Large-scale comparison of 3d and 4d transition metal complexes illuminates the reduced effect of exchange on second-row spin-state energetics

Abstract

Density functional theory (DFT) is widely used in transition-metal chemistry, yet essential properties such as spin-state energetics in transition-metal complexes (TMCs) are well known to be sensitive to the choice of the exchange–correlation functional. Increasing the amount of exchange in a functional typically shifts the preferred ground state in first-row TMCs from low-spin to high-spin by penalizing delocalization error, but the effect on properties of second-row complexes is less well known. We compare the exchange sensitivity of adiabatic spin-splitting energies in pairs of mononuclear 3d and 4d mid-row octahedral transition-metal complexes. We analyze hundreds of complexes assembled from four metals in two oxidation states with ten small monodentate ligands that span a wide range of field strengths expected to favor a variety of ground states. We observe consistently lower but proportional sensitivity to exchange fraction among 4d TMCs with respect to their isovalent 3d TMC counterparts, leading to the largest difference in sensitivities for the strongest field ligands. The combined effect of reduced exchange sensitivities and the greater low-spin bias of most 4d TMCs means that while over one-third of 3d TMCs change ground states over a modest variation (ca. 0.0–0.3) in exchange fraction, almost no 4d TMCs do. Differences in delocalization, as judged through changes in the metal–ligand bond lengths between spin states, do not explain the distinct behavior of 4d TMCs. Instead, evaluation of potential energy curves in 3d and 4d TMCs reveals that higher exchange sensitivities in 3d TMCs are likely due to the opposing effect of exchange on the low-spin and high-spin states, whereas the effect on both spin states is more comparable in 4d TMCs.

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1. Introduction

Approximate density functional theory (DFT) is widely used in studying the catalytic^1–6 and materials^7–13 properties of open-shell transition-metal complexes.¹⁴ The well-localized d or f electrons of open-shell transition-metal centers impart unique properties but also can lead to a significant number of low-energy spin and oxidation states that are challenging to describe on equal footing using approximate electronic structure methods. For DFT in particular, presently available exchange–correlation (xc) approximations suffer from one- and many-electron self-interaction errors,^15–19 commonly referred to as delocalization error^20–22 (DE).

The high earth abundance and prevalence of 3d transition metals in enzymatic systems has motivated the widespread study^16,23–41 of electronic structure method accuracy in first-row transition-metal complexes. For 3d transition-metal chemistry, DE can lead to pronounced errors in calculated bond dissociation energies,^16,23–27 barrier heights,^28,29 and properties of the density^30–33 within a given spin state as well as the relative energetic ordering of spin states.^29,34–41 While in some cases, it is possible to use higher accuracy methods such as correlated wavefunction theory, such methods have their own challenges in open-shell transition-metal complex (TMC) property prediction accuracy^42–45 and remain cost-prohibitive for large-scale discovery of new TMCs.^46–51

Although less earth abundant than 3d metals, 4d transition metals often form catalysts with superior catalytic activity and turnover number (e.g., Ru^52–54 for water oxidation and Mo⁵⁵ for hydrogen evolution) and are excellent photosensitizers.⁵⁶ Despite this importance of 4d transition metals, comparatively less is known about the relative accuracy of electronic structure methods for these TMCs.^57–59 While we can anticipate that there should be some transferable observations between 3d and 4d metals, a more detailed understanding of the relationship between 3d and 4d TMC sensitivity to method choice is needed.

Broadly, the need to balance cost and accuracy has motivated a number of strategies aimed at efficiently eliminating DE in approximate DFT by recovering the derivative discontinuity⁶⁰ lacking from pure (i.e., generalized gradient approximation or GGA) functionals.^15,61–66 Approaches have included self-interaction^67–69 and closely related DFT+U^70–73 corrections, tailoring higher order terms in the xc functional,^{37,40,45,74–85} or incorporating an admixture^{32,33,86–102} of Hartree–Fock (HF) exchange globally or with range-separation in a GGA. All such strategies generally behave similarly in TMCs by decreasing covalency³³ or dative bonding^32,103–105 and localizing density away from the metal.^30,37 Functional tuning for either 3d or 4d transition-metal chemistry is challenged¹⁰⁶ both by the limited availability of reference data and the highly system- and property-specific nature of optimal functional choice,^{29,35–37,107–112} although physically motivated tuning approaches have been developed for select cases.^113–115 Still, tuning the global amount of HF exchange (i.e., a_HF) in a GGA hybrid remains one of the most widely used approaches to improving approximate DFT errors.

For 3d midrow, octahedral TMCs, it is well known that the ground state is highly sensitive to the fraction of HF exchange due to the near-degeneracy of multiple spin states in these complexes^{36,82,116–119} as well as the close relationship between spin-state stabilization and DEs in approximate DFT.^30,111 Pure, semi-local GGA functionals^120,121 consistently stabilize overly-delocalized,^18,30 strongly covalent states,³³ tending to favor the increased bonding in low-spin (LS) over high-spin (HS) states.^{29,39,70,77,122–125} This manifestation of DE leads GGAs to predict incorrect ground states as well as relative energies^14,124,125 between spin states (e.g., adiabatic HS to LS spin splitting, ΔE_H–L). Hybrid functionals counteract^{35–37,118,126–129} the bias for LS states, but the appropriate fraction of HF exchange is strongly system dependent.^{35–37,108–111,130} For 3d TMCs, conflicting proposals of low (as little as a_HF = 0.0)^{35,117,131,132} and high (a_HF = 0.4–0.5)^{31,36,40,133,134} fractions have been recommended, but 4d TMCs do not have similar heuristics.

In this work we apply our pragmatic approach of understanding the linearized^{29–31,37,39} exchange sensitivity of spin-splitting energies^{35–37,108,117,131,132} simultaneously to 3d and 4d TMC spin-state energetics. In 3d TMCs, this spin-state ordering exchange sensitivity is correlated to the ligand field strength of coordinating ligands,^37,108 with bare ions exhibiting reduced sensitivities³⁹ in comparison to strong-field ligands. For common ligands in mid-row, 3d TMCs, this exchange sensitivity can be significant, leading to a change in ΔE_H–L on the order of 10–20 kcal mol⁻¹ for a change in HF exchange from a_HF = 0.15 (e.g. in B3LYP*¹³²) to a_HF = 0.25 (e.g., in PBE0¹³⁵). While the form of the pure GGA xc being tuned (e.g., BLYP vs. PBE) can influence ΔE_H–L, the HF exchange sensitivity has been shown to be invariant to GGA choice in 3d TMCs.^37,39,40 These strong relationships between chemical structure and method sensitivity in 3d TMCs have enabled the development of machine learning models to predict exchange sensitivity and exchange-dependent properties, enlarging understanding of how changes in functional definition can influence large-scale discovery efforts.^136–139

Although these relationships have been established and fruitfully applied for 3d TMCs, the sensitivity of second-row (i.e., 4d) TMCs to HF exchange is not well known. In this work, we carry out a large-scale study of hundreds of mononuclear octahedral TMCs to understand broad relationships among first- and second-row spin-state ordering and exchange sensitivity. In Section 2, we describe how we construct a data set in which both the 3d and an isovalent 4d metal complex's properties and sensitivity to exchange are known for mid-row transition metals in low-, intermediate-, and high-spin states. In Section 3, we discuss our results on quantifying distinct behavior in ground state preference and sensitivity to HF exchange fraction for the second-row complexes. We identify sources for this difference by noting distinct effects of exchange on each spin state's potential energy surface in 3d and 4d TMCs. Finally, in Section 4, we provide our conclusions.

