Spin States and Other LigandâField States of Aqua Complexes

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Quantum Electronic Structure

Spin States and Other Ligand–Field States of Aqua Complexes Revisited with Multireference Ab Initio Calculations Including Solvation Effects Mariusz Radon, and Gabriela Drabik J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00200 • Publication Date (Web): 26 Jun 2018 Downloaded from http://pubs.acs.org on June 27, 2018

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Journal of Chemical Theory and Computation

Spin States and Other Ligand–Field States of Aqua Complexes Revisited with Multireference Ab Initio Calculations Including Solvation Effects Mariusz Radon´ ∗ and Gabriela Drabik E-mail: [email protected]

Abstract High-level multireference (CASPT2, NEVPT2) calculations are reported for transition metal aqua complexes with electronic configurations from (3d)1 to (3d)8 . We focus on the experimentally evidenced excitation energies to their various ligand–field states, including different spin states. By employing models accounting for both explicit and implicit solvation, we find that solvation effect may contribute up to 0.5 eV to the excitation energies, depending on the charge of ion and character of the electronic transition. We further demonstrate that with an adequate choice of the active space and the energetics extrapolated to the complete basis set limit, the presently computed excitation energies are in a good agreement with the experimental data. This allows us to conclusively resolve significant discrepancies reported in earlier theory works [e.g., J. Phys. Chem. C, 2014, 118, 29196–29208]. For the benchmark set of 19 spin-forbidden and 24 spin-allowed transitions (for which experimental data are unambiguous), we find the mean absolute error of 0.15 or 0.13 eV, and the maximum error of 0.56 or 0.42 eV for CASPT2 or NEVPT2 calculations, respectively. For the particularly challenging ∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment

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sextet–quartet gap for [Fe(H2 O)6 ] 3+ , we support our interpretation by additional calculations with multi-reference configuration interaction (MRCI) and coupled cluster theory up to the CCSDT(Q) level. By underlining a rather subtle interplay between the solvation and correlation effects, the findings of this Article are relevant not only for modeling and interpretation of optical spectra of transition metal complexes, but also in further benchmarking of theoretical methods for the challenging problem of spin–state energetics.

1 Introduction Owing to significant electron correlation effects and multitude of close-lying energy levels, transition metal (TM) complexes are widely recognized as computationally demanding systems. 1–7 It is a challenge for both DFT (density functional theory) and ab initio WFT (wave function theory) methods, to accurately describe relative energies of TM spin states 2,8–11 and other ligand–field excited states, 12–15 as well as metal–ligand bond energies. 16–20 In DFT, there is a well known problem with the strong dependence of results on the choice of exchange–correlation functional. 2,21–23 Therefore, independently from improvements of DFT methods, 24–26 computationally more expensive ab initio WFT methods are applied to challenging TM systems with the hope of obtaining conclusive, benchmark-quality results. 9,27–33 The following WFT methods are particularly promising in the field of (bio)inorganic chemistry: (1) multireference methods based on second-order perturbation theory (CASPT2, 34 RASPT2, 35 and NEVPT2 36 ), recently also in combination with the DMRG (density matrix renormalization group) approach; 29,37 (2) a multi-reference configuration interaction (MRCI) method, most notably the spectroscopy–oriented formulation (SORCI) by Neese 38 , and the new internally-contracted implementation of MRCI with singles and doubles (MRCISD) by Werner and co-workers; 32 (3) the single-reference coupled cluster method with singles, doubles, and approximate triples, CCSD(T). 39 Novel promising methods, including quantum Monte Carlo 11,28 and other approaches 31,40–42 are also actively developed with the focus on challenging TM systems. Still, however, it is not trivial to know how accurate are predictions of approximate WFT meth2


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ods, like CASPT2, NEVPT2 or CCSD(T), when applied to complicated TM systems, whose electronic structure involve a delicate balance of dynamic and nondynamic correlation effects. 10,41,43 In fact, contradictory results have occasionally been reported when using different WFT methods and the discrepancies are not explained well enough. 27,44,45 Moreover, to make advanced WFT calculations manageable, a substantial compromise must often be made with respect to the basis set and active space used, the size of molecular model, and the way of accounting for solvation or other environmental effects. Thus, even when employing potentially very accurate methods, it may be difficult to reach the level of accuracy needed for chemical prediction and discovery in (bio)inorganic chemistry and material science. In view of the methodological uncertainties, it becomes highly relevant to benchmark theory against reliable and (preferably) quantitative experimental data for well-characterized TM complexes. In the context of spin–state energetics useful experimental data can be obtained from optical spectra of many TM complexes, in which vertical energies of various d–d excitations, including spin–forbidden ones, are directly evidenced. 46 A parallel source of experimental data for spin–state energetics are thermochemical parameters of spin crossover complexes, 47 but this is beyond the scope of the present work. A number of authors drew attention to optical spectra of aqua complexes, [M(H2 O)6 ]n+ (where Mn+ is a 3d-electron transition metal ion), as valuable methodology benchmarks. 1,13,15,48,49 Indeed, the spectra of aqua complexes exhibit a number of spectral features due to ligand–field transitons, either spin-allowed (∆S = 0) or spin-forbidden (∆S 6= 0) ones, giving direct information about vertical energy differences between the ground state and an electronically excited state with the same or different spin multiplicity. Most of the experimental data for aqua complexes have been established and interpreted (by means of phenomenological ligand–field theory, LFT) already many decades ago, 50 but these systems are still of experimental 51 and theoretical interest (see below). We note that aqua complexes are not only small “toy molecules” to benchmark theory, but also they are receiving attention as cluster models representing transition metal sites in oxide materials, properties of which are discussed in the context of solar energy conversion. 15,48,52

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Although aqua complexes may be perceived as very simple ones, severe controversies were encountered in recent theory works when trying to explain their optical spectra by means of WFT calculations. The study by Yang, Ratner, and Schatz, 15 where ligand–field excitation energies were calculated for aqua complexes of 3d-electron metal ions from Ti III (3d1 ) to Cu II (3d9 ), seemed to indicate that even the best multireference methods available (CASPT2, MRCI) give rather large errors with respect to the experimental data. For instance, the lowest d–d band of Fe III complex, attributed by LFT to the 6 A1g → 4 T1g transition, was predicted at almost twice larger excitation energy (2.85–3.02 eV) than found in the experiment (1.56 eV). Since the computed excitation energy matched some other bands observed in the spectrum around 3 eV, it was even speculated that the lowest d–d transition might be, indeed, ∼ 3 eV, whereas the experimental band at 1.56 eV originates from impurities, like traces of Fe II aqua complex present due to redox equilibrium. 15 The same d–d band of Fe III complex was also found problematic in earlier studies by Ghosh and Taylor 1 (CASPT2 and CCSD(T) calculations) and Neese et al. 13 (SORCI calculations), who also found discrepancies with respect to the experiment on the order of 0.5–1 eV. In our most recent study on this topic, 53 we found that the lowest d–d band for Fe III aqua complex is strongly affected by solvation (0.5 eV). This effect was neglected in the previous theoretical studies, in line with a widespread, but apparently incorrect presumption that solvation effects should be small for d–d excitations due to their localized character. To estimate the solvation effect in ref 53, we explicitly included a layer of 12 second-sphere water molecules, in the energetically most stable conformation proposed by Uudsemaa and Tamm 54 and by Markham et al. 55 for description of hydrated metal ions. Upon including the solvation effect and also refining the electron correlation treatment, we were able to improve the computed excitation energy and, thereby, to partly resolve the previous controversies. However, it still remains unclear whether the large solvation shift for the 6 A1g → 4 T1g transition energy of Fe III aqua complex is an anomaly or rather a typical effect for TM aqua complexes. Interestingly, in ref 15 noticeable and still unresolved discrepancies (i.e., more than 0.3 eV error or bands unexplained by theory) were observed also for aqua complexes of other TM ions (e.g., Mn II ,

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Co III , Co II , V III ). These discrepancies may be partly rooted in the neglect of solvation effects, but—as suggested in ref 15—they may also indicate problems with certain spectral data for aqua complexes. Indeed, aqua complexes are rather complicated in experimental handling due to their propensity to hydrolysis and/or redox instability, and some spin-forbidden bands reported in the literature are extremely weak (e.g., molar attenuation coefficient for d–d bands of [Mn(H2 O)6 ] 2+ is ∼ 0.01 M−1 cm−1 ). 56 Given the importance of aqua complexes as long-standing benchmarks for theory and models of oxide materials, all the above discrepancies call for a thorough reinvestigation. It is also intriguing that for Ni II aqua complex, an almost quantitative agreement with experiment was obtained in ref 15 despite ignoring solvation. As our previous paper suggested, one may not rule out a cancellation of the model error (lack of solvation) with the method error. 53 Therefore, it becomes mandatory to study all interesting aqua complexes and their numerous ligand–field states using one consistent methodology: accounting for solvation and accurately treating electron correlation effects. In this Article, we put under scrutiny aqua complexes of the first-row transition metals ions: Ti III , V III , V II , Cr III , Cr II , Mn III , Mn II , Fe III , Fe II , Co III , Co II , and Ni II (note that Cu II is intentionally excluded from this benchmark study in view of recent controversies whether Cu II aqua complex is six- or five-coordinate 57,58 , hindering unambiguous interpretation of the experimental data.) For each complex, a number of ligand–field states whose energies are evidenced in the experimental spectrum will be computed, accounting for solvation effects. (Thus, due to the large number of TM complexes and electronic excited states studied here, the present Article is a considerable extension with respect to our previous work, 53 which was limited to the lowest spin states of Fe III and Ru III complexes). Different levels of introducing the solvation effects into the calculations will be compared (no solvation, implicit or explicit solvation, or combination of both approaches). In regard to quantum chemical methodology, we focus on two multireference methods which are particularly promising for large-scale applications in (bio)inorganic chemistry: CASPT2 and NEVPT2. We would like to see whether a consistent interpretation of the experimental spectra of aqua complexes can be obtained by using these methods combined with the

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appropriate solvation treatment. After critically comparing theory with the experimental data of aqua complexes on a case-by-case basis, we will be able to reach solid and statistically relevant conclusions on the accuracy of both multireference methods.

