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Letter
Multiconfiguration Pair-Density Functional Theory Predicts Spin-State Ordering in Iron Complexes with the Same Accuracy as Complete Active Space Second-Order Perturbation Theory at a Significantly Reduced Computational Cost Liam Wilbraham, Donald G. Truhlar, Laura Gagliardi, and Ilaria Ciofini J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b00570 • Publication Date (Web): 04 Apr 2017 Downloaded from http://pubs.acs.org on April 8, 2017
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Multiconfiguration Pair-Density Functional Theory Predicts Spin-State Ordering in Iron Complexes with the Same Accuracy as Complete Active Space Second-Order Perturbation Theory at a Significantly Reduced Computational Cost Liam Wilbraham,a Donald G. Truhlar,b* Laura Gagliardi,b*and Ilaria Ciofinia* a
PSL Research University, Institut de Recherche de Chimie Paris IRCP, CNRS – Chimie ParisTech, 11 rue Pierre et Marie Curie, F-75005 Paris, France b Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
ABSTRACT: The spin-state orderings in nine Fe(II) and Fe(III) complexes with ligands of diverse ligand-field strength were investigated with multiconfiguration pair-density functional theory, MC-PDFT. The performance of this method was compared to that of complete active space second-order perturbation theory, CASPT2, and Kohn-Sham density functional theory, KSDFT. We also investigated the dependence of CASPT2 and MC-PDFT results on the size of the active-space. MC-PDFT reproduces the CASPT2 spin-state ordering, the dependence on the ligand field strength, and the dependence on active space at a computational cost significantly reduced as compared to CASPT2.
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Accurate description of spin-state energetics in molecular complexes containing transition metals is essential for the understanding and prediction of catalytic reactivity,1,2 including biological reactions, and of magnetic properties, such as spin-crossover.3,4 The quantitative calculation of spin-state ordering in transition metal complexes has been highlighted repeatedly as a significant challenge for modern computational methods.5,6 Furthermore, in many cases the key issue is not the quantitative magnitude of the spin state splitting but simply the correct ordering of the spin states, which is important for understanding the reactivity and catalytic power of transition metal complexes, because these properties are strongly dependent on the spin state of the system. Within this context, Kohn-Sham density functional theory (KS-DFT) suffers from sensitivity to the chosen approximation of the exchange-correlation density functional. Local approximations, such as the generalised gradient approximation (GGA), are biased towards lowspin ground states due to their over-delocalization of bonding molecular orbitals.7,8 This can be countered by the admixture of Hartree-Fock (HF) exchange hybrid density functionals, which are known to favor high-spin ground states;9,10 the optimal fraction of HF exchange, however, is system-dependent and may depend on the ligand field strength and,11,14 due to the inherently multiconfiguration nature of many of the spin states, a balanced and controlled description of static and dynamic correlation is required. Consequently some commonly-used functionals, such as B3LYP12 and PBE0,13 sometimes fail to correctly predict the ground spin state. Another method that has been applied to this problem16–18 is the complete active space selfconsistent field15 (CASSCF) method. However, CASSCF calculations recover only a small portion of dynamic correlation energy. Dynamic correlation energy may be added by multireference perturbation theory, such as CASPT2.19 The CASPT2 method, however, is generally limited to small- to medium-sized systems because its computer time and computer memory requirements increase rapidly with system size. The more-recently-developed multiconfiguration pair-density functional theory20,21 (MCPDFT) seeks to combine the explicitly multiconfigurational CASSCF with the low cost of KSDFT; and it provides a promising alternative to CASPT2 in terms of the required computational resources.22 The formulation of MC-PDFT has two key differences from KS-DFT: (i) the reference wave function is multiconfigurational, and (ii) the density functional employed (called the on-top density functional) depends on the total density and the on-top pair density (in contrast to the exchange-correlation density functionals of KS-DFT, which depend on the up-spin and the down-spin densities). Furthermore, MC-PDFT has recently been extended to work with generalised-active-space self-consistent field (GASSCF) wave functions,24,25,26 which have lower cost than CASSCF.
