Article pubs.acs.org/JPCA
Spin Propensities of Octahedral Complexes From Density Functional Theory Sara R. Mortensen and Kasper P. Kepp* DTU Chemistry, Technical University of Denmark, Building 206, Anker Engelunds Vej 1, 2800 Kongens Lyngby, Denmark S Supporting Information *
ABSTRACT: The fundamental balance between high- and low-spin states of transition metal systems depends on both the metal ion and the ligands surrounding it, as often visualized by the spectrochemical series. Most density functionals do not reproduce this balance, and real spin state propensities depend on orbital pairing and vibrational entropies absent in the spectrochemical series. Thus, we systematically computed the tendency toward high or low spin of “text-book” octahedral metal complexes versus ligand and metal type, using eight density functionals. Dispersion effects were generally 0 imply that low spin is favored.
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ligand bonds in high-spin states, which again arises from the occupation of eg orbitals. To include these corrections to the energies, the harmonic frequencies of both high-spin and lowspin states of the optimized complexes were computed numerically at the BP86-def2-SVP level, the same level as the geometry optimization, in Cosmo water, using the NumForce script, and the ZPEs of high-spin and low-spin states were subtracted and added to the high-spin low-spin energy differences. Furthermore, the high-spin states of mononuclear octahedral coordination complexes typically contain 10−30 kJ/mol more vibrational entropy than the low-spin states, due to the longer and weaker metal−ligand bonds of the high-spin state.12,13 The vibrational entropy obtained from the harmonic frequency state function provides a fairly accurate estimate for this purpose and was included in the analysis by using the freeh script of Turbomole, which can perform thermochemistry calculations.
RESULTS AND DISCUSSION
Zero-Point Energy Effects versus Ligand Type. Vibrational ZPEs contribute to the relative energies of states: the ZPE is usually smaller for high-spin states than for low-spin states, due to weaker metal−ligand bonds, causing the differential ZPE to favor high-spin states.13 Computed ZPEs and their effect on the high-spin low-spin energy difference can be seen in Figure 1A (numerical data are compiled in Supporting Information, Table S11A−F). The terms “strong field” and “weak field” strictly refer to the spectroscopically derived ligand-field splitting parameter Δo. Thus, in the following, we use the words weak and strong ligands in the sense of favoring high or low spin, not in terms of Δo. As we will see, there are important distinctions between the spectrochemical series and the true spin state preference resulting not only from orbital pairing but also from vibrational and thermodynamic effects, which should be relevant to thermal-equilibrium chemistry such as experimental and 4043
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bond strength, which relates to the differential LFSE. The identified grouping of Fe(II) ∼ Mn(II) ∼ Co(III) ∼ Fe(III) versus Co(II) ∼ Mn(III) thus follows the differential LFSE. Upon spin transition, two pairings occur for Fe(II), Mn(II), Co(III), and Fe(III), and one pairing occurs for Co(II) and Mn(III). The differential LFSE is 2 −2P for complexes containing the first four metal ions, and 1 − P for the Co(II) and Mn(III) complexes, where P is the pairing energy. The particularly large vibration effects of Fe(II) and Mn(II) result because the divalent metal ions exhibit larger bond changes during spin transition as the ionic component of the bond strength is smaller, and hence, the LFSE components are larger. Finally, Co(II) complexes behave irregularly due to the strong Jahn−Teller distortion of low-spin states. Together, these features explain our observed trends in metal- and liganddependent vibration effects that favor high-spin states. Spin Propensities of Octahedral Complexes: General Observations. Figure 2 shows the computed energy difference between high-spin and low-spin states for the 48 different octahedral complexes (six metal ions with any of eight ligands in homoleptic complexes), divided into ligand type. Parts A and B of Figure 2 display the Mn(II) and Mn(III) complexes, parts C and D of Figure 2 display the Fe(II) and Fe(III) complexes, and parts E and F of Figure 2 show the Co(II) and Co(III) complexes, respectively. Also, results with each of the eight used functionals are shown by different colors: the cold colors (blue/ green) correspond to nonhybrid functionals, whereas the warm colors (yellow, orange, red) correspond to hybrid functionals. A negative number implies that high spin is lower in energy than low spin, whereas a positive number implies that low spin is lower in energy. The energies in Figure 2 are without correction for dispersion, ZPE, and entropy, to isolate the effect of density functional clearly. Corrections for entropy and ZPE can be found in Figure 1. As seen from the data in Supporting Information (Tables S7−S10), dispersion changes the high-spin low-spin energy difference by maximally 5 kJ/mol except for the hexabromocobalt(II) complexes, where the hybrid functionals provide larger corrections (13−23 kJ/mol) relating to the large geometric changes associated with the strongly Jahn− Teller distorted low-spin state and the weak ligand. Thus, dispersion effects will not be discussed in further detail as they, absent this special case, do not change the spin propensities significantly for the simple complexes studied here (for other, sterically crowded coordination complexes, dispersion corrections in favor of low spin are sometimes substantial30). First, in all cases except Co(II), the two π-acceptor ligands cause substantially larger low-spin preference than other ligands, which is also seen for the spectrochemical series. For the Co(II) complexes, in particular the hybrid functionals have very little spread in preference, relating to the fact that the lowspin Co(II) complexes are strongly Jahn−Teller distorted (see below); this is to some, although weaker, extent also seen for Mn(III) complexes where the high-spin states are distorted. Second, CN− is consistently weaker than CO for the M(II) complexes, but stronger than CO for the M(III) complexes. This observation is new and is observed with all eight functionals and all six metal ions, so this is a statistically significant finding. The relative preferences of Mn(III)/CN− and M(II)/CO should be relevant to the discussion whether cyanide is a strong ligand or not.59 Electrostatic attraction is enhanced in the CN−/M(III) combination, thus strengthening the bonding to favor low spin further. In the M(II) systems,
theoretical studies of spin crossover, catalysis with multiple spin states, etc. From Figure 1A, ZPEs are found to be generally larger for the low-spin states and for the stronger ligands, and smaller for the high-spin states and weaker ligands, producing a wide range of differential ZPEs of 1−33 kJ/mol. This is understood from the stronger ligands and the low-spin states with less eg orbital set occupation giving stronger coordinative bonds with steeper potential energy surfaces and, hence, larger ZPEs. This means that ZPE corrections are particular important for complexes with moderate and strong ligands (e.g., N-ligands, CO, and CN−). In contrast, for weak halide ligands, the change in relative energies of high spin and low spin amounts to 1−4 kJ/ mol. Thus, ZPEs cannot be ignored when estimating the spin state of a coordination complex, in particular with stronger ligands. Vibrational Entropy Contributions to High-Spin Preferences. As discussed previously, vibrational entropy is typically 10−30 kJ/mol larger in high-spin than low-spin states of mononuclear complexes because occupation of the eg set of orbitals leads to elongated, weaker metal−ligand bonds.12,13 At thermal equilibrium relevant to most studies of coordination chemistry, neglecting this systematic entropy contribution leads to erroneous appraisal of higher HF exchange fractions, since HF exchange “mimics” the neglected entropy difference.13 Both direct comparison to experimental enthalpies of spin crossover and account of entropy in estimates of observed ground states accordingly found TPSSh with only 10% HF exchange to perform accurately in several benchmarks.13,17,30,53,58 The differential entropy is shown in Figure 1B (numerical data in Supporting Information, Table S12A−F). Consistent with thermochemical experiments on spin-crossover complexes,12 vibrational entropy is found to favor high-spin states by up to 40 kJ/mol. Importantly, again the vibrational entropy correction is larger for strong ligands subject to larger changes in vibrational