $$. Then, 2〈Ŝ_A · Ŝ_B〉 is equal to 〈²〉 – S_A(S_A + 1) – S_B(S_B + 1) [see eq. (13)]. As mentioned above, the local spin quantum numbers S_A and S_B are equal for the hs and the BS determinant, so twice the difference between the 〈Ŝ_A · Ŝ_B〉 values of the hs and the BS determinant reduces to the difference of their total spin expectation values. This yields the denominator in Yamaguchi's expression, that is, both expressions are equal then.

Local Spin

To replace the somewhat arbitrary assignment of chemically intuitive local spins that is often employed when calculating Heisenberg coupling constants [see eq. (3) above], Clark and Davidson introduced a more rigorous quantum chemical method to calculate the expectation values of products of local spin operators Ŝ_A · Ŝ_B, which occur in the Heisenberg Hamiltonian.31-34 This technique has been applied to the calculation of Heisenberg coupling constants in polynuclear manganese complexes.50 The calculation of local spin expectation values relies on unitary local one-electron projection operators p̂_A(i) onto molecular fragments A such as metal centers in transition metal complexes. When the local projection operators onto all molecular fragments are summed up, this sum will be equal to the unity operator if the local projection operators are defined properly,

(9)

so that the sum of all local spin operators Ŝ_A defined by

(10)

will be equal to the total spin operator Ŝ,

(11)

Other local quantities such as the local contributions _zA and N_A to the z-component of the total spin operator, _z, and to the total electron number N, can also be formulated in terms of these local projection operators,51 as has been first proposed for population analyses by Clark and Davidson.52 The expectation values of the local z-component _zA of the total spin, and the local population of α and β electrons, N$$ and N$$, are related to the local 〈_zA〉 value, which is half the spin density on center A,

(12)

Equation (12) provides the connection to general population analyses. Usually the Mulliken analysis53 is applied for this purpose. However, the Mulliken projector does not fulfill the conditions required for true projection operators (see the discussion in ref.51). Local properties are not quantum chemical observables, but somewhat arbitrarily defined quantities that serve for a qualitative understanding of a complicated and unintuitive wave function. Please recall that there is no unique definition of local projection operators. Instead, a variety of partitioning schemes was developed for population analyses. The oldest of these schemes are the Mulliken53 and the Löwdin analyses.54 Clark and Davidson introduced in their work on local spin a modified Löwdin projection operator, where the basis set was first orthonormalized within the functions centered on one atom and then subjected to the usual Löwdin orthonormalization procedure. This local projection scheme, denoted here by Löwdin*, yielded local _zA values close to those obtained with the formally appealing Atoms-in-Molecules projectors55 for small molecules.51 It is therefore employed in this work for the calculation of the local spin expectation values 〈_zA〉 and 〈Ŝ_A · Ŝ_B〉. A derivation of expressions for these expectation values is given in refs.34 and51. Whereas the 〈Ŝ_A · Ŝ_B〉 expectation value is useful for the calculation of coupling constants, the 〈_zA〉 expectation value can be used for assigning local spin states to metal atoms in transition metal complex.

Calculation of Ideal Local Spin Values

For later convenience we should note here that a state with ideal spin values (for the applicability of the Heisenberg model) should be an eigenfunction of the total spin operator ² and of the local spin operators $$ and $$. Then, its ideal local spin eigenvalue 〈Ŝ_A · Ŝ_B〉 can be calculated from

(13)

(14)

(15)

Experimental Results for MMOH

As far as they are relevant for our calculations, the experimental results known for the intermediate Q of MMOH shall be summarized here. These data only serve as an orientation, because differences between calculated and measured values may originate for various reasons.

It is known from Mössbauer spectroscopy that there are two antiferromagnetically coupled iron(IV) atoms in Q with a coupling constant of J_Fe1Fe2 lower than −30cm⁻¹.65, 66 Furthermore, according to EXAFS spectra, the iron–oxygen bridges are asymmetric with Fe–O distances of 1.77 and 2.06 Å, and the Fe–Fe distance is between 2.46 and 2.52 Å.67, 68 XAS data propose a coordination number of five or lower for the iron atoms.67 For more details on how the geometry of Siegbahn's model compares to experiment see ref.29.