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In this light, we decided to test the performance of MC-PDFT on the prediction of spinstate ordering. To this end we tested MC-PDFT on nine Fe(II) and Fe(III) complexes with five ligands of diverse ligand-field strength, as shown in Figure 1. We shall compare the performance of this method with both CASPT2 and KS-DFT results for selected density functionals. Additionally, we investigated the dependence of CASPT2 and MC-PDFT on the size of the active-space and on the formulation of the on-top density functional. for the latter purpose we carried out calculations using four on-top density functionals, two of which are based on the PBE27 exchange-correlation functional with the other two based on revPBE28, a modified form of PBE where one parameter is recalibrated based on atomic exchange energies. These exchangecorrelation functionals are converted to on-top functionals by translation protocols explained previously, in particular by the original translation20 that does not include the gradient of the ontop pair density and by the full translation29 that does include it. The original translation is indicated by the prefix “t” and the full translation by the prefix “ft”. The resulting on-top density functionals are tPBE, ftPBE, trevPBE, and ftrevPBE. These functionals are examples of firstgeneration on-top density functionals that, in principle, could be further improved. In the main text we shall discuss results obtained with tPBE and trevPBE, with the fully translated results available in supporting information (SI). For clarity, we shall provide a brief review of MC-PDFT here; a more detailed description can be found elsewhere.20,21 MC-PDFT involves two steps: (1) a multiconfiguration selfconsistent-field calculation and (2) a post-SCF calculation with the on-top density functional. The MC-PDFT energy is , Π = ∑ ℎ + ∑ + , Π +
(1)
where is the MCSCF density; Π is the MCSCF on-top pair density; p, q, r, and s are orbital indices; ℎ are the one-electron kinetic energy, nuclear attraction, and effective core potential integrals, is an element of the MCSCF one-electron density matrix, are the twoelectron integrals, is the on-top energy, and is the internuclear repulsion. The on-top energy, akin to the exchange-correlation energy in KS-DFT, constitutes exchange and correlation additions to the MCSCF classical Coulomb energy (which is given by the four-index sum in eq 1). We note that the correlation energy explicitly accounts for the effect of dynamic correlation to minimize the electron-electron interaction energy, and it also implicitly contains a correction to the MCSCF kinetic energy. Details of the calculations are given in the SI. We calculate the spin-state ordering of the aforementioned complexes using a CASSCF wave function as the MCSCF wave function used as a basis for post-SCF energy calculations using both CASPT2 and MCPDFT. We use basis sets analogous to those previously suggested by Pierloot and co-workers.17 An exploratory
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investigation of the effect of active space size was performed, and we shall refer to the active spaces considered as (m/m-1, n), where m [m-1] is the number of active electrons in the active space for Fe(II) [Fe(III)], and n is the number of active orbitals. Five choices of active space are considered (the orbitals are pictured in Figure S2 in the SI): •
(6/5, 5) – Fe(II) valence d orbitals
•
(10/9, 7) – Fe(II) valence d orbitals plus two Fe-ligand -bonding orbitals
•
(6/5, 10) – Fe(II) valence d orbitals plus a set correlating d orbitals
•
(10/9, 12) – (6/5, 10) plus two -bonding, eg-like Fe-ligand orbitals
•
(12/11, 14) – (10/9, 12) plus Fe 3s and a correlating orbital for 3s.