state functions when undergoing high-spin lowspin transition. Thus, we find that both entropy and ZPE effects are substantial in particular for strong and moderate ligands, but even for a relatively weak ligand such as water, the combined effects favor high spin by 15−30 kJ/mol, depending on metal ion. Since the two effects both depend on the strength of the metal−ligand bond, they correlate strongly (R2 = 0.58, see Supporting Information, Figure S1). It is worth noting that the total energy correction depends on the full vibration state function and tends to be approximately half the size of the ZPE (numerical data in Supporting Information, Table S13A−F). These energy corrections counteract the vibrational entropy, as typically observed in interaction-bonding-related entropy−energy compensation, due to the softer, energywise weaker bonds of high-spin states having more vibrational entropy. Thus, the total vibrational energy and entropy corrections toward high spin will be typically about 5 (but in some cases up to 15) kJ/mol smaller than the sum of the contributions shown in Figure 1, parts A and B. We find that the computed ZPEs and the vibrational entropies also follow a trend in metal ion Fe(II) ∼ Mn(II) ∼ Co(III) ∼ Fe(III) > Co(II) ∼ Mn(III) that can be rationalized in the same way as the trend in ligand: the metal ions with the largest ligand field stabilization energies (LFSE, which depends on both the splitting and the orbital pairing) produce the strongest metal−ligand bonds. The vibrational corrections to the spin state propensity depend on the change in metal−ligand 4044
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Figure 3. Spin state propensity ordered according to metal ion for all eight studied ligands: (A) Br−; (B) Cl−; (C) H2O; (D) SCN−; (E) NCS−; (F) NH3; (G) CN−; (H) CO.
higher spin states.14,15 For the four nonhybrids, the nature of the exchange and correlation functionals is less important and almost has no effect on the relative order of ligands in the series, and in most cases, the spin gaps produced by TPSS, PBE, BP86, and BLYP are quite similar. The important exceptions are the π-acceptor ligands CO and CN−, which give rise to larger spin gap differences for these methods than commonly observed. This tendency has, as far as we know, not
back-bonding is more pronounced to the neutral CO ligand, as observed before,59 giving stronger interaction and favoring low spin relative to the hexacyano complexes. Density Functional Differences. From Figure 2, the nonhybrid functionals consistently provide higher energies, i.e., are shifted in favor of low-spin states, compared to hybrid functionals. This is well-known consequence of HF exchange producing polarized electron densities with energy bias toward 4045
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Similarly to the magnetochemical series of quantum-admixed porphyrins described by Reed and co-workers,32,60 the series in Figure 2 is a series of ground-state spin propensities for normal “text-book” octahedral complexes. The series by Reed and coworkers follows the order Br− < H2O < Cl− < NCS− < F− ≪ CO for ligands studied here, with the order H2O < OH− ∼ Cl− being distinct from the spectrochemical series, Cl− < OH− < H2O.60 The spectrochemical series, originally based on Co(III) systems,9 is commonly listed for the ligands studied here as28 Br− < SCN− < Cl− < H2O < NCS− < NH3 < CN− < CO. From Figure 2, the order Br− < Cl− < H2O is observed as in the spectrochemical series. The similarities are due to the fact that the spin pairing and orbital relaxation effects are similar for the same metal ions with the same nuclear charge and dq configurations, in particular for the weak ligands. In contrast, SCN− is generally a stronger ligand than H2O when orbital effects are included as in Figure 2. The difference is likely due to the spin pairing being less expensive in SCN− that contains some d-electron delocalization by back-bonding, making this ligand more prone to low spin than would be assumed simply from a spectrochemical series of Δo. Also, the spectrochemical series based on Co(III) complexes lists the order CN− < CO, but when all electronic effects are included, we find that this order is CN− < CO in the M(II) complexes (Figure 2, parts A, C, and E) but CO < CN− in the M(III) complexes (Figure 2, parts B, D, and F). Since this tendency is independent of dq configuration, it is mainly due to charge, i.e., stronger binding of the negatively charged ligand to the higher oxidation state. The work by Ichii et al. produced a remarkable order of CN− < NH3 < CO for Δo based on DFT calculations of M(III) complexes.28 They explained this as due to reduced back-bonding into CN− due to its negative charge bringing it even below NH3. Such an effect is not observed in any of our calculations and is possibly due to the spin pairing energy being smaller in the back-bonded (d-electron delocalized) CN− ligand compared to NH3. Effect of Metal Ions on Spin State Propensity. As discussed, there are major effects of the metal ion, both in terms of dq configuration and effective nuclear charge, on the spin state propensities. To visualize this more clearly, the effect of metal ion on spin state propensity is shown in Figure 3. All the studied functionals produce the most favorable lowspin states in the Co(III) complexes, consistent with their t2g6 low-spin states being particularly stable. For weaker ligands Br− (Figure 3A), Cl− (Figure 3B), H2O (Figure 3C), and SCN− (Figure 3D), Co(II) tends to follow Co(III) in low-spin preference, whereas for the stronger N-ligands (Figure 3, parts E and F) and CN− and CO (Figure 3, parts G and H), the Fe(II) complexes are the second-most low-spin prone, after Co(III). Thus, the general view that low-spin d6 complexes are particularly stable is evident from the data, but the data clearly show interplay between ligand and metal type. This insight arises from the systematic comparison and is consistent across the used methods. The next metal ion in line, Fe(III), favors low spin typically by ∼100 kJ/mol less than Fe(II) for the strong-ligand complexes. The two most important trends in low-spin propensity from Figure 3 are the underlying trend in increased effective nuclear charge, which follows the order Mn(II) < Fe(II) < Co(II) < Mn(III) < Fe(III) < Co(III), and the special effect of LFSE, notably of low-spin Fe(II), producing peaks for the moderate and strong ligands. The LFSE plays out mainly in the strong ligands contributing the most to the low-spin state propensity,
been observed before. For the strong binding and dense electron density situation, differences between functionals will be more pronounced than in the weak-binding (weak ligand) limit. Thus, care should be taken in testing distinct performances of these methods when studying π-acceptor ligands, whereas for other ligands, the nonhybrids perform similarly. After the nonhybrids, the TPSSh functional follows in the middle of the range (light orange), consistent with its 10% HF exchange. Interestingly, the B97-D functional behaves irregularly by changing relative position in the functional order depending on ligand strength. B97-D includes the strongly binding D2 dispersion correction and therefore tends to favor low-spin states more than the B97 functional itself. In case of Co(II) complexes, the B97-D energies are actually more in favor of low spin than TPSSh. In general, B97-D favors low spin relatively more with the strong ligands as these have tighter electronic structures favored by dispersion corrections, and this causes B97-D to resemble the 10% HF exchange hybrid TPSSh for the strong ligands CO and CN− but less in favor of low spin for the weak ligands and, hence, falls below other functionals in the weak-ligand limit. Dispersion corrections are quite liganddependent and grow with the tightness of the electronic structure,13 in particular for the strongly attractive D2 correction but less so for the more recent and moderate D3 corrections that are generally Fe(II) > Fe(III) ∼Co(III) > Mn(III) > Co(II) is seen, with the distorted low-spin Co(II) complexes again resulting in an anomaly, as these already have partially elongated metal−ligand bonds in the low-spin states. As seen from Supporting Information Table S15, the geometries of the low-spin Co(II) complexes beautifully capture the tetragonal distortion with large standard deviations in bond lengths resulting from two elongated axial bonds. There is also a clear trend in the change in bond length as a function of ligand: the strong ligands cause larger changes in
Figure 4. Differences (angstroms) between average metal−ligand bond lengths of high-spin and low-spin states for geometries optimized at the BP86/def2-SVP level in a Cosmo model of water.