2 Computational Details 2.1

Molecular Structures, DFT Calculations

All ground state (GS) structures were optimized at the DFT-D3 level (BP86-D3/def2-TZVP) as implemented in Turbomole 7. 59 Except for the GS of Co III (S = 0), all calculations were spin– unrestricted, but with no significant spin contamination. To account for hydrogen-bonding with solvent (water), the bare [M(H2 O)6 ]n+ clusters were surrounded by 12 H2 O molecules in the second coordination sphere, leading to {[M(H2 O)6 ] · (H2 O)12 }n+ models, abbrev. as [M(H2 O)18 ]n+ . Long–range solvation effects were treated by the COSMO model 60 (with the default selection of atomic radii, rsolv=1.3 Å, and ε = 80 corresponding to water). To quantify solvation effects in Table 2, additional comparative calculations were performed for selected metal ions employing the following models: the [M(H2 O)6 ]n+ cluster in gas phase, the same cluster with the COSMO model, and the [M(H2 O)18 ]n+ cluster in gas phase. Regarding the optimum conformation of [M(H2 O)18 ]n+ models, we considered various arrangements of second-sphere waters proposed in the literature 54,55,58,61 (see Figure 1; more details can be found in Supporting Information). We found that for all complexes studied here, the most stable conformation is consistently the one shown in Figure 1(a), which was originally proposed (in different context) in refs 54 and 55, and later used by us in ref 53 for Fe III and Ru III . While the adopted model represents the most stable minimum energy structure, it clearly does not allow to describe the dynamics of the solvation layer and exchange of water molecules with the bulk solvent. Although the preferred conformation of [M(H2 O)18 ]n+ models may possibly have as high symmetry as S6 , it must be reduced to Ci to work with the COSMO model as implemented in Turbomole. 53 The choice of Ci symmetry is also necessary to describe Jahn-Teller (JT) distortion for 6


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(a)

(b)

0.0

12.2

(c)

(d)

23.6

7.4

Figure 1: Considered structures of aqua complexes on the example of [Fe(H2 O)18 ] 2+ : Panel (a) – the most stable structure with approximately trigonal symmetry S6 (the main axis perpendicular to the plane of figure) which was used for all calculations below; Panels (b),(c),(d) – less stable alternative structures. Annotated numbers are relative energies in kcal/mol at the BP86-D3/def2TZVP + COSMO level. Geometry of type (b) is the alternative structure proposed in ref 54; type (c) is the one proposed (as “model C”) in ref 61; type (d) is based on the one proposed for Cu II aqua complex (as “model 18w-a”) in ref 58. Optimized Cartesian coordinates of all structures and their relative stabilities can be found in Supporting Information.

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complexes with a degenerate GS, for which the optimization resulted in three unequal equilibrium M–O distances. We also observed that in calculations employing the COSMO model, the Ci structures still show tiny imaginary frequencies (|ν| < 100 cm−1 ) and the total energy can be slightly lowered (by 0.3–0.9 kcal/mol) when removing the inversion symmetry. Note that this Ci → C1 symmetry breaking was not observed in test gas-phase optimizations and cannot be rationalized in terms of JT effect (e.g., it also occurs for complexes with non-degenerate GS). Both Ci and C1 and structures were considered for computation of vertical excitation energies (see below).

2.2

Multireference Calculations

Single-point calculations of vertical excitation energies were carried out at the CASPT2 level (default IPEA shift 62 of 0.25 a.u., imaginary shift of 0.1 a.u.) using Molcas 8 (service pack 1) 63 and at the NEVPT2 level (partially contracted) using Molpro 2012. 64, 65 Scalar relativistic effects were accounted for by second-order Douglas-Kroll Hamiltonian. 66 Core electrons (except metal 3s3p) were frozen. We covered all excited states necessary to explain the bands observed in the experimental spectra. As an example, in the case of Mn II (d5 ) aqua complex, we are interested in the sextet GS (arising from the 6 A1g term) and the excited states arising from the following terms 1 4 T1g , 1 4 T2g , 1 4 Eg , 4 A1g , 2 4 T2g , 2 4 Eg , 2 4 T1g and 2 T2g . Accounting for degeneracies of the terms, this gives one sextet, 17 quartet (i.e., 3 + 3 + 2 + 1 + 3 + 2 + 3), and three doublet states. All states with a given multiplicity were computed together in state-average CASSCF calculations. Single-state (SS) CASPT2 energies are reported. The choice of active space was made consistently with standard rules for TM complexes. 12,67 The standard active space includes five orbitals with predominant TM 3d character, two σ bonding orbitals describing covalent metal–ligand linkage and five double-shell d orbitals (the latter only for metal ions with more than 3 electrons on the 3d shell). For a metal ion with dk configuration, the standard active space is thus obtained by distributing (k + 4) active electrons in 7 (if k ≤ 3) or 12 (if k > 3) active orbitals. We also performed comparative calculations with minimal active space, i.e., obtained by distributing k active electrons in 5 orbitals with the predominant metal 3d 8


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character; such active spaces were used for most calculations in ref 15. Except for Fe III and Co III complexes, we encountered the following problem with orbital rotations: during CASSCF calculations with the standard active space one of the σ -bonding rotated into 3s on metal and one of the double-shell d orbitals (if present) rotated into 4s on metal, leading to an unbalanced active space. The problem could be solved by enlarging the active space with the mentioned orbitals, 3s (k ≤ 3) or 3s4s (k > 3). However, since resulting active spaces with 14 active orbitals would be quite large and computationally demanding (especially for subsequent NEVPT2) and differential correlation effects related to 3s,4s orbitals being either active or not, are negligible (Tables S4 and S5, Supporting Information), we adopted the following procedure to solve the orbital rotation problem. We first performed CASSCF calculations with the large active space containing the additional 3s/3s4s orbital(s). We subsequently removed the additional orbital(s) (by making 3s inactive and 4s virtual) and repeated CASSCF calculations where the removed 3s/3s4s orbital(s) was/were prevented from mixing with other orbitals. The active spaces resulting from this procedure will be identified by the appended suffix “fs” (“fixed s orbitals”). For instance, in the case of Mn II , the standard active space is denoted as (9,12)fs. It means that CASSCF (11,14) with additional 3s4s orbitals were performed first and the resulting natural orbitals—after removing 3s4s from the active space—were used as initial guess to perform the CASSCF (9,12) calculations during which the 3s4s orbitals were fixed. The minimal and standard active spaces employed for the studied aqua complexes are summarized in Table S3, whereas the contour plots of active orbitals for an illustrative case can be found in Figure S1, Supporting Information. The accuracy of the fs approximation was checked for [Fe(H2 O)18 ] 3+ and [Mn(H2 O)18 ] 2+ (see Tables S4 and S5, Supporting Information). The difference between the CASPT2 results obtained from the standard and minimal active space is strongly system dependent. For the Fe III complex, the sextet–quartet excitation energies obtained for the two choices differ by up to 0.3 eV (6.9 kcal/mol). By contrast, for the isoelectronic Mn II complex, the analogous effect is only 0.03 eV (0.7 kcal/mol). Therefore, in some cases one can rely on the result from the minimal active space, but not in general. It is noteworthy that the

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Table 1: Basis Sets Used for CASPT2 and NEVPT2 Calculations. abbreviation T(D) T Q(T) a ref

metal

first-sphere O atoms

other atoms

cc-pwCVTZ-DKa cc-pwCVTZ-DKa cc-pwCVQZ-DKa

cc-pVTZ-DKb cc-pVTZ-DKb cc-pVQZ-DKb

cc-pVDZ-DKb cc-pVTZ-DKb cc-pVTZ-DKb

68; b ref 69.

results from standard and minimal active spaces may differ significantly even for complexes for which we had to used the above fs procedure to stabilize the standard active space. For instance, in the case of Cr III complex, the excitation energies computed with the minimal (3,5) and standard (7,7)fs active spaces differ by up to 0.14 eV (3.2 kcal/mol). Finally, it is worth noting that correlation effects related to extending the minimal active space are strongly non-additive with respect to grouping orbitals. For [Fe(H2 O)18 ] 3+ , when the minimal active space, (5,5) is extended with only double-shell orbitals [to give (5,10) active space] or with only σ -bonding orbitals [to give (9,7) active space], the result vary by much less than when both group of orbitals are added simultaneously yielding the standard active space (9,12) (Table S4, Supporting Information). Moreover, double-shell d orbitals are important for Fe III complex even though their occupation numbers are below 0.001 (Table S9, Supporting Information). We used three basis sets defined in Table 1. Already the T(D) basis set (of mixed triple-ζ / double-ζ quality), earlier used in ref 53, provides reasonably accurate results for the studied aqua complexes. The other two basis sets, T (full triple-ζ ) and Q(T) (mixed quadruple-ζ / triple-ζ quality) served to estimate the complete basis set (CBS) limit using the extrapolation formula 70 En = ECBS + B(n + 12 )−4 ,

(1)

where n = 3 for basis set T and n = 4 for Q(T) one. Test calculations for [Fe(H2 O)18 ] 3+ revealed that the basis set incompleteness error (BSIE) for the T(D) basis is almost the same for CASPT2 and NEVPT2 calculations, and practically independent on the choice of active space (Table S8). Therefore, the CBS extrapolation in eq. (1) was applied only to the CASPT2 results with minimal

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active spaces to yield an additive correction for BSIE with respect to the T(D) basis set CASPT2(min) CASPT2(min) δCBS = ∆ECBS − ∆ET(D) .

(2)

The correction is on average 0.04 eV, maximum 0.11 eV, median 0.04 eV (absolute values, statistics for all excitation energies studied in this work). We also checked, on the example of [Fe(H2 O)18 ] 3+ , that adding diffuse function to the T(D) basis set would affect the excitation energies by only 0.01– 0.02 eV (Table S8). Unless mentioned otherwise, all calculations are performed for the structures optimized under the Ci symmetry, although there is a small symmetry breaking (Ci → C1 ) effect in COSMO optimizations (Section 2.1). The impact of this effect on the excitation energies was estimated at the CASPT2 level with the minimal active space and basis set T(D): CASPT2(min) CASPT2(min) (Ci ). (C1 ) − ∆ET(D) δCi →C1 = ∆ET(D)

(3)

The resulting corrections turned out to be negligible except in a few cases (maximum 0.10 eV for M = Cr II , average value 0.02 eV, median 0.01 eV). To obtain the final set of results reported below (Tables 3–15), the excitation energies were computed as method ∆E method = ∆ET(D) + δCBS + δCi →C1 ,

(4)

method is the excitation energy from CASPT2 or NEVPT2 calculations with the desired where ∆ET(D)

active space for the Ci structure and using basis set T(D); δCBS and δCi →C1 are the small corrections defined above. Detailed results, including contributions in eq. (4) and total energies, can be found in Supporting Information (Tables S10–S21). Total and relative CASSCF energies are also included for comparison. As can be seen, the second-order energy corrections (i.e., CASPT2 or NEVPT2 vs CASSCF) for relative excitation energies are highly system- and transition-dependent, varying from rather negligible (0–0.1 eV) to very significant (0.5–1.2 eV).

