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Dynamical magnetostructural properties of Anabaena ferredoxin

Edited by Louis Noodleman, The Scripps Research Institute, La Jolla, CA, and accepted by the Editorial Board November 12, 2007 (received for review July 19, 2007)
Abstract
A mixed quantum/classical investigation of the dynamical magnetostructural properties, that is, “magnetodynamics,” of oxidized Anabaena PCC7119 ferredoxin is carried out at room temperature in two distinct conformational states. This protein hosts a [2Fe–2S] cluster in which two iron centers are antiferromagnetically coupled to an overall lowspin electronic ground state that has a genuine multireference character. To study the magnetodynamics of this prosthetic group, an approximate spin projection method is formulated in the framework of density functional theory that allows for multideterminant ab initio molecular dynamics simulations to be carried out efficiently. By using this scheme, the influence of both thermal fluctuations and conformational motion on the structure of the [2Fe–2S] cluster and on the dynamics of the antiferromagnetic coupling constant, J(t), has been investigated. In addition to demonstrating how sensitively the shape of the [2Fe–2S] core itself is affected by hydrogen bonding, the analyses reveal a complex dynamical coupling of J to both local vibrations and largeamplitude motion. It is shown that this interplay can be understood in terms of specific vibrational modes and distinct hydrogenbonding patterns between the iron–sulfur cluster and the protein backbone, respectively. This implies going beyond the Goodenough–Kanamori rules for angular magnetostructural correlations of oxidized iron–sulfur prosthetic groups.
Iron–sulfur clusters are among the most ubiquitous, functionally versatile, and ancient prosthetic groups in proteins (1–4). One of the major classes of mobile electron carriers in contemporary biology is ferredoxin (Fd). Its fourfold cysteinecoordinated [2Fe–2S] core mediates electron transfer from photosystem I to ferredoxinNADP^{+}oxidoreductase, which reduces NADP^{+} to NADPH. A particularly interesting system is Fd from cyanobacterium Anabaena PCC7119 whose crystal structure was recently obtained at 1.3 and 1.17 Å resolution in the oxidized and the reduced forms (5), respectively. Most interestingly, it was found that two alternative conformations of Cys46 exist, which is one of the four cysteinyl ligands of the [2Fe–2S] cluster (5). In the oxidized state Cys46 adopts the socalled COin conformation, where the peptide oxygen points in the direction of the cluster, whereas, on reduction, the backbone flips to the COout conformation.
Spin Coupling in Iron–Sulfur Clusters.
In the ground state of the oxidized form of bioinorganic [mFe–nS] clusters, the Fe^{3+} metal centers are typically in a local highspin d ^{5} configuration [i.e., S _{Fe} = 5/2 for ferric Fe(III) centers], whereas they themselves are antiferromagnetically coupled to yield an overall lowspin state due to dominant superexchange interactions (6). In particular, the electronic ground state of [2Fe–2S] clusters in Fd has genuine multireference character and makes computational studies challenging (6). The electronic structure is described most conveniently by the brokensymmetry (BS) state. At the heart of this approach is a onedeterminant spinpolarized (unrestricted) representation where the two sets of electrons localized on the two transition metal centers are kept in an antiparallel spin arrangement. Clearly, this state features enormous spin contamination and, because it is not an eigenstate of the total (coupled) spin operator, it does not represent the true lowspin antiferromagnetic ground state (7). Alternatively, sophisticated quantumchemical methods can be used to describe the coupled state, which suffer, however, from having to take into account both “static” (multireferenceinduced) and “dynamic” (interelectron cuspinduced) electron correlation (8–10). Computationally this implies severe limitations in terms of system size and coupling to molecular dynamics (MD). A well established compromise in terms of cost versus accuracy are spinprojection schemes (11, 12) that rely on an approximate purification of the spin state (13) by carrying out independent total energy calculations for several uncoupled spin configurations (14, 15). Such approaches allow the calculation of the exchange coupling constant J directly and have been proven to perform particularly well for binuclear homovalent clusters in static calculations (16, 17).
