Per E. M. Siegbahn

The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule

The complete active space (CAS) SCF method is presented in detail with special emphasis on computational aspects. The CASSCF wave function is formed from a complete distribution of a number of active electrons in a set of active orbitals, which in general constitute a subset of the total occupied space. In contrast to other MCSCF schemes, a CASSCF calculation involves no selection of individual configurations, and the wave function therefore typically consists of a large number of terms. The largest case treated here includes 10 416 spin and space adapted configurations. To be able to treat such large CI expansions, a density‐matrix oriented formalism is used. The Newton–Raphson scheme is applied to calculate the orbital rotations, and the secular problem is solved with recent developments of CI techniques. The applicability of the method is demonstrated in calculations on the HNO molecule in ground and excited states, using a triple‐zeta basis and different sizes of the active space. With a reasonable ch...

A Comparison of the Super-CI and the Newton-Raphson Scheme in the Complete Active Space SCF Method

A density matrix formulation is presented of the super-C I and Newton-Raphson methods in complete active space SCF (CASSCF) calculations. The CASSCF method is a special form of the MC-SCF method, where the C I wave function is assumed to be complete in a subset of the orbital space (the active space), leaving the remaining orbitals doubly occupied in all configurations. Explicit formulas are given for all matrix elements in the super-C I method and the first and second derivatives in the Newton-Raphson formulation. The similarities between the two methods are pointed out and the differences in the detailed formulations are discussed. Especially interesting is the fact, that while the second derivatives can be expressed in terms of first and second order density matrices, the matrix elements between the super-C I states involve also the third order density matrix in some cases.

Structures and Energetics for O2 Formation in Photosystem II

Water oxidation, forming O(2) from water and sunlight, is a fundamental process for life on earth. In nature, the enzyme photosystem II (PSII) catalyzes this reaction. The oxygen evolving complex (OEC), the complex within PSII that catalyzes the actual formation of the O-O bond, contains four manganese atoms and one calcium atom connected by oxo bonds. Seven amino acid side chains in the structure, mostly carboxylates, are ligated to the metal atoms. In the study of many enzyme mechanisms, theoretical modeling using density functional theory has served as an indispensable tool. This Account summarizes theoretical research to elucidate the mechanism for water oxidation in photosynthesis, including the most recent findings. The development of successively larger models, ranging from 50 atoms in the active site up to the present model size of 170 atoms, has revealed the mechanism of O(2) formation with increasing detail. The X-ray crystal structures of PSII have provided a framework for optimizing the theoretical models. By constraint of the backbone atoms to be at the same positions as those in the X-ray structures, the theoretical structures are in good agreement with both the measured electron density and extended X-ray absorption fine structure (EXAFS) interpretations. By following the structural and energetic changes in those structures through the different steps in the catalytic process, we have modeled the oxidation of the catalytic complex, the binding of the two substrate water molecules, and the subsequent deprotonations of those substrate molecules. In these models, the OEC forms a basin into which the water molecules naturally fit. These findings demonstrate that the binding of the second water molecule causes a reconstruction, results that are consistent with earlier EXAFS measurements. Most importantly, this Account describes a low-barrier mechanism for formation of the O-O bond, involving an oxygen radical that reacts with a mu-oxo ligand of the OEC. Further research revealed that the oxygen radical is bound in the Mn(3)Ca cube rather than to the outside manganese. This Account provides detailed diagrams of the energetics of the different S-transitions both without and with a membrane gradient. An interesting detail of these reactions concerns the role of the tyrosine (Tyr(Z)), which appears as an intermediate radical in the oxidation of the OEC. By simple electrostatic arguments, these results show that the initial oxidation of Tyr(Z) is downhill for the first two transitions but uphill for the final ones. In these later transitions, the oxidation of the OEC is coupled to deprotonations of water.

Gaussian basis sets for the first and second row atoms

Gaussian basis sets consisting for first row atoms of 7 s-type and 3 p-type and for second row atoms of 10 s-type and 6 p-type functions with optimized exponents are reported. These basis sets consists of at least two functions per atomic orbital.ZusammenfassungEs werden für die Atome der ersten und zweiten Reihe Basissätze aus Gaußfunktionen mitgeteilt, die aus 7 Funktionen vom s-Typ und 3 Funktionen vom p-Typ für die Elemente der ersten Reihe und 10 Funktionen vom s-Typ und 6 Funktionen vom p-Typ für die Elemente der zweiten Reihe mit optimierten Exponenten bestehen. Diese Basissätze bestehen aus wenigstens zwei Funktionen pro Atomorbital.RésuméUne base de 7 gaussiens du type s et 3 du type p est presenté pour les éléments du premier rang et de 10 gaussiens du type s et 6 du type p pour des éléments du deuxième rang; les exposants sont optimisés. Les bases consistent au moins en deux fonctions par orbital atomique.

