## Introduction

Polynuclear transition metal complexes play an important role as catalysts not only in technical and synthetical applications, but also serve as essential building blocks in the active centers of various enzymes.1–3 Prominent examples of transition metal containing sites in enzymes that are essential for life on earth are the iron–molybdenum cofactor (FeMoco) of nitrogenase,4–6 which catalyzes the conversion of atmospheric dinitrogen to ammonia in nitrogen-fixating bacteria, the Mn_{4}Ca site within photosystem II,7 where the photosynthetic oxidation of water takes place, and the diiron center in the hydroxylase component (MMOH) of methane monooxygenase (MMO) found in methanotropic bacteria that convert methane to methanol.8, 9

To understand their functionality, the geometric and the electronic structure of these enzymes, in particular of their active sites, needs to be known. The experimental determination of the geometric structure is possible via X-ray crystallography. However, in cases where a protein cannot be crystallized, where the state under study is too labile or too short-lived, or where for other reasons the spatial and/or time resolution is not sufficient, X-ray crystallography cannot provide a detailed picture of the active site structures, so one has to rely on information from other spectroscopic techniques such as extended X-ray absorption fine structure (EXAFS) or EPR/ENDOR. Further information on the electronic structure can be extracted from temperature-dependent magnetic susceptibility measurements,10 which yield qualitative and quantitative information on the energetics of different spin states expressed by the sign and the magnitude of the Heisenberg coupling constants *J*.

It is not always possible to deduce without ambiguity from the sum of all avaliable spectroscopic data which geometric structures are present in the active site of an enzyme during reaction. Quantum chemical calculations can help to distinguish between different suggested structures by identifying the one that agrees best with all experimental data. For one recent example of EPR calculations on an oxygen-bridged manganese complex that may serve as a Mn_{4} photosystem II model, see ref.11. Once confidence in the theoretical methods is established by such comparisons, theoretical studies may be used in a predictive way for tackling mechanistic aspects.

Several factors affect the accuracy with which quantum chemical calculations reproduce or predict experimental results. First, one either has to build a truncated model of the active site or restrict the quantum chemical description of the system under study to a limited region within an embedding scheme like a QM/MM treatment (see, e.g., ref.12), because a complete enzyme is not accessible by current first principles methods. Second, the results are affected by the ability of the chosen quantum chemical method to describe the complicated electronic structure of transition metal complexes correctly. At present, the only method that is suited for the treatment of such systems of a reasonable size is density functional theory (DFT). Within DFT, the choice of the basis set and of the approximate exchange–correlation functional as well as the degree of control over the state to which the self-consistent-field (SCF) algorithm converges have considerable influence onto the results. Because it is straightforward to improve the basis set systematically, the focus of this study is on the effect of the exchange–correlation functional on various molecular quantities such as total energies, Heisenberg coupling constants, and local spins, as well as on the stability of SCF convergence. Because systematic investigations are very rare, we aim at a detailed account that concentrates on the essential parameters.

The identification of essential parameters for such a study is guided by the observation that the absolute and the relative total energies of various mononuclear transition metal complexes in different spin states depend linearly on the exact exchange admixture,13 that is, on the admixture of Hartree–Fock-type exchange interaction. It is the exact exchange contribution in present-day density functionals that makes the difference rather than a specific set of optimized parameters or functional contributions. A reparametrization of the exact exchange admixture of 20% in the well-known B3LYP functional by comparison to experimental data suggested to use a smaller exact exchange admixture of 15% (or less)13, 14 (see also ref.15 by Harvey for an up-to-date discussion). The performance of this B3LYP* functional for standard test sets is the same as for the original B3LYP functional.16 Consequently, the energies of the spin states of polynuclear clusters are also expected to vary strongly with the amount of exact exchange in the density functional. Because Heisenberg coupling constants are constructed as a measure for the energy difference between different spin states of a polynuclear transition metal cluster, they are also likely to be sensitive to exact exchange. In fact, dependencies of spin-state energy splittings17, 18 as well as of calculated coupling constants19–21 in dinuclear complexes on the density functional have been noticed in previous studies. But due to the lack of a systematic connection between different density functionals, no systematic investigation is possible. However, owing to the pronounced dependence of energy splittings on exact exchange, this Hartree–Fock-type contribution appears to be the only but promising contribution in the functionals to be investigated systematically. In this work we close the gap of a missing systematic study and conduct an investigation of the dependence of local spin expectation values and other quantities on the exact exchange admixture.

We aim at an overview of the energetic situation of all energetically close lying different spin states [not only the high-spin (hs) and Broken-Symmetry (BS) states] as predicted by DFT using contemporary density functionals. It is important to note that the term “different spin states” shall also refer not only to the total spin, but also to the spatial distribution of the spin excess in a Kohn–Sham determinant of given total spin. Because it is not known *a priori* which state is the ground state and which other states are close in energy and therefore may become important for the reactivity of the system under study (compare the two-state reactivity concept22), we must aim to converge all possible close lying spin states. For this it is necessary to screen a large part of the total energy hypersurface in the parameter space of molecular orbital (MO) coefficients. Therefore, we are adviced to investigate convergence of MO coefficients starting from low-quality guesses.

To summarize, the goal of this study is to answer the following questions:

How does a modification of the exact exchange admixture affect the qualitative and quantitative properties of the converged SCF solutions?

How large is the influence of the exact exchange on the Heisenberg coupling constant calculated with three different mathematical expressions known from the literature? (The relation between these expressions shall also be explicated.)

How stable is the convergence of standard initial guesses for different total spin states with respect to to a varying exact exchange admixture?

As a test case, we have chosen a model for the intermediate **Q** in the MMOH reaction cycle, which has been proposed by Siegbahn23 (see also ref.24). This dinuclear complex is small enough to carry out a large number of comparative calculations. Furthermore, a large number of both experimental and theoretical results on MMO as well as on the structurally related Ribonucleotide Reductase (RNR)25, 26 has been published, which allows for comparisons to established density functionals, to other model systems,12, 27, 28 and to experiment. Quantum chemical studies on selected enzymes and catalytic synthetic systems in general and on MMO in particular have been recently reviewed by Noodleman et al.29 and by Baik et al.30 These reviews also include an overview on the reaction cycle of MMOH and the role of the intermediate **Q**, as well as extensive lists of references to the original articles.

This work is organized as follows: first, we briefly recall the theoretical background of spin contamination in Slater determinants, Clark and Davidson's local spin expectation values,31–34 and the calculation of Heisenberg coupling constants with Noodleman's BS approach35 to set the stage. In Sections 3 and 4, details on the computational methodology and the model cluster under study are presented. The following sections summarize total energies as well as total and local spin expectation values for different spin states. The results obtained employing standard density functionals are contained in Section 5, those obtained with B3LYP-based functionals with modified exact exchange admixture in Section 6. In Section 7, Heisenberg coupling constants calculated with expressions based on different assumptions on the interacting local spins and with different density functionals are investigated. A brief discussion of SCF convergence, some aspects of which are already adressed in Section 6, is summarized in Appendix A, which compares the convergence behavior of two standard ways of creating initial guess MOs for different spin states.