bond length, whereas the weaker ligands and in particular the halide ligands exhibit minor changes in geometry and the order of metal ions no longer follows the order described above; for these weak negatively charged ligands, the role of electrostatics is more important than LFSE, and hence, M(III) complexes exhibit the largest changes in geometry upon spin state transition. Thus, the trends observed in spin state propensity in the previous sections, since these were rationalized from differential LFSE or changes in metal−ligand bond strengths, can be directly related to the changes in the bond lengths as well. Realism of Density Functionals. The systematic study performed here allows an assessment of the realism of the eight studied functionals, by comparison to the experimentally known spin states of some of the studied complexes. Due to the systematic approach required to identify fundamental trends, many of the studied complexes are not stable in the investigated form (e.g., they hydrolyze or oxidize under standard conditions in aqueous solution). However, many other of the complexes are well-known experimentally. One of the main strengths of computational chemistry is to establish physical trends across series where some species are experimentally inaccessible. To compare with experimental ground states, one should include the two major systematic effects that favor high spin, ZPE and vibrational entropy (relativistic corrections and solvent effects are generally smaller and less systematic for first row of the d-block).13 Thus, as seen from our results for these corrections, a functional performs well if the energies are 20−40 kJ/mol shifted toward low spin, but these corrections are quite ligand-dependent and also vary with metal ion, as discussed above. From Figure 2, all the functionals comfortably produce low spin for CO except for Mn(II) where the gap is small. Also, for the Co(III) complexes and Fe(II) complexes of strong ligands, the low spin is so favored that all functionals comfortably produce low spin, as exemplified by the well-known diamagnetic [Co(NH3)6]3+, and consistent with the general knowledge.61 However, already for CN−, the B3LYP and PBE0 functionals produce nearly isoenergetic high- and low-spin states for Mn(II) and Co(II) complexes, while the M(III) complexes and the t2g6 low-spin Fe(II), as discussed above, comfortably remain low spin also for these functionals. The average and standard deviation of entropy and ZPE corrections for the hexacyano complexes are 29 ± 7 and 19 ± 7 kJ/mol, both in favor of high spin. Thus, a conservative correction of 40 kJ/mol, more 4047
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CONCLUSIONS The need to understand spin state propensity and transitions between spin states of transition metal systems inspired us to study the spin state propensity of a range of text-book octahedral coordination complexes with combinations of simple ligands and the metal ions Mn(II), Fe(II), Co(II), Mn(III), Fe(III), and Co(III). Due to the systematic nature of such study, many complexes that are not experimentally accessible can be explored, and those complexes that are known serve as benchmarks for the eight studied density functionals. Dispersion effects are generally small but favor low-spin states and are more important to include for strong ligands that produce tighter electron densities. Zero-point energies favor high-spin states on average by 10 kJ/mol, but very dependent on ligand: for stronger ligands up to 33 kJ/mol correction in favor of high spin is sometimes seen, whereas for weak ligands it changes spin state propensity by only a few kilojoules per mole. Vibrational entropy also favors high-spin states on average by 16 kJ/mol, but up to 40 kJ/mol for strong ligands. Both entropy and zero-point energy effects follow a metal ion order of Fe(II) ∼ Mn(II) ≫ Co(III) ∼ Fe(III) > Co(II) ∼ Mn(III) resembling the differential ligand field stabilization energy. Thus, vibrational energy and entropy corrections cannot be ignored when studying transition metal chemistry in thermal equilibrium (up to ∼60 kJ/mol in total) except for weak ligands. We find systematic orders of low-spin propensity for both ligands and metal ions that resemble the spectrochemical series of the ligand-field splitting parameter as commonly stated in text books but notably differ in a number of situations, as discussed in the paper. Our derived “thermochemical spin series” includes electronic effects such as orbital pairing. The study reproduces the general notion of particular stability and back-bonding capability of t2g6 configurations and finds that CN− is consistently weaker than CO for M(II), but stronger than CO for the M(III). SCN− and NCS− change order in M(II) versus M(III) complexes, and interesting anomalies relating to Jahn−Teller distortion of low-spin Co(II) complexes and back-bonding in low-spin Fe(II) complexes are observed.