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3 Results and Discussion 3.1

Solvation Effects

In sharp contrast to assumptions made in earlier studies, 1,13,15 we have shown previously in ref 53 that relative energies of the lowest spin states for Fe III and Ru III aqua complexes are substantially affected by solvation. Now, in order to generalize these findings and obtain more systematic view into these solvation effects, we quantify them for aqua complexes of various metal ions with different numbers of 3d electrons and in different electronic states (Table 2). We also aim to compare different ways of introducing solvation effects into the calculations. Starting from bare [M(H2 O)6 ]n+ cluster in gas phase, one may use an implicit solvation model, like COSMO, or add explicit water molecules leading to [M(H2 O)18 ]n+ models, or both approaches can be combined. Regarding the use of implicit solvation, it is crucial to stress at the very beginning that it can be applied to: (a) geometry optimization of the ground state, (b) calculation of vertical excitation energies using non-equilibrium approach (i.e., for an excited state only the fast component of the reaction field is optimized), or (c) both. Concerning (b), we used the non-equilibrium PCM model as implemented in Molcas, but we found that it has negligibly small effect (below 0.01 eV) on the excitation energies; see Table S2, Supporting Information. We thus conclude that (in agreement with analogous results in our previous work 53 ) the main effect of implicit solvation is mediated through the change of geometry (a), whereas the direct effect on the calculated vertical excitation energies (b) is negligibly small and, hence, it will be neglected in all further calculations. For all entries in Table 2, the excitation energies are compared for [M(H2 O)6 ]n+ and [M(H2 O)18 ]n+ models, each one in either gas-phase or COSMO geometry. The difference between the excitation energy for [M(H2 O)18 ]n+ model in the COSMO geometry and that for [M(H2 O)6 ]n+ model in the gas-phase geometry defines the total solvation effect (the last column in Table 2). Comparison of results for different models in Table 2 reveals that it is usually indispensable to include both short-range effects (by adding explicit waters) and long-range ones (by means of COSMO model) in order to reproduce the total solvation effect on the excitation energies. The

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Table 2: Selected Vertical Excitation Energies of Aqua Complexes Computed for [M(H2 O)6 ]n+ and [M(H2 O)18 ]n+ Models (using either gas-phase or COSMO geometries) and Resulting Estimates of Solvation Effects.a,b,c metal excitationc

typec

[M(H2 O)6 ]n+

[M(H2 O)18 ]n+

gasd

COSMOe

gasd

COSMOe

solvation effect f

Ti III

2T 2g

→ 2 Eg

t2g → eg

2.03

2.37

2.31

2.39

0.36

V III

1 3 T1g → 3 T2g 1 3 T1g → 2 3 T1g

t2g → eg t2g → eg

1.68 2.96

1.96 3.23

2.06 3.34

2.18 3.40

0.50 0.44

Cr III

4A

→ 4 T2g 4A → 4T 2g 1g 4A → 2E g 2g 4A → 2T 2g 1g

t2g → eg t2g → eg spin-flip spin-flip

1.99 2.81 2.03 2.13

2.24 3.09 2.02 2.37

2.08 3.10 1.96 2.31

2.15 3.18 1.96 2.39

0.16 0.37 −0.08 −0.04

Mn II

6A

2g

6A

→ 4 T1g 2 1g → T2g

eg → t2g eg → t2g

2.76 3.93

2.68 3.78

2.53 3.73

2.50 3.67

−0.26 −0.26

Fe III

6A

1g

→ 4 T1g 6A → 4T 1g 2g 6A → 4A 1g 2g 6A → 4E g 1g

eg → t2g eg → t2g spin-flip spin-flip

2.39 2.93 3.42 3.49

2.18 2.76 3.39 3.46

1.97 2.63 3.32 3.39

1.89 2.54 3.29 3.35

−0.50 −0.39 −0.13 −0.14

Fe II

5T 2g

t2g → eg

1.02

1.23

1.28

1.31

0.29

Co III

1A

1g

→ 3 T1g 1A → 3T 1g 2g 1A → 1T 1g 1g 1A → 1T 1g 2g

t2g → eg t2g → eg t2g → eg t2g → eg

0.36 0.93 1.52 2.57

0.67 1.26 1.79 2.88

0.38 0.99 1.57 2.63

0.46 1.03 1.60 2.66

0.11 0.10 0.08 0.09

Ni II

3A

2g

t2g → eg t2g → eg spin-flip

0.86 1.66 1.87

0.97 1.78 1.86

0.92 1.75 1.86

0.92 1.75 1.86

0.06 0.08 −0.01

1g

→ 5 Eg

→ 3 T2g 3A → 3T 2g 1g 3A → 1E g 2g

a All

values in eV. b Calculations performed at the CASPT2/T(D) level, Ci symmetry, and the following choice of active space: Ti III (1,5); V III (2,5); Cr III (7,7)fs; Mn II (9,12)fs; Fe III (9,12); Fe II (6,5); Co III (10,12); Ni II (8,5); note that details of these calculations (like basis set, number of states included, and active space) are different than in the final set of calculations reported below. c Excitation energies are averaged for multiplets of close-lying states and electronic states are orbitals and customarily labelled under the Oh point group although the actual computational symmetry is lower (see Section 3.2.1). d Geometry optimized in gas phase. e Geometry optimized within the COSMO model (water). f Defined as the difference between excitation energy for the [M(H2 O)18 ]n+ model (COSMO geometry) the [M(H2 O)6 ]n+ model (gas phase geometry).

13



simplest possible approximation—i.e., combining small cluster [M(H2 O)6 ]n+ with the implicit solvation model—is generally insufficient, as it may either strongly underestimate (Mn II , Fe III , V III ) or overestimate (Co III ) the total solvation effect. (Note, however, that changes observed with respect to the same cluster in gas phase are in the correct direction and on the right order of magnitude. Moreover, for a few cases, like Ti III complex, even this simple approach reproduces the solvation effect accurately.) When comparing different metals and different electronic transitions, solvation effects vary from rather negligible 0.04–0.05 eV up to very significant 0.3–0.5 eV. This variation can be rationalized based on two factors: the type of electronic excitation and the charge of ion. Regarding the first factor, it is useful to differentiate between two types of transitions: (a) t2g → eg and eg → t2g transitions, which involve redistribution of electrons between the t2g and eg levels; (b) spin-flip transitions, which do not involve such redistribution. It is clear from data in Table 2 that solvation effects are far more pronounced for t2g → eg and eg → t2g transitions than for spin-flip transitions. For instance, in the case of Fe III , the first two excitations (6 A1g → 4 T1g , 6 A1g → 4 T2g ) relocate one electron from eg into t2g orbitals; accordingly, solvation effects for these transitons are large, about 0.4–0.5 eV. By contrast, the higher two transitions (6 A1g → 4 A1g , 4 Eg ) are spin-flips within the t2g and eg orbital levels (with no significant redistributions of electrons between the two levels); accordingly, the solvation effects are several times smaller. The similar trends can be observed for Cr III aqua complex: small solvation effects for 4 A2g → 2 Eg , 2 T1g (spin-flip) transitions and larger effects for 4 A2g → 4 T2g , 4 T1g (t2g → eg ) transitions. The second important factor is the charge of ion. Comparing between isoelectronic Fe III and Mn II complexes, the solvation effect on the corresponding (6 A1g → 4 T1g ) excitation energies is larger for the higher-charged iron complex. In general, the most significant solvation effects are observed for complexes of trivalent ions (Ti III , V III , Cr III ). A somewhat exceptional case of Co III deserves separate discussion below. The larger influence of solvation on the t2g ↔ eg excitation energies, is intuitive and can be traced back to the effects of solvation on the equilibrium ground-state (GS) geometry, specifically the metal–oxygen (M–O) bond distance. The variation of the M–O bond distance for different

14


Page 14 of 57

●

●

MnII

●

2.15

●

2.10

FeII

2.05

NiII TiIII

●

FeIII 2.00

°) M−O bond distance (A

●

●

●

VIII

●

●

CrIII

●

1.95

●

CoIII

1.90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.20

Page 15 of 57

gas

COSMO

[M(H2O)6]

n+

gas

COSMO

[M(H2O)18]n+

Figure 2: Average metal–oxygen distances computed for [M(H2 O)6 ]n+ and [M(H2 O)18 ]n+ models and geometries optimized either in gas phase or within the COSMO solvation model.

15



models is illustrated in Figure 2 showing that, in general, the M–O bond lengths are longer for bare unsolvated [M(H2 O)6 ]n+ model than for solvated [M(H2 O)18 ]n+ one. As found in our previous study of Fe III aqua complex, the observed contraction of the M–O bonds is coupled to elongation of the O–H bonds in water ligands, suggesting that the ligand O atoms become more basic in the presence of solvent, and thus they coordinate the metal ion more strongly than in gas phase. 53 Because the eg orbitals have σ -antibonding character with respect to the M–O bond and the t2g orbitals are approximately nonbonding, the shortening of the M–O distance by solvation always increases the t2g –eg splitting, i.e., the effective ligand field. This effect leads to either destabilization or stabilization of the excited state with respect to the GS, depending on the whether the excitation is of the t2g → eg or eg → t2g type. The analysis of M–O distances also helps to understand why solvation effects are small for Ni II and Co III complexes. For Ni II complex, the Ni–O distance is hardly affected by the treatment of solvation (cf Figure 2), and hence there is almost no effect on the excitations energies (cf Table 2). The situation is slightly more complicated for Co III complex. All four ligand-field excitations for Co III involve redistribution of electrons between t2g and eg levels and thus their energies should be sensitive to the Co–O distance. The Co–O distance is also quite sensitive to the treatment of solvation; for instance, it shrinks by almost 0.05 Å when [Co(H2 O)6 ] 3+ geometry is reoptimized within the COSMO model (cf Figure 2), and the excitation energies, indeed, vary as expected (cf Table 2). However, it turns out that the optimized Co–O distance is very similar for unsolvated [Co(H2 O)6 ] 3+ model as for the COSMO-solvated [Co(H2 O)18 ] 3+ one (cf Figure 2); this helps to rationalize why the total solvation effect predicted by our calculations is far smaller than it might be expected for this triply-charged ion. Although correlations with the M–O distance are clearly useful, not all results in Table 2 can be rationalized in this simple way. For instance, when small cluster [M(H2 O)6 ]n+ is optimized within the COSMO model, the contraction of the M–O bond is always overestimated compared with the more reliable solvation model (cf Figure 2), but the resulting effect on the excitation energy can be either underestimated (the lowest transitions for Fe III , Mn II ) or overestimated (Co III ). Moreover,

16


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although comparable contractions of the M–O distances (by 6–7 × 10−3 Å) are observed for Mn II and Co III complexes when going from the [M(H2 O)6 ]n+ (gas-phase) to [M(H2 O)18 ]n+ (COSMO) models, the resulting solvation effects on the excitation energies differ by a factor of two. Therefore, it would be oversimplified to reduce the observed solvation effects only to variations of the M–O distance.