Iron–Sulfur Clusters in Proteins.
When it comes to [2Fe–2S] embedded in a protein such as Fd, quantumchemical calculations have been performed to estimate redox potentials (18, 19) or reorganization energies (20). Recently, there were attempts to understand the conformational changes (5) of Anabaena PCC7119 Fd by using density functional theory (DFT) calculations (21). All of these calculations were performed by using static structures. However, it has been shown most recently for rubredoxin (Rd) that a molecular description of the environment is crucial (22); note that Rd features only a singleFe center and thus no magnetic interactions. For large systems like proteins, mixed quantum/classical MD offers a particularly suited and efficient way to take such effects into account (23, 24).
Magnetodynamics of Iron–Sulfur Proteins.
Thus, when aiming at understanding iron–sulfur proteins it is inevitable to consider temperature effects, protein conformational motions, solvent interactions, and, most importantly, it is mandatory to properly describe the antiferromagnetic electronic ground state. Here, we introduce a general computational technique by which the antiferromagnetic groundstate structure and the dynamics of (binuclear homovalent) transition metal clusters in proteins at realistic conditions can be obtained efficiently in the framework of mixed quantum/classical MD in combination with an extended spinprojection formalism. By using this procedure, dynamical magnetostructural properties, that is, “magnetodynamics,” of the oxidized [2Fe–2S] cluster in fully solvated Anabaena PCC7119 Fd have been investigated in its COin and COout conformations at room temperature. Detailed insights into the relationships of magnetic properties and structural dynamics, including the conformational influence of the protein environment, are thus obtained. In particular, it is possible to go beyond the Goodenough–Kanamori rules (6), which fail to predict angular magnetostructural properties of such d ^{5}–d ^{5} binuclear clusters.
The Extended Broken Symmetry Formalism.
Phenomenologically, the exchange coupling in homovalent magnetic dimers can be modeled by the Heisenberg Hamiltonian (6) where Ŝ _{A} and Ŝ _{B} are effective local spin operators for the coupled sites A and B with associated quantum numbers S _{A} and S _{B}, respectively, Ŝ = Ŝ _{A} + Ŝ _{B} is the total (coupled) spin operator, and J is the exchange (magnetic) coupling constant. This Hamiltonian generates a spinladder of eigenstates with total (coupled) spin quantum number S ranging from S _{max} = S _{A}+ S _{B} (highspin, HS) down to S _{min} = S _{A}−S _{B} (lowspin, LS). The total energy E^{S} of a spincoupled state with total spin quantum number S is For given S _{A} and S _{B} this gives rise to Landé's interval rule, E^{S} − E^{S} ^{−1} = −2JS, which allows one to calculate the energies of all spin states if J and the total energy of one state is known. To calculate these quantities one needs the exact Clebsch–Gordancoupled eigenstates S _{A} S _{B;} S M〉 of the used many electron Hamiltonian. If, however, the coupled eigenstates are not exactly known, as often encountered in computational approaches, it is very useful to evaluate the expectation value of the total spin operator using approximately coupled states.