The largest of these is the active space proposed by Hauser18 and co-workers. For comparison, and to examine the effects of Hartree-Fock exchange on spin-state ordering, we also performed calculations at KS-DFT level using PBE26 and PBE013 functionals, the latter containing 25% Hartree-Fock exchange. In all cases, D2h symmetry was imposed for both geometries and energy calculations (except for complexes 4 and 9, for which Ci symmetry was used). We used symmetry mainly because it retains consistency of the active space across different complexes with ligands of varying ligand-field strength. Generally, we discuss results in terms of the energy difference between high- and low-spin forms of each complex (Δ/ ). Positive (negative) values of Δ/ indicate that the low (high) spin state is energetically favored. We consider the difference in Δ/ for different ligands, and we note that strong-field ligands are expected to produce higher (more positive or less negative) values of Δ/ than weak-field ligands. We consider five different ligands with the ordering expected from the spectrochemical series being H2O < NH3 < NCH < CO < CNH, in terms of Δ/ . Based on diffusion Monte Carlo results from Droghetti31 and on ligand field theory, we expect Fe complexes with CNH and CO will usually have low-spin ground states, and complexes with NCH, NH3 and H2O will usually have high-spin ground states. Figure 2 shows ΔEH/L values calculated by KS-DFT (using PBE and PBE0), by MC-PDFT (using tPBE and trevPBE), and by CASPT2. For MC-PDFT and CASPT2, the (12/11, 14) active space was used. For both Fe(II) and Fe(III) complexes, we note that PBE overestimates ΔEH/L for all ligands with respect to CASPT2. To stress the absolute spin-state ordering obtained in each case, an analogous plot is given in the SI. As anticipated above, based on previous observations for GGA exchange-correlation functionals,11,32 this is interpreted as stemming from an overdelocalization of bonding orbitals, and we find that it severely overstabilizes low-spin states in
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the case of strong-field ligands. Introducing HF exchange in the form of PBE0 generally stabilizes high-spin states, reducing the value of ΔEH/L for all complexes with respect to PBE. However, the effect of HF exchange is not uniform for ligands of different ligand field strengths, with the induced high-spin stabilization of strong-field ligands (CNH, CO) more severe than for weak field ligands (NCH, NH3, H2O), echoing the previous investigation of the effect of HF exchange on spin-state splitting of octahedral Fe complexes by Ioannidis and Kulik.11This effect is exemplified by the performance of PBE0, which underestimates (overestimates) ΔEH/L for strong-field (weak-field) ligands. The imbalance in the PBE0 treatment of ligands of different ligand field strengths can be rationalized by considering the different types of ligand–metal bonding present in each complex. Ligands that are good π-acceptors (CNH, CO) are most affected by introducing HF exchange. As the self-interaction error32 is reduced in orbitals of π-symmetry, these orbitals are destabilized and the overall ligand-field strength is reduced, resulting in lower values of Δ/ . Conversely, ligands that are good π-donors (NCH, H2O) are less affected due to the increased population of more localized, antibonding d-orbitals upon ligand-to-metal donation. Lastly, ligands which only participate in σ-donation (NH3) are the least affected by the introduction of HF exchange, due to the more local nature of the σ-symmetry bonding orbitals. For both Fe(II) and Fe(III) complexes, the difference in Δ/ computed between each of the two strong-field ligands remains largely constant, irrespective of the functional used. The same holds true for weak-field ligands. This indicates the need for an effective and balanced description of different types of metal-ligand bonding. In analogous work, a recent analysis of the effects of density localization in octahedral metal complexes computed with GGA, global hybrid, and range-separated hybrid density functionals compared with CASPT2 reference values indicated the need for a separate treatment of density delocalization and energetic errors to recover properties that rely on simultaneous treatment of density and orbital energies.33 In this context, we shall now discuss the performance of the MC-PDFT approach compared to CASPT2. Despite the systematic overestimation of Δ/ by tPBE with respect to CASPT2 (Figure 2), we see that the trend of Δ/ values with respect to ligand-field strength at MCPDFT level traces that of CASPT2. Consequently, and in contrast to the results obtained at KSDFT level, the difference in Δ/ as a function of ligand type is largely preserved when using MC-PDFT, demonstrating that using an on-top density functional does not compromise a balanced treatment of strong- and weak-field ligand bonding provided by CASPT2. This is especially striking when one considers that the CASSCF step that precedes the Post-SCF step in both CASPT2 and MC-PDFT does not yield good results, as shown in Fig. S3 of the SI.