3.2

Ligand–Field Excitation Energies

3.2.1 General Comments Below, we discuss all experimentally evidenced ligand–field excitation energies for first-row TM aqua complexes in the sequence from Ti III (d1 ) to Ni II (d8 ); see Tables 3–15. The experimental data are taken from the Jørgensen’s book 50 and original experimental works, 71–73 including also recent ones. 51,74 Note that the experimental data of M III aqua complexes were obtained in strongly acidic solution to prevent their hydrolysis. When comparing theory with experiment, we customarily label the electronic states by irreps of the Oh point group—as in the LFT and most experimental studies—even though lower symmetry (Ci or C1 ; see Section 2.1) is used in the computations. It is obvious that the degeneracies of states predicted by the LFT under Oh symmetry will be lifted for Ci /C1 models, but an approximate quasi-degeneracy is usually maintained in our calculations for groups of states arising from a common term in LFT. For instance, in the case of Fe III aqua complex, the six quartet states arising from the 4 T1g and 4 T2g terms appear in two multiplets (triplets) separated by 0.65 eV, whereas the splitting of energy levels within each multiplet does not exceed 0.15 eV (cf Table 10). Note also that the splitting between electronic states within a given multiplet is on the same order of magnitude as typical widths of the experimental d–d bands (∼ 1–2 × 103 cm−1 , i.e., ∼ 0.12–0.25 eV). Therefore, it is usually possible to match a multiplet of close-lying computed states to the experimental band position and its designation within the LFT (using Oh irreps). For the sake of comparison with the experimental band position, we take the average energy for the multiplet of close-lying computed levels, which will be indicated by curly braces in the tables below. (The 17



lowest component arising from the ground state term is always taken as the zero level for computation of vertical excitation energies.) Note that precise matching of the computed levels to the LFT terms becomes not possible (and not useful either) if two or more LFT terms are lying close in energy, so that the resulting multiplets interpenetrate and mix with each other. Such groups of states will be designated, collectively, as originating from several involved LFT terms (e.g., dense manifold of S = 1 states for Fe II aqua complex; see Table 12). For comparison between theory and experiment we assume that the position of band maximum coincides with the vertical energy (average value for multiplets of close-lying energy levels) computed for the equilibrium GS geometry. The association of the band maximum with the vertical excitation energy is the standard approximation in the spirit of the Franck–Condon principle. It is also well established in the literature for interpretation of optical spectra of large systems in condensed phases (where vibrational fine structure is usually blurred), including TM complexes and aqua complexes in particular. 13,15 The accuracy of this approximation may be inferred from model studies in which the vibrational effects were approximately accounted for. 75,76 In a relevant study by Landry-Hum et al. vibrationally-resolved spectra of [V(H2 O)6 ] 3+ and [Ni(H2 O)6 ] 2+ models were computed in gas phase under the simplifying approximation of only one effective vibrational mode. 75 The results obtained show that the maxima of spectral envelope (surrounding the simulated vibronic spectrum) coincide with the computed vertical excitation energies to within 500 cm−1 (0.06 eV). 77 Also, it is known from experimental studies 78 that thermal vibrational effects may change the positions of band maxima by 200 to 500 cm−1 . Overall, we believe that positions of band maxima obtained from the experimental spectra should approximate the vertical electronic energy differences with an estimated maximum error bar of ∼ 0.1 eV. Note, however, that complexes showing Jahn–Teller (JT) effect in the excited state (Ti III 2 Eg , Fe II 5 Eg ) requires special treatment when comparing our static computations with the experimental data (see below).

18


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Table 3: Electronic Energies of Ligand–Field States for d1 [Ti(H2 O)18 ] 3+ .a,b,c CASPT2 (5,7)fs

NEVPT2 (5,7)fs

expd

0 0.09 0.45

0 0.09 0.45

0

2T 2g

2E

2.37 2.41

g

2.39

2.39 2.42

2.41

2.33e

a All

values in eV. b Energies from eq. (4), Section 2.2. annotated by curly braces indicate average energies for multiplets of close-lying excited states (see Section 3.2.1). d Experimental data from ref 50. e Average of 2.52 eV and 2.14 eV (see text). c Values

3.2.2 Ti III , d1 configuration For Ti III (d1 ) complex, only one ligand-field transition, 2T2g →2 Eg , is expected under the octahedral symmetry. Under lower symmetry the involved terms are split into more energy levels, like in our calculations for [Ti(H2 O)18 ] 3+ model, whose results are given in Table 3. The pattern in which the 2 T2g - and 2 Eg -based energy levels split in our calculations is reminiscent of almost trigonal symmetry of the model (cf Figure 1(a)), even if the real computational symmetry is lower. The 2T 2g

level splits into three levels, but the first two remain almost degenerate (to within 0.1 eV)

compared with the splitting between any of them and the third level (0.4 eV). The energy levels of the 2 Eg origin remain almost degenerate (to within 0.05 eV). Comparably small splitting of the two 2 Eg -based energy levels was also obtained in previous computational studies for [Ti(H2 O)6 ] 3+ model 15,79,80 , consistently with the expectation of only weak JT distortion for the GS of a d1 complex. Experimentally, the 2T2g →2 Eg transition appears as a broad, asymmetric band which can be resolved into 2.52 and 2.16 eV Gaussian peaks. 50,79b, 80 At first sight, such a large splitting (0.36 eV) might be regarded 15 as inconsistent with the calculations. However, the experimentally observed splitting is caused by JT effect in the excited state (2 Eg ), not in the GS. 76,80 In such a case, the calculation of vertical energies for the equilibrium geometry of the GS is fundamentally un-

19



able to explain the observed splitting of the absorption band. 76 However, as was also proposed by Neese et al., 13 we can use the arithmetic mean of positions of the two experimental peaks in order to estimate the vertical excitation energy for the GS geometry. This leads to the experimental estimate of 2.33 eV given in Table 3. Note that in order to describe the shape of the experimental 2T 2g

→ 2 Eg band one should take vibrations into explicit account, 76 which is clearly beyond the

scope of this work. However, model considerations of this excited-state JT problem given by Reber and Zink (section 3.4 in ref 76) show that the above averaging procedure should yield a very good approximation of the vertical energy difference. Both CASPT2 and NEVPT2 calculations, giving the excitation energy of ∼ 2.4 eV, agree quite well with the (averaged) experimental estimate. The agreement is much better than in the previous study by Yang et al. (calculated vertical energy 1.7–1.8 eV). 15 Part of the discrepancy can be traced back to the neglect of solvation effect in ref 15. As shown in Section 3.1, solvation effects are quite important for Ti III aqua complex, contributing almost 0.4 eV to the the excitation energy (see Table 2). In another related work, Maurelli et al. applied time-dependent DFT method with the B3LYP functional, yielding the excitation energies between 2.2 and 2.5 eV (depending on the precise choice of geometry for the model). 80 Since all their calculations were performed for bare [Ti(H2 O)6 ] 3+ ion in gas phase, the remarkably good agreement with experiment may be caused by a cancellation between the neglect of solvation and an intrinsic error of the DFT:B3LYP method. 3.2.3 V III , d2 configuration Previous multireference calculations for hydrated V III ion severely underestimated the lowest transition energy, 1 3 T1g → 3 T2g , experimentally at 2.21 eV. 50 The calculations by Landry-Hum et al. yielded 1.4 eV 75 and those by Yang et al. 1.5–1.6 eV. 15 By contrast, the present CASPT2 and NEVPT2 calculations agree up to 0.1 eV with the experimental value (Table 4). The improvement over the previous studies is mainly due to the inclusion of solvation effects which contribute ∼ 0.5 eV to the excitation energy (cf Table 2). A slightly worse agreement is obtained for the second transition, to the 2 3 T1g state, whose energy is overestimated by 0.18 / 0.27 eV, by CASPT2 / NEVPT2

20


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Table 4: Electronic Energies of Ligand–Field States for d2 [V(H2 O)18 ] 3+ .a,b,c

1 3T1g

3T 2g

2 3T1g

CASPT2 (6,7)fs

NEVPT2 (6,7)fs

expd

0 0.37 0.39

0 0.39 0.41

0

 2.03  2.19 2.14  2.22  3.20  3.45 3.37  3.46

 2.20  2.35 2.30  2.37  3.32  3.54 3.46  3.54

2.21

3.19

a All

values in eV. b Energies from eq. (4), Section 2.2 c Values annotated by curly braces indicate average energies for multiplets of close-lying excited states (see Section 3.2.1). d Experimental data from ref 50.

calculations, respectively. For comparison, the calculations in ref 75 underestimated this excitation energy by more than 0.5 eV (presumably due to missing solvation effects). The calculations in ref 15 gave very good agreement with the experimental estimate, but since the first excitation energy was not reproduced well enough (see above), a mere cancellation of errors cannot be excluded. 3.2.4 V II and Cr III , d3 configuration Aqua complexes of both metals have the 4 A2g GS arising from (t2g )3 (eg )0 configuration. Their optical spectra are dominated by transitions to other quartet states. The first two spin-allowed transitions, to 4 T2g - and 1 4 T1g -based excited states, are well resolved for both metals 50 and their energies are reproduced very well by our calculations (Tables 5 and 6). The errors do not exceed 0.15 eV, except for NEVPT2 calculations of the 1 4 T1g excitation for Cr III (0.24 eV discrepancy). This is to be compared with errors as large as 0.2–0.4 eV on the same set of transition energies from CASPT2 and MRCI calculations reported in ref 15. The situtation is slightly more complicated for the third spin-allowed transition, 4 A1g → 2 4T1g . First, one should notice that this transition is experimentally well resolved only for V II complex, 81 whereas for Cr III , it overlaps with the edge of an intense LMCT (ligand-to-metal charge transfer) 21



Page 22 of 57

Table 5: Electronic Energies of Ligand–Field States for d3 [V(H2 O)18 ] 2+ .a,b,c CASPT2 (7,7)fs

NEVPT2 (7,7)fs

expd

0

0

Quartet states 4A

2g

0

4T 2g

1.43 1.43 1.59

1 4T1g

2.34 2.35 2.44

2 4T1g

3.49 3.49 3.97

  1.48    2.38    3.65 

Doublet states 1.60 2E 1.61 g 1.67 and 2T 1.68 1g 1.78

      1.67     

1.53 1.53 1.74

3.53 3.53 3.96

  1.60    2.43    3.67 

1.56 1.58 1.68 1.69 1.78

      1.66     

2.40 2.41 2.48

a All

1.53

2.29

3.46

1.62

values in eV. b Energies from eq. (4), see Section 2.2 c Values annotated by curly braces indicate average energies for multiplets of closelying excited states (see Section 3.2.1). d Experimental data from ref 50.