Possible Spin States Obtained by Chemical Intuition

The two iron centers of the MMO model complex have the formal oxidation number IV, which corresponds to a d⁴ configuration. The total spin state, which is usually referred to as hs is therefore the one with four unpaired electrons on each of the two iron atoms, which all have the same spin, yielding a total S quantum number of 4. Because the total number of electrons in the model complex is even, the lowest possible spin state is characterized by S = 0. This total spin can result from several local spin distributions. The two extrema are an equal distribution of the α and the β spin density over the molecule (closed-shell), and two locally hs iron centers that are coupled antiferromagnetically (M_SFe = ±2). The latter case is represented by one KS determinant here, which corresponds to Noodleman's BS solution.35

The two determinants that represent local hs states, that is, the BS and the total hs determinant, correspond to states that can be described by a Heisenberg model with two interacting S = 2 spins. If the energy of other spin states, however, is close enough to the ground state, they should not be neglected. Therefore, we also consider intermediate spin states.

Construction of Slater Determinants and Notation

To achieve the goal to present a complete picture of the energetics of relevant spin states as predicted by DFT, the full range of low-, intermediate-, and high-spin states has to be converged. We are aware that usually only the BS and hs states are converged by setting up a quite sophisticated MO guess, which then may converge more quickly to the states with desired total and local spin properties. However, there is no formal reason why fast convergence may always be expected. Therefore, we investigate convergence from standard initial guesses. It should be emphasized, however, that all states obtained after the SCF optimization are, of course, stationary points in the parameter space of the MO coefficients used for the calculation of the total electronic energy. A BS-like guess may only converge faster to some of these solutions.

Two different standard initial guesses for the MOs have been employed. MOs obtained from an Extended Hückel calculation (E) and the converged KS orbitals of the hs (S = 4) state (H) using the same functional and structure as for the actual calculation with lower spin. Both initial guesses have been used for calculating MOs with the BP86, the standard B3LYP, and modified B3LYP functionals with a varying fraction of exact exchange, for a BP86 optimized (1) and a B3LYP optimized (2) hs structure. The resulting determinants have been labeled by a special notation that is explained in Table 1.

Optimized Cluster Structures

To evaluate the influence of the density functional on the geometry, and the influence of this geometry difference onto the spin state energies, the molecular structures optimized with two functionals are compared. These are the pure density functional BP86 and the hybrid density functional B3LYP, which contains 20% of exact exchange. The triple-zeta basis set with polarization functions TZVP was employed. Because the choice of the density functional can influence the local spin properties of the KS determinant to which a given initial guess converges in the SCF algorithm, the hs state (S = 4), which features only one chemically reasonable local spin distribution (M$$ = M$$ = S/2), has been chosen for the optimization. The conclusions on the influence of the exact exchange in the density functional and the SCF convergence of standard initial guesses can be expected not to be affected by using structures obtained by optimization of other spin states. The optimization has been carried out in C₁ symmetry. The optimized molecular structures of the MMO model obtained with B3LYP and BP86 (values in parentheses) are compared in Figure 1.

Whereas the Fe–Fe distance is the same in both structures, the symmetry of the oxygen bridge depends on the density functional. The Fe–O bond lengths to Fe1 and Fe2 differ by 0.197 and 0.166 Å for the bridging oxygen atoms O1 and O2, respectively, in the B3LYP optimized structure. The result is a rhomboid-like geometry for the quadrangle consisting of Fe1, O1, Fe2, and O2. In contrast to this, in the BP86 optimized structure all Fe–O distances are equal and about in the middle between the two different values for the B3LYP structure. They range from 1.802 to 1.872 Å, differing thus by at most 0.025 Å. Because the geometry optimization has been performed in C₁ symmetry, that is, without symmetry, the structures are not perfectly symmetric. However, this slight distortion of the perfect symmetry is in the spirit of this work as it allows us to search for two BS solutions of slightly different energy.

