A DFT Based Ligand Field Theory

Structure and Bonding, Vol. 106 (2004): 97–125 DOI 10.1007/b11308HAPTER 1

A DFT Based Ligand Field Theory M. Atanasov1, 2 · C. A. Daul2, 3 · C. Rauzy2 1 2 3

Bulgarian Academy of Sciences, Institute of General and Inorganic Chemistry, Acad. G. Bontchev Str. Bl.11, 1113 Sofia, Bulgaria Université de Fribourg, Département de Chimie, Perolles, 1700 Fribourg, Switzerland E-mail: [email protected] Author to whom correspondence should be addressed

Abstract A general and user-oriented ligand field (LF) theory – LFDFT with parameters ad-

justed to DFT energies of separate Slater Determinants (SD) of the partly filled dn shell [n=2(8), 3(7), 4(6) and 5] of transition metals (TM) complexes – is developed and tested using 22 well documented examples from the literature. These include CrIII, d3 and CoII d7 in octahedral and CrIV, MnV, FeVI d2, CoII d7, MnII d5, NiII d8 in tetrahedral complexes for which reliable values of d-d transition energies available from high-resolute ligand field spectra of complexes with halogenide (F, Cl, Br and I), oxide and cyanide ligands have been reported. The formalism has been implemented and consists of three steps allowing provision of geometries, ligand field Kohn-Sham orbitals and SD-energies in a way consistent with the LF phenomenology. In a fourth step LF parameters are utilized to yield multiplet energies using a full CI LF program. Comparing SD energies from DFT with those calculated using the LF parameter values, we can state for all considered cases, that the LF parameterization scheme is remarkably compatible with SD energies from DFT; standard deviations between DFT SD-energies and their LFDFT values being calculated between 0.016 and 0.124 eV. We find that, when based on the average of configuration with n/5 occupancy of each MO dominated by TM d-orbitals and on geometries with metal-ligand bond lengths from experiment, the 10Dq parameter values (cubic symmetry) are very close to the ones obtained from a fit to reported ligand field transitions. In contrast, when using common functionals such as LDA or gradient corrected ones (GGA) we find that the parameters B and C deduced from a fit to the SD energies are systematically lower than experimental. Thus spin-forbidden transitions which are particularly sensitive to B and C are calculated to be by 2000 to 3000 cm–1 at lower energies compared to experiment. Based on DFT and experimental B and C values we propose scaling factors, which allow one to improve the agreement between DFT and experimental transition energies, or alternatively to develop a DFT theory based on effective LF functionals and/or basis sets. Using a thorough analysis of the dependence of the Kohn-Sham orbital energy on the orbital occupation numbers, following Slater theory, we propose a general LFDFT scheme allowing one to treat, within the same formalism low symmetric ligand fields as well. Test examples, which illustrate the efficiency of this approach, include Cs distorted CrO44– and D2d distorted MnO43– chromophores. Finally, for cubic LF we propose a hybrid LFDFT model (HLFDFT) which leads to an improvement of the existing DFT-multiplet theories. We show, taking low-spin Co(CN)63– as an example, that the new model yields better results as compared to time-depending DFT (TDDFT). A discussion of the LFDFT method in the context of other CI-DFT approaches is given. Keywords Ligand field theory · Density functional theory · Electronic transition energies · Ligand field parameters

© Springer-Verlag Berlin Heidelberg 2004

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1

Introduction

1.1 1.2 1.3

The Conventional Ligand Field Theory CFT [3] and AOM [4] . . . . 99 The Electron Repulsion Parameterization of Griffith [12] . . . . . . 100 Calculations of Multiplet Energies Using DFT . . . . . . . . . . . . . 101

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The LFDFT Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.1 2.2 2.3 2.4 2.5

Cubic d2(8) Systems . . . . . . . . . . . . . . . . . . . . Cubic d3(7), d4(6) and d5 Systems . . . . . . . . . . . . . . Generalization to Low Symmetric LF and Poly-Nuclear TM Complexes . . . . . . . . . . . . A Hybrid LFDFT Model for Cubic Symmetry (HLFDFT) Computational Details . . . . . . . . . . . . . . . . . .