22


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Table 6: Electronic Energies of Ligand–Field States for d3 [Cr(H2 O)18 ] 3+ .a,b,c CASPT2 (7,7)fs

NEVPT2 (7,7)fs

expd

0

0

Quartet states 4A

2g

0

4T 2g

2.07 2.07 2.24

1 4T1g

3.09 3.09 3.30

2 4T1g

4.81 4.81 5.17

Doublet states 1.93 2E 1.93 g 2.07 and 2T 2.07 1g 2.10 2T 2g

  2.13    3.16    4.93 

1.93   2.08   2.88  3.06 3.00  3.07

2.20 2.21 2.43 3.23 3.22 3.41 4.87 4.88 5.21

  2.28    3.29    4.99 

1.86 1.86 1.86  2.02  2.02 2.03  2.05  2.95  3.06 3.02  3.07

a All

2.16

3.05

∼ 4.7e

1.86 –

∼ 2.6e

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of closelying excited states (see Section 3.2.1). d Experimental data from ref 50. e Accuracy and reliability of these data is questionable, see text.

23



band 78 and hence the excitation energy cannot be precisely determined. Although the energy of this transition for Cr III is occasionally given in some literature reviews, 50,82 we found no proof for the existence of a detectable band by referring to the source works quoted therein. Second important difference is that the energy levels arising from the 2 4T1g term are predicted to be more noticeably split than those arising from 4 T2g and 1 4 T1g terms. (The larger sensitivity of 2 4T1g to symmetry breaking effects may be rationalized by principal contribution of the doubly excited configuration, (t2g )1 (eg )2 .) Comparison with the average energy of the three computed 2 4T1g based levels, leads to rather significant discrepancies with respect to the experimental data (0.2 eV for V II , 0.3 eV for Cr III , but note that the experimental value for Cr III is uncertain). Making an ad hoc assertion that the experimental band arise from the two lower energy levels, whereas the third one is not observed (e.g., beyond the energy scale for V II , 81 obscured by the LMCT band for Cr III 78 ), would lead to a much better agreement. However, we believe that the splitting of the 2 4T1g -based energy levels may be somewhat overestimated by the present static calculations and taking the average of the three energy levels is methodologically more sound. Other d–d transitions observed for these d3 aqua complexes are spin–forbidden ones: 4 A1g → 2E

g,

2T 1g

and 2 T2g . Note that the 4 A1g → 2 T2g excitation energy was only reported for Cr III , but

the existence of band is not evident from the original spectrum and only appears through Gaussian analysis, whose details were not given in ref 50 and in the original works quoted therein. Thus, the experimental datum for this excitation energy is elusive and we are not surprised by its relatively poor agreement with the calculated values (2.6 vs 3.0 eV; cf Table 6). Note that Neese et al. do not consider this suspicious transition in ref 13. The agreement with experiment is much better for the 2 Eg and 2 T1g excited states, for which the experimental excitation energies are also more certain (cf Tables 5 and 6). Both 2 Eg and 2 T1g are spin-flip excited states originating from the same electronic configuration as the GS, (t2g )3 , and hence their energies are only weakly affected by the ligand field strength, resulting in sharp bands (known also for other Cr III complexes 56 ). According to the LFT, the doublet states 2 Eg and 2T 1g

are almost degenerate under the Oh symmetry, but this is not necessarily the case in a lower-

24


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symmetry environment. Although Jørgensen in his influential book 50 only reports one energy for both states, it is noteworthy that Cr III sites in glasses show usually two spin-forbidden bands in this spectral region 83 and the same holds true for Cr III solutions in water–sulfuric acid mixtures of varying concentration. 84 Interestingly, while the 2 Eg , 2 T1g -based states are computed to remain almost degenerate for V II (cf Table 5), they are predicted to be noticeably split by 0.15 eV for Cr III (cf Table 6). Out of the two computed energy levels, the lower-energy one (1.86–1.93 eV) is very close to the 1.86 eV feature in the experimental spectrum of Co III . The higher energy one (2.03–2.08 eV) is presumably too close to the the spin-allowed band 4 A1g → 4 T2g , to be resolved in the spectrum. Comparing our calculations with the earlier work, 15 we note that Yang et al. were unable to explain the 1.86 eV band of Cr III , since the lowest doublet excitation obtained from their CASPT2 and MRCI calculations was 2.23 and 2.35 eV. For V II complex, likewise, the lowest doublet excitation energy was noticeably overestimated (1.9–2.0 eV). 15 The present calculations give a much better agreement with experiment for these spin-forbidden bands. 3.2.5 Cr II and Mn III , high-spin d4 Formally, both complexes have the 5 Eg GS arising from the (t2g )3 (eg )1 configuration. This leads to a strong tetragonal distortion of the GS geometry due to the JT effect (see Figure 3) and noticeable splitting of energy levels with the 5 Eg and 5 T2g origin. The lowest ligand–field transition is between the two components of the 5 Eg origin. The higher energy transitions have 5 Eg → 5 T2g character. Some care must be taken when comparing with experimental results from the literature. It was reported 71,72 that the band positions for these d4 aqua complexes are particularly sensitive to experimental conditions. For instance, the position of the intra-5 Eg transition for Mn III aqua complex may vary by 2 × 103 cm−1 when comparing spectra of different alums containing the same [Mn(H2 O)6 ] 3+ cation, and up to 3 × 103 cm−1 when comparing the solid state with solution spectra. 72 It is thus very important to use the solution (not crystal) spectrum for comparison with the computational results for the present model, which contains only water molecules in the second

25



Figure 3: Structure of [Cr II (H2 O)18 ] 2+ , type (a) conformation optimized without symmetry, showing considerable tetragonal distortion of the first coordination sphere due to JT effect. The Cr−O bond lengths are 2.485 and 2.491 Å (dashed lines) and 2.054–2.055 Å (solid lines). coordination sphere and thus may not be appropriate to describe the spectra of [Mn(H2 O)6 ] 3+ in alums with different composition of the second coordination sphere. Table 7: Electronic Energies of Ligand–Field States for d4 [Cr(H2 O)18 ] 2+ .a,b,c

5E

g

5T 2g

CASPT2 (8,12)fs

NEVPT2 (8,12)fs

0 1.31

0 1.33

1.65 2.02 2.04 2.06

1.66 2.04 2.06 2.07

expd

1.18 1.75 2.11

a All

values in eV. b Energies from eq. (4), see Section 2.2. annotated by curly braces indicate average energies for multiplets of close-lying excited states (see Section 3.2.1). d Experimental data from ref 71. c Values

As shown in Tables 7 and 8, our calculations give generally good agreement with the experimental solution spectra 71,72 regarding both the intra-5 Eg and the 5 Eg → 5 T2g transitions. It is remarkable that the 5 T2g term is predicted to be split by ∼ 0.3 eV for aqua complexes of both metals. This is in agreement with the experimental spectrum for Cr II aqua complex where, indeed, the higher energy band is asymmetric and can be resolved into two Gaussian peaks (centered at 1.75 and 2.11 eV; see figure 3 in ref 71). Unfortunately, for Mn III aqua complex studied in ref 72, the 26


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Table 8: Electronic Energies of Ligand–Field States for d4 [Mn(H2 O)18 ] 3+ .a,b,c

5E

g

5T 2g

CASPT2 (8,12)fs

NEVPT2 (8,12)fs

expd

0 1.30

0 1.34

0 1.55

 2.57  2.93 2.82  2.97

 2.56  2.95 2.83  2.99

2.58

a All

values in eV. b Energies from eq. (4), Section 2.2. annotated by curly braces indicate average energies for multiplets of close-lying excited states (see Section 3.2.1). d Experimental data from ref 72 (solution spectrum in 6M H2 SO4 ). c Values

solution spectrum was not reproduced graphically and only one number was reported to describe the postion of the 5 Eg → 5 T2g band. Therefore, we take the number given in ref 72 for comparison with the average energy of the three transitions with the 5 Eg → 5 T2g origin (cf Table 8). This leads to a moderate error of 0.25 eV. The error would be much smaller if the experimental value were compared with the lowest excitation energy arising from our calculations, like in the case of Cr II (see above). Whereas we have not enough experimental data for Mn III aqua complex in solution to verify this hypothesis, the present calculations indicate that the 5 Eg → 5 T2g band should be also resolvable into two Gaussian peaks. Although quantitative comparison between solid state data and our solution model should be avoided, we note that the 5 Eg → 5 T2g band for related CsMn III alum is, indeed, broad and asymmetric. 85 3.2.6 Mn II and Fe III , high-spin d5 Both complexes have the high-spin GS (6 A1g ), and all ligand–field transitions are spin-forbidden. In the spectrum of Fe III aqua complex one may distinguish transitions to the following excited states (in the order of increasing energy): 4 T1g , 4 T2g , 4 Eg and 4 A1g (the latter two being almost degenerate). All higher ligand–field states (above 3 eV) are obscured by an onset of the LMCT band (with maximum at 5.2 eV). 50 However, for Mn II the analogous LMCT occurs at higher energy and additional ligand–field states can be observed (in the order of increasing energy): 2 4 T2g , 27