Generalizations of the direct CI method based on the graphical unitary group approach. II. Single and double replacements from any set of reference configurations

The direct CI method is generalized to the case of all single and double replacements from an arbitrary set of reference configurations. This is a continuation of the work and ideas presented in an earlier paper on first order wave functions. The analysis is done using the unitary group formulation of the correlation problem, and the resulting method is a combination of the direct CI method and the unitary group approach as formulated particularly by Paldus and Shavitt. The main idea in the present work is the factorization of the coupling coefficients appearing in the direct CI formalism, into a complicated internal part and a simple external part. The general philosophy is like in all direct CI methods to allow long CI expansions by avoiding the storage and retrieval of a large formula tape. The longest CI expansion treated in this paper is an application on the system CH2(3B1)+H2→CH3+H with five reference states, resulting in 16 096 configurations. The barrier height for the reaction is calculated to b...

Singlet and triplet energy surfaces of NiH2

Contracted CI calculations have been performed for the concerted dissociation of NiH2 into Ni and H2. All low‐lying states have been considered. The most important result of the calculations is that the 1A1 state is much lower in energy than the other states for bent geometries. This state even forms a slightly stable complex with an H–Ni–H angle of 49°. The binding is between Ni sd‐hybridized orbitals and a weakened H2 molecule. The ground state of NiH2 is, however, found to be a linear 3Δg state, with the lowest singlet almost 1.5 eV higher in energy at this geometry. The forbidden dissociation reactions for the triplets and singlets with the lowest barriers proceed in steps with the occupation of each symmetry changing by only one unit in each step. The lowest triplet barrier is 1.8 eV, and for this state a concerted dissociation is slightly preferred over a stepwise loss of a hydrogen at a time. The mechanism for the concerted triplet dissociation may also be through spin‐orbit coupling to the 1A1 sta...

