Article pubs.acs.org/JCTC
Effects of Static Correlation between Spin Centers in Multicenter Transition Metal Complexes Shibing Chu,†,‡ Daniele Bovi,†,‡ Francesco Cappelluti,§ Alberto Giacomo Orellana,‡ Henry Martin,† and Leonardo Guidoni*,† †
Dipartimento di Scienze Fisiche e Chimiche, Università degli studi dell’Aquila, Via Vetoio (Coppito), 67100 L’Aquila, Italy Dipartimento di Fisica and §Dipartimento di Chimica, Sapienza - Università di Roma, P.le Aldo Moro, 2, 00187 Roma, Italy
‡
S Supporting Information *
ABSTRACT: Multicenter transition metal complexes are the key moieties of many processes in chemistry, biochemistry, and materials science such as in the active sites of enzymes, molecular catalysts, and biological electron carriers. Their electronic structure, often characterized by high-spinpolarized metal sites, is a challenge for theoretical chemists because of their high degree of dynamical and static correlation. Static correlation is necessary both for the appropriate description of the metal−ligand bonding and for a correct description of the multideterminant character arising from the magnetic interactions between spin centers. Density functional theory (DFT) is usually applied using a single-determinant broken-symmetry state that is lacking the correct spin symmetry when the ground state has total low-spin character. To alleviate this drawback, we use the extended brokensymmetry (EBS) approach to derive approximate ground-state energies and, for the first time, forces for the correctly symmetric ground state of an arbitrary number of spin centers within the framework of the Heisenberg−Dirac−van Vleck Hamiltonian. Remarkably, the proposed procedure supplies relaxed geometries that are fully consistent with the calculated J-coupling constants. We apply the method to investigate the relaxed geometrical structure of the low-spin ground state of iron−sulfur clusters with two, three, and four iron centers. We observed significant differences in both geometrical parameters and coupling constant J between the symmetrized ground state, the high-spin, and the broken-symmetry optimized structures. These changes are often comparable with the differences observed by using different functionals, and the use of EBS always improves the description of the studied systems. It will be therefore important to include it in any DFT attempt to quantitatively describe multicenter transition metal complexes in the future.
1. INTRODUCTION Multicenter transition metal (TM) complexes have a number of intriguing applications in different fields of chemistry, materials science, and biology. They are present in a large variety of catalysts such as molecules,1 surfaces,2 confined spaces,3,4 and nanostructured materials.5 In biology, multicenter TM complexes are present as catalytic and redox centers in many metalloenzymes6 and electron-transport proteins such as iron− sulfur proteins,7−9 hydrogenases,10 copper-based enzymes,11 and photosynthetic complexes.12 They also play an important role in the research of biomimetic inorganic molecular catalysts for water oxidation13 and hydrogen production.14 Among them, iron−sulfur clusters perform diverse functions in many proteins as mediators for the charge transfer, ranging from nitrogen fixation to photosynthesis and respiration.15 The electronic structure of these clusters is characterized by the presence of spin-polarized Fe(II) or Fe(III) ions that are antiferromagnetically coupled through sulfur bridges. The lowspin ground state (GS) and its corresponding low-lying spin states have different electronic and magnetic characters, which is a key point of their rich chemistry. Investigation of such states is a © XXXX American Chemical Society
current challenge for both experimental and theoretical methods. A direct assignment of these electronic energy levels from experimental methods in large clusters is often not possible because their low-energy levels are embedded within the vibrational modes of the clusters,7,16 although several investigations on geometry and J-coupling of iron−sulfur clusters [2Fe-2S],17−19 [3Fe-4S],20 and [4Fe-4S]21−24 have been carried out experimentally. On the theoretical side, post-Hartree−Fock correlated methods such as CASSCF,25 CASPT2,26 CCSD, DMRG,9 and QMC27 are still a challenge to describe both the static and dynamical correlated nature of the ground state for large multicenter TM complexes. Concerning the static correlation, we may distinguish two different effects: the local static correlation necessary to correctly describe the nature of the bonds between each metal atom and its ligands and the global static correlation among the spin centers due to the interactions between localized unpaired electrons. This latter contribution, which is crucial to guarantee the correct overall spin symmetry of Received: March 25, 2017 Published: August 1, 2017 A
DOI: 10.1021/acs.jctc.7b00316 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX
Article