A way to estimate the coupling constant J by using Landé's rule in conjunction with E ^{HS} and E ^{BS} was put forward a long time ago (11, 12, 25), which is concisely reformulated and extended here. The HS and BS states are singledeterminant states characterized by parallel and antiparallel alignment of spins on the two coupled paramagnetic sites, respectively. Their total energies, E ^{HS} and E ^{BS}, are easily accessible by using a single Slater or Kohn–Sham spinpolarized (unrestricted) determinant and are written as: in the spirit of Eq. 2 . By using Eq. 3 together with Landé's rule it can be shown readily that, because of J can be obtained from theses energies (25). Based on Löwdin's exact expression (26) for the total spin of an Nelectron system (of N ^{α} and N ^{β} spin up and down electrons) using the twoparticle density matrix Γ and its formulation (27) in terms of offdiagonal elements one obtains with Here, N ^{α} + N ^{β} = N with N ^{α} ≥ N ^{β}, N _{nmag} ^{σ} + N _{mag} ^{σ} = N ^{σ}, N _{nmag/mag} ^{σ} is the number of paired/unpaired (i.e., “nonmagnetic/magnetic”) σelectrons, and where x _{i} = (r _{i} , σ _{i} ) is the combined spatial and spin coordinate of electron i; Eq. 5 has been given in terms of N ^{α} and N ^{β} in ref. 27. The physical meaning of the exact expression Eq. 5 , which does not rely on any orbital picture, is best revealed in the approximation to θ in the framework of unrestricted Hartree–Fock theory (28) and noting that θ = 0 and thus 〈Ŝ ^{2}〉 = S(S + 1) when the exact solution of Eq. 1 , that is, the pure spin state of the coupled system, is used; f_{i} ^{σ} = {0, 1} is the occupation number of spin orbital φ_{i} ^{σ}. Thus, θ is a “correction term” that takes into account the overlap of magnetic (unpaired) orbitals and spin contamination when 〈Ŝ ^{2}〉^{HS/BS} is evaluated in expressions like Eq. 4 . In addition, this view opens up a simple and unified pathway to derive systematically further approximations to turn the scheme into a computationally practical approach, vide infra. Importantly, θ can also be formulated approximately within DFT (27) and thus it can be used as an alternative in the formulae for J and c given below.
Putting all this together a general expression for J can thus be written as in terms of the two determinants. For an equal number of α and βelectrons, and assuming strong localization (i.e., no overlap of magnetic orbitals), θ is zero so that Eq. 7 reduces to the well known formula introduced by Noodleman (12). However, other formulae used in the literature (29, 30) can easily be derived as special cases of strong/weak localization and the total number of unpaired electrons. Last but not least, finite overlap could also be taken into account approximately in the spirit of a systematic improvement.
Crucial for our purpose is the observation that, based on Eq. 2 , the total energy of the lowspin ground state itself, E ^{LS}, can be reformulated with the help of Eq. 7 in terms of the HS and BS total energies as where projects the energy of the lowspin state from the energies of the two singledeterminant BS and HS states and c is given by for homovalent binuclear clusters. Next, having obtained such a closed expression for the total LS energy, forces, and thus an ab initio MD (31) approach can be introduced for antiferromagnetically coupled spin dimers in terms of a twodeterminant expression of its total energy. The gains of this extended brokensymmetry (EBS) scheme are twofold. It is possible, first of all, to optimize the structure of such antiferromagnetic clusters and, second, to study their dynamics by means of MD, which is our main interest. Finally, we note in passing that these expressions can be generalized to heterovalent and polynuclear magnetic centers.
Next, neglecting spin contamination and assuming strong localization of magnetic orbitals, thus neglecting their mutual overlap, Eq. 9 simplifies to with the corresponding LS total energy Eq. 8 . This special case is identical to that exploited by Noodleman and coworkers (19), but can be improved systematically depending on the approximation to θ. At this stage, we can formulate ab initio MD very efficiently in the framework of a generalized Car–Parrinello Lagrangian (31, 32) by which classical nuclear dynamics can be performed on the antiferromagnetically coupled lowspin ground state energy hypersurface. Here, M_{I} is the nuclear mass of the Ith nucleus and μ is the fictitious electron mass parameter. Note that the BS and HS determinants, and thus the two sets of orbitals {ψ_{i} ^{BS}} and {ψ_{i} ^{HS}}, are simultaneously propagated under the constraint that orthonormality is satisfied individually.
The EBS scheme so far allows the propagation of an isolated [2Fe–2S] cluster, thus calling for a coupling to force fields to take the protein environment into account. The coupling approach used here is based on the CPMD/Gromos QM/MM interface (33). Because the LS charge density is not known, it is crucial to obtain all contributions to hierarchical electrostatics (33) consistently with the chosen projection scheme (Eq. 8 ).