2 4 Eg , and 2 4 T1g . 50 For Mn II there is also a noteworthy band at 3.29 eV (not described in the Jørgensen’s book 50 ) which was assigned to the 6 A1g → 2 T2g transition. 73 This is a rare example of a doubly spin-forbidden transition (|∆S| = 2) conclusively observed for a TM complex, making it a valuable benchmark for theory! As shown in Tables 9 and 10, our results for these high-spin d5 complexes give good agreement with the experimental data. With maximum discrepancies of 0.15 eV (NEVPT2) or 0.3 eV (CASPT2), the present calculations greatly improve over the previous CASPT2 and MRCI ones, 15 where discrepancies with the experimental data were as large as 1.3 eV for the lowest transition in Fe III complex and 0.6 eV for analogous transition in Mn II complex. Based on the presently calculated excitation energies, it is possible to directly match all the experimental d–d bands, and there is no need for any of the previously suggested controversial reinterpretations of the experimental data 1,15, 86 (see Introduction). A noteworthy detail is the assignment of the characteristic sharp and double-head band found in the experimental spectra of these high-spin d5 complexes at 3 − 3.1 eV (cf figures 4 and 5 in ref 56). As is well recognized based on the LFT considerations, this band must originate from excitations to the pair of quasi-degenerate states 4 Eg and 4 A1g because only for these spin-flip states the d–d excitation energies are (almost) insensities to the strength of the ligand field, and thus could generate two sharp and almost coincident absorption peaks. Our present results are consistent with this assignment, which was not the case in many previous theory works. 1,15,86 A considerable improvement with respect to the previous studies is not only due to accounting for solvation effects, but also due to a more balanced treatment of correlation effects (proper active space, correlating the outer–core electrons; see Methodology). Indeed, even for bare [Fe(H2 O)6 ] 3+ cluster in gas phase (cf Table 2) our calculated excitation energies are closer to the experimental values than the corresponding results from ref 15. This being said, we note that for these d5 complexes, compared with other complexes studied in this work, we have somewhat larger discrepancies between the computational and experimental results. Moreover, the difference between CASPT2 and NEVPT2 results is larger than for the

28


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Table 9: Electronic Energies of Ligand–Field States for d5 [Mn(H2 O)18 ] 2+ .a,b,c

Sextet state 6A 1g

CASPT2 (9,12)fs

NEVPT2 (9,12)fs

expd

0

0

0

Quartet states 1 4T1g

1 4T2g

1 4Eg and 4A1g

2 4T2g 2 4Eg 2 4 T1g

 2.35  2.46 2.42  2.46  2.82  2.92 2.89  2.93  3.14  3.14 3.14  3.15  3.52  3.53 3.53  3.54 3.80 3.80 3.80  4.26  4.31 4.30  4.32

 2.15  2.24 2.21  2.25  2.66  2.77 2.73  2.78  3.00  3.00 3.01  3.02  3.42  3.43 3.43  3.44 3.70 3.71 3.71  4.17  4.20 4.19  4.22

 3.48  3.60 3.58  3.61

 3.05  3.20 3.15  3.20

2.34

2.86

3.12e

3.47

3.69

4.09

Doublet states 2T 2g a All

3.29 f

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of closelying excited states (see Section 3.2.1). d Experimental data from ref 50, unless noted otherwise. e Sharp peak with the maximum at 3.09 eV and a shoulder at 3.14 eV. f Ref 73.

29



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Table 10: Electronic Energies of Ligand–Field States for d5 [Fe(H2 O)18 ] 3+ .a,b,c

Sextet state 6A 1g

CASPT2 (9,12)

NEVPT2 (9,12)

expd

0

0

0

Quartet states 4T 1g

4T 2g

4A

1g

and 4Eg

 1.76  1.88 1.84  1.89  2.39  2.52 2.48  2.54  3.21  3.27 3.25  3.28

 1.44  1.54 1.51  1.55  2.08  2.22 2.18  2.22  2.98  3.04 3.02  3.04

1.56

2.29

3.03e

a All

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of closelying excited states (see Section 3.2.1). d Experimental data from ref 50. e Sharp peak with a shoulder, resolved into 3.05 and 3.01 eV.

other studied complexes, and the results of both methods for Fe III complex are more sensitive to the choice of active space (see Table S4, Supporting Information). This is consistent with the earlier opinion by Neese et al. that these high spin d5 systems are particularly challenging for computational treatment. 13 Therefore, we decided to further investigate the lowest transition energy 6A

1g

→ 4 T1g for Fe III aqua complex with other methods, including MRCI and high-level coupled

cluster methods. In order to facilitate computationally more expensive methods, this part of calculations was performed for bare [Fe(H2 O)6 ] 3+ cluster in gas phase, but the experimental reference was back-corrected by subtracting the estimated solvation effect (−0.50 eV, from Table 2). Except otherwise noted, all calculations were performed with the T(D) basis set. Basis set incompleteness error (BSIE) can be estimated as −0.05 to −0.10 eV upon comparison with CASPT2-CBS, CCSD(T)-CBS, and CCSD(T)-F12 results. The comparison of methods provided in Table 11 may be considered an extension of an analogous comparison in ref 53, where the highest level of theory was CCSDT. Now, we are able to

30


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Table 11: Excitation Energy 6 A1g → 4 T1g for [Fe(H2 O)6 ] 3+ Model.a method

∆E (eV)

Multireference CASPT2(9,12)b CASPT2(9,12)-CBSc NEVPT2(9,12)b MRCISD(9,12)d MRCISD+Q(9,12)d,e SORCI(5,5) f CASPT2(5,5) NEVPT2(5,5) MRCISD+Q(5,5)c,d

2.40 2.35 1.99 2.51 2.32 2.07 2.65 2.30 2.63

Single-reference CCSD(T)b CCSD(T)-CBSc CCSD(T)-F12b CCSDT(Q) estimateg

2.21 2.16 2.11 2.16

Exptl (back-corrected)h

2.06

a All

calculations except SORCI (from ref 13) were performed with the T(D) basis set, DK Hamiltonian, 3s3p electrons correlated. b From ref 53 (including supporting information). c Extrapolation using eq (1). d Internally-contracted MRCI with singles and doubles. 32 e With the Davidson correction, relaxed reference (fixed reference gives the same results to within 0.01 eV) f From ref 13. g Estimated by correcting the ROHF-CCSD(T) value (2.21 eV, above) for the difference between UHFCCSDT(Q) (2.48 eV) and UHF-CCSD(T) (2.53 eV) estimated using the def2SVP basis set, nonrelativistic Hamiltonian and without the 3s3p correlation; UHFCCSDT(Q) calculations were performed with the MRCC code 87 h Experimental value (1.56 eV) back-corrected for the estimate of solvation effect (0.50 eV) from Table 2.

31



compare even with the estimate from very expensive CCSDT(Q) calculations. Due to the high computational cost, calculations at this level of theory are very rare for TM complexes. The presently obtained CCSDT(Q) estimate approaches the (back-corrected) experimental value very closely, especially after accounting for the remaining BSIE (see above). Moreover, we note that the CCSDT(Q) estimate differ by only 0.05 eV (1 kcal/mol) from the CCSD(T) result, assuring about the high accuracy of the CCSD(T) approximation even for this challenging TM complex. Paradoxically, the situation is more problematic for multireference calculations. Among them, NEVPT2(9,12) (this work) and SORCI(5,5) (quoted from ref 13) give the best agreement with experimental estimate (back-corrected for solvation effect). Interestingly, NEVPT2(9,12) is the only method in Table 11 which slightly underestimates the experimental value (and it would even more if the BSIE was compensated). Other interesting observations can be made regarding the MRCISD method. First of all, the Davidson correction (+Q) makes a relatively important contribution to the excitation energy (−0.2 eV), showing that size-extensivity is crucial for these spin-forbidden excitations. Secondly, even MRCISD+Q(9,12) is not much more accurate than CASPT2(9,12) and it appears to be less accurate than NEVPT2(9,12). Finally, from a difference of 0.3 eV between the MRCISD+Q(9,12) and MRCISD+Q(5,5) results, we conclude that variational MRCISD+Q results are comparably sensitive to the choice of active space as the perturbational ones (CASPT2, NEVPT2). Neither of these methods can account for correlation effects in [Fe(H2 O)6 ] 3+ accurately when using the minimal active space. The SORCI(5,5) calculations from ref 13 appear to give almost perfect agreement with the back-corrected experimental estimate. Unfortunately, the calculations with this method were only performed with the minimal active space (5,5), which we found insufficient for other multireference methods (see above). It might be that the SORCI method—due to its rather complicated construction—is balancing the differential correlation effects in a more efficient way than MRCISD+Q, CASPT2, and NEVPT2 methods do, so that SORCI is capable of producing a very good excitation energy even with the minimal active space. However, we note that the SORCI calculations in ref 13 were carried out for the crystalline Fe–O distance (not the one optimized in gas

32


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phase), which allowed them to partly capture the environmental effects discussed above. Since the solvation effect is roughly comparable to the effect of enlarging the active space, some error cancellation cannot be excluded. 3.2.7 Fe II , high-spin d6 The Fe II aqua complex has the quintet GS 5 T2g arising from (t2g )4 (eg )2 configuration. The spectrum is dominated by a spin–allowed t2g → eg excitation to the 5 Eg excited state. The resulting band is split into two peaks (centered at 0.99 and 1.29 eV) due to the excited-state JT effect. 88 Like for the related case of of Ti III aqua complex (Section 3.2.2), the calculation of vertical excitation energies are not supposed to reproduce the splitting of the experimental band, and we take the average energy of the two observed peaks to approximate the vertical energy. The resulting estimate (1.14 eV) is reproduced by all calculations in Table 12 up to 0.13 eV. The spectrum of Fe II aqua complex also contains a number of spin-forbidden transitions, attributed to various triplet excited states. Jørgensen reports the following triplet bands: 1.79, 2.45, 2.62, 2.75, and 3.21 eV. 50 In the recent experimental study by Fontana et al. additional bands at 1.98, 2.23, and 2.85 eV were identified. 51 Note that it is impossible (in most cases) to firmly assign these bands to individual triplet terms of the LFT due to the high number of close-lying triplet states, resulting in a very complicated shape of the experimental spectrum. 51 Our calculations are able to explain all the triplet bands identified in Jørgensen’s book. 50 The groups of three lowest and three highest triplet states are well isolated from other triplet states and can be matched to the experimental bands at 1.79 and 3.21 eV, respectively. The calculations also predict a group of triplet states around 2.0–2.1 eV matching the bands at 1.98 and 2.23 eV reported by Fontana et al. The dense manifold of triplet state predicted between 2.3 and 3.0 eV is consistent with the complicated shape of the spectrum in this region and we, obviously, make no attempt to firmly match the computed states to LFT terms or individual experimental bands. Here, it must be noticed that due to large number of triplet states to be considered in the calculations, we were not able to carry out NEVPT2 calculations with the standard active space, (10,12)fs, and thus NEVPT2 results are only

