Louis Noodleman

Valence bond description of antiferromagnetic coupling in transition metal dimers

A single configuration model containing nonorthogonal magnetic orbitals is developed to represent the important features of the antiferromagnetic state of a transition metal dimer. A state of mixed spin symmetry and lowered space symmetry is constructed which has both conceptual and practical computational value. Either unrestricted Hartree–Fock theory or spin polarized density functional theory, e.g., Xα theory, can be used to generate the mixed spin state wave function. The most important consequence of the theory is that the Heisenberg exchange coupling constant J can be calculated simply from the energies of the mixed spin state and the highest pure spin multiplet.

Ligand spin polarization and antiferromagnetic coupling in transition metal dimers

Abstract Starting with spin polarized determinants for an antiferromagnetic transition metal dimer and spin projected states obtained from them, we show that both superexchange coupling and ligand spin polarization contribute to the Heisenberg coupling constant J describing a ladder of spin states. A single broken symmetry UHF calculation when combined with an independent calculation for the high-spin state yields J from the equation, E(Smax) − EB = − S2maxJ. Ligand spin polarization affects the average value of both spin-independent one-electron operators like the electron density at the metal site and spin-dependent operators such as A tensors at metal and ligand sites. The theoretical analysis here shows a number of new physical effects and generalizes the results of our previous paper which dealt solely with the effects of superexchange coupling. The limitations of this approach are also examined. A more comprehensive theory is readily formulated for computational purposes, but closed form equations as a function of S are very complicated. In general, the spin hamiltonian is not of Heisenberg form when all ligand spin polarization terms are included. However, the major ligand spin polarization terms and the superexchange terms are of Heisenberg form. Both of these are included in the broken symmetry approach.

Orbital interactions, electron delocalization and spin coupling in iron-sulfur clusters

Abstract The interconnections among orbital interactions, electron delocalization and spin coupling in iron-sulfur clusters are reviewed, with special attention to the complex nature of spin and orbital states in 4Fe4S complexes. We summarize the uses of broken symmetry density functional calculations and spin projection methods for extracting Heisenberg spin coupling and electron delocalization parameters, as well as for understanding charge distributions and orbital aspects of electronic structure. The value of spin projection coefficients for sorting out spin coupling patterns in complex systems is also emphasized. Among the systems examined are oxidized, high-potential, iron-sulfur proteins, 4Fe ferredoxin proteins and related synthetic complexes. By analysis of experimental hyperfine parameters, a detailed model of spin coupling for the “double cubane” P cluster of nitrogenase has been proposed in recent work based on Mossbauer, electron paramagnetic resonance (EPR) and X-ray structural data; there is one pairwise valence delocalized and one trapped valence cubane in the P (oxidized) state. In the area of electron transfer energetics, we have found that Heisenberg spin coupling and electron delocalization both contribute substantially to the redox potentials of 4Fe4S complexes, and Heisenberg coupling contributes to the difference in redox potential between 1Fe and 2Fe2S complexes, based on recent density functional calculations for model systems in solvents.

Density-Functional Theory of Spin Polarization and Spin Coupling in Iron—Sulfur Clusters

Publisher Summary This chapter discusses recent progress toward the development of a unified picture of the electronic structures and spin interactions of iron–sulfur and related systems; these concepts provide a close connection between a spin Hamiltonian description and a more detailed orbital picture of the electron distribution. The iron–sulfur proteins and synthetic analogs are challenging systems for quantum mechanical methods because they contain a large number of electrons and because spin polarization and spin coupling are essential features of the complexes. Standard approaches of ab initio quantum chemistry start from a spin-restricted picture, which is poorly adapted to problems involving high-spin transition metal centers. For this reason, a combination has been developed of broken symmetry and spin-unrestricted methods that is particularly well adapted to study spin-polarized and spin-coupled systems. These ideas are well adapted for use with density-functional methods. In the chapter, the basic ideas are developed of this approach, using a perturbation theory formalism to rationalize the spin Hamiltonian and energy-splitting formulas that should be appropriate for spin-coupled transition metal clusters.

The Xα valence bond theory of weak electronic coupling. Application to the low‐lying states of Mo2Cl84−

We show that valence bond (VB) concepts can be introduced into Xα theory. The resulting Xα–VB model yields energy states which either are pure multiplets or can be combined by straightforward projection to give pure multiplets. The new theory should be more computationally efficient than Hartree–Fock‐based CI models. A preliminary study of the δ→δ* transition in Mo2Cl84− yields an excitation energy closer to experiment than previous theoretical values, including those obtained to date from GVB–CI calculations.

Density functional calculation of p K(a) values and redox potentials in the bovine Rieske iron-sulfur protein.

Abstract. The redox potential of the Rieske iron-sulfur protein depends on pH. It has been proposed that the histidines coordinating one of the irons are responsible for this pH dependence, but an experimental proof for this proposal is still lacking. In this work, we present a density functional/continuum electrostatics calculation of the pKa values of the histidines in the Rieske iron-sulfur center. The calculated apparent pKa values are 6.9 and 8.8 in the oxidized state, which are in good agreement with the corresponding experimental values of 7.5 and 9.2 and the measured pH dependence of the redox potential. Neither of these two pKa values can, however, be assigned to only one of the histidines. We find that both histidines titrate over a wide pH range in the oxidized state. Reduction of the iron-sulfur center shifts the pKa values to 11.3 and 12.8, thus above 10.0 as found experimentally. The results provide a complete picture of the coupling of proton and electron binding, showing strongly cooperative binding of protons at electrode potentials near the redox midpoint potential of the cluster. The potential biological function of the low pKa value of the histidines and the shift upon reduction are briefly discussed. Electronic supplementary material to this paper (comprising tables with the partial charges of the Rieske iron-sulfur cluster, optimized geometries of the 12 different states, and pK1/2 values for the non-coordinating residues of the Rieske protein in the oxidized and reduced state) can be obtained by using the Springer Link server located at http://dx.doi.org/10.1007/s00775-002-0342-6.