reasonably 50 kJ/mol, in favor of high spin should be added to the energies of the hexacyano complexes in Figure 2. The red [Co(CN)6]4− can be produced in the lab but oxidizes to the Co(III) complex and is likely low spin like the related Mn(II) complex.62 Since dispersion corrections are smaller than 5 kJ/ mol, it shows that B3LYP, even with dispersion, is not correct for the hexacyano complexes. This is consistent with benchmarks against experimental enthalpies of spin crossover of cobalt and iron complexes,17 where B3LYP also substantially overstabilized high spin when thermal corrections were accounted for. Complexes with known spin states and moderately strong ligands provide the best test case for the functionals, as failures will become more apparent when the spin balance is subtle. Nligands such as thiocyanate are often encountered in spincrossover systems due to their moderate ligand field strength. The salts of the [Fe(NCS)6]3− complex anion exist as high-spin ground states:63 the functionals spread around zero energy difference for this complex anion (Figure 2D). When entropy and ZPE corrections are added (40 kJ/mol, see Supporting Information) all methods correctly predict high spin. The hexammine complexes constitute a relevant benchmark, because of the changes in spin states seen experimentally, depending on metal ion. Notably, [Co(NH3)6]3+ is low spin, but divalent metal complexes such as [Co(NH3)6]2+ and [Fe(NH3)6]2+ are high spin.61 From Figure 2, parts C and E, the nonhybrids PBE, BP86, BLYP, and TPSS produce low-spin [Fe(NH3)6]2+ and [Co(NH3)6]2+ by margins of ∼10−40 kJ/ mol. The vibrational and entropy corrections are relatively small (∼7 and 8 kJ/mol) for the Co(II) complex due to Jahn− Teller distortion but sum to ∼41 kJ/mol for [Fe(NH3)6]2+ (see Supporting Information) due to its loss of two eg electrons upon spin transition and associated larger differences in metal− ligand bonds, as discussed above. This complex and the hexaaquairon(II) complex were previously studied with CASPT2, B3LYP, and PBE0,30 giving slightly smaller thermal and vibrational corrections and 10−30 kJ/mol more high-spin preference, but otherwise similar results to ours; the differences are attributable to our use of BP86-Cosmo-optimized structures due to previous benchmarks.52 Even with vibrational and thermal corrections, the nonhybrids do not produce the experimental high-spin states comfortably, viz., the consensus that some HF exchange is required to obtain a balanced description of electron correlation energy in such systems.14,15,30 The hexaaqua complexes also serve as an insightful benchmark: notably, it is known experimentally that [Co(H2O)6]3+ is low spin, whereas [Mn(H2O)6]3+, like most other hexaaqua complexes, is high spin.64 From Figure 2F, PBE0, B97-D, and B3LYP produce energies for [Co(H2O)6]3+ that are only 5−30 kJ/mol or less in favor of low spin, and when corrected for entropies and ZPEs, they produce high-spin states. In contrast, [Mn(H2O)6]3+ is predicted high spin with all methods as all other studied hexaaqua complexes except Co(III). Thus, aqua complexes indicate that 20% HF exchange is too much, consistent with the emerging consensus.13−15,17,30,52 Finally, all halide complexes are comfortably high spin except the Co(III) complexes which produce energies near zero for these ligands. When adding entropy and ZPE corrections, all Co(III) complexes are predicted to be low spin by all eight methods as seen experimentally (one known exception is highspin [CoF6]which was not studied here).
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ASSOCIATED CONTENT
S Supporting Information *
Tables S1−S6, electronic energies for computed high-spin and low-spin states of all complexes with all methods; Tables S7− S10, dispersion-corrected electronic energies; Table S11A−F, computed zero-point energies; Table S12A−F, computed entropy and difference in entropy between high-spin and low-spin states at 300 K; Table S13A−F, total thermal corrections to high-spin low-spin energy difference, from vibrational state functions; Table S14, optimized metal−ligand bond lengths of high-spin complexes; Table S15, optimized metal−ligand bond lengths of low-spin complexes; Figure S1, correlation between TΔS and ΔZPE corrections to spin state propensity. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +045 45 25 24 09. E-mail: [email protected]. Notes
The authors declare no competing financial interest. 4048
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ACKNOWLEDGMENTS This research has been supported by the Danish Center for Scientific Computing (Grant No. 2012-02-23).
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(1) Scheidt, W. R.; Reed, C. A. Spin-State/Stereochemical Relationships in Iron Porphyrins: Implications for the Hemoproteins. Chem. Rev. 1981, 81, 543−555. (2) Jensen, K. P.; Ryde, U. How O2 Binds to Heme: Reasons for Rapid Binding and Spin Inversion. J. Biol. Chem. 2004, 279, 14561− 14569. (3) Harvey, J. N.; Poli, R.; Smith, K. M. Understanding the Reactivity of Ttransition Metal Complexes Involving Multiple Spin States. Coord. Chem. Rev. 2003, 238, 347−361. (4) Lomont, J. P.; Nguyen, S. C.; Harris, C. B. Ultrafast Infrared Studies of the Role of Spin States in Organometallic Reaction Dynamics. Acc. Chem. Res. 2014, 47, 1634−1642. (5) Gütlich, P.; Goodwin, H. A. Spin CrossoverAn Overall Perspective. Top. Curr. Chem. 2004, 233, 1−47. (6) Létard, J.-L.; Guionneau, P.; Goux-Capes, L. Towards Spin Crossover Applications. Top. Curr. Chem. 2004, 235, 221−249. (7) Soari, M.; Nakano, M.; Miyazaki, Y. Calorimetric Investigation of Phase Transitions Occurring in Molecule-Based Magnets. Chem. Rev. 2006, 106, 976−1031. (8) Fajans, K. Struktur und Deformation der Elektronenhüllen in ihrer Bedeutung für die Chemischen und Optischen Eigenschaften Anorganischer Verbindungen. Naturwissenschaften 1923, 11, 165−172. (9) Tsuchida, R. Absorption Spectra of Co-ordination Compounds. I. Bull. Chem. Soc. Jpn. 1938, 13, 388−400. (10) Moffitt, W.; Ballhausen, C. J. Quantum Theory. Annu. Rev. Phys. Chem. 1956, 7, 107−136. (11) Ballhausen, C. J. Introduction to Ligand Field Theory; McGrawHill Book Company, Inc.: New York, 1962. (12) Toftlund, H. Spin Equilibrium in Solutions. Monatsh. Chem. 2001, 132, 1269−1277. (13) Kepp, K. P. Consistent Descriptions of Metal−Ligand Bonds and Spin-Crossover in Inorganic Chemistry. Coord. Chem. Rev. 2013, 257, 196−209. (14) Reiher, M. Theoretical Study of the Fe(Phen)(2)(NCS)(2) Spin-Crossover Complex with Reparametrized Density Functionals. Inorg. Chem. 2002, 41, 6928−6935. (15) Paulsen, H.; Duelund, L.; Winkler, H.; Toftlund, H.; Trautwein, A. X. Free Energy of Spin-Crossover Complexes Calculated with Density Functional Methods. Inorg. Chem. 2001, 40, 2201−2203. (16) Daku, L. M. L.; Vargas, A.; Hauser, A.; Fouqueau, A.; Casida, M. E. Assessment of Density Functionals for the High-Spin/Low-Spin Energy Difference in the Low-Spin Iron(II) Tris(2,2′-Bipyridine) Complex. ChemPysChem 2005, 6, 1393−1410. (17) Jensen, K. P.; Cirera, J. Accurate Computed Enthalpies of Spin Crossover in Iron and Cobalt Complexes. J. Phys. Chem. A 2009, 113, 10033−10039. (18) Hughes, T. F.; Friesner, R. A. Correcting Systematic Errors in DFT Spin-Splitting Energetics for Transition Metal Complexes. J. Chem. Theory Comput. 