Results and Discussion
Before addressing the central issue of magnetodynamics, average structures obtained from ab initio MD of the iron–sulfur cluster in vacuo and as a prosthetic group are compared in Table 1; concerning equilibrium structures in vacuo, it is noted in passing that using the approximately spinprojected lowspin total energy, that is, Eq. 8 with Eq. 11 , for structure optimization, instead of the heavily spincontaminated singledeterminant E ^{BS} potential energy surface, leads to slightly more compact Fe_{2}S_{2}(SH)_{4} ^{2−} and to 10% larger J values. The average structure of the [2Fe–2S] moiety is fairly symmetric and planar in vacuo, whereas it is best described as a nonplanar and distorted lozenge within the biomolecular environment. The observed asymmetry of the [2Fe–2S] core in the COin conformer is in accordance with experimental xray data (5) where it has been connected to an asymmetry in the hydrogen bond network. Our simulations also show different bond distances within the [2Fe–2S] cluster that can clearly be traced back to the hydrogenbonding topology involving the iron–sulfur cluster and peptide hydrogen atoms as seen in Fig. 1 a. Although it has been suggested that the S2HN(Cys46) hydrogen bond exists only in the reduced state of Fd (5), the present simulations clearly show that this is also the case for the oxidized state.
In the COout conformation the pattern of hydrogen bonds between the core and the backbone differs significantly (see Fig. 1 b). Each of the sulfur atoms has three hydrogen bonds, but of considerably different lengths. The asymmetry in the bond distances within the Fe1S1Fe2 and the Fe1S2Fe2 moieties is also present in the COout conformation, although to a lesser extent than in the COin case. Overall, these data show how intimately the structure of such iron–sulfur cofactors is coupled to the particular hydrogen bond pattern spanned by the protein.
Furthermore, the data compiled in Table 1 demonstrate that the cluster is less compact in Fd and that the average value of J is decreased therein, thus showing a direct influence of the protein environment. At the same time, the difference in the average value of J for the COin and COout conformations reveals the influence of different protein environments. It is also interesting to observe that the amplitude of J fluctuations is larger in the COout conformation than in the COin (see Fig. 2 a).
Experimental values of J for Anabaena Fd are not available, but similar proteins like spinach Fd yield values between −183 cm^{−1} (34, 35) and −145 cm^{−1} (36). Overestimation of the absolute value of J by about a factor of two is a well known problem in DFT when using generalized gradient approximations (7). It is a direct result of the selfinteraction error leading to artificially delocalized magnetic orbitals, which should not be confused with the consequences of proper spinprojection as lucidly discussed in ref. 7. To cure this deficiency on a theoretically sound basis both spinprojection and selfinteraction correction must be taken into account (7), that is, by using hybrid functionals (37, 38) or by employing constrained DFT procedures (39). But in view of the additional computational complexity we presently do not follow this route; instead, we compare the dynamical EBS data with very elaborate, complete activespace configuration interaction (CASCI) calculations to assess their quality. To this end, approximately 100 structures of the ab initio fragment are sampled from the trajectory of the COin conformer and, for each of them, J has been computed by both EBS and CASCI. First, CASCI confirms explicitly the validity of the phenomenological spin Hamiltonian Eq. 1 by computing explicitly its energy levels. Thus, even in the fluctuating Fd environment, the spin eigenstates are found to obey Landé's pattern within 2% (data not shown), thus yielding Heisenberg's spinladder rather than imposing it a priori. Furthermore, according to Fig. 2 b, the behavior of J(t) provided by EBS is extremely well reproduced by CASCI. However, this highlevel wavefunctionbased calculation now underestimates J compared with experiment, which is a well understood phenomenon because the orbitals determined in the Hartree–Fock method underlying the CASCI calculations underestimate the Fe–S covalency and overestimate the energies of the d ^{4}–d ^{6} charge transfer states.