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Applications of the Theory . . . . . . . . . . . . . . . . . . . . . . . 110

3.1

Tetrahedral d2 Oxoanions of CrIV, MnV, FeVI and CrX4 (X=F–, Cl–, Br–, I–) . . . . . . . . . . . . . . . . . . Low-Symmetry Distortions: CrO44– (Cs) and MnO43–(D2d) . . Octahedral CrIII d3 and CoII d7 Complexes . . . . . . . . . . . Octahedral MnIII d4 and CoIII and FeII d6 Complexes . . . . . Applications to Tetrahedral d5 MnCl42– and FeCl41– Complexes Performance of the Existing Functionals Interpreting and Predicting LF Spectra of TM Complexes . . . . . . . . .

3.2 3.3 3.4 3.5 3.6

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110 114 115 118 119

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Conclusions and Outlook

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Appendix

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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1 Introduction In this contribution we present a general theory of transition metal (TM) complexes which is based on DFT calculations of the manifold of all Slater determinants (SD) originating from a given dn configuration of a transition metal complex. These results are used as a data base allowing one to reformulate the ligand field problem within the DFT framework and to determine and interpret its parameters using DFT data prior to experiment.We study the performance of the presently available functionals and we explore the possibility of accounting for both the dynamic and the non-dynamic correlation by a combined use of DFT and configuration interaction (CI) treatments, respectively. A comparison of some well characterized systems in the literature with other developments for calculating ground and excited states properties using DFT, such as time-depending DFT [1] and CI-DFT [2], is used to demonstrate the impact of this new approach. Ligand field theory (LFT) has been and still is a useful tool for interpreting optical and magnetic properties of TM complexes. The predominantly ionic picture,

A DFT Based Ligand Field Theory

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based on metal ions perturbed by surrounding anions (ligands) encountered in crystal field theory (CFT) [3], later supplemented by metal-ligand covalency in the Angular-Overlap Model (AOM) [4], provides an adequate description of the optical d-d transition, paramagnetism, magnetic interactions and electronic spin resonance (ESR). Yet, LFT is an empirical approach gaining its parameters by comparison with existing experimental data and its justification lies in its ability to reproduce these data. Thus, all LF approaches, both CFT and AOM are tools for interpreting rather than tools for predicting electronic phenomena in TM complexes. The demand for a predictive theory in this respect is mostly tangible for systems, such as active sites of enzymes, where little or no structural and/or spectroscopic data in high resolution are known. The need for a model which is parameter free on one site but compatible with the usual ligand field formalism on the other site becomes increasingly pronounced in view of the new developments in bio-inorganic chemistry [5, 6]. Density Functional Theory (DFT) is such a model. In an extension of previous studies [7] it has been shown using numerous examples by C. Daul [8], that the multiplet theory of TM ions and complexes is compatible with the DFT formalism. In turn, recipes have been proposed [9–11] to link the latter formalism in the form of SD energies obtained within DFT with the energies of various multiplets of the free TM atom or ion or TM complex. In this introduction, we will briefly summarize, for readers less familiar with LF theory, the existing LF approaches covering the CFT and AOM and reaching up-to-date DFT based multiplet theory developments. Later we will describe our theory and, following that, extensive applications will be given. 1.1 The Conventional Ligand Field Theory CFT [3] and AOM [4]

The physical background of the existing LF models stems from the rather localized character of the wavefunction of the d-electrons allowing one to consider a dn-configuration as well defined. It gives rise to a number of SD (i.e. 45, 120, 210 and 252 SD for n=2(8), 3(7), 4(6) and 5). They are used as a basis for expanding the many electron wavefunction for the ground and the excited states. The expectation values of the energy within this basis are obtained from the solution of the eigenvalue problem using the operator Eq. (1): HLF = S (i) h(i) + S (i, j) G(i, j)

(1)

Equation (1) includes the one electron, ligand field Hamiltonian h(i) and the twoelectron G(i,j) operator which takes account of the Coulomb interactions between d-electrons (via the 1/rij operator); summation is carried out over the d-electrons i