2011, 7, 19−32. (19) Baranovic, G. Thermochemistry of Spin-Crossover Fe(II) Complexes Calculated with Density Functional Methods. Chem. Phys. Lett. 2003, 369, 668−672. (20) Neese, F. A Critical Evaluation of DFT, Including TimeDependent DFT, Applied to Bioinorganic Chemistry. J. Biol. Inorg. Chem. 2006, 11, 702−711. (21) Zein, S.; Borshch, S. A.; Fleurat-Lessard, P.; Casida, M. E.; Chermette, H. Assessment of the Exchange-Correlation Functionals for the Physical Description of Spin Transition Phenomena by Density Functional Theory Methods: All The Same? J. Chem. Phys. 2007, 126, 014105. (22) Paulsen, H.; Schünemann, V.; Wolny, J. A. Progress in Electronic Structure Calculations on Spin-Crossover Complexes. Eur. J. Inorg. Chem. 2013, 628−641. 4049
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The Journal of Physical Chemistry A (45) Perdew, J. P. Density-Functional Approximation for the Correlation Energy of the Inhomogeneous Electron Gas. Phys. Rev. B 1986, 33, 8822−8824. (46) Becke, A. D. Density-Functional Thermochemistry. V. Systematic Optimization of Exchange-Correlation Functionals. J. Chem. Phys. 1997, 107, 8554−8560. (47) Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction. J. Comput. Chem. 2006, 27, 1787−1799. (48) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (49) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110, 6158−6169. (50) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the Density Functional Ladder: Nonempirical Meta-Generalized Gradient Approximation Designed for Molecules and Solids. Phys. Rev. Lett. 2003, 91, 146401. (51) Perdew, J. P.; Tao, J.; Staroverov, V. N.; Scuseria, G. E. MetaGeneralized Gradient Approximation: Explanation of a Realistic Nonempirical Density Functional. J. Chem. Phys. 2004, 120, 6898− 6911. (52) Jensen, K. P.; Roos, B. O.; Ryde, U. Performance of Density Functionals for First Row Transition Metal Systems. J. Chem. Phys. 2007, 126, 014103. (53) Jensen, K. P. Bioinorganic Chemistry Modeled with the TPSSh Density Functional. Inorg. Chem. 2008, 47, 10357−10365. (54) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (55) Furche, F.; Perdew, J. P. The Performance of Semilocal and Hybrid Density Functionals in 3d Transition-Metal Chemistry. J. Chem. Phys. 2006, 124, 044103. (56) Zhao, Y.; Truhlar, D. G. Comparative assessment of density functional methods for 3d transition-metal chemistry. J. Chem. Phys. 2006, 124, 224105. (57) Matouzenko, G. S.; Borshch, S. A.; Schünemann, V.; Wolny, J. A. Ligand Strain and Conformations in a Family of Fe(II) Spin Crossover Hexadentate Complexes Involving the 2-Pyridylmethylamino Moiety: DFT Modelling. Phys. Chem. Chem. Phys. 2013, 15, 7411−7419. (58) Kepp, K. P.; Dasmeh, P. Effect of Distal Interactions on O2 Binding to Heme. J. Phys. Chem. B 2013, 117, 3755−3770. (59) Nakamura, M. Is Cyanide Really a Strong-Field Ligand? Angew. Chem., Int. Ed. 2009, 48, 2638−2640. (60) Evans, D. R.; Reed, C. A. Reversal of H2O and OH− Ligand Field Strength on the Magnetochemical Series Relative to the Spectrochemical Series. Novel 1-equiv Water Chemistry of Iron(III) Tetraphenylporphyrin Complexes. J. Am. Chem. Soc. 2000, 122, 4660− 4667. (61) Garcia, Y.; Gütlich, P. Thermal Spin Crossover in Mn(II), Mn(III), Cr(II) and Co(III) Coordination Compounds. Top. Curr. Chem. 2004, 234, 49−62. (62) Baumgärtel, N.; Flambard, A.; Köhler, F. H.; Lescouëzec, R. Paramagnetic Hexacyanometalates. The Diversity of Spin Distribution Studied by 13C and 15N MAS NMR Spectroscopy. Inorg. Chem. 2013, 52, 12634−12644. (63) Addison, A. W.; Butcher, R. J.; Homonnay, Z.; Pavlishchuk, V. V.; Prushan, M. J.; Thompson, L. K. The Hexakis(thiocyanato)ferrate(III) Ion: a Coordination Chemistry Classic Reveals an Interesting Geometry Pattern for the Thiocyanate Ligands. Eur. J. Inorg. Chem. 2005, 2404−2408. (64) Johnson, D. A.; Nelson, P. G. Ligand Field Stabilization Energies of the Hexaaqua 3+ Complexes of the First Transition Series. Inorg. Chem. 1999, 38, 4949−4955.
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Spin Propensities of Octahedral Complexes From Density Functional Theory Sara R. Mortensen and Kasper P. Kepp* DTU Chemistry, Technical University of Denmark, Building 206, Anker Engelunds Vej 1, 2800 Kongens Lyngby, Denmark S Supporting Information *
ABSTRACT: The fundamental balance between high- and low-spin states of transition metal systems depends on both the metal ion and the ligands surrounding it, as often visualized by the spectrochemical series. Most density functionals do not reproduce this balance, and real spin state propensities depend on orbital pairing and vibrational entropies absent in the spectrochemical series. Thus, we systematically computed the tendency toward high or low spin of “text-book” octahedral metal complexes versus ligand and metal type, using eight density functionals. Dispersion effects were generally 0 imply that low spin is favored.
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ligand bonds in high-spin states, which again arises from the occupation of eg orbitals. To include these corrections to the energies, the harmonic frequencies of both high-spin and lowspin states of the optimized complexes were computed numerically at the BP86-def2-SVP level, the same level as the geometry optimization, in Cosmo water, using the NumForce script, and the ZPEs of high-spin and low-spin states were subtracted and added to the high-spin low-spin energy differences. Furthermore, the high-spin states of mononuclear octahedral coordination complexes typically contain 10−30 kJ/mol more vibrational entropy than the low-spin states, due to the longer and weaker metal−ligand bonds of the high-spin state.12,13 The vibrational entropy obtained from the harmonic frequency state function provides a fairly accurate estimate for this purpose and was included in the analysis by using the freeh script of Turbomole, which can perform thermochemistry calculations.