After having demonstrated that the EBS approach leads to a good description of the dynamics of the exchange coupling constant, J(t), its connections to structural motions and thus the dynamical magnetostructural properties can be analyzed. It is gratifying that the calculated vibrational power spectrum for the iron–sulfur cluster as well as its spectral decomposition are in agreement with recent resonance Raman data (40). A true magnetodynamical quantity, which is accessible with our technique, is the power spectrum J(ω) in Fig. 3 a, as obtained for the iron–sulfur cluster in vacuo as well as for the COin and COout conformers of Fd. Apparently, the sharp single peak present at ≈160 cm^{−1} in vacuo becomes broad and blueshifted in the protein. Additionally, in the COin conformation the spectrum between 260 and 450 cm^{−1} is more complex than in vacuo, whereas for the COout case the J spectrum exhibits one asymmetric peak at ≈370 cm^{−1}. Moreover, in both Fd conformers we observe some highfrequency components in J(ω) at ≈3,320 cm^{−1}.
To understand this complex dynamics we relate the features of J(ω) to specific vibrational modes of the [2Fe–2S] core. Taking into account the approximate D _{2h} symmetry of the [2Fe–2S] average structure in vacuo, it is appropriate to use symmetryadapted orthonormal modes as basis. Apart from overall rotations and translations of all four atoms in space, six internal modes exist, but only a subset is found to drive the magnetodynamics of the prosthetic group of Fd. The coupling is revealed in Fig. 3 b–d by presenting the frequencydependent crosscorrelations between J and the relevant normal modes in addition to the power spectra. In vacuo the two totally symmetric modes that describe the innercore vibrations in terms of angles (A_{g,A}) and distances (A_{g,D}) are sufficient to explain the dynamics of J. Whereas the former accounts for the lowfrequency part, the latter can be associated with the region between 300 and 450 cm^{−1}. The appearance of a double peak for the A_{g,D} breathing mode and thus also in J(ω) is attributed to a coupling of A_{g,D} to the Fe–SH ligand vibrations.
In the protein, Fd, the situation is much more complicated. In the region between 70 and 260 cm^{−1}, the J spectrum becomes broad and is blueshifted. Analysis shows that in addition to the A_{g,A} mode (≈190 cm^{−1}) the B_{1g,oop} outofplane vibration strongly correlates with J at ≈160^{−1}. At first sight, the appearance of the latter contribution may be due to the asymmetry of the cluster in Fd because of hydrogen bonding. However, this mode shows no strong correlation to J in the COout case, although the cluster is also asymmetric there. Thus, the correlation of B_{1g,oop} with J must be due to the coupling of other modes to B_{1g,oop}.
Between 260 and 450 cm^{−1}, we observe contributions from B_{2u} and B_{1g,ip} inplane vibrations apart from those of A_{g,D}. Naively, one would not expect J to be affected by these modes (6): motion along the B_{2u} mode enhances one superexchange pathway while destroying the other one, whereas both pathways are destroyed along the B_{1g,ip} mode. This picture indeed holds very well for the cluster in vacuo. However, because the structure is asymmetric in Fd, the above argumentation does not apply, at least not for B_{2u}. To resolve this situation we also analyzed the mutual coupling of the different modes. It turns out that the B_{2u} mode is only very weakly coupled to A_{g,D}, which is reflected in the small feature in the A_{g,D}J correlation at ≈400 cm^{−1}. In contrast to that, the B_{1g,ip} vibration shows strong coupling to both A_{g} modes which is reflected in their frequencydependent J crosscorrelations. It is interesting that, for the COout conformation, the B_{1g,ip} contribution disappears completely and also B_{2u} affects J(ω) only to a very small extent, indicating a different coupling pattern among the modes.
Finally, there are the intriguing highfrequency components of J(ω) at ≈3,320 cm^{−1} that must stem from an ultrafast modulation of J(t). Obviously, none of the core vibrations themselves can account for this behavior. As shown in Fig. 1 the iron–sulfur cluster in Fd is connected to the protein matrix by various hydrogen bonds in addition to the strong covalent cysteine links. In particular, hydrogen bonding between the bridging sulfur atoms of the cluster, S1 and S2, and hydrogen atoms of the peptide groups of the backbone influences the electronic structure of the iron–sulfur cluster. Indeed, a correlation analysis suggests a direct relationship between the corresponding S–H distances and the exchange coupling constant J. In particular, we find that in the COin conformation of Fd the interactions between Cys46 and S1 most notably influence J. Additionally, Cys49 forms a hydrogen bond that oscillates between S1 and S2. None of these hydrogen bonds seems to be related to J in the COout structure where instead Ser40 and Ser47 NH groups play the decisive role by directly interacting with S1 and S2, respectively.