33



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Table 12: Electronic Energies of Ligand–Field States for [Fe II (H2 O)18 ] 2+ .a,b,c CASPT2 (10,12)fs

CASPT2 (6,5)

NEVPT2 (6,5)

expd

0 0.04 0.28

0 0.04 0.26

0

Quintet states 5T 2g

0 0.04 0.27

5E

1.23 1.27

g

1.25

1.25 1.29

1.27

1.21 1.25

1.23

1.14e

Triplet states  1.65  1.67 1.68  1.73  2.09  2.10 2.11  2.15

 1.69  1.72 1.72  1.75  2.10  2.12 2.14  2.19

 1.45  1.47 1.49  1.56  1.97  1.97 1.98  1.99

mixed: 2 3T1g , 3E , g 3 2 T2g , and 3 3 T1g

2.41 2.53 2.55 2.65 2.68 2.73 2.74 2.82 2.84 2.91 2.93

2.39 2.52 2.54 2.63 2.67 2.73 2.74 2.81 2.81 2.88 2.90

2.31 2.43 2.45 2.58 2.62 2.68 2.74 2.75 2.75 2.86 2.89

3 3T2g

 3.27  3.28 3.29  3.31

 3.31  3.32 3.33  3.36

 3.28  3.29 3.30  3.31

2.39

1.75

1 3 T1g

1 3 T2g

Singlet state 1A 2.24 1g a All

1.79

1.98 f , 2.23 f                 2.45,   2.62,  2.75,     2.85 f             3.21

0.74 f ,g

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of close-lying excited states (see Section 3.2.1). d Experimental data from ref 50, unless noted otherwise. e Average of the two maxima at 1.29 and 0.99 eV attributed to excited-state JT effect. f Ref 51. g The assignment of this band is questionable (see text).

34


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reported for the minimal active space (6,5). At CASPT2 level, both choices of the active space lead to similar results. Comparing with the earlier work by Yang et al., the 5 T2g → 5 Eg excitation was predicted at considerably lower energy (0.8–0.9 eV) than found here. Since the minimum active space is appropriate for this excitation energy (cf Table 12), the difference must be rooted mainly in the neglect of solvation effects in ref 15. Also, Yang et al. were unable to reproduce the lowest triplet band, experimentally at 1.79 eV. Other triplet states were matched with comparable accuracy as in the present study, but obviously the interpretation of spectral region 2.3–3.0 eV must be highly speculative due to the reasons discussed above. 89 Undoubtedly, the most intriguing feature of the new experimental spectrum reported by Fontana et al. is a small band at 0.75 eV (6 × 103 cm−1 ), which was tentatively assigned 51 to the doubly spin-forbidden transition 5 T2g → 1 A1g . We note that this intriguing band has never been observed in other experimental studies or considered in previous computations. Our calculations strongly oppose the assignment of the 0.75 eV band to the singlet state because the computed excitation energy is higher by 1–1.5 eV (cf Table 12) and it is unlike that such big discrepancies can be explained by shortcomings of our computational protocol. We also performed comparative CCSD(T) calculations of the quintet–singlet splitting for gaseous [Fe(H2 O)6 ] 2+ , yielding the energy difference of 2.11 eV with the T(D) basis set. After accounting for the solvation effect of 0.31 eV (value not included in Table 2, but obtained analogously), the CCSD(T) estimate of the quintet–singlet energy gap is 1.80 eV. This estimate is very close to the NEVPT2(6,5) result and still by ∼ 1 eV too high compared with the excitation energy suggested in ref 51. In our opinion, the assignment of the 0.75 eV band to the 5 T2g → 1 A1g transition is also problematic on purely experimental grounds. Such a doubly-spin forbidden transition, if indeed observed, should generate not only a much weaker, but also a broader band than that reported in ref 51 (as the 5 T2g –1 A1g gap is strongly affected by the ligand field strength, the band should be subject to a considerable vibrational broadening). Since the problematic band falls in the near IR region, it may be a vibrational overtone or a combination band of H2 O.

35



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3.2.8 Co III , low-spin d6 Co III is the only first-row transition ion which gives a low-spin aqua complex. The 1 A1g GS corresponds to the (t2g )6 (eg )0 configuration, whereas the two lowest singlet and triplet excited states (1,3 T1g , 1,3 T2g ) originate from the (t2g )5 (eg )1 configuration. Jørgensen only reports two singlet transitions at 2.05 eV and 3.10 eV, 50 but more recently Wangila and Jordan also identified the spin–forbidden transition 1 A1g →3 T2g at 1.56 eV. 74 Our computations (Table 13) allow to reproduce all three experimentally resolved transitions with moderate accuracy (see below) and confirm that the 1A1g →3 T1g transition occurs at too low energy to be observed. Table 13: Electronic Energies of Ligand–Field States for d6 [Co(H2 O)18 ] 3+ .a,b,c CASPT2 (10,12)

NEVPT2 (10,12)

expd

0

0

Singlet states 1A

1g

0

1T 1g

1.51 1.51 1.64

1T 2g

2.54 2.56 2.67

  1.55    2.59 

Triplet states  0.40  3T 0.41 0.45 1g  0.53  0.94  3T 0.96 1.00 2g  1.09

1.75 1.76 1.92 2.89 2.89 2.98

  1.81    2.92 

 0.69  0.71 0.76  0.87  1.26  1.30 1.31  1.38

2.05

3.10

–

1.56e

a All

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of closelying excited states (see Section 3.2.1). d Experimental data from ref 50, unless noted otherwise. e Ref 74.

Relatively significant discrepancies with experimental data (up to 0.56 eV for CASPT2 and 0.25 eV for NEVPT2) and between the two PT2 methods are observed here. This resembles the Fe III case (Section 3.2.6) and also here the discrepancies may be rooted in deficient description 36


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of correlation effects. Indeed, for transitions from the closed-shell GS to open-shell (singlet or triplet) excited states, the excitation energies are heavily influenced by dynamic correlation effects, describing of which may be challenging for CASPT2 and NEVPT2. Intriguingly, the PT2 corrections from both methods differ considerably and those from CASPT2 actually worsen the agreement with experimental results compared with the CASSCF values (Table S19). Unfortunately, in this case we cannot clarify the discrepancies by comparing with CC calculations since they can be only performed for the lowest triplet state, for which no experimental data are available. Alternatively, the discrepancies with experiment may be caused by still imperfect description of solvation. Although the total solvation effect estimated in Table 2 appears to be small, the excitation energies are, in fact, very sensitive to the Co–O bond lengths. Calculations for bare [Co(H2 O)6 ] 3+ with COSMO model point to much shorter Co–O distances than those obtained for the larger model used in this work (cf Figure 2) and to better excitation energies (cf Table 2). We note in passing that also the calculations of redox potentials in ref 54 for analogous [Co(H2 O)18 ]2+/3+ models, pointed to rather significant discrepancy with experiment. While this does not disprove the quality of our model, it might suggest that the second solvation sphere of Co III is still not perfectly described. We stress that for this Co III complex, correlation and solvation effects drive in the same direction, namely to stabilization of the closed-shell GS with respect to open-shell excited states. It is presently unclear which of these two factors (i.e., deficient description of correlation or solvation effects) matters more for the observed discrepancies, calling for further studies of related Co III complexes in different coordination environments. 3.2.9 Co II , d7 Co II aqua complex has 4 T1g ground state arising from (t2g )5 (eg )2 configuration. The experimental spectrum 50,90 contains intense absorptions at 1.02, 2.41, and 2.67 eV (the 2.67 eV peak was resolved from a shoulder of the 2.41 eV one) as well as weaker features at 1.40 and 1.98 eV. The assignment of bands in the literature is somewhat uncertain. Whereas the intense bands are due

37



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Table 14: Electronic Energies of Ligand–Field States for d7 [Co(H2 O)18 ] 2+ .a,b,c CASPT2 (11,12)fs Quartet states 0 4 0.17 1 T1g 0.17

NEVPT2 (11,12)fs

expd

0 0.17 0.17

0

4T 2g

 0.90  1.01 0.98  1.02

 0.97  1.06 1.03  1.06

1.02

4A

2.15

2.12

1.98

2.43 2.62 2.62

2.53 2.69 2.70

2.41 2.67

2g

2 4T1g

Doublet states 1.51 2E 1.52 g 1.53  2.14    2.16   2T   1g 2.17 2.20 and 2.21  2T   2g 2.25     2.26

1.32 1.34 2.10 2.14 2.16 2.21 2.24 2.24

1.33         2.18       

a All

1.40

–

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of closelying excited states (see Section 3.2.1). d Experimental data from ref 50.