Quantum Chemical Studies of Mechanisms for Metalloenzymes

ing a hydrogen atom from substrates with relatively weak X−H bonds (Figure 56). Such a situation was found in the catalytic cycle of ACCO, where cleavage of the N−H bond is facilitated by the ferrous ion capable of (partly) reducing the resulting N-based radical to an anion. In the catalytic cycle of HEPD and in the reaction of HppE with an enantiomer of the native substrate, a C−H bond at the carbon hosting a deprotonated alcohol group is severed by the superoxide. In these cases transfer of a hydrogen atom to the superoxide is coupled to one-electron reduction of Fe(III) to Fe(II), which yields aldehyde or ketone products. In a similar way a thioaldehyde is produced during the initial steps of the IPNS catalytic cycle. Figure 53. Reaction catalyzed by NDO. Figure 54. X-ray structure of the non-heme iron cofactor in the NDO active site (PDB 1O7G). Figure 55. Reaction mechanism for the dihydroxylation reaction catalyzed by NDO suggested on the basis of the results of a DFT study. Relative energy values in kilocalories per mole are given beneath the structures shown and above the arrows for the transition states connecting them. Chemical Reviews Review dx.doi.org/10.1021/cr400388t | Chem. Rev. 2014, 114, 3601−3658 3636 The most often encountered reaction of the Fe(III)−O2 species is an electrophilic attack on an electron-rich (co)substrate that yields an Fe(II) intermediate with a peroxide bridge between the ferrous ion and an organic molecule (Figure 57). In all examples shown in the figure the reactive spin state is a quintet with high-spin Fe(III) antiferromagnetically coupled to the superoxide radical. This particular electronic structure allows for a smooth two-electron reaction, leading to a ferrous intermediate featuring a high-spin Fe(II) ion, i.e., also lying on a quintet PES. When the substrates are easily one-electron-oxidized, as, for example, carotenoids or catecholates, already the ternary enzyme−dioxygen−organic substrate complex may contain an organic radical along with the superoxide anion stabilized by coordination to the metal (Figure 58). With proper (antiferromagnetic) alignment of the spins of the unpaired electrons on the two radicals, direct coupling between them proceeds with a straightforward formation of a new C−O bond. With the O2 ligand already protonated, several different reaction scenarios are usually plausible, and one of them, arguably the simplest, involves a transfer of the HOO group from the metal ion to the organic substrate (Figure 59). In the reaction of HEPD, it is the distal, i.e., originally protonated, oxygen atom that attacks the aldehyde group and the proton is transferred to the second (proximal) oxygen atom with the help of a phosphonic group of the substrate. In this reaction HOO acts as a nucleophile. In the case of HGD, the HOO ligand has a partial radical character, and hence, in its attack on the aromatic ring, it behaves as an electrophilic reagent. Moreover, in the HGD and ACO cases, it is the proximal oxygen atom that is directly transferred from the metal ion to the organic radical, i.e., the proton remains on the same oxygen atom throughout the reaction. When a catalytic reaction involves a homolytic O−O bond cleavage, one end of the peroxo group is in contact with the metal ion, and it is reduced to HO− or RO− when the O−O bond breaks (Figure 60). The electron required for the reduction is provided usually by the ferrous ion; in intradiol dioxygenases, which host Fe(III) in the active site, a tyrosinate ligand can serve as a reductant. Heterolytic cleavage of the O−O bond typically yields highvalent iron(IV)−oxo species, and such a reaction requires Fe(II), a deprotonated proximal oxygen atom, and usually also that the distal oxygen atom has a chance to develop a second covalent bond when the O−O gets broken (Figure 61). The bonding partner for the distal oxygen can be a hydrogen (PDHs, IPNS, ACCO) or a (co)substrate’s carbon (αKAOs, ACO, Dke1) atom. Heterolysis to Fe(IV)O usually proceeds on the quintet PES. Finally, heterolytic O−O bond cleavage may proceed without changing the oxidation state of the metal but instead with coupled two-electron oxidation of an organic substrate (Figure Figure 56. Examples of Fe(II)/Fe(III)−O2 reactions involving hydrogen atom abstraction. Only the most relevant fragment of the substrate is shown. Figure 57. Examples of Fe(III)−O2 reactions involving electrophilic attack and two-electron oxidation of an organic (co)substrate. Figure 58. Examples of Fe(II)/Fe(III)−O2 reactions involving radical coupling between the superoxide and an organic radical. Only the most relevant fragment of the substrate is shown. Chemical Reviews Review dx.doi.org/10.1021/cr400388t | Chem. Rev. 2014, 114, 3601−3658 3637 62). In both cases depicted in the figure, a hydroperoxo group is bound to high-spin Fe(III), and when theO−Obond cleaves, the OH group remains on iron and the other oxygen atom forms two covalent bonds with the organic substrate. This kind of heterolytic O−O cleavage requires that the donor orbital of the organic substrate, i.e., the one that provides two electrons for reduction of the peroxo group, has a good overlap with the O−O σ* orbital. 5.2. Dinuclear Non-Heme Iron Enzymes 5.2.1. Methane Monooxygenase. MMO is an enzyme which inserts one oxygen from O2 into methane to form methanol. The active site of the soluble form of MMO is shown in Figure 63. It contains an iron dimer complex linked by oxygen-derived ligands and has four glutamates and two histidines. Due to the presence of a well-resolved X-ray structure, and the technical importance of the reaction catalyzed, this was actually the first redox-active enzymemechanism that was treated with the cluster model using modern DFT functionals in 1997. Several groups were active at an early stage, and the most essential parts of the reaction mechanism were determined more than a decade ago. A comprehensive review of this development was written by Friesner et al. In short, the active species (compound Q) has a diamond core structure with two bridging oxo groups and is in an Fe2(IV,IV) state, one of themost oxidized species in nature. One of the oxo groups of compound Q activates nethane by an abstraction of one hydrogen atom. The TS is linear in C···H···O to form an Fe2(III,IV) state and amethyl radical. The loss of entropy at the TS is a large part of the barrier. In the second step, the methyl radical recombines with the bridging hydroxide formed in the first step. The TS for this step was first located by Basch et al. This rebound mechanism was criticized by interpretations of radical clock experiments, which appeared to show that there could not be a sufficiently long-lived alkyl radical to be consistent with a two-step mechanism. A concerted mechanism with simultaneous cleavage of the C−H bond and formation of the C−O bond was therefore suggested. Several explanations were suggested to resolve this apparent discrepancy between experiment and theory. In one of them, it was concluded that the radical clock probe molecules Figure 59. Examples of HOO transfer reactions. Only the most relevant fragment of the substrate is shown. Figure 60. Examples of O−O bond homolysis facilitated by oneelectron reduction of the proximal oxygen atom. Only the most relevant fragment of the substrate is shown. Figure 61. Examples of heterolytic O−O bond cleavage yielding iron(IV)−oxo species. Figure 62. Examples of heterolytic O−Obond cleavage proceeding with direct two-electron oxidation of the organic molecule. Chemical Reviews Review dx.doi.org/10.1021/cr400388t | Chem. Rev. 2014, 114, 3601−3658 3638 were chemically so different from methane that different mechanisms were likely. The probes are much easier to ionize and could form cations instead of radicals. Another suggestion was a so-called two-channel mechanism of the dynamics involving a bound radical intermediate. A third possibility could be something analogous to the two-state reactivity mechanism suggested for P450 to explain similar discrepancies in that case, but this has not been tested. In summary, a concerted mechanism, as suggested by the experiments, has never been found in DFT modeling calculations for methane hydroxylation by MMO, at least when reasonable models have been tried, and the suggestion has therefore not survived. In the initial phase of the MMO studies, there were problems converging to proper electronic states. These problems were solved by Friesner et al., who were able to obtain the correct antiferromagnetic coupling of compound Q with two high-spin irons. A state of key importance is also the first intermediate after Q, with an electronic structure characterized as Fe2(III,IV)− O•, discussed further below in connection with mixedMn−Fe dimers. It was found that already at the TS for hydrogen abstraction the iron dimer is in an Fe2(III,IV) state as it is in the product of this step. The bridging oxygen radical would then act as a hydrogen atom abstractor. At the TS, the spins are divided between a bridging oxo ligand and the methyl, while the iron spins stay essentially constant from the Fe2(III,IV)−O state to the product. The O−O bond cleavage to reach compound Q is also a significant step in the catalytic cycle. Again, several groups were involved in studying this step at an early stage. There was essential agreement among these studies on the mechanism. First, a peroxide (compound P) is formed between the two irons in an Fe2(III,III) state. Several different structures of P are nearly degenerate. In the TS for the O−O cleavage, the oxygens are symmetric, but only one of the irons is redox-active. In the final Figure 63. Active site of methane monooxygenase. Figure 64. R1 and R2 proteins in RNR. Chemical Reviews Review dx.doi.org/10.1021/cr400388t | Chem. Rev. 2014, 114, 3601−3658 3639 stage, the other iron also changes its oxidation state to IV, and compound Q is formed. In the study by Friesner et al.,