2. Computational details

a. Data set construction

We studied the effect of HF exchange on the spin-state ordering of octahedral TMCs with a single mid-row transition-metal center. We compared properties of TMCs comprised of first-row (i.e., 3d valence) Cr, Mn, Fe, and Co to second-row (i.e., 4d valence) Mo, Tc, Ru, and Rh (Fig. 1). In all cases, we calculated properties of metal centers in formal M(II) or M(III) oxidation states to ensure differences in spin state correspond to differences in d orbital occupations (Fig. 1). The TMCs were evaluated in up to three spin states: low-spin (LS), intermediate-spin (IS), and high-spin (HS), where we defined the IS and HS states as those that differ from the LS state (i.e., with ls unpaired electrons) by two more (i.e., ls + 2) or four more (i.e., ls + 4) unpaired electrons, respectively. We then computed the gas phase, adiabatic spin-splitting energy: between the LS and HS states, ΔE_H–L, as well as between the IS state and either the HS or the LS state (i.e., ΔE_H–I and ΔE_I–L). The nominally d³ Cr(III)/Mo(III) and d⁷ Co(II)/Rh(II) were evaluated only in LS doublet and IS quartet spin multiplicities (i.e., also the highest accessible spin state), and only ΔE_I–L was computed (Fig. 1). The d⁵ metals (i.e., Mn(II)/Tc(II) or Fe(III)/Ru(III)) were studied in LS doublet, IS quartet, and HS sextet states (Fig. 1). Analogously, d⁴ (i.e., Mn(III)/Tc(III) or Cr(II)/Mo(II)) and d⁶ (i.e., Fe(II)/Ru(II) or Co(III)/Rh(III)) metals were calculated in LS singlet, IS triplet, and HS quintet states (Fig. 1).

Fig. 1 (top) Qualitative diagrams of electron configurations in low-spin (LS), intermediate-spin (IS), and high-spin (HS) states for the mononuclear octahedral transition-metal complexes studied in this work (schematically shown at left). For both d³/d⁷ and d⁴/d⁶ M(II) or M(III) complexes, the additional electrons for the later transition metal are shown in red, and the electrons that apply to both states are shown in blue. The d³ or d⁷ complexes do not have a defined HS state. (bottom) The ten main monodentate ligands studied in this work ordered by their increasing ligand field strength, which tunes the octahedral field splitting (schematically shown at left). Atoms in the ball-and-stick representation are colored as follows: H in white, C in gray, N in blue, O in red, F in light blue, P in orange, S in yellow, and Cl in green.

We calculated properties of complexes formed from combinations of ten small, monodentate ligands that spanned ligand field strengths and coordinating element identities. Negatively charged halides (Cl⁻ and F⁻) ions are known¹⁴⁰ to have among the weakest field strengths, while several others (i.e., phosphine, carbonyl, and cyanide) have among the highest field strengths (Fig. 1). Intermediate behavior is expected of the remaining (i.e., water, ammonia, hydrogen sulfide, acetonitrile, and methyl isocyanide) ligands (Fig. 1). In addition to homoleptic complexes, heteroleptic complexes were formed from up to two ligands (i.e., L₁ and L₂). Both M(L₁)₄(L₂)₂ TMCs with the two minority L₂ ligands either trans (i.e., aligned 180° in the TMC) or cis (i.e., 90° in the TMC) were studied along with M(L₁)₅(L₂) TMCs.

b. Electronic structure calculations

Calculations on mononuclear, octahedral TMCs studied in this work followed an established protocol.^141,142 All initial structures were generated using molSimplify,^48,143,144 which employs OpenBabel^145,146 as a backend for ligand structure generation. These calculations were automated and checked for fidelity with molSimplify automatic design (mAD).^119,141 All TMCs were geometry optimized with DFT using a development version of TeraChem.^147,148 For the geometry optimizations, the standard B3LYP^149–151 global hybrid functional was employed along with modified forms in which the Hartree–Fock exchange fraction (a_HF) was varied from its default value of 0.20 to as low as a_HF = 0.00 (i.e., a pure BLYP GGA) or as high as a_HF = 0.30 in increments of 0.05 while holding the LDA/GGA exchange ratio fixed.^37,136 As in 3d TMCs,^37,39,40 sensitivities from a_HF variations using PBE as the GGA xc functional are comparable on a representative 4d TMC (ESI,† Table S1). All calculations employed the LANL2DZ¹⁵² effective core potential for transition metals and the 6-31G* basis for all other atoms. Only singlet calculations were carried out in a spin-restricted formalism, with all other spin multiplicities carried out unrestricted. Level shifting¹⁵³ was employed to aid self-consistent field (SCF) convergence with the majority spin and minority spin virtual orbitals each shifted by 0.25 Ha. The default SCF convergence threshold of 3 × 10⁻⁵ for the largest component of the DIIS vector was employed. Geometry optimizations were carried out in translation rotation internal coordinates¹⁵⁴ using the L-BFGS algorithm to default tolerances for the gradient of 4.5 × 10⁻⁴ Hartree Bohr⁻¹ and energy difference between steps of 10⁻⁶ Hartree. Geometric properties from a representative case optimized with a larger triple-ζ basis set and the resulting exchange sensitivities preserves trends (ESI,† Table S1).

For the mAD calculation workflow, calculations were run for 24-hour increments and resubmitted for up to five additional runs. At each resubmission, mAD applies loose geometric criteria¹⁴¹ and abandons any calculations that fail these checks (ESI,† Table S2). In this workflow, the B3LYP (a_HF = 0.20) geometry optimization was carried out first. If the B3LYP geometry optimization converged, we used the converged structure and wavefunction to initialize geometry at the adjacent increased (i.e., 0.25) and decreased (i.e., 0.15) a_HF values, as in prior work.¹³⁶ If these calculations converged, their structures and wavefunctions were then used for the next adjacent (e.g., increased to 0.30 or decreased to 0.10) a_HF value geometry optimizations. However, if an optimization failed to converge, the next a_HF value was not attempted.

For all converged calculations, automated data fidelity checks were employed based on refinements of prior geometric¹⁴¹ and electronic structure criteria^141,142,155 (ESI,† Table S2). Specifically, complexes were retained if their structure was deemed to be intact based on tighter geometric criteria than were employed during the optimization (ESI,† Table S3). The electronic structure criteria required that the deviation of the Ŝ² expectation value from its anticipated value (i.e., S(S + 1)) was below 1 and the Mulliken metal atomic spin density was within 1 μ_B of the total spin of the molecule (ESI,† Table S3).

Linearized exchange sensitivities, S, were obtained from linear fits of the dependence of the relevant property (e.g., ΔE_H–L) on a_HF. The resulting sensitivity (e.g., S(ΔE_H–L)) is reported as the change in property over the range from a_HF = 0.0 to 1.0, which we refer to as HFX as in prior work.^29,37,136 As long as a single, qualitatively consistent electronic state has been converged over all points, this linear approximation is known to be good for a range of properties, both energetic (e.g., spin splitting^{29,35–37,131,136} and reaction energies^29,156) and electronic^30,31,157 in nature. To ensure that linearized exchange sensitivities could be quantitatively obtained from collected data points, we applied a series of constraints and filtering steps; this procedure and its results are detailed in ESI,† Table S4. All raw data, resulting R² values, computed sensitivities, and reasons for eliminating points or sensitivities are provided in the ESI.†

Potential energy curves (PECs) of 3d Fe(II) and 4d Ru(II) in LS singlet and HS quintet states were obtained for homoleptic complexes of He atoms and CO ligands also using TeraChem and B3LYP/LACVP* with modified a_HF values. These PECs were obtained by rigidly shifting all six He atom (CO) ligands from distances as short as 1.70 Å (1.80 Å) to as long as 2.90 Å (2.80 Å) in 0.01 Å (0.02 Å) increments, with calculations at longer bond lengths starting from the converged wavefunction at the shorter bond lengths. The He atom was selected following recent work⁴³ that showed it is a representative weak field ligand, and a single He atom is simpler to translate during PEC evaluation than a typical non-linear weak-field ligand (i.e., H₂O or NH₃). The CO bond length was fixed to 1.125 Å, its value in relaxed TMCs. These calculations were repeated for a_HF fractions increasing from 0.0 to 0.45 in increments of 0.05, with wavefunctions always initializing from shorter bond lengths and lower a_HF values. All total energies and computed sensitivities are provided in the ESI.†