Density functional methods applied to metalloenzymes

Abstract Density functional calculations for structures, spin states, redox energetics and reaction pathways are presented for some selected metalloenzymes. The specific enzymes examined are: (1) Fe and Mn superoxide dismutase for redox energetics and the role of second shell residues; (2) galactose oxidase (Cu enzyme) and (3) glyoxalase I (Zn enzyme) for reaction pathways, mechanisms, intermediates, and transition states (reaction barriers); (4) iron-oxo dimer enzymes methane monooxygenase and ribonucleotide reductase for characterizing the oxidized and reduced forms in terms of structures and protonation states, and for a proposed structure for the high-valent intermediate Q in MMO. The interaction of the active site with the surrounding protein environment is also explored in a number of cases either by using expanded quantum mechanically treated clusters, or by using electrostatic/dielectric representations of the protein–solvent environment.

Structure, redox, pKa, spin. A golden tetrad for understanding metalloenzyme energetics and reaction pathways

After a review of the current status of density functional theory (DFT) for spin-polarized and spin-coupled systems, we focus on the resting states and intermediates of redox-active metalloenzymes and electron transfer proteins, showing how comparisons of DFT-calculated spectroscopic parameters with experiment and evaluation of related energies and geometries provide important information. The topics we examine include (1) models for the active-site structure of methane monooxygenase intermediate Q and ribonucleotide reductase intermediate X; (2) the coupling of electron transfer to proton transfer in manganese superoxide dismutase, with implications for reaction kinetics; (3) redox, pKa, and electronic structure issues in the Rieske iron–sulfur protein, including their connection to coupled electron/proton transfer, and an analysis of how partial electron delocalization strongly alters the electron paramagnetic resonance spectrum; (4) the connection between protein-induced structural distortion and the electronic structure of oxidized high-potential 4Fe4S proteins with implications for cluster reactivity; (5) an analysis of cluster assembly and central-atom insertion into the FeMo cofactor center of nitrogenase based on DFT structural and redox potential calculations.

Toward a chemical mechanism of proton pumping by the B-type cytochrome c oxidases: application of density functional theory to cytochrome ba3 of Thermus thermophilus.

A mechanism for proton pumping by the B-type cytochrome c oxidases is presented in which one proton is pumped in conjunction with the weakly exergonic, two-electron reduction of Fe-bound O 2 to the Fe-Cu bridging peroxodianion and three protons are pumped in conjunction with the highly exergonic, two-electron reduction of Fe(III)- (-)O-O (-)-Cu(II) to form water and the active oxidized enzyme, Fe(III)- (-)OH,Cu(II). The scheme is based on the active-site structure of cytochrome ba 3 from Thermus thermophilus, which is considered to be both necessary and sufficient for coupled O 2 reduction and proton pumping when appropriate gates are in place (not included in the model). Fourteen detailed structures obtained from density functional theory (DFT) geometry optimization are presented that are reasonably thought to occur during the four-electron reduction of O 2. Each proton-pumping step takes place when a proton resides on the imidazole ring of I-His376 and the large active-site cluster has a net charge of +1 due to an uncompensated, positive charge formally associated with Cu B. Four types of DFT were applied to determine the energy of each intermediate, and standard thermochemical approaches were used to obtain the reaction free energies for each step in the catalytic cycle. This application of DFT generally conforms with previously suggested criteria for a valid model (Siegbahn, P. E. M.; Blomberg, M. A. R. Chem. Rev. 2000, 100, 421-437) and shows how the chemistry of O 2 reduction in the heme a 3 -Cu B dinuclear center can be harnessed to generate an electrochemical proton gradient across the lipid bilayer.

Density functional studies of the ground- and excited-state potential-energy curves of stilbene cis - trans isomerization

Using spin-unrestricted density functional theory (the VWN Becke-Perdew potential), including broken-symmetry and spin-projection methods, we have obtained the potential-energy curves as a function of the central torsional angle of stilbene in the ground (S0), the first excited triplet (T1), the first excited singlet (S1), and the doubly excited singlet (S2) states. The thermal trans-->cis isomerization of stilbene passes through a diradical broken-symmetry electronic structure around the twisted conformation (90 degrees central torsional angle) in the ground state. Our calculations support the proposed triplet mechanism for sensitized cis [symbol: see text] trans photoisomerization and the nonadiabatic singlet mechanism proposed by Orlandi and Siebrand. On the T1 potential-energy curve, the rotation of the C=C bond for both trans- and cis-stilbene will lead stilbene to the twisted conformation, from which the twisted stilbene will decay to the ground-state surface that is nearly isoenergetic with the T1 surface and has diradical electronic structure in the twisted region. On the S1 potential-energy curve, the energy increases in the direction from trans- to the twisted stilbene, and crosses with the neutral doubly excited S2 potential-energy curve, which has a minimum at the twisted structure and is lower in energy than the zwitterionic doubly excited state. The twisted stilbene around the energy minimum of the neutral doubly excited S2-state will decay onto the ground-state surface from where the rotation of the C=C bond leads the twisted stilbene to either the trans or cis configuration and the isomerization of stilbene is then completed. Similar studies have also been performed on a stilbene derivative with a substituent group, NHCOCH3.