Journal of Chemical Theory and Computation
2. METHODS In multicenter TM compounds the single-determinant character of the Kohn−Sham formulation of DFT is lacking the necessary multideterminant character to describe low-spin states. In strongly magnetic clusters such as the iron−sulfur complexes, the ground state is generally of low-spin character and the description through an open-shell single-determinant wave function breaks the spin symmetry. As we have shortly discussed in the Introduction, an alternative proposal to go beyond this approach is the EBS method, in which the GS energy surface of a multicenter complex is reconstructed using separated calculations performed on broken-symmetry (BS) and high-spin (HS) states through the evaluation of the J-coupling constants. In the present section first we will review the computational method to evaluate the J-coupling constants for TM complexes using HDvV spin Hamiltonian.35−37 Later we will present a general formula for calculating the ground-state gradient for an arbitrary number of spin centers. For the general case of N spin centers, the HDvV spin Hamiltonian, Ĥ , can be decomposed into a sum of two spin center operators as following:
the wave function, will be the subject of the present work. The most used technique for medium and large size systems remains density functional theory (DFT).28 On the high-spin state, the single determinant of DFT can provide an adequate description of the global static contribution; on the contrary, when studying low-spin states, multiple determinants are required. The simplest approach with DFT is to use the so-called broken-symmetry (BS) wave function, namely, a single unrestricted determinant, which cannot represent a pure spin state because it is not an eigenstate of the S2̂ total spin operator. This approach has been applied to study two-center29,30 and multicenter31,32 TM complexes. To avoid or limit the effects of the spin contamination of such single-determinant state, different spinprojection schemes have been proposed at first for wavefunction-based methods like HF29,33 and later extended to DFT.30,34 The basic idea is to describe the energetic of the GS through the Heisenberg or Heisenberg−Dirac−van Vleck (HDvV) empirical Hamiltonian35−37 and to estimate the magnetic J-coupling using independent total energy calculations of single-determinant unrestricted spin configurations. With such an EBS method, two-center38−41 and multicenter42 TM complexes have been investigated. The energetics of the ground and all other spin states constituting the “spin ladder” can be therefore calculated, offering a good compromise in terms of cost versus accuracy with respect to post-Hartree−Fock correlated methods. Because the coupling constant J strongly depends on the metal−ligand (M−L) distances and on the M−L−M angles, an important ingredient in the quantitative evaluation of the spin energetics is represented by the possibility to perform geometry optimization on the ground-state energy surface. Such calculations might shed some light on structural variability7,15,43 and on cases where the experimental structures are less clear.38,44 Recently, efforts45,46 have been made for obtaining the gradient of an energy calculated using Yamaguchi’s47 spinprojection correction, to perform geometry optimizations. For dinuclear complexes, the EBS approach has been proposed for this same purpose,38,39,41 successfully applied to investigate the geometry of the low-spin states of Fe2S2 ferredoxin complexes by DFT38,40 and QMC.27 Following the same ideas, the procedure has been extended to the calculation of vibrational frequencies.40 However, so far the gradient information on geometries with the EBS method has been limited to dinuclear TM complexes, which represent only a limited portion of the many interesting prosthetic groups in metalloenzymes. In the present work we have extended for the first time the EBS geometry optimization to an arbitrary number of spin centers, allowing us to fully optimize multicenter TM complexes as large as desired. On the basis of this extension, we have calculated the electronic properties, the coupling constants J between each pair of spin centers, and the optimized geometries of three complexes: [Fe2S2(SH)4]2−, [Fe3S4(SH)4]3−, and Fe4S4(N(Si(CH3)3)2)4. The obtained results for all the above complexes were compared with the X-ray crystal structures available in the literature. The present paper is organized as follows: In section 2 we review the EBS formalism and we derive the formalism to obtain the energy gradient on GS by linear combination of BS states to perform geometry optimizations. At the end of section 2, all detailed information on calculations are listed. In section 3 we present the magnetic and electronic characters of [Fe2S2(SH)4]2−, [Fe3S4(SH)4]3−, and Fe4S4(N(Si(CH3)3)2)4 clusters. Geometry optimization information on these systems was also reported in this section.
̂ + H13 ̂ + ... + Ĥ ij Ĥ = H12 = −2 ∑ Jij sî ·sĵ i