3. Results and discussion

a. Ligand field sensitivity trends in homoleptic TMCs

To understand the extent to which previous observations^{35–38,40,126,131} in first-row transition metals are likely to be generalizable to the second row, we first compared spin-splitting properties for a subset of homoleptic complexes. Depending on the field strength of the ligand, 3d Fe(II) TMCs are known to favor distinct ground states.¹⁴⁰ For example, d⁶ Fe(II) TMCs with intermediate-field nitrogen-coordinating ligands commonly exhibit spin crossover (SCO)¹⁵⁸ phenomena by changing from an LS singlet to HS quintet state with increasing temperature. The ΔE_H–L for these complexes is sensitive to HF exchange fraction, becoming more negative with increasing a_HF. The combination of this physical phenomenon with sensitivity to DFT functional parameters means that different a_HF values can predict the same complex to have either an LS or HS ground state.^{35–38,40,126,131}

Indeed, for the complexes studied in this work, homoleptic complexes of Fe(II) with acetonitrile or cyanide have near-degenerate LS and HS states when evaluated with B3LYP and thus are predicted to be LS with lower HF exchange (i.e., a_HF = 0.1) but HS with higher fractions (i.e., a_HF = 0.3, Fig. 2). Consistent with prior observations,^36,37,136 the magnitude of this typically negative exchange sensitivity, S, generally increases with field strength. This means that a hexa-carbonyl Fe(II) complex that is strongly LS with B3LYP can become HS with higher exchange fractions (ca. a_HF = 0.4), whereas weak-field hexa-aqua Fe(II) is uniformly HS for all a_HF values (Fig. 2).

Fig. 2 The HS–LS adiabatic spin splitting (ΔE_H–L, in kcal mol⁻¹) for isovalent 3d (in green) and 4d (in blue) homoleptic TMCs: Tc(III) and Mn(III) (top) or Fe(II) and Ru(II) (bottom). Ligands have been ordered on the x-axis roughly according to ligand field strength with the metal-coordinating atom listed first in the chemical name. The points (circles for 3d, squares for 4d) correspond to values at a_HF = 0.2, the inner translucent shaded regions correspond to the a_HF = 0.1–0.3 range, and the outer translucent shaded regions correspond to a_HF = 0.0–0.4. A zero axis is shown to indicate where spin state ordering changes between HS and LS states. A dashed line for Tc(III) indicates that no data is available for NH₃ and CN⁻ complexes.

In comparison to their first-row counterparts, the isovalent 4d Ru(II) TMCs are considerably low-spin shifted, with almost all Ru(II) complexes strongly favoring LS states over HS states except for the weakest field (i.e., hexa-fluoride) cases (Fig. 2). The Ru(II) homoleptic complexes also exhibit uniformly reduced exchange sensitivities (Fig. 2). Thus, complexes with modest B3LYP ΔE_H–L values (e.g., hexa-aqua Ru(II), ca. 25 kcal mol⁻¹) are not predicted to change their ground state with varied a_HF in contrast to Fe(II) homoleptic complexes (Fig. 2). Despite these differences, qualitative trends with ligand field strength appear preserved from Fe(II) to Ru(II), with increasing exchange sensitivity apparent with increasing ligand field (Fig. 2).

Given the strong LS shift of 4d Ru(II) TMCs with respect to their 3d Fe(II) TMC counterparts, we identified d⁴ Tc(III) as a possible metal/oxidation state where TMCs could be expected to favor HS or LS states depending on field strength (Fig. 2). The 4d Tc(III) singlet LS and quintet HS states are in fact close in energy with B3LYP for weak-field (i.e., hexa-aqua) and intermediate-field (i.e., hexa-acetonitrile) complexes (Fig. 2). The isovalent Mn(III) complexes are comparatively HS shifted, with only the homoleptic methyl-isocyanide with a pure BLYP (i.e., a_HF = 0.0) GGA coming close to spanning the LS–HS transition (Fig. 2). The exchange sensitivity of all Mn(III) homoleptic TMCs is reduced with respect to the Fe(II) TMCs but follows a consistent trend of increasing with increasing ligand field strength (Fig. 2). Consistent with 3d/4d observations from Fe(II)/Ru(II), Tc(III) homoleptic TMCs have reduced exchange sensitivity compared to isovalent Mn(III) (Fig. 2). This means that even though a number of 4d Tc(III) TMCs have modest B3LYP ΔE_H–L (<25 kcal mol⁻¹) values, the only ground state assignment change for Tc(III) in this series would be for the B3LYP HS hexa-aqua complex to become LS when a_HF is 0.1 or less (Fig. 2). Overall, reduced exchange sensitivities in these 4d TMCs (i.e., whether with Ru(II) or Tc(III)) would require the B3LYP ΔE_H–L to be within ±10 kcal mol⁻¹ as opposed to ±25 kcal mol⁻¹ for comparable 3d TMCs for the 4d TMC spin-state preference to change with HF exchange.

Both d⁶ Fe(II)/Ru(II) and d⁴ Mn(III)/Tc(III) TMCs have an IS triplet state. We thus also calculated the IS state to obtain ΔE_H–I and ΔE_I–L as well as their sensitivities (i.e., S(ΔE_H–I) and S(ΔE_I–L)) for select homoleptic TMCs. We identified hexa-carbonyl TMCs as representative strong-field complexes and hexa-aqua as representative weak-field complexes (Fig. 3). All 4d TMCs favor IS states over HS states with B3LYP (i.e., ΔE_H–I > 0) unlike the isovalent 3d TMCs (Fig. 3). Spin-state orderings change with a_HF due to high S for several of the 3d TMCs (i.e., HS–IS/HS–LS Fe(II)(CO)₆, IS–LS Fe(II)(H₂O)₆, and HS–IS Mn(III)(CO)₆) but only one of the 4d TMCs (i.e., HS–LS Tc(III)(H₂O)₆) in part due to the reduced exchange sensitivity of 4d TMCs (Fig. 3).

Fig. 3 Comparisons of adiabatic spin splitting (ΔE, in kcal mol⁻¹) for HS–LS (green lines and shading), HS–IS (blue lines and shading) and IS–LS (red lines and shading) for pairs of homoleptic TMCs grouped first by the isovalent 3d and 4d metals (i.e., Fe(II)/Ru(II) vs. Mn(III)/Tc(III)) and then by ligand (i.e., CO vs. H₂O), as indicated on the x-axis. The solid lines correspond to values at a_HF = 0.2, the inner translucent shaded regions correspond to the a_HF = 0.1–0.3 range, and the outer translucent shaded regions correspond to a_HF = 0.0–0.4. A zero axis is shown to indicate where ordering changes for any pair of states. The Ru(II)(H₂O)₆ IS state was eliminated during filtering steps, and so its HS–IS or IS–LS data is unavailable.