Compared to experiment, the B3LYP optimized structure reproduces the asymmetric oxygen bridge quite well, whereas the BP86 structure does not. For both functionals, the iron–iron distance is about 0.1 to 0.2 Å longer than experiments suggest. One might expect this to change when using antiferromagnetically coupled BS MOs for the geometry optimization, because Mössbauer data indicate that this is the ground state. However, a structure optimization with the hs geometries as starting point yielded a geometry even further away from experiment with iron–iron distances of 2.610 (2.590) Å and oxygen bridges which are symmetric with respect to the bridge, but asymmetric with respect to on which side the bridge is. On the imidazole side, the Fe–O distances turned out to be 1.806 (1.800) and 1.800 (1.802) Å, whereas on the acetate side, they are equal to 1.773 (1.763) and 1.777 (1.763) Å. The values for B3LYP and BP86 (in parentheses) do not differ much. Siegbahn obtained structural parameters much closer to experiment23 for this model with B3LYP. This might be either due to different stationary points on the potential energy surface reached in this and in Siegbahn's work, or to the smaller double-zeta basis set and the ECP Siegbahn employed. We have chosen the hs geometry obtained with B3LYP for all further investigations of the spin state energies and spin properties, because it is closest to the experimental data. There is no reason to expect large deviations for the dependency of the SCF convergence behavior and the spin state energetics on the functional when changing the geometry for a few tenth of an Ångstrøm.

Total Energies

The total energies obtained with KS-DFT for different total spin states using the standard density functionals BP86 and B3LYP are summarized in Figure 2. The energy of the closed-shell singlet determinant obtained with an Extended Hückel guess and the corresponding density functional and structure is taken as the reference energy.

Figure 2 shows that the absolute values of the total energies depend strongly on the functional. The BP86 energies for a given structure are spread over a range of at most 155 kJ/mol, whereas for the B3LYP energies, the range is more than twice as large (up to 341 kJ/mol). The order of the spin states on the energy scale is the same for the following determinants regardless of functional and geometry: the BS determinant (with S = 0) has always the lowest energy and the restricted closed-shell determinant the highest one. In between, there are always a S = 1, a S = 3 and a S = 2 determinant with the same local spin properties for both functionals and both structures, and with the energy increasing in this order. The hs determinant, which has the highest energy of all unrestricted determinants when calculated with the pure functional BP86, is favored so much in the hybrid functional B3LYP calculation that it has the second-lowest energy after the BS determinant. For a more detailed analysis, knowledge on the local spins is necessary, so the discussion will be continued in the next section.

Local Spins

A comparison of total energies is only sensible if the corresponding determinants optimized with different density functionals describe the same electronic structure, that is, possess similar local spins. The relevant total and local spin expectation values of all determinants obtained with the different initial guesses, geometries and functionals are summarized in Table 2. These are the total spin expectation value 〈²〉, which is equal to S(S + 1) for total spin eigenfunctions, and the portion of the z-component of the total spin that can be attributed to the iron centers and to the bridging oxygen atoms, _zFe1, _zFe2, _zO1, and _zO2. The 〈Ŝ_Fe1 · Ŝ_Fe2〉 expectation value is also listed, which will be needed below for the calculation of Heisenberg coupling constants according to the equation by Clark and Davidson. The values for the closed-shell determinants have not been included, since their total and local spin expectation values (except 〈Ŝ_Fe1 · Ŝ_Fe2〉) are all equal to zero. The Löwdin* local projection scheme was used throughout for formal reasons (see earlier). Although usually employed in the literature, the (not rigorously defined) corresponding Mulliken local 〈_zA〉 values are not reported because they are very close to the Löwdin* ones. This is different from population analysis, where the Löwdin and the Mulliken partitioning scheme may yield quite different results.69, 70 Obviously, the dependence of the absolute α and β populations on the projection operator cancel for the example under study here when differences between those absolute populations are considered. As far as the local decomposition of the ² operator is concerned, on the other hand, no standard local projection operator has been established so far. Because their dependence on the local projection scheme has already been investigated,31-33,51 we restrict this work to the formally consistent Löwdin* projectors for the calculation of 〈Ŝ_Fe1 · Ŝ_Fe2〉 values.