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Figure 2 also includes the trevPBE results. We see that the over-stabilization of low-spin states with tPBE is reduced, yet – as before – the trend in Δ/ with respect to ligand field strength is retained. This is consistent with the formulation of the underlying revPBE functional, in which one parameter of the PBE functional has been recalibrated to improve agreement of atomic exchange energies with known exact results. The stability of the good results obtained with MC-PDFT to functional variation extends to other on-top density functionals (ftPBE, ftrevPBE) as well, as seen in the SI. Crucially, a change in the functional formulation shows no difference in the treatment of strong- and weak-field ligands. We also emphasise that these results represent the application of first-generation on-top density functionals and are, in this respect, especially encouraging. The MC-PDFT results are also stable with respect to the size of the chosen active space, as is shown in Figures S3 and S4 of the SI, underlining the even treatment of different metal-ligand interactions by the CASSCF wave function. This is very encouraging. In order to substantiate the claim that MC-PDFT is significantly less computationally expensive than CASPT2, we measured the wall-time taken for each approach. Figure 3 shows the results of this analysis for the Fe(II) complexes, expressed as a ratio between calculation times for CASPT2 and MC-PDFT with respect to active space size. Analogous plots for Fe(III) complexes as well as the full set of absolute wall clock times for each complex are given in the SI. For very small active spaces, we see that there is a negligible increase in speed only in the case of ligands with a reduced number of heavy atoms – i.e. a smaller basis set (H2O, NH3). For all other complexes (CNH, CO, NCH) we observe a significant decrease in calculation time even at the smallest active spaces. The time-saving benefit of MC-PDFT becomes more evident with increasing active space size and for complexes with a larger basis sets. The most striking speed improvement is for [Fe(CO)6]2+, using the (12/11, 14) active space, with MC-PDFT energies for the quintet state calculated approximately 18 times faster than at CASPT2 level. Especially when combining with GASSCF spaces, as mentioned above, we anticipate being able to treat much larger problems with MC-PDFT than with CASPT2. By testing the performance of MC-PDFT when calculating the spin-state ordering of nine Fe(II) and Fe(III) complexes, we have found that this method able to qualitatively reproduce the spin-state ordering calculated at CASPT2 level; the ordering is reproduced in all but one case, and the relative treatment of ligands with diverse ligand field strengths is retained. We conclude that MC-PDFT is able to take advantage of the apparently even-handed even treatment of different ligand-metal bonding types by the underlying CASSCF wave function, and at the same time it benefits from the reduced computational cost associated with calculating exchange and correlation energy using a density functional. MC-PDFT also reacts similarly to CASPT2 as a
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function of active space size, and shows modest, yet consistent, sensitivity to the on-top density functional. ASSOCIATED CONTENT Supporting Information. Supporting Information. Computed structural details; active space dependence; on-top density functional dependence; computational details. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Authors *E-mail:
[email protected] (IC) *E-mail:
[email protected] (LG) *E-mail:
[email protected] (DGT) ORCiD
Liam Wilbraham: 0000-0003-4522-6380 Donald G. Truhlar: 0000-0002-7742-7294 Laura Gagliardi: 0000-0001-5227-1396
Ilaria Ciofini: 0000-0002-5391-4522 Notes The authors declare no competing financial interest. ACKNOWLEDGMENT The authors are grateful to Pragya Verma for helpful interactions. This work was supported in part (L.G. and D.G.T.) by the U. S. National Science Foundation by Grant CHE14-64536. I.C. received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 648558, STRIGES CoG grant).
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Figure 1. Complexes considered in this study.
Figure 2. ΔEH/L calculated for Fe(II) (left) and Fe(III) (right) complexes in ascending order of computed ligand field strength. Results obtained using PBE, PBE0, MC-PDFT (with two different on-top functionals tPBE and trevPBE), and CASPT2 are shown, with the CASPT2 and MC-PDFT results based on the (12/11, 14) active space. Positive (negative) values of ΔEH/L correspond to a low-spin (high-spin) ground state.
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Figure 3. Speed increase of MC-PDFT (tPBE) calculations with respect to CASPT2, expressed as the ratio between wall clock times of CASPT2 (tCASPT2) and MC-PDFT (tMC-PDFT) calculations. Squares (triangles) indicate singlet (quintet) spin states. Only results for Fe(II) complexes are shown, with analogous plots for Fe(III) complexes given in the supporting information.
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