For all spin-splitting energies, the hexa-aqua exchange sensitivities are lower than those for hexa-carbonyl in both 3d and 4d TMCs (Fig. 3). As can be expected,^39,40S(ΔE_H–I) and S(ΔE_I–L) are reduced with respect to S(ΔE_H–L), with S(ΔE_I–L) consistently the smallest of the three (Fig. 3). The trend holds across the over 150 pairs of 3d/4d heteroleptic or homoleptic TMCs for which all three spin-state energies have been evaluated. On average, S(ΔE_I–L) and S(ΔE_H–I) values are 38% and 62% of S(ΔE_H–L), respectively (ESI,† Table S5). That is, when the 4d TMC S value is reduced with respect to its 3d TMC counterpart, this reduction is proportional across both spin states and ligand chemistry.

b. Global comparison of 3d and 4d TMCs

We next expanded our comparison of HS–LS energies and their sensitivities to include heteroleptic complexes for an overall set of more than 200 d⁴–d⁶ TMC pairs. Over all of these 3d/4d TMC pairs, the 4d S(ΔE_H–L) was consistently reduced with respect to that for 3d TMCs (Fig. 4). The largest (i.e., most negative) 3d S(ΔE_H–L) values are larger than those of 4d TMCs by around 50 kcal mol⁻¹ HFX⁻¹ (i.e., around 40% larger, ESI,† Table S6). For each 3d metal/oxidation state, average exchange sensitivities also vary, i.e., highest for Mn(II) or Fe(II) and lowest for Cr(II), with similar trends for the isovalent 4d metal/oxidation states (ESI,† Table S6). The overall reduced sensitivities of 4d TMCs become more apparent the larger the 3d TMC S values are (Fig. 4). Based on prior analysis^36,37,136 and our observations on homoleptics, this should mean that strong-field 3d TMCs are much more sensitive to HF exchange than 4d TMCs while weak-field 3d and 4d TMCs complexes have more similar exchange sensitivity (Fig. 4). Indeed, the sensitivities of the prototypical strong-field hexa-carbonyl and weak-field hexa-aqua homoleptics largely reside at extremes of the overall relationship between 3d and 4d TMC S values (Fig. 4).

Fig. 4 Sensitivities, S, of 3d vs. 4d ligand-matched TMCs (in kcal mol⁻¹ HFX⁻¹) for ΔE_H–L (left), ΔE_I–L (top, right), and ΔE_H–I (bottom, right). Translucent symbols are colored by element: Cr in gray, Mn in orange, Fe in red, and Co in blue. The formal electron configuration of each point is indicated by the symbol shape: d³ right-pointing triangles, d⁴ up-pointing triangles, d⁵ circles, d⁶ squares, and d⁷ diamonds. The d³ and d⁷ data is only available for ΔE_I–L. All hexa-aqua complexes are filled solid and outlined in green, and all hexa-carbonyl complexes are filled solid and outlined in black. The zero axis for both sensitivities is shown as a dashed black line, and a black dotted parity line is also shown.

There are a few noteworthy exceptions where the 4d TMC S exceeds that of its 3d TMC counterpart, namely: (i) in weak-field, hexafluoro complexes of Mn(III)/Tc(III) (S(3d) ca. −35 vs. S(4d) −50 to −70 kcal mol⁻¹ HFX⁻¹) and (ii) in stronger fields (e.g., Mn(II)(NCCH₃)₄(CNCH₃)₂) where the 3d TMC sensitivity is typical but the Tc(II) value is among the largest evaluated for 4d TMCs (ESI,† Table S7). The strong net negative charge on the halide complexes also results in other types of outliers: Ru(III) hexafluoride and a Cl⁻-containing Mo(II) complex are the only two 4d TMC cases for which the S(ΔE_H–L) is weakly (ca. 20–45 kcal mol⁻¹ HFX⁻¹) positive (ESI,† Table S8). In both 4d TMCs, the corresponding 3d TMC still has a negative, albeit small S value (ESI,† Table S8).

Trends for the 2-electron ΔE_H–I are largely consistent with those for the 4-electron ΔE_H–L. Although overall S(ΔE_H–I) values for both 3d and 4d TMCs are lower than S(ΔE_H–L), 3d TMC sensitivities nearly always exceed their 4d TMC values (Fig. 4 and ESI,† Table S6). The 3d/4d metal/oxidation states with the highest average S(ΔE_H–L) (i.e., Mn(II)/Tc(II) and Fe(II)/Ru(II)) also have the highest S(ΔE_H–I) values (ESI,† Table S6). On average, best-fit lines relating S(4d) vs. S(3d) for ΔE_H–L and ΔE_H–I have comparable ratios of change in S(4d) per change in S(3d) of 0.55 and 0.65, respectively (ESI,† Table S6). As in the case of ΔE_H–L, a small number of 4d TMC ΔE_H–I sensitivities exceed their 3d TMC counterparts and typically correspond to Mn(II) complexes with strong-field equatorial ligands (CO or NCCH₃) expected to have strong Jahn–Teller distortion (ESI,† Table S7).

We also analyzed 2-electron ΔE_I–L pair trends, which additionally include d³ Cr(III)/Mo(III) and d⁷ Co(II)/Rh(II) TMCs in their doublet LS and quartet IS states (see Fig. 1). As in the other two cases of spin-splitting energetics, 4d TMCs exhibit reduced exchange sensitivity with respect to 3d TMCs (Fig. 4 and ESI,† Table S6). For the d³/d⁷ TMCs for which only ΔE_I–L is defined, S(4d) : S(3d) ratios from best-fit lines of 0.42 for d³ and 0.64 for d⁷ are consistent with the ratios observed in d⁴–d⁶ TMCs (ESI,† Table S6). Several more 4d TMC outliers that are close to parity are observed, typically Mn(III)/Tc(III) complexes with equatorial chlorides or Cr(II)/Mo(II) complexes with equatorial ammonia ligands (Fig. 4 and ESI,† Table S7). Far more positive 4d TMC sensitivities are observed for several Rh(III) complexes with a range of weak- and strong-field ligands (ESI,† Table S9). A handful of 3d TMCs (e.g., in Fe(II) complexes with CN⁻ ligands) also have positive sensitivities, but the equivalent, isovalent Ru(II) TMCs have weakly negative sensitivities (ESI,† Table S9). These observations on ΔE_I–L are distinct from ΔE_H–L and ΔE_H–I where only one cis Mo(II)(NH₃)₄(Cl⁻)₂ complex had positive 4d TMC sensitivities for both quantities (ESI,† Table S8).

c. Analysis of a heteroleptic complex series

Given the diversity of TMC chemistry in these larger data sets, we simplified our analysis to understand the extent to which trends in ΔE and S(ΔE) were additive in a narrower subset of representative 3d and 4d TMC complexes. To assess the additive nature of ligand contributions, we chose a subset of the data consisting of homoleptic complexes of a weak-field ligand (H₂O), the homoleptic complexes of a strong-field ligand (CO), and the heteroleptic complexes with both ligands present. In addition to their differing field strengths, H₂O and CO were chosen for their neutral charge, eliminating the need to compare TMCs of differing net charge. For a fixed metal, oxidation state, and principal quantum number, the full set can contain up to six mixed H₂O/CO complexes between the homoleptic hexa-aqua and hexa-carbonyl limits. This includes two cases where a single ligand of the minority type is in the complex (i.e., 5 + 1: (CO)₅(H₂O) or (H₂O)₅(CO)) as well as four cases where two ligands of the alternate type are included in either a cis (i.e., equatorial/axial adjacent) or trans (i.e., axial) conformation (e.g., (CO)₄(H₂O)₂ or (H₂O)₄(CO)₂). Although the exchange sensitivity of homoleptic 3d TMCs has been widely studied,^{35–37,118,126–129} the expected exchange sensitivity of heteroleptic TMCs is not well established for either 3d or 4d TMCs.