RESULTS AND DISCUSSION
Zero-Point Energy Effects versus Ligand Type. Vibrational ZPEs contribute to the relative energies of states: the ZPE is usually smaller for high-spin states than for low-spin states, due to weaker metal−ligand bonds, causing the differential ZPE to favor high-spin states.13 Computed ZPEs and their effect on the high-spin low-spin energy difference can be seen in Figure 1A (numerical data are compiled in Supporting Information, Table S11A−F). The terms “strong field” and “weak field” strictly refer to the spectroscopically derived ligand-field splitting parameter Δo. Thus, in the following, we use the words weak and strong ligands in the sense of favoring high or low spin, not in terms of Δo. As we will see, there are important distinctions between the spectrochemical series and the true spin state preference resulting not only from orbital pairing but also from vibrational and thermodynamic effects, which should be relevant to thermal-equilibrium chemistry such as experimental and 4043
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bond strength, which relates to the differential LFSE. The identified grouping of Fe(II) ∼ Mn(II) ∼ Co(III) ∼ Fe(III) versus Co(II) ∼ Mn(III) thus follows the differential LFSE. Upon spin transition, two pairings occur for Fe(II), Mn(II), Co(III), and Fe(III), and one pairing occurs for Co(II) and Mn(III). The differential LFSE is 2 −2P for complexes containing the first four metal ions, and 1 − P for the Co(II) and Mn(III) complexes, where P is the pairing energy. The particularly large vibration effects of Fe(II) and Mn(II) result because the divalent metal ions exhibit larger bond changes during spin transition as the ionic component of the bond strength is smaller, and hence, the LFSE components are larger. Finally, Co(II) complexes behave irregularly due to the strong Jahn−Teller distortion of low-spin states. Together, these features explain our observed trends in metal- and liganddependent vibration effects that favor high-spin states. Spin Propensities of Octahedral Complexes: General Observations. Figure 2 shows the computed energy difference between high-spin and low-spin states for the 48 different octahedral complexes (six metal ions with any of eight ligands in homoleptic complexes), divided into ligand type. Parts A and B of Figure 2 display the Mn(II) and Mn(III) complexes, parts C and D of Figure 2 display the Fe(II) and Fe(III) complexes, and parts E and F of Figure 2 show the Co(II) and Co(III) complexes, respectively. Also, results with each of the eight used functionals are shown by different colors: the cold colors (blue/ green) correspond to nonhybrid functionals, whereas the warm colors (yellow, orange, red) correspond to hybrid functionals. A negative number implies that high spin is lower in energy than low spin, whereas a positive number implies that low spin is lower in energy. The energies in Figure 2 are without correction for dispersion, ZPE, and entropy, to isolate the effect of density functional clearly. Corrections for entropy and ZPE can be found in Figure 1. As seen from the data in Supporting Information (Tables S7−S10), dispersion changes the high-spin low-spin energy difference by maximally 5 kJ/mol except for the hexabromocobalt(II) complexes, where the hybrid functionals provide larger corrections (13−23 kJ/mol) relating to the large geometric changes associated with the strongly Jahn− Teller distorted low-spin state and the weak ligand. Thus, dispersion effects will not be discussed in further detail as they, absent this special case, do not change the spin propensities significantly for the simple complexes studied here (for other, sterically crowded coordination complexes, dispersion corrections in favor of low spin are sometimes substantial30). First, in all cases except Co(II), the two π-acceptor ligands cause substantially larger low-spin preference than other ligands, which is also seen for the spectrochemical series. For the Co(II) complexes, in particular the hybrid functionals have very little spread in preference, relating to the fact that the lowspin Co(II) complexes are strongly Jahn−Teller distorted (see below); this is to some, although weaker, extent also seen for Mn(III) complexes where the high-spin states are distorted. Second, CN− is consistently weaker than CO for the M(II) complexes, but stronger than CO for the M(III) complexes. This observation is new and is observed with all eight functionals and all six metal ions, so this is a statistically significant finding. The relative preferences of Mn(III)/CN− and M(II)/CO should be relevant to the discussion whether cyanide is a strong ligand or not.59 Electrostatic attraction is enhanced in the CN−/M(III) combination, thus strengthening the bonding to favor low spin further. In the M(II) systems,
theoretical studies of spin crossover, catalysis with multiple spin states, etc. From Figure 1A, ZPEs are found to be generally larger for the low-spin states and for the stronger ligands, and smaller for the high-spin states and weaker ligands, producing a wide range of differential ZPEs of 1−33 kJ/mol. This is understood from the stronger ligands and the low-spin states with less eg orbital set occupation giving stronger coordinative bonds with steeper potential energy surfaces and, hence, larger ZPEs. This means that ZPE corrections are particular important for complexes with moderate and strong ligands (e.g., N-ligands, CO, and CN−). In contrast, for weak halide ligands, the change in relative energies of high spin and low spin amounts to 1−4 kJ/ mol. Thus, ZPEs cannot be ignored when estimating the spin state of a coordination complex, in particular with stronger ligands. Vibrational Entropy Contributions to High-Spin Preferences. As discussed previously, vibrational entropy is typically 10−30 kJ/mol larger in high-spin than low-spin states of mononuclear complexes because occupation of the eg set of orbitals leads to elongated, weaker metal−ligand bonds.12,13 At thermal equilibrium relevant to most studies of coordination chemistry, neglecting this systematic entropy contribution leads to erroneous appraisal of higher HF exchange fractions, since HF exchange “mimics” the neglected entropy difference.13 Both direct comparison to experimental enthalpies of spin crossover and account of entropy in estimates of observed ground states accordingly found TPSSh with only 10% HF exchange to perform accurately in several benchmarks.13,17,30,53,58 The differential entropy is shown in Figure 1B (numerical data in Supporting Information, Table S12A−F). Consistent with thermochemical experiments on spin-crossover complexes,12 vibrational entropy is found to favor high-spin states by up to 40 kJ/mol. Importantly, again the vibrational entropy correction is larger for strong ligands subject to larger changes in vibrational state functions when undergoing high-spin lowspin transition. Thus, we find that both entropy and ZPE effects are substantial in particular for strong and moderate ligands, but even for a relatively weak ligand such as water, the combined effects favor high spin by 15−30 kJ/mol, depending on metal ion. Since the two effects both depend on the strength of the metal−ligand bond, they correlate strongly (R2 = 0.58, see Supporting Information, Figure S1). It is worth noting that the total energy correction depends on the full vibration state function and tends to be approximately half the size of the ZPE (numerical data in Supporting Information, Table S13A−F). These energy corrections counteract the vibrational entropy, as typically observed in interaction-bonding-related entropy−energy compensation, due to the softer, energywise weaker bonds of high-spin states having more vibrational entropy. Thus, the total vibrational energy and entropy corrections toward high spin will be typically about 5 (but in some cases up to 15) kJ/mol smaller than the sum of the contributions shown in Figure 1, parts A and B. We find that the computed ZPEs and the vibrational entropies also follow a trend in metal ion Fe(II) ∼ Mn(II) ∼ Co(III) ∼ Fe(III) > Co(II) ∼ Mn(III) that can be rationalized in the same way as the trend in ligand: the metal ions with the largest ligand field stabilization energies (LFSE, which depends on both the splitting and the orbital pairing) produce the strongest metal−ligand bonds. The vibrational corrections to the spin state propensity depend on the change in metal−ligand 4044
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Figure 3. Spin state propensity ordered according to metal ion for all eight studied ligands: (A) Br−; (B) Cl−; (C) H2O; (D) SCN−; (E) NCS−; (F) NH3; (G) CN−; (H) CO.