Having demonstrated that magnetodynamical quantities such as P(J) and J(ω) provide a wealth of insights into the magnetic interactions, it remains to be discussed how they could be measured. Highresolution inelastic neutron scattering is a powerful technique to directly measure intensity profiles due to exchange splittings in complex molecular clusters (41, 42), which can be directly related to P(J) of Fig. 2 a for a given splitting if the isotropic Heisenberg model holds. Second, the frequencydependent modulation of J due to dynamical motion giving rise to rich power spectra J(ω) in Fig. 3 can be probed in the spirit of action spectroscopy. The change of P(J) (measured by inelastic neutron scattering) that is induced by concurrently exciting vibrational modes with strong electromagnetic radiation is recorded by scanning the excitation frequency ω in the infrared regime. This gives rise to resonances only at those frequencies ω where the modulation is most pronounced, which are the peaks of J(ω). Alternatively, the ωdependent excitation could be achieved by nuclear inelastic scattering of synchrotron radiation at Mössbaueractive nuclei which is already established for intramolecular vibrational spectroscopy that is sensitive to exclusively those modes that involve Fe nuclei (43, 44).
Summary and Conclusions.
To understand the properties of antiferromagnetically coupled binuclear transition metal clusters, we introduced an efficient twodeterminant Car–Parrinello ab initio MD scheme based on a spin projection technique that extends the traditional description by using the brokensymmetry approach. Using this method, we investigated the magnetostructural dynamics of an oxidized binuclear iron–sulfur cluster in Anabaena PCC7119 Fd in two conformational states, COin and COout, at room temperature.
The cluster is symmetric in vacuo, whereas in the protein matrix the local hydrogen bond network distorts its structure to a higher degree in the COin than in the COout form. Importantly, the protein environment not only affects the magnitude of the exchange coupling but also modulates its dynamics.
Thermal fluctuations are found to be able to change J by ≈50% from its average value. In all cases investigated, the dynamics of the exchange coupling constant J(t) and thus the gross structure of the resulting power spectrum J(ω) can be understood in terms of A_{g,A} and A_{g,D} symmetryadapted normal modes due to cluster vibrations. However, there is conformationspecific fine structure: within the COin conformation in Fd the B_{2u} symmetric mode also directly affects J and, in addition, the two B_{1g} modes modulate J(t) by coupling to the A_{g} modes, which is in contrast to the COout conformer. Furthermore, there is a highfrequency component at J(ω) ≈3,300 cm^{−1} in the protein that can be traced back to very specific hydrogen bonds, namely, Cys46S1 and Cys49S2 in the COin conformer and Ser40S1 and Ser47S2 in the COout Fd.
To conclude, the results presented lead to a deep understanding of the dynamic magnetostructural properties of oxidized iron–sulfur prosthetic groups. They confirm the expected dependence of J on the Fe–S distance within the core, but allow one to transcend the well established Goodenough–Kanamori rules (6) for the angular behavior. In particular, the dependence of J(t) on essentially only two symmetryadapted modes, A_{g,A} and A_{g,D}, suggests the possibility to parameterize the exchange coupling constant as a function of these most crucial internal coordinates.
Methods
Force Field and System Setup.