There is a smooth, nearly monotonic increase in ΔE_H–L with increasing number of strong-field CO ligands for Fe(II) or Ru(II) complexes (Fig. 5). For the Fe(II) TMCs, B3LYP LS states become stabilized over HS states for four or more CO ligands, whereas all Ru(II) complexes are uniformly LS (Fig. 5). The trend of increasing ΔE_H–L with increasing CO number is largely consistent between Ru(II) and Fe(II), but B3LYP ΔE_H–L for Ru(II) is more sensitive to the number of CO ligands (Fig. 5). Interestingly, the only exception to the monotonic trend is the trans (H₂O)₄(CO)₂ isomer, with an ΔE_H–L closer to that for the mono-carbonyl complexes (Fig. 5 and ESI,† Tables S10, S11). The cis (H₂O)₂(CO)₄ isomer has been filtered from the dataset, prohibiting broader conclusions about cis/trans isomers (see ESI†). For the Fe(II) TMCs, all mixed H₂O/CO complexes span the LS/HS transition between a_HF = 0.0 and 0.4 (Fig. 5). While Fe(II) TMC S values are significantly larger than for Ru(II) TMCs, both 3d and 4d complexes appear to have a monotonic increase in sensitivity with the addition of more strong-field ligands (Fig. 5 and ESI,† Table S12).

Fig. 5 Properties of complexes formed by mixing H₂O and CO ligands. Variation of 4-electron ΔE_H–L (in kcal mol⁻¹) with increasing number of CO ligands for isovalent Mn(III)/Tc(III) or Fe(II)/Ru(II). The points (circles for 3d, squares for 4d) correspond to values at a_HF = 0.2, the inner translucent shaded regions correspond to the a_HF = 0.1–0.3 range, and the outer translucent shaded regions correspond to a_HF = 0.0–0.4. A zero axis is shown to indicate where spin-state ordering changes. A spline has been fit through the a_HF = 0.2 values excluding the open, lighter colored symbol, which corresponds to an outlier trans configuration with two CO ligands. No cis data is available for the case with four CO ligands, and no calculations were attempted for 3 CO ligands, as indicated by vertical translucent bar.

For the series of d⁴ Mn(III)/Tc(III) TMCs, reduced overall ligand field dependence of both ΔE_H–L and S is observed with respect to Fe(II)/Ru(II) TMCs (Fig. 5 and ESI,† Tables S10, S12). The 3d Mn(III) TMCs are uniformly HS-favored regardless of a_HF value (Fig. 5). Both ΔE_H–L and S values appear to reach a maximum for Mn(III) complexes once all four equatorial ligands are CO with no effect of added CO axial ligands (Fig. 5). This likely occurs both because of the reduced ligand field sensitivity of Mn(III) and non-monotonic geometric changes for the most CO-saturated Mn(III) complexes (see ESI† data). Overall trends are consistent between Tc(III) and Mn(III), although the 4d TMCs are again marked by reduced S values with respect to the isovalent 3d TMCs (Fig. 5). Given the reduced sensitivities of 4d TMCs, only the homoleptic aqua Tc(III) complex spans the LS-to-HS transition, whereas even the modest ΔE_H–L (<15 kcal mol⁻¹) in the mono-carbonyl Tc(III) complex remains LS-favored over all a_HF values (Fig. 5). Across an expanded series of TMCs with more metals (i.e., d⁵ Fe(III)/Ru(III) and Mn(II)/Tc(II)), addition of strong-field carbonyl ligands has a largely monotonic, linearly increasing effect on S values starting from the lower extreme of homoleptic hexa-aqua TMCs and moving toward the upper extreme of hexa-carbonyl TMCs (Fig. 6 and ESI,† Table S12). The most significant deviations in S between 3d/4d TMC pairs occurs for the Fe(II)/Ru(II) cases (i.e., up to a 60 kcal mol⁻¹ HFX⁻¹ difference) but remains substantial for most metals, excluding only the low-S Mn(III)/Tc(III) pairs (Fig. 6). Over this broad set, we confirm the cis (H₂O)₄(CO)₂ complexes have significantly higher sensitivities for both 3d and 4d metals than their trans counterparts, with the exception of Mn(III)/Tc(III) (Fig. 6).

Fig. 6 Properties of complexes formed by mixing H₂O and CO ligands. S(3d) vs. S(4d) for ΔE_H–L (in kcal mol⁻¹ HFX⁻¹) of Mn/Tc and Fe/Ru TMCs with CO or H₂O ligands in both oxidation states, colored by element (Mn/Tc in orange or Fe/Ru in red) and with symbols corresponding to formal electron configuration (d⁴ in up triangles, d⁵ in circles, and d⁶ in squares). All hexa-aqua complexes are outlined in green, all hexa-carbonyl complexes are outlined in black, and the remaining symbols are outlined in dark gray. A gray arrow shows the path from each cis to trans (H₂O)₄(CO)₂ sensitivity. A dotted parity line is shown for reference.

We also analyzed this ligand series for ΔE and S(ΔE) trends in the 2-electron HS–IS and IS–LS spin state splitting. For ΔE_H–I, increasing the number of CO ligands shifts the B3LYP preference from HS to IS for most 3d metals (i.e., excluding only Mn(III) and Fe(II)), but the 4d TMC preference remains uniformly IS regardless of complex composition (ESI,† Tables S13 and S14). Although S(ΔE_H–I) is lower than S(ΔE_H–L), the 3d TMC spin-state preference can still shift depending upon a_HF value (e.g., trans Fe(II)(H₂O)₄(CO)₂ is IS-favoring for a_HF = 0.15 but HS for B3LYP, ESI,† Table S13). The relationships of 3d and 4d TMC S(ΔE_H–I) values are similar to that for S(ΔE_H–L): (i) the smallest and largest S magnitudes are typically bounded by the homoleptic hexa-aqua and hexa-carbonyl complexes, respectively, and (ii) mixed complexes with more CO ligands have larger sensitivities especially in 3d versus 4d TMCs (ESI,† Fig. S1 and Table S15). For ΔE_I–L, multiple d³ Cr(III)/Mo(III) and d⁷ Co(II)/Rh(II) complexes could also be compared in their LS doublet and IS quartet (i.e., ls + 2) states (ESI,† Tables S16 and S17). Reduced exchange sensitivity for IS–LS pairs mean that few ground state assignments change in either 3d or 4d TMCs with change in functional (ESI,† Table S15). Across all metals, the trend of increasing deviation of 3d and 4d S values with increasing ligand field strength is consistent with HS–LS and HS–IS behavior for mixed-ligand complexes, but non-monotonic behavior with CO ligand number is observed and several of the Rh(III) complexes have positive S(ΔE_I–L) values (ESI,† Fig. S2). The S values are also relatively comparable and modest for early d³ Cr(III)/Mo(III) and d⁴ Mn(III)/Tc(III) complexes (ESI,† Fig. S2).

As a possible explanation for the distinct behavior of IS–LS spin splitting trends among complexes, we considered differences in closed-shell LS states (i.e., for d⁴ and d⁶) and open-shell ones (i.e., for d⁵) but observe no significant difference (ESI,† Fig. S2). The greater variability is likely due to noise in evaluating the lower IS–LS S values, in line with prior observations,^29,136 and the larger range of metals and oxidation states. Despite some differences for IS–LS in comparison to HS–IS or HS–LS spin-splitting energies, the ligand series analysis suggests overall that there are consistently greater differences in the effect of exchange on 3d compared to 4d TMCs that grow with ligand field strength.

d. Effect of exchange on spin-state ordering

TMC spin-state splitting sensitivities depend both on row (3d vs. 4d) and electron configuration (e.g., d⁴vs. d⁶) of the metal atom as well as on the strength of the ligand field. The combined effect of these differences can be further understood by evaluating how the distribution of ΔE_H–L shifts with HF exchange fraction. For this comparison, we chose 64 Fe(II)/Ru(II) and 51 Tc(III)/Mn(III) metal/oxidation state pairs where 3d TMCs and 4d TMCs span the HS–LS transition, respectively (Fig. 7). For Fe(II) TMCs, increasing HF exchange from a pure GGA to 30% has a strong effect on the distribution of ΔE_H–L values (Fig. 7). Because complexes with strong-field ligands have more negative sensitivities than complexes with weak-field ligands, the width of the Fe(II) ΔE_H–L distribution decreases monotonically from a_HF = 0.0 (−75 to 80 kcal mol⁻¹) to a_HF = 0.3 (−90 to 30 kcal mol⁻¹, Fig. 7). We note that a complex with |ΔE_H–L| ≤ 5 kcal mol⁻¹ could be a spin crossover (SCO) complex. Using this definition, the number of potential Fe(II) SCOs is maximal for moderate, a_HF = 0.1–0.2 exchange and decreases again at higher exchange values (Fig. 7). From a_HF = 0.0 to 0.3, the population of HS-favored Fe(II) TMCs also increases dramatically from 12 to 49 of 64 complexes (Fig. 7 and ESI,† Table S18).