Recent developments of the quantum chemical cluster approach for modeling enzyme reactions

The quantum chemical cluster approach for modeling enzyme reactions is reviewed. Recent applications have used cluster models much larger than before which have given new modeling insights. One important and rather surprising feature is the fast convergence with cluster size of the energetics of the reactions. Even for reactions with significant charge separation it has in some cases been possible to obtain full convergence in the sense that dielectric cavity effects from outside the cluster do not contribute to any significant extent. Direct comparisons between quantum mechanics (QM)-only and QM/molecular mechanics (MM) calculations for quite large clusters in a case where the results differ significantly have shown that care has to be taken when using the QM/MM approach where there is strong charge polarization. Insights from the methods used, generally hybrid density functional methods, have also led to possibilities to give reasonable error limits for the results. Examples are finally given from the most extensive study using the cluster model, the one of oxygen formation at the oxygen-evolving complex in photosystem II.

Manganese Oxyl Radical Intermediates and O−O Bond Formation in Photosynthetic Oxygen Evolution and a Proposed Role for the Calcium Cofactor in Photosystem II

Spin state considerations are proposed to sharply limit the possible O−O bond-forming steps in water oxidation by the oxygen evolving center of Photosystem II. A series of intermediates are proposed for the Kok S states on the basis of quantum chemical studies on simple model complexes; these are also consistent with the main biophysical data. Only one Mn atom in the active site cluster is thought to be redox-active and mediate O−O bond formation. A key concept is the formation of an unreactive MnO oxo at the S2 state, followed by its conversion to a reactive Mn−O• oxyl form at the S3 level, with radical character on the oxyl oxygen, at which point O−O bond formation can occur by a coupling between the oxyl and an outer-sphere water molecule. An MnOOH intermediate at S3 is proposed to lose a hydrogen atom to give O2. The role of the Ca cofactor is to bring about a 5- to 6-coordination change at S2, necessary for formation of a reactive oxo in S3. The chloride cofactor is assigned the role of charge neutra...

Water oxidation mechanism in photosystem II, including oxidations, proton release pathways, O―O bond formation and O2 release ☆

The present status of DFT studies on water oxidation in photosystem II is described. It is argued that a full understanding of all steps is close. In each S-transition, the manganese that is oxidized and the proton released are strongly implicated, and structures of all intermediates have been determined. For the S2-state, recent important experimental findings support key elements of the structure and the mechanism. In this mechanism, the O-O bond is formed between an oxyl radical in the center of the cluster and an Mn-bridging μ-oxo ligand, which was suggested already in 2006. The DFT structure of the oxygen evolving complex, suggested in 2008, is very similar to the recent high-resolution X-ray structure. Some new aspects of the interaction between P680 and the OEC are suggested. This article is part of a Special Issue entitled: Metals in Bioenergetics and Biomimetics Systems.