higher spin states.14,15 For the four nonhybrids, the nature of the exchange and correlation functionals is less important and almost has no effect on the relative order of ligands in the series, and in most cases, the spin gaps produced by TPSS, PBE, BP86, and BLYP are quite similar. The important exceptions are the π-acceptor ligands CO and CN−, which give rise to larger spin gap differences for these methods than commonly observed. This tendency has, as far as we know, not
back-bonding is more pronounced to the neutral CO ligand, as observed before,59 giving stronger interaction and favoring low spin relative to the hexacyano complexes. Density Functional Differences. From Figure 2, the nonhybrid functionals consistently provide higher energies, i.e., are shifted in favor of low-spin states, compared to hybrid functionals. This is well-known consequence of HF exchange producing polarized electron densities with energy bias toward 4045
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Similarly to the magnetochemical series of quantum-admixed porphyrins described by Reed and co-workers,32,60 the series in Figure 2 is a series of ground-state spin propensities for normal “text-book” octahedral complexes. The series by Reed and coworkers follows the order Br− < H2O < Cl− < NCS− < F− ≪ CO for ligands studied here, with the order H2O < OH− ∼ Cl− being distinct from the spectrochemical series, Cl− < OH− < H2O.60 The spectrochemical series, originally based on Co(III) systems,9 is commonly listed for the ligands studied here as28 Br− < SCN− < Cl− < H2O < NCS− < NH3 < CN− < CO. From Figure 2, the order Br− < Cl− < H2O is observed as in the spectrochemical series. The similarities are due to the fact that the spin pairing and orbital relaxation effects are similar for the same metal ions with the same nuclear charge and dq configurations, in particular for the weak ligands. In contrast, SCN− is generally a stronger ligand than H2O when orbital effects are included as in Figure 2. The difference is likely due to the spin pairing being less expensive in SCN− that contains some d-electron delocalization by back-bonding, making this ligand more prone to low spin than would be assumed simply from a spectrochemical series of Δo. Also, the spectrochemical series based on Co(III) complexes lists the order CN− < CO, but when all electronic effects are included, we find that this order is CN− < CO in the M(II) complexes (Figure 2, parts A, C, and E) but CO < CN− in the M(III) complexes (Figure 2, parts B, D, and F). Since this tendency is independent of dq configuration, it is mainly due to charge, i.e., stronger binding of the negatively charged ligand to the higher oxidation state. The work by Ichii et al. produced a remarkable order of CN− < NH3 < CO for Δo based on DFT calculations of M(III) complexes.28 They explained this as due to reduced back-bonding into CN− due to its negative charge bringing it even below NH3. Such an effect is not observed in any of our calculations and is possibly due to the spin pairing energy being smaller in the back-bonded (d-electron delocalized) CN− ligand compared to NH3. Effect of Metal Ions on Spin State Propensity. As discussed, there are major effects of the metal ion, both in terms of dq configuration and effective nuclear charge, on the spin state propensities. To visualize this more clearly, the effect of metal ion on spin state propensity is shown in Figure 3. All the studied functionals produce the most favorable lowspin states in the Co(III) complexes, consistent with their t2g6 low-spin states being particularly stable. For weaker ligands Br− (Figure 3A), Cl− (Figure 3B), H2O (Figure 3C), and SCN− (Figure 3D), Co(II) tends to follow Co(III) in low-spin preference, whereas for the stronger N-ligands (Figure 3, parts E and F) and CN− and CO (Figure 3, parts G and H), the Fe(II) complexes are the second-most low-spin prone, after Co(III). Thus, the general view that low-spin d6 complexes are particularly stable is evident from the data, but the data clearly show interplay between ligand and metal type. This insight arises from the systematic comparison and is consistent across the used methods. The next metal ion in line, Fe(III), favors low spin typically by ∼100 kJ/mol less than Fe(II) for the strong-ligand complexes. The two most important trends in low-spin propensity from Figure 3 are the underlying trend in increased effective nuclear charge, which follows the order Mn(II) < Fe(II) < Co(II) < Mn(III) < Fe(III) < Co(III), and the special effect of LFSE, notably of low-spin Fe(II), producing peaks for the moderate and strong ligands. The LFSE plays out mainly in the strong ligands contributing the most to the low-spin state propensity,
been observed before. For the strong binding and dense electron density situation, differences between functionals will be more pronounced than in the weak-binding (weak ligand) limit. Thus, care should be taken in testing distinct performances of these methods when studying π-acceptor ligands, whereas for other ligands, the nonhybrids perform similarly. After the nonhybrids, the TPSSh functional follows in the middle of the range (light orange), consistent with its 10% HF exchange. Interestingly, the B97-D functional behaves irregularly by changing relative position in the functional order depending on ligand strength. B97-D includes the strongly binding D2 dispersion correction and therefore tends to favor low-spin states more than the B97 functional itself. In case of Co(II) complexes, the B97-D energies are actually more in favor of low spin than TPSSh. In general, B97-D favors low spin relatively more with the strong ligands as these have tighter electronic structures favored by dispersion corrections, and this causes B97-D to resemble the 10% HF exchange hybrid TPSSh for the strong ligands CO and CN− but less in favor of low spin for the weak ligands and, hence, falls below other functionals in the weak-ligand limit. Dispersion corrections are quite liganddependent and grow with the tightness of the electronic structure,13 in particular for the strongly attractive D2 correction but less so for the more recent and moderate D3 corrections that are generally Fe(II) > Fe(III) ∼Co(III) > Mn(III) > Co(II) is seen, with the distorted low-spin Co(II) complexes again resulting in an anomaly, as these already have partially elongated metal−ligand bonds in the low-spin states. As seen from Supporting Information Table S15, the geometries of the low-spin Co(II) complexes beautifully capture the tetragonal distortion with large standard deviations in bond lengths resulting from two elongated axial bonds. There is also a clear trend in the change in bond length as a function of ligand: the strong ligands cause larger changes in
Figure 4. Differences (angstroms) between average metal−ligand bond lengths of high-spin and low-spin states for geometries optimized at the BP86/def2-SVP level in a Cosmo model of water.