Fig. 7 Normalized histograms of ΔE_H–L (in kcal mol⁻¹) (bin width: 5 kcal mol⁻¹) for 51 pairs of 3d Mn(III) and 4d Tc(III) TMCs (top) and 64 pairs of 3d Fe(II) and 4d Ru(II) TMCs (bottom). Histograms are filled translucent for 3d (green) and 4d (dark blue) TMCs to show regions of overlap. A zero line is indicated as black solid where HS–LS spin state ordering changes, and a range of 5 kcal mol⁻¹ around that line is indicated with black dotted lines. The panes correspond to increasing % HF exchange from left to right, as labeled in top inset.

The isovalent Ru(II) TMCs span nearly as wide a range for a_HF = 0.0 (30 to 150 kcal mol⁻¹) as the 3d TMCs, but none are HS at any exchange fraction (Fig. 7 and ESI,† Table S18). Modest differences in exchange sensitivity across Ru(II) complexes have a limited effect on the shape of the distribution in comparison to Fe(II) TMCs (Fig. 7). The range of the Ru(II) ΔE_H–L distribution for a_HF = 0.3 (20 to 125 kcal mol⁻¹) is reduced, but this reduction (12%) is smaller than was observed for the Fe(II) TMCs (19%).

For the Tc(III)/Mn(III) isovalent pairs, the 4d TMCs have a more even distribution of HS and LS ground states (Fig. 7). This is particularly true (18 LS and 33 HS) at a_HF = 0.3, while LS is generally favored by Tc(III) TMCs (44 LS and 7 HS) at a_HF = 0.0, and all a_HF values have a significant number of potential Tc(III) HS–LS SCOs (Fig. 7 and ESI,† Table S18). In contrast to Fe(II), most (49 of 51) Mn(III) TMCs are HS even for a_HF = 0.0, and all are significantly HS (i.e., outside of the SCO region) for a_HF = 0.1 and higher (Fig. 7 and ESI,† Table S18). The Mn(III) ΔE_H–L values span a smaller range than the Fe(II) TMCs, meaning that even though the width of the Mn(III) ΔE_H–L distribution narrows only slightly with increasing HF exchange (a_HF = 0.0: −50 to 5 kcal mol⁻¹vs. a_HF = 0.3: −25 to −70 kcal mol⁻¹), this 18% reduction is roughly comparable to that for Fe(II) (Fig. 7). For the Tc(III) ΔE_H–L distribution, there is no change in the width due to sensitivities of outliers in the distribution tails (Fig. 7). Nevertheless, both Tc(III) and Mn(III) distributions become more peaked at intermediate values of the ΔE_H–L range, and the effect of a_HF on the shape of both distributions is more modest than for Fe(II) but more significant than for Ru(II) TMCs (Fig. 7).

Over all 3d/4d metal/oxidation state pairs for which ΔE_H–L was calculated, trends are similar to those observed for Fe(II)/Ru(II) and Tc(III)/Mn(III) (ESI,† Table S18). The HS–LS preference of 3d TMCs is highly sensitive to exchange, with an even number of LS-favoring vs. HS-favoring 3d TMCs at a_HF = 0.0 becoming predominantly (90%) HS at a_HF = 0.3 or higher (ESI,† Table S18). The 4d TMCs almost exclusively favor LS states at a_HF = 0.0, and a much smaller number (i.e., 28) of TMCs, typically with Tc(III) and to a lesser extent Tc(II) or Mo(II), becoming HS-favored at a_HF = 0.3 (ESI,† Table S18). Thus, based on this analysis, the 4d TMC LS-bias with respect to 3d TMCs combined with reduced exchange sensitivity also makes the 4d TMC HS/LS preferences significantly less sensitive to a_HF values.

Ultimately, to quantify the effect of HF exchange on 3d/4d TMC ground state (GS) assignment, we extended the comparison of spin-state energetics to include IS states. There are 155 d⁴–d⁶ 3d/4d TMC pairs for which all three (i.e., HS, IS, and LS) spin states and the spin-splitting energy sensitivities are computed. We combine these pairs with the 92 d³ or d⁷ 3d/4d TMC pairs for which only two states were accessible (i.e., IS and LS), and their energies are computed. Over this 247-complex set, significant differences in the sensitivity of GS assignment to a_HF are apparent between 3d and 4d TMCs (Fig. 8). Namely, the number of HS ground states rises from around one in five at a_HF = 0.0 to over half at a_HF = 0.3 for 3d TMCs, whereas none of the 4d TMCs have a HS GS at any a_HF value (Fig. 8 and ESI,† Tables S19, S20). The proportion of IS states remains roughly constant for 3d TMCs over this range, although their identities shift with added exchange to predominantly d³/d⁷ TMCs (i.e., excluding only two Mn(III) and one Cr(II) complex, ESI,† Table S19). The 4d TMCs differ from 3d TMCs because most have either an IS (ca. 35%) or LS (ca. 65%) GS, with almost no change in this distribution with a_HF value (Fig. 8 and ESI,† Table S19). While all 3d metal/oxidation states have at least one GS change from a_HF = 0.0 to 0.3, only a small number (i.e., 5 of 45) of d³ Mo(III) 4d TMCs change from LS to IS over this range (Fig. 8 and ESI,† Table S19). This change in GS is somewhat larger than for the Cr(III) TMCs, which are all IS for a_HF = 0.2 or above (Fig. 8 and ESI,† Table S19). The d⁴ Mo(II) and Tc(III) TMCs favor IS ground states exclusively, whereas most of the 3d isovalent Cr(II) or Mn(III) TMCs are HS-favoring especially with increasing a_HF values (ESI,† Table S19). Thus, overall 4d TMC GS assignment is significantly less likely to be affected by a change in functional than an equivalent 3d TMC although the sensitivity of predicted energies to exchange fraction is only on average reduced by around one third from the first- to second-row TMCs.

Fig. 8 Ground state assignment for the 247 for the 247-complex set of 3d and 4d TMC pairs: LS (red bars and circles), IS (gray bars and circles), and HS (blue bars). (top) Number of 47 pairs that are LS or IS for d³ Cr(III) (left) or Mo(III) (right) with a_HF fraction. (bottom) Stacked bar plot of HS, IS, and LS ground states for all 3d (left) and 4d (right) TMCs with a_HF fraction.

e. Relationships between energetic sensitivity and structure

In 3d TMCs, increasing spin multiplicity typically leads to longer metal–ligand bond lengths as antibonding states become preferentially occupied over states with bonding character, and this effect is more significant with increasing ligand field. Bond lengths do not change significantly with a_HF value,¹³⁸ and thus structural differences between spin states are also invariant. To determine if structural differences between 3d and 4d TMC pairs could explain distinct exchange sensitivities due to increased diffuseness of the 4d orbitals, we compared the B3LYP structures (i.e., metal–ligand bond lengths) of isovalent complexes. In both 3d and 4d TMCs, increasing spin multiplicity leads to longer metal–ligand bond lengths, consistent with our expectations, but bond lengths differ primarily due to the different sizes of 3d and 4d metals (ESI,† Fig. S3 and Table S21).