Table 2. Selected Total and Local Spin Expectation Values Obtained with the Löwdin* Local Decomposition Scheme

Spin state	Functional	〈2〉	zFe1	zFe2	zO1	zO2	〈Fe1·Fe2〉
S4 E	1 BP86	20.06	1.55	1.56	0.23	0.17	2.30
	1 B3LYP	20.13	1.67	1.68	0.18	0.11	2.76
	2 BP86	20.06	1.52	1.55	0.26	0.18	2.32
	2 B3LYP	20.13	1.66	1.70	0.20	0.12	2.70
S3 E	1 BP86	12.10	1.35	1.08	0.19	0.18	1.31
	1 B3LYP	12.19	1.04	1.62	0.14	0.11	1.62
	2 BP86	12.07	1.47	0.89	0.18	0.20	1.26
	2 B3LYP	12.13	1.63	0.93	0.14	0.17	1.40
S3 H	1 BP86	12.10	1.34	1.09	0.19	0.18	1.31
	1 B3LYP	12.19	1.04	1.61	0.15	0.11	1.62
	2 BP86	12.07	1.47	0.89	0.17	0.20	1.26
	2 B3LYP	12.13	1.63	0.93	0.14	0.16	1.40
S2 E	1 BP86	6.12	0.81	0.81	0.17	0.14	0.44
	1 B3LYP	6.55	0.67	1.02	0.24	0.10	0.62
	2 BP86	6.18	0.80	0.82	0.17	0.16	0.54
	2 B3LYP	6.37	0.80	0.87	0.20	0.17	0.58
S2 H	1 BP86	6.12	0.81	0.81	0.17	0.14	0.44
	1 B3LYP	6.62	1.09	0.62	0.25	0.06	0.61
	2 BP86	6.18	0.80	0.81	0.17	0.15	0.54
	2 B3LYP	7.03	1.64	−0.10	0.13	0.13	−0.28
S1 E	1 BP86	3.67	−0.68	1.41	0.16	−0.03	−1.12
	1 B3LYP	3.15	0.56	0.46	0.34	−0.33	0.20
	2 BP86	3.97	−0.70	1.44	−0.10	−0.15	−1.09
	2 B3LYP	3.48	−0.31	0.90	0.30	0.15	−0.4
S1 H	1 BP86	3.68	1.42	−0.68	0.17	−0.04	−1.13
	1 B3LYP	4.14	1.65	−0.97	0.15	−0.05	−1.65
	2 BP86	3.75	1.44	−0.70	0.20	−0.10	−1.09
	2 B3LYP	3.23	0.77	−0.28	0.26	0.26	−0.34
	*2 B3LYP	4.09	1.64	−0.89	0.17	−0.12	−1.51
S0 E	1 BP86	3.33	−1.41	1.42	0.01	−0.01	−2.19
	1 B3LYP	2.22	0.39	0.35	−0.35	−0.32	0.08
	2 BP86	3.38	1.39	−1.41	0.09	−0.09	−2.07
	2 B3LYP	2.21	−0.37	0.33	0.33	−0.29	−0.24
S0 H	1 BP86	3.33	−1.42	1.42	−0.01	0.01	−2.19
	1 B3LYP	2.19	−0.38	−0.35	0.35	0.32	0.08
	2 BP86	3.38	−1.39	1.41	−0.08	0.09	−2.07
	2 B3LYP	2.30	−0.34	−0.29	0.32	0.25	−0.02

Ideal Local Spin Values

To be able to judge to which degree the determinants are close to eigenfunctions of the total and local spin operators, the ideal values for the corresponding expectation values are given in Table 3 for a selection of ideal spin states that are closest to the values in Table 2. The ideal 〈Ŝ_Fe1·Ŝ_Fe2〉 expectation values are calculated according to eq. (13). The total 〈_z〉 expectation values are not included in Tables 2 and 3, because all Slater determinants are eigenvalues of _z by construction. The states which are obtained by interchanging the ideal 〈_zFe〉 values are not included either, because the labeling of the iron atoms is not essential.