bond length, whereas the weaker ligands and in particular the halide ligands exhibit minor changes in geometry and the order of metal ions no longer follows the order described above; for these weak negatively charged ligands, the role of electrostatics is more important than LFSE, and hence, M(III) complexes exhibit the largest changes in geometry upon spin state transition. Thus, the trends observed in spin state propensity in the previous sections, since these were rationalized from differential LFSE or changes in metal−ligand bond strengths, can be directly related to the changes in the bond lengths as well. Realism of Density Functionals. The systematic study performed here allows an assessment of the realism of the eight studied functionals, by comparison to the experimentally known spin states of some of the studied complexes. Due to the systematic approach required to identify fundamental trends, many of the studied complexes are not stable in the investigated form (e.g., they hydrolyze or oxidize under standard conditions in aqueous solution). However, many other of the complexes are well-known experimentally. One of the main strengths of computational chemistry is to establish physical trends across series where some species are experimentally inaccessible. To compare with experimental ground states, one should include the two major systematic effects that favor high spin, ZPE and vibrational entropy (relativistic corrections and solvent effects are generally smaller and less systematic for first row of the d-block).13 Thus, as seen from our results for these corrections, a functional performs well if the energies are 20−40 kJ/mol shifted toward low spin, but these corrections are quite ligand-dependent and also vary with metal ion, as discussed above. From Figure 2, all the functionals comfortably produce low spin for CO except for Mn(II) where the gap is small. Also, for the Co(III) complexes and Fe(II) complexes of strong ligands, the low spin is so favored that all functionals comfortably produce low spin, as exemplified by the well-known diamagnetic [Co(NH3)6]3+, and consistent with the general knowledge.61 However, already for CN−, the B3LYP and PBE0 functionals produce nearly isoenergetic high- and low-spin states for Mn(II) and Co(II) complexes, while the M(III) complexes and the t2g6 low-spin Fe(II), as discussed above, comfortably remain low spin also for these functionals. The average and standard deviation of entropy and ZPE corrections for the hexacyano complexes are 29 ± 7 and 19 ± 7 kJ/mol, both in favor of high spin. Thus, a conservative correction of 40 kJ/mol, more 4047
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CONCLUSIONS The need to understand spin state propensity and transitions between spin states of transition metal systems inspired us to study the spin state propensity of a range of text-book octahedral coordination complexes with combinations of simple ligands and the metal ions Mn(II), Fe(II), Co(II), Mn(III), Fe(III), and Co(III). Due to the systematic nature of such study, many complexes that are not experimentally accessible can be explored, and those complexes that are known serve as benchmarks for the eight studied density functionals. Dispersion effects are generally small but favor low-spin states and are more important to include for strong ligands that produce tighter electron densities. Zero-point energies favor high-spin states on average by 10 kJ/mol, but very dependent on ligand: for stronger ligands up to 33 kJ/mol correction in favor of high spin is sometimes seen, whereas for weak ligands it changes spin state propensity by only a few kilojoules per mole. Vibrational entropy also favors high-spin states on average by 16 kJ/mol, but up to 40 kJ/mol for strong ligands. Both entropy and zero-point energy effects follow a metal ion order of Fe(II) ∼ Mn(II) ≫ Co(III) ∼ Fe(III) > Co(II) ∼ Mn(III) resembling the differential ligand field stabilization energy. Thus, vibrational energy and entropy corrections cannot be ignored when studying transition metal chemistry in thermal equilibrium (up to ∼60 kJ/mol in total) except for weak ligands. We find systematic orders of low-spin propensity for both ligands and metal ions that resemble the spectrochemical series of the ligand-field splitting parameter as commonly stated in text books but notably differ in a number of situations, as discussed in the paper. Our derived “thermochemical spin series” includes electronic effects such as orbital pairing. The study reproduces the general notion of particular stability and back-bonding capability of t2g6 configurations and finds that CN− is consistently weaker than CO for M(II), but stronger than CO for the M(III). SCN− and NCS− change order in M(II) versus M(III) complexes, and interesting anomalies relating to Jahn−Teller distortion of low-spin Co(II) complexes and back-bonding in low-spin Fe(II) complexes are observed.
reasonably 50 kJ/mol, in favor of high spin should be added to the energies of the hexacyano complexes in Figure 2. The red [Co(CN)6]4− can be produced in the lab but oxidizes to the Co(III) complex and is likely low spin like the related Mn(II) complex.62 Since dispersion corrections are smaller than 5 kJ/ mol, it shows that B3LYP, even with dispersion, is not correct for the hexacyano complexes. This is consistent with benchmarks against experimental enthalpies of spin crossover of cobalt and iron complexes,17 where B3LYP also substantially overstabilized high spin when thermal corrections were accounted for. Complexes with known spin states and moderately strong ligands provide the best test case for the functionals, as failures will become more apparent when the spin balance is subtle. Nligands such as thiocyanate are often encountered in spincrossover systems due to their moderate ligand field strength. The salts of the [Fe(NCS)6]3− complex anion exist as high-spin ground states:63 the functionals spread around zero energy difference for this complex anion (Figure 2D). When entropy and ZPE corrections are added (40 kJ/mol, see Supporting Information) all methods correctly predict high spin. The hexammine complexes constitute a relevant benchmark, because of the changes in spin states seen experimentally, depending on metal ion. Notably, [Co(NH3)6]3+ is low spin, but divalent metal complexes such as [Co(NH3)6]2+ and [Fe(NH3)6]2+ are high spin.61 From Figure 2, parts C and E, the nonhybrids PBE, BP86, BLYP, and TPSS produce low-spin [Fe(NH3)6]2+ and [Co(NH3)6]2+ by margins of ∼10−40 kJ/ mol. The vibrational and entropy corrections are relatively small (∼7 and 8 kJ/mol) for the Co(II) complex due to Jahn− Teller distortion but sum to ∼41 kJ/mol for [Fe(NH3)6]2+ (see Supporting Information) due to its loss of two eg electrons upon spin transition and associated larger differences in metal− ligand bonds, as discussed above. This complex and the hexaaquairon(II) complex were previously studied with CASPT2, B3LYP, and PBE0,30 giving slightly smaller thermal and vibrational corrections and 10−30 kJ/mol more high-spin preference, but otherwise similar results to ours; the differences are attributable to our use of BP86-Cosmo-optimized structures due to previous benchmarks.52 Even with vibrational and thermal corrections, the nonhybrids do not produce the experimental high-spin states comfortably, viz., the consensus that some HF exchange is required to obtain a balanced description of electron correlation energy in such systems.14,15,30 The hexaaqua complexes also serve as an insightful benchmark: notably, it is known experimentally that [Co(H2O)6]3+ is low spin, whereas [Mn(H2O)6]3+, like most other hexaaqua complexes, is high spin.64 From Figure 2F, PBE0, B97-D, and B3LYP produce energies for [Co(H2O)6]3+ that are only 5−30 kJ/mol or less in favor of low spin, and when corrected for entropies and ZPEs, they produce high-spin states. In contrast, [Mn(H2O)6]3+ is predicted high spin with all methods as all other studied hexaaqua complexes except Co(III). Thus, aqua complexes indicate that 20% HF exchange is too much, consistent with the emerging consensus.13−15,17,30,52 Finally, all halide complexes are comfortably high spin except the Co(III) complexes which produce energies near zero for these ligands. When adding entropy and ZPE corrections, all Co(III) complexes are predicted to be low spin by all eight methods as seen experimentally (one known exception is highspin [CoF6]which was not studied here).
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ASSOCIATED CONTENT
S Supporting Information *
Tables S1−S6, electronic energies for computed high-spin and low-spin states of all complexes with all methods; Tables S7− S10, dispersion-corrected electronic energies; Table S11A−F, computed zero-point energies; Table S12A−F, computed entropy and difference in entropy between high-spin and low-spin states at 300 K; Table S13A−F, total thermal corrections to high-spin low-spin energy difference, from vibrational state functions; Table S14, optimized metal−ligand bond lengths of high-spin complexes; Table S15, optimized metal−ligand bond lengths of low-spin complexes; Figure S1, correlation between TΔS and ΔZPE corrections to spin state propensity. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +045 45 25 24 09. E-mail: [email protected]. Notes
The authors declare no competing financial interest. 4048
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ACKNOWLEDGMENTS This research has been supported by the Danish Center for Scientific Computing (Grant No. 2012-02-23).
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