As in previous work,¹³⁸ we facilitate comparison by focusing on relative metal–ligand bond lengths, d_rel:

obtained by computing the ratio of the bond length, d, obtained from B3LYP with respect to the substituent metal (M) or ligand (L) atoms’ covalent radii, r (ESI,† Table S21). The d_rel values averaged over all six metal–ligand bonds have a similar range of values for both 3d and 4d TMCs; this averaging works better for the more symmetric LS states than HS or IS states (ESI,† Fig. S4–S6 and Tables S22, S23). For all TMCs, the d_rel values increase on average by 0.1 from LS to HS states and increase by about half (ca. 0.05) that amount from LS to IS states (ESI,† Fig. S4 and Tables S22, S23).

For 3d TMCs, the d_rel difference, Δd_rel, between HS and LS states is well correlated (R² = 0.86) with S(ΔE_H–L) values (Fig. 9 and ESI,† Table S24). The Fe(II)(CO)₆ Δd_rel is among the largest, consistent with the large magnitude (ca. −175 kcal mol⁻¹ HFX⁻¹) of its S(ΔE_H–L), while Mn(III)(H₂O)₆ has both a much lower S(ΔE_H–L) (ca. −50 kcal mol⁻¹ HFX⁻¹) and Δd_rel of 0.04 (Fig. 9). In line with these observations, the overall largest HS–LS Δd_rel values are observed for the cases with the most positive ΔE_H–L values (ESI,† Fig. S7). As with HS–LS, a similar relationship can be observed for S(ΔE) and Δd_rel for the 3d TMC HS–IS or IS–LS splittings (ESI,† Fig. S8 and Table S24). In fact a single good correlation (R² = 0.81) can be fit through all three sets of S(ΔE) and Δd_rel values for 3d TMCs (Fig. 9 and ESI,† Fig. S8 and Table S24). Taken together with our prior findings,²⁹ these observations suggest that increasing a_HF penalizes the delocalized, highly bonded lower-spin states more strongly than higher-spin states. When this bonding difference is larger, the sensitivity to a_HF is also higher.

Fig. 9 Trends of exchange sensitivity of spin-splitting energies, S(ΔE) (in kcal mol⁻¹ HFX⁻¹), with averaged relative distance, d_rel, quantities: (left) difference in d_rel between spin states, Δd_rel, and (right) LS d_rel. The Δd_relvs. S(ΔE) trends (left) are shown for 155 pairs of 3d TMCs in the case of 4-electron ΔE_H–L (green translucent circles), 2-electron ΔE_H–I (dark gray translucent circles), and 2-electron ΔE_I–L (light gray translucent circles, with an additional 92 I–L-only pairs). The results for 4d TMCs for the 4-electron ΔE_H–L case only are also shown (blue translucent squares). Best-fit lines are shown for all 3d (green dotted line) or 4d data (blue dotted line). The LS d_rel trends vs. S(ΔE_H–L) only are shown at right and follow the same coloring scheme for both symbols and best-fit lines as indicated in left inset legend. For both plots, representative hexa-aqua Mn(III)/Tc(III) HS–LS complex pairs are shown as solid symbols with red outline, and representative hexa-carbonyl Fe(II)/Ru(II) HS–LS complex pairs are shown as solid symbols with orange outline.

For the second-row transition metals, there is also a good correlation (R² = 0.71) between S(ΔE) and Δd_rel for the HS–LS states of the 4d TMCs (Fig. 9 and ESI,† Fig. S9, Table S24). The slope of the relationship however is reduced with respect to the value for 3d TMCs (Fig. 9 and ESI,† Table S24). Indeed, Δd_rel values across all spin-state pairs in 4d TMCs are as large as those for the 3d TMCs. Thus, more diffuse 4d orbitals do not change Δd_rel value trends and cannot be used to explain the reduced exchange sensitivity of 4d TMCs (ESI,† Fig. S4). A weaker, but somewhat unified, relationship can be observed between the LS state d_rel, which is generally longer in some but not all 4d TMCs, and S(ΔE_H–L) across both 3d and 4d TMCs (Fig. 9 and ESI,† Table S24, Fig. S4).

To understand the origins of this relationship, we obtained potential energy curves (PECs) with varied metal–ligand bond length and varied a_HF values for 3d Fe(II) and 4d Ru(II) TMCs in their quintet HS and singlet LS states. We contrasted homoleptic helium-atom 3d/4d complexes as a simplified example of the weak-field limit with strong-field, hexa-carbonyl 3d/4d TMCs, in which we rigidly displaced the CO ligands (see Section 2). The closed-shell He atoms act as very weak field ligands, with the 3d/4d TMCs both strongly favoring the HS state over the LS state (B3LYP ΔE_H–L: Fe(II) ca. −64 kcal mol⁻¹ and Ru(II) ca. −31 kcal mol⁻¹, Fig. 10 and see ESI,† Table S25). The S(ΔE_H–L) values are both small and comparable (Fe(II): −47 vs. Ru(II): −43 kcal mol⁻¹ HFX⁻¹), consistent with small S(ΔE_H–L) values (hexa-aqua Fe(II): −53 vs. Ru(II): −38 kcal mol⁻¹ HFX⁻¹) from other weak-field TMCs (ESI,† Tables S12 and S25). Despite these comparable exchange sensitivities, the dependence on a_HF of individual PECs of Ru(II) and Fe(II) TMCs is distinct (Fig. 10). Both HS and LS Ru(II)(He)₆ PECs are stabilized with increasing a_HF, but the HS relative energy is more sensitive to exchange (Fig. 10). For Fe(II)(He)₆, on the other hand, the HS PEC is stabilized with increasing a_HF while the LS is destabilized (Fig. 10). For individual LS or HS states, Ru(II) TMCs PECs have higher sensitivities than the Fe(II) counterparts (Fig. 10 and ESI,† Table S25). The negative S(ΔE_H–L) for Fe(II)(He)₆ arises from additive sensitivities of the HS and LS states, while for Ru(II)(He)₆ the simultaneous LS and HS stabilization leads to a comparable net S(ΔE_H–L) (Fig. 10 and ESI,† Table S25).

Fig. 10 Potential energy curves (PECs, in kcal mol⁻¹) with M–He distance (in Å) for Fe(II)(He)₆ (left) and Ru(II)(He)₆ (right) in LS singlet (red) and HS quintet (blue) spin states. Solid center lines (red for LS and blue for LS, as indicated in inset legend) correspond to the B3LYP (a_HF = 0.2) PEC, the outer solid line (maroon for LS and dark blue for HS) corresponds to a_HF = 0.0, and the outer dashed line corresponds to a_HF = 0.4 (as indicated also in inset legend). The inner shaded regions correspond to the a_HF = 0.1–0.3 range, and the outer shaded region to a_HF = 0.0–0.4. The relative energies have been aligned to set the minimum of the B3LYP PEC to zero, and a zero line is shown. A circle indicates the minimum of the PECs for a_HF = 0.0, 0.2, and 0.4.