Table 3. Selected Ideal Total and Local Spin Expectation Values

Spin state	〈2〉	zFe1	zFe2	zO1	zO2	〈zFe1zFe2〉
S4	20.00	2.00	2.00	0.00	0.00	4.00	↑↑↑↑ - ↑↑↑↑
S3	12.00	2.00	1.00	0.00	0.00	2.00	↑↑↑↑ - ↑↑
S2	6.00	2.00	0.00	0.00	0.00	0.00	↑↑↑↑ - 0
	6.00	1.00	1.00	0.00	0.00	1.00	↑↑ - ↑↑
S1	2.00	2.00	−1.00	0.00	0.00	−3.00	↑↑↑↑ - ↓↓
	2.00	1.00	0.00	0.00	0.00	0.00	↑↑ - 0
S0	0.00	2.00	−2.00	0.00	0.00	−6.00	↑↑↑↑ - ↓↓↓↓
	0.00	1.00	−1.00	0.00	0.00	−2.00	↑↑ - ↓↓

• In the last column, the local spins on the iron atoms are depicted. Excess α and β electrons are marked by arrows pointing up and down, respectively.

Drawing conclusions from local spin expectation values on local S_A quantum numbers in KS determinants within the collinear KS-DFT applied here is not straightforward. The 〈_A²〉 expectation values yield nonzero local spin quantum numbers for closed-shell molecules with zero spin density at every point of space,51 and are therefore not a reliable measure for the local spin quantum numbers S_A on the magnetic centers for any molecule that contains a “closed-shell” contribution to chemical bonds of the magnetic centers. Note that this case cannot be excluded for the transition metal complex under investigation here. The local 〈_zA〉 expectation values describe closed-shell molecules in accordance with chemical intuition, but the local M$$ quantum numbers they yield are not necessarily equal to the local S_A quantum numbers. However, in accordance with total spin quantum numbers of Slater determinants, where we use M_S as a means to produce a well-defined total spin quantum number for S, we will interpret the 〈_zA〉 values as local S_A quantum numbers.

Total Spin

The spin contamination is, in general, low for the higher spin states (S = 2 to S = 4), but increases the more the local 〈_zA〉 values on the two iron atoms differ. Such differences occur more frequently for the lower-spin states (S = 1 and S = 0). The 2(S2,H,B3) determinant with ideal local iron spins of 2 and 0, for example, has a 〈²〉 expectation value of 7.03, whereas the same determinant obtained from an Extended Hückel guess, 2(S2,E,B3), with ideal spins of 1 on both iron atoms, has only 6.37, which is fairly close to the ideal value of 6. The BS determinants (with S = 0), with 〈²〉 between 3.33 and 3.38, all deviate strongly from the ideal total spin expectation value of zero. For determinants with comparable total and local spins, the hybrid functional B3LYP always yields a slightly higher 〈²〉 expectation value.

Total Energies

The dependence of the total energies of the dinuclear MMO model complex on the exact exchange admixture in the hs B3LYP optimized geometry (structure 2) is depicted in Figure 3.

Figure 3 is divided into two plots. The upper one contains results for all determinants whose energy depends continuously on c₃ (including the higher spin states with S = 4, S = 3, and S = 2). The lower one contains those determinants for which the energy plot is discontinuous (with the lower spins S = 1 and S = 0). The discontinuities indicate qualitative changes in the unrestricted determinants (the reference closed-shell state possesses, of course, a perfectly continuous behavior with varying c₃). If the initial guess was modified appropriately and/or the state to which the SCF algorithm converges could be controlled during the MO optimization, it might be possible to obtain several continuous lines corresponding to all these different states as indicated by dotted lines in Figure 3. For the S = 0 state, this continuation is carried out explicitely by taking the molecular orbitals from the determinant 2(S0,H,B3[0.04]) as an initial guess for electronic structure calculations with S = 0 and with c₃ parameters ranging from 0.08 to 0.24. This determinant was chosen because it is closest to the lower energy side of the discontinuity of the 2(S0,H) plot.