The protein model is based on the oxidized Anabaena PCC7119 Fd (5) (PDB ID code 1qt9; chain B) assuming standard protonation states and solvation with 13265 TIP3P water molecules while retaining the water molecules from the crystal structure and adding 23 Na^{+} and 5 Cl^{−} to establish a neutral system. Protein interactions are described by using AMBER94 (45) and the partial charges of the [2Fe–2S] core were determined by Bader analysis (46) of the BS density of [Fe_{2}S_{2}(SH)_{4}]^{2−} in vacuo, yielding 0.9385 and −0.7303 for Fe and S, respectively. To maintain charge neutrality in the QM/MM setup, the partial charges of the C_{α}, H_{α}, and S_{γ} atoms were adapted accordingly. The nonbonding interactions for Fe were obtained from a published heme group (47), whereas cysteinyl parameters were used for S.
Hybrid Simulations.
The hybrid QM/MM simulations (23, 24) were carried out by using the CPMD/Gromos interface (33) within the CPMD program package (31, 48; see also www.cpmd.org), both extended by the EBS scheme. The QM fragment consisted of the [2Fe–2S] cluster core supplemented by the S_{γ}, C_{β}, and H_{β} atoms of the four cysteinyl ligands; the dangling bonds at C_{β} were saturated by using capping H atoms constrained to the C_{α}–C_{β} connecting line. The QM fragment was described within spinunrestricted Kohn–Sham DFT in its plane wave/pseudopotential formulation (31). The PBE functional (49, 50) was chosen and the core electrons were taken into account by ultrasoft pseudopotentials (51) at a cutoff of 30 Ry containing additional dprojectors in case of S as well as scalar relativistic corrections and semicore states for Fe. Finite cluster boundary conditions (52) were imposed on a large cubic simulation box of 40 a.u. with a total charge of −2e. Car–Parrinello propagation (32) was performed with a time step of 4 a.u. (≈0.12 fs) by using a fictitious orbital mass parameter of 500 a.u. and substituting the H masses by D. Separate Nosé–Hoover chain thermostats (53) at 300 K were coupled to the nuclei and atoms in the QM and MM systems, respectively, in addition to thermostating separately the set of HS and BS orbitals. After equilibration of ≈3 ps trajectories of ≈16 and 12 ps were collected for the COin and COout conformations, respectively. Reference simulations of [Fe_{2}S_{2}(SH)_{4}]^{2−} where carried out in vacuo for approximately 5 + 18 ps by using similar parameters.
Power spectra of vibrations and J(ω) were computed from Fourier transforms of velocity and J timeautocorrelation functions, respectively, and were quantumcorrected by using the “harmonic quantum correction factor” (54). Crosscorrelations were obtained after projecting the nuclear coordinates on orthogonal modes normalized to unity variance. These latter modes were calculated by transforming the D _{2h} symmetryadapted modes such as to minimize the crosscorrelation of the corresponding projections of the trajectory.
Configuration Interaction Calculations.
Close to 100 configurations of the QM subsystem were sampled (every 60 fs from the EBSQM/MM trajectory of the COin conformer) for which CASCI calculations within an active space spanned by the 10 singly occupied Fe dorbitals were carried out by using orbitals from restricted openshell Hartree–Fock calculations of the HS determinant. To compensate for the known underestimation of superexchange by CASCI, J was obtained by formula 11 in ref. 55. with α = 2.93 calculated by using the scheme of refs. 55 and 56. The basis set had approximately triplezeta quality, was augmented by various diffuse functions (augVTZPP), and consisted of 500 functions.
Acknowledgments
We thank Christian Boehme, Axel Kohlmeyer, Gerald Mathias, and Bernd Meyer for fruitful discussions and technical help and RechnerverbundNRW and Bovilab@RUB for providing computational resources. This work was supported in part by Deutsche Forschungsgemeinschaft Grant DFG MA 1547/7 and Fonds der Chemischen Industrie (FCI).
Footnotes
 ^{†}To whom correspondence should be addressed. Email: eduard.schreiner{at}theochem.rub.de

Author contributions: D.M. designed research; E.S., N.N.N., R.P., and V.S. performed research; E.S., N.N.N., V.S., and D.M. analyzed data; and E.S., N.N.N., V.S., and D.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. L.N. is a guest editor invited by the Editorial Board.
 © 2007 by The National Academy of Sciences of the USA
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