The He complexes are weakly bound, with the small covalent radius of the He atom leading to a very long estimated d_rel values in LS states for both 3d and 4d TMCs (Fe(II): d_rel = 1.08 vs. Ru(II): d_rel = 1.11) despite moderate absolute bond lengths of 1.84 and 1.93 Å, respectively (Fig. 10 and ESI,† Tables S21, S25). The HS states are even more weakly bound, with large Δd_rel values of 0.16 for Fe(II) and 0.24 for Ru(II), and these observations are insensitive to a_HF (Fig. 10 and ESI,† Table S25). Since we can obtain the exchange sensitivity over a wide range of M–He bond lengths in these TMCs, we also evaluated the extent to which S values vary at points away from equilibrium (Fig. 10). The vertical HS–LS sensitivity evaluated at the LS geometry for Ru(II) or Fe(II) is comparable to that for the adiabatic case, despite the fact that the LS–HS ordering is inverted at the LS geometry for Ru(II) but unchanged for Fe(II) (Fig. 10 and ESI,† Table S25). When stretched to the HS geometry, the S values decrease significantly for both Ru(II) and Fe(II) (Fig. 10 and ESI,† Table S25).

While the ground state preferences for model strong-field complexes of both Fe(II) and Ru(II) are uniformly LS (B3LYP ΔE_H–L: Fe(II) ca. 30 kcal mol⁻¹ and Ru(II) ca. 116 kcal mol⁻¹), the exchange sensitivity trends are comparable to the weak-field He case (Fig. 11 and ESI,† Table S26). That is, larger total energy sensitivity in the HS Ru(II) PEC still leads to a lower overall S(ΔE_H–L) magnitude compared to Fe(II) because exchange destabilizes the LS 3d TMC while weakly stabilizing the LS 4d TMC (Fig. 11 and ESI,† Table S26). The only significant distinction observed for the strong-field CO ligands is that S for each PEC and the splitting is much more dependent on bond length, with the effect of significantly reducing (>50%) S values for vertical spin-state energetics with respect to adiabatic S(ΔE_H–L) values (Fig. 11 and ESI,† Table S26).

Fig. 11 Potential energy curves (PECs, in kcal mol⁻¹) with M–C distance (in Å) for Fe(II)(CO)₆ (left) and Ru(II)(CO)₆ (right) in LS singlet (red) and HS quintet (blue) spin states. Solid center lines (red for LS and blue for LS, as indicated in inset legend) correspond to the B3LYP (a_HF = 0.2) PEC, the outer solid line (maroon for LS and dark blue for HS) corresponds to a_HF = 0.0, and the outer dashed line corresponds to a_HF = 0.4 (as indicated also in inset legend). The inner shaded regions correspond to the a_HF = 0.1–0.3 range, and the outer shaded region to a_HF = 0.0–0.4. The relative energies have been aligned to set the minimum of the B3LYP PEC to zero, and a zero line is shown. A circle indicates the minimum of the PECs for a_HF = 0.0, 0.2, and 0.4.

From the set of all 3d/4d TMC pairs, over 25 3d TMCs have positive sensitivities of the LS total energies whereas only one of the 4d TMCs does (ESI,† Table S27). While average sensitivities of the total energy with respect to a_HF are higher for both states in 4d TMCs, they are more likely to be similar in magnitude between LS and HS states. Thus, distinct geometric and electronic structure of the LS states of 3d TMCs likely contributes to their higher exchange sensitivity (ESI,† Table S27).

4. Conclusions

We have carried out a comprehensive study of the extent to which established trends in 3d TMC exchange sensitivity and spin-state ordering have analogies in equivalent, isovalent 4d TMCs. From ten ligands covering a wide range of ligand field strengths, we generated hundreds of homoleptic and heteroleptic TMCs with a range of mid-row M(II)/M(III) 3d and 4d metal centers to interpret both 4-electron (i.e., HS–LS) and 2-electron (i.e., HS–IS or IS–LS) spin-splitting energies and sensitivities. We observed consistently increasing exchange sensitivity of ΔE_H–L with increasing ligand field in both 3d and 4d homoleptic TMCs. Over the larger data set, we made a number of surprising observations:

(1) Strong-field ligand effects were found to be additive in heteroleptics for both ΔE_H–L and S(ΔE_H–L). Similar trends were observed for HS–IS and IS–LS spin-state energetics, albeit with all magnitudes and ligand field sensitivities reduced.

(2) The 4d TMC exchange sensitivities were consistently smaller in magnitude than the equivalent 3d TMC sensitivities for all metals and oxidation states considered, and this deviation between 3d and 4d TMCs increased with increasing ligand field strength.

(3) Despite 4d TMC energetics being less sensitive than their 3d TMC counterparts, variations by as much as 10 kcal mol⁻¹ per 10% change in HF exchange were still observed, which could have a significant effect on energetic predictions.

(4) Combined with the significant LS-shifting of 4d TMCs with respect to their equivalent 3d TMCs, the reduced 4d TMC exchange sensitivity led to a much lower likelihood of change in spin-state ordering than for their 3d TMC counterparts. While over the range of a_HF = 0.0 to 0.3 many 3d TMCs changed ground states from LS to IS or HS states, a very small number of 4d TMCs shifted from LS to IS and none were HS.

(5) Delocalization does not explain the reduced exchange sensitivity of 4d TMCs. The 3d and 4d bond length differences between spin states were comparably large but could not explain the reduced sensitivity of 4d TMCs.

(6) Analysis of PECs of representative complexes indicated the higher spin-splitting exchange sensitivities of 3d TMCs is likely due to the opposing effects of exchange on the two spin states, despite higher exchange-sensitivity of the total energy in 4d TMCs.

Overall, we find that while spin-state energetics of 4d TMCs are roughly two-thirds as sensitive as those of 3d TMCs, this study suggests it is unlikely for HF exchange tuning to alter ground state predictions for 4d TMCs. This does not guarantee that common DFT functionals are always capable of predicting the ground state spin but does suggest that conventional tuning approaches that work in 3d TMCs to reproduce experimental spin state ordering will not be applicable in 4d TMCs.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

The authors acknowledge primary support by the Office of Naval Research under grant numbers N00014-17-1-2956 and N00014-18-1-2434 (for D. H., N. A., and C. D.) and the National Science Foundation under grant number CBET-1704266 (for A. N.). This work was also supported by the Department of Energy under grant number DE-SC0018096 (for Y. C. and D. B. K. C.). A. N. was partially supported by a National Science Foundation Graduate Research Fellowship under Grant #1122374. This work was carried out in part using computational resources from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. H. J. K. holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund and an AAAS Marion Milligan Mason Award, which supported this work. The authors thank Adam H. Steeves for providing a critical reading of the manuscript.

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Footnotes

† Electronic supplementary information (ESI) available: Details of calculation criteria, filtering steps, and eliminated complexes; statistics of sensitivities and relative sensitivities for different spin-splitting energies; sensitivities of pairs of 3d and 4d TMCs including outliers for different spin splittings; spin splitting and sensitivity properties of CO/H₂O complexes; ground state preference and HS vs. LS preference for 3d and 4d TMCs vs. exchange fraction; absolute averaged M–L distances for LS and HS states; covalent radii for computing relative distances; histograms and statistics of relative distances; comparison of equatorial and axial relative distances; relationship between relative distances and exchange sensitivity; properties of potential energy curves for hexa-helium and hexa-carbonyl complexes; and total energy sensitivities of HS and LS states in 239 TMC pairs (PDF). Structures and total energies of all 3d and 4d TMCs studied in this work; reasons complexes or sensitivities were not included in the data set; individual metal–ligand bond lengths for all converged complexes at all exchange fractions; total energy sensitivities in HS–LS pairs of TMCs; individual metal–ligand bond lengths for B3LYP and the relative metal–ligand bond length calculation for all pairs of 3d/4d TMCs; spin splitting values and sensitivities for all pairs of 3d/4d TMCs; total energies and analysis of sensitivities for PECs of Fe(II)/Ru(II) homoleptic TMCs in LS and HS states with CO and He atom ligands (ZIP). See DOI: 10.1039/d0cp02977g

‡ These authors contributed equally.

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