The slope of all graphs in Figure 3 is negative. Except for the BS determinant 2(S0,H), the graphs show a nonlinear behavior, which is most pronounced for the hs determinant 2(S4,E). This nonlinear behavior is atypical when compared to the findings for mononuclear transition metal complexes, where the dependence of the total energy difference ΔE between two spin states of uncharged complexes on c₃ is strictly linear.13, 14, 16 Such a strict linear dependence is to be expected when the molecular orbitals ψ_i(c₃) do not change significantly with varying c₃. In that case, the contributions to the total energy that results from the kinetic energy of the electrons, the external potential and the classical Coulomb interaction of the electrons will be independent from c₃. The remaining contribution, the exchange–correlation functional E$$, depends linearly on c₃ when the orbitals do not change [see eq. (16)]. Thus, for largely c₃-independent orbitals, the energy difference ΔE is a sum of a term which varies linearly with c₃ and a constant,

(17)

The constant sums up the differences of the energy contributions from the kinetic energy, the external potential, the Coulomb integrals, and the quantities in E$$ which hardly depend on c₃. The deviation from strict linearity observed for the dinuclear MMO model complex must therefore be due to a nonnegligible dependence of the molecular orbitals {ψ_i(c₃)} on the c₃ parameter. This qualitative change in the molecular orbitals can be seen in the dependence of the local spin expectation values on c₃, which is discussed in detail later.

In addition to the absolute energies, the relative values for different spin states also change when c₃ is varied. The hs (S = 4) determinant possesses the highest energy of the four determinants plotted in the upper part of Figure 3 when the exact exchange admixture is equal to zero, but has the lowest energy of the four when the exact exchange admixture is higher than 12%. The BS determinant 2(S0 HB/EB) in the lower part, which depends continuously on c₃, has always the lowest energy of all spin states under investigation. Furthermore, Figure 3 shows that the energies cover a broader range of values when the c₃ parameter is increased. These qualitative findings are in agreement with the corresponding energy scheme for the pure BP86 functional and the original B3LYP functional in Figure 2.

Furthermore, it can be concluded that to which state a determinant converges strongly depends on the exact exchange admixture in the density functional when the total spin state is low (S = 0 or S = 1). These determinants obtained with either Extended Hückel or hs guess MOs suddenly converge to higher-energy states when c₃ is higher than 0.04 to 0.12.

Total Spin

The change in the molecular orbitals with varying c₃ discussed in the previous section is reflected in the total and local spin values, which are plotted in Figures 4-7.

For all spin states, the total 〈²〉 expectation values in Figure 4 increase with c₃. For the S = 4, S = 3, and S = 2 determinants, the dependence on c₃ is very weak. The values cover a range of 0.1 (S = 4, S = 3) or 0.2 (S = 2) atomic units, respectively. For the lower spin determinants based on standard initial guesses (Extended Hückel or hs MOs), with S = 1 and S = 0, all plots reflect the discontinuous behavior, which has been discussed earlier with respect to the change of the total energy. The dependence of 〈²〉 on c₃ is much more pronounced for the lower spin than for the higher spin determinants. For the plot of the BS determinant 2(S0,EB/HB), whose continuity parallels the behavior of the same determinant in the total energy plot in Figure 3, the total 〈²〉 expectation value raises from 3.3 to 4.1 a.u. as c₃ is increased. It can be concluded from eq. (2) that this indicates a gradually decreasing overlap of the magnetic orbitals of the 2(S0,EB/HB) determinant with increasing c₃ parameter (provided a separation of the MOs into “closed-shell” orbitals (denoting pairs of α and β orbitals with an spatial overlap close to unity) and “magnetic” orbitals (which hardly overlap spatially with an orbital of the other spin quantum number) can be made as described earlier).

Local Spins

Figures 5 and 6 summarize the local 〈_zA〉 expectation values obtained with the Löwdin* partitioning scheme, which are equal to half the difference of the α and β electrons located on the atom A, for one of the Fe atoms (Fe1 in Fig. 1) and one of the bridging oxygen atoms (O1). The 〈_z〉 fraction on all other atoms except the two iron centers and the bridging oxygen atoms is below a value of 0.1 atomic units.

The local spin analysis shows that there are approximately three unpaired electrons located at Fe1 for the hs (S = 4), the S = 3 and one for the S = 2 determinants [2(S2,H)]. The corresponding 〈_zFe1〉 values increase from 1.51 to 1.67, from 1.45 to 1.65, and from 1.33 to 1.66, respectively. On the other iron atom, which is not included in the plot, the local spins correspond to those given for B3LYP/structure 2 in Table 2. The remaining unpaired electrons are distributed mainly onto the oxygen bridges.

The spin delocalization from the metal centers onto the bridging oyxgen atoms is reduced for S = 4, S = 3 and 2(S2,H) as c₃ increases. The 〈_zA〉 values for these determinants show that there is a small sudden qualitative change in the MOs between c₃ = 0.16 and c₃ = 0.20. For S = 2, the two standard intitial guess MOs have converged to determinants with different local spin properties. The MOs based on an Extended Hückel guess have a 〈_zFe1〉 value, which is quite constant around 0.8, and thus corresponds best to ideal local spins of 1 on both iron atoms. The MOs obtained starting from the hs MOs have most of the spin excess localized on the first iron atom, as indicated by the 〈_zFe1〉 value ranging from 1.32 to 1.66 (see also Table 2). This spin distribution is equal in energy to the symmetric one for c₃ = 0.0, but gets more favored at larger c₃.

The S = 1 and S = 0 curves show the same discontinuities as the plots discussed in the preceding sections. The 2(S0,EB/HB) curve, which was constructed from the 2(S0,H,B3[0.04]) MOs as initial guess, has local 〈_zFe1〉 values dropping from −1.36 to −1.68 a.u., which corresponds to an increase of local spin polarization with c₃ as also observed for the higher spin determinants. On the other iron atom, which has not been plotted, the values are the same as for Fe1 with the signs inverted. This determinant is therefore the closest one can get to a BS determinant with a standard SCF guess for this MMO model complex. It effectively has three unpaired α electrons on one iron atom and three unpaired β electrons on the other one. The somewhat strange convergence behavior of the S = 1 guesses to solutions with simply interchanged iron spins, but energy differences of up to 40 kJ/mol, has already been adressed.

The 〈Ŝ_Fe1 · Ŝ_Fe2〉 expectation values can be employed for the calculation of the Heisenberg coupling constants according to eq. (7). Because only the hs and the BS determinant are needed for this calculation, only those two plots in Figure 7 shall be considered explicitely.

As far as the other curves are concerned, they display the already described qualitative discontinuities for the S = 1 and S = 0 determinants, but also additional ones for S = 2 and S = 3, which have not been observed at this c₃ value in the other local spin, total spin, and energy plots. Such sudden qualitative changes of the orbitals that are not visible in the energy plots are also revealed by the plots of the local 〈_zFe1〉 and 〈_zO1〉 expectation values (for the S = 4 and S = 3 determinants in the former and the S = 3 and S = 2 determinants in the latter case). The important hs and BS 〈Ŝ_Fe1 · Ŝ_Fe2〉 curves [2(S4,E) and 2(S0,EB/HB)], however, are continuous. The hs values are close to the negative values of the BS case. The magnitude of both is increasing with c₃. With values of maximal magnitude of 2.82 and −2.92, both are quite far form the values for ideal local spin eigenfunctions of 4 and −6.