38


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to spin-allowed transitions and the weaker band at 1.40 eV is due to a spin-forbidden one, the band at 1.98 eV may be either due to a spin-allowed or spin-forbidden transition. 50 Therefore, in our calculations we included a sufficient number of quartet and doublet states to account for both possibilities (note that close-lying states with the 2 T1g and 2 T2g origin are treated jointly). Our calculated results in Table 14 confirm that the band at 1.40 eV is due to the spin-forbidden transition, 1 4T1g →2 Eg . The results also demonstrate that the band at 1.98 eV can be explained by the spin-allowed transition, 1 4T1g →4 A2g . (The spin-forbidden transitions to 2 T1g , 2 T2g states are also predicted at around ∼ 2 eV and may potentially contribute to the absorption in this region.) Interestingly—and in accord with the original experimental interpretation 91 —our calculations suggest that the spectral features at 2.41 eV and 2.67 eV are associated with the same, 1 4T1g → 2 4T1g transition. This is because the calculations for solvated [Co(H2 O)18 ] 2+ model (with approximately trigonal symmetry) predict a noticeable splitting of ∼ 0.2 eV for the 2 4T1g -based energy levels. By contrast, for small model [Co(H2 O)6 ] 2+ the three energy levels originating from the 2 4T1g term remain degenerate to within 0.01 eV (see Table S6, Supporting Information). The negligible splitting of the 2 4 T1g level was also predicted by Atanasov et al. who employed the same small model and explained the presence of shoulder at 2.67 eV by vibronic coupling. 14 However, our results suggest that the splitting of the 1 4T1g → 2 4T1g transition can be satisfactorily explained by static distortion (trigonal) of the ground state geometry when the explicitly solvated model is used. By contrast, in ref 15 (employing the small unsolvated model), Yang et al. assigned the feature at 2.67 eV to a doublet state, inconsistently with the experimental interpretation 91 and the other theoretical study. 14 Moreover, the band at 1.40 eV remained unexplained in ref 15 (the doublet states of 2 Eg origin were predicted by ∼ 0.5 eV too high in energy). Our present calculations resolved both problems and helped to confirm the assignment of the 1.98 eV band. 3.2.10 Ni II , d8 Ni II aqua complex has the 3 A2g GS arising from (t2g )6 (eg )2 configuration. The experimental spectrum is dominated by three relatively intense bands due to spin–allowed transitions (to 3 T2g , 1 3 T1g ,

39



Page 40 of 57

and 2 3 T1g excited states) on the top of which there are less intense features attributed to spin– forbidden transitions (to 1 Eg and 1 T2g excited states). 50,90,92 The calculated excitation energies are given in Table 15. (Again, like for Fe II complex above, we were unable to complete NEVPT2 calculations with the standard active space due to the high number of triplet states. Hence, NEVPT2 results are given only for the minimal active space, but CASPT2 results for both choices of the active space are very similar.) The agreement with experimental data is very good, with most discrepancies below 0.1 eV and all of them below 0.2 eV (Table 15). Table 15: Electronic Energies of Ligand–Field States for d8 [Ni(H2 O)18 ] 2+ .a,b,c CASPT2 (12,12)fs Triplet states 3A 0 2g 3T 2g

0.95 0.97 1.07

1 3T1g

1.68 1.71 1.76

2 3T1g

2.92 3.05 3.44

  1.00    1.71    3.14 

Singlet states 1.82 1E 1.83 g 1.84  2.69  1T 2.75 2.76 2g  2.85

CASPT2 (8,5)

NEVPT2 (8,5)

expd

0

0

0

  0.93    1.75    3.16 

1.02 1.07 1.18

1.78  2.60  2.65 2.67  2.75

1.85 1.86

0.91 0.91 0.99 1.75 1.75 1.76 2.97 3.06 3.44 1.78 1.78

1.77 1.80 1.85 3.06 3.19 3.54

  1.09    1.81    3.26 

1.86  2.80  2.88 2.90  3.02

1.05

1.67

3.14

1.91

2.73

a All

values in eV. b Energies from eq. (4), Section 2.2. c Values annotated by curly braces indicate average energies for multiplets of close-lying excited states (see Section 3.2.1). d Experimental data from ref 50.

A somewhat intriguing feature is a noticeable splitting of the 2 3 T1g -based energy levels predicted by our calculations, as also observed for certain excited states of Cr III and V II aqua complexes, and presumable caused by considerable contribution of double excitations (see above). We

40


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believe that the splitting is exaggerated by static calculations for the present model. Dynamical simulation of the spectra would be necessary to predict the realistic shapes of the bands. 93 But even then, we suggest that the 3 A2g → 2 3 T1g band might have some asymmetry on the high energy side, a possibility which was unexplored so far in the interpretation of experimental data. Except for this little controversy, the agreement with experimental data for Ni II aqua complex is excellent, as was also the case in the previous study by Yang et al. 15 The good agreement of our results with ref 15 is not surprising since for Ni II aqua complex: (a) solvation effects are negligible (cf Table 2) and (b) extension of the active space beyond the minimal one is not necessary (cf Table 15). These two factors make Ni II aqua complex a particularly “easy case” for theoretical treatment. One should, however, not extrapolate this conclusion to other aqua complexes.

3.3

Accuracy of CASPT2 and NEVPT2

The accuracy of CASPT2 and NEVPT2 calculations for the studied ligand–field excitations is summarized in Figure 4, where correlation with experimental data and error statistics are given. Histograms of absolute errors can be found in Figure S2, Supporting Information. Upon excluding the dense manifold of triplets around 2.3–2.9 eV for Fe II complex (where assignment of individual bands is impossible) and the suspicious 5 T2g → 1 A1g transition for the same complex (which, we believe, was misassigned), our benchmark set counts 24 spin-allowed (∆S = 0) and 19 spinforbidden transitions (18 with |∆S| = 1, one with |∆S| = 2); see Table S7, Supporting Information. For this benchmark set, as is clearly shown in Figures 4 and S2, most of the computed results agree to within ±0.15 eV with the experimental data. (Here, one should recall from Section 3.2.1 that the implied uncertainty of the experimental data is ∼0.05–0.1 eV). The resulting values of mean absolute error (MAE) are 0.15 eV for CASPT2 and 0.13 eV for NEVPT2. The maximum errors are 0.56 eV for CASPT2 and 0.42 eV for NEVPT2. Just a few outliers are responsible for these maximum errors. Two of them (in the case of both CASPT2 and NEVPT2 methods) are the two transition for Cr III aqua complex (4 A2g → 24 T2g , 2 T2g ) for which the experimental record is very scarce (see Section 3.2.4). The other most 41



5 III 4

4

Cr ( A2g → 2 T1g)

CASPT2 NEVPT2 y = x ± 0.15

4.5

4

3.5 Computed (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 57

III 4

2

Cr ( A2g → T2g)

3

2.5

III 1

III 6

1

Co ( A1g → T2g)

4

Fe ( A1g → T1g)

2 III 1

1

Co ( A1g → T1g)

1.5

III 1

3

Co ( A1g → T2g)

1 1

1.5

2

CASPT2: MAE 0.15, Max 0.56 eV NEVPT2: MAE 0.13, Max 0.42 eV

2.5 3 3.5 Experimental (eV)

4

4.5

5

Figure 4: Correlation diagram showing CASPT2 and NEVPT2 results versus experimental data for the test set containing 19 spin-forbidden and 24 spin–allowed transitions of aqua complexes. Values of MAE (mean absolute error) and maximum error (Max) are annotated. Several outliers (discussed in the text) are indicated by arrows. significant discrepancies are obtained with CASPT2 for the 6 A1g → 4 T1g transition for Fe III aqua complex and for all three experimentally assigned (1 A1g → 1 T1g , 1 T2g , 3 T2g ) transitions for Co III aqua complex. As discussed in Section 3.2.6, comparison with high-level CC calculations indicates that excellent agreement with experimental data for Fe III complex can be obtained by improving the treatment of dynamic correlation effects. The situation is less clear for Co III aqua complex due to the lack of similar CC benchmark. As discussed in Section 3.2.8, the discrepancies observed for this complex may either indicate shorcomings of CASPT2 in description of dynamic correlation effects or limitations of the present solvation model. A more detailed analysis presented in Figure S2, Supporting Information, reveals that the error statistics are very comparable for spin–allowed and spin–forbidden transitions. Moreover, for the 42


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problematic Co III complex, equal discrepancies are observed for the singlet–triplet and singlet– singlet transitions (cf Section 3.2.8). These arguments suggest that describing the change of spin state itself is not particularly challenging for theory. The challenge is rather situated in describing relative energies of states with different occupations of the t2g and eg orbitals, due to strong influence of the correlation effects intrinsic to covalent metal–ligand bonding (as well as radial correlation in the 3d shell). 12 Regarding the key role of metal–ligand bonding, we made a related observation earlier in the context of DFT calculations. 94

4 Conclusions Multireference ab initio calculations (CASPT2 and NEVPT2) were performed for aqua complexes of the first-row TM ions from Ti III (d1 ) to Ni II (d8 ), with the focus on their experimentally evidenced ligand–field (d–d) transitions, both spin–allowed and spin–forbidden ones, the latter providing useful benchmark data in the context of spin–state energetics. As a considerable improvement over previous studies, 1,13–15 we accounted for both explicit and implicit solvation effects on the excitation energies. We also extrapolated the energetics to the complete basis set limit and used the active spaces describing metal–ligand covalency and the double-shell effect. We systematically studied the variation of solvation effects for a representative series of complexes with different ions and different types of excited states, and found that they vary from rather negligible (e.g., for Ni II ) up to very significant 0.3–0.5 eV (certain transitions in Fe III , V III , Ti III complexes), depending on the charge of ion and character of the electronic transition. These solvation effects were rationalized in terms of structural changes and redistribution of electrons between the t2g - and eg -like orbital levels. Concerning the use of implicit solvation models, we confirmed our previous conclusion 53 that for these d–d transitions, the main solvation effect is mediated through the change of geometry, whereas the direct effect on the excitation energy for fixed geometry is negligible. Our study also demonstrated a rather subtle interplay between solvation and correlation effects. For studied aqua complexes both factors drive in the same direction, namely to

43



an increase of the effective t2g –eg gap, thus usually to stabilization of the lower-spin with respect to higher–spin states. Thus, for benchmark studies of TM complexes one should take into account the possibility of partial cancellation between the error due to neglect or imperfect description of solvation and the one due to deficiencies of a given electron correlation method. The excitation energies obtained in this Article agree well with the published experimental data, except for a few cases where resolution of the experimental data is poor, so that either existence of bands or their precise assignment are doubtful (see Section 3.2). Notably, all previously mentioned interpretational controversies 1,15,86 regarding the assignment of bands for high-spin d5 and other aqua complexes have been resolved leading us to consistent interpretations of spectra for all complexes studied in this work. This also confirmed (indirectly) that the bands observed in the experimental spectra of aqua complexes are, indeed, caused by d–d transitions in their [M(H2 O)6 ] chromophores. Such a fundamental assumption has been challenged in the earlier theory works, 1,15,86 where it was suggested (in our opinion, incorrectly) that some of these bands are due to chemically different impurities or experimental errors. We further note that analogous reinterpretations have been put forward in a CASPT2 study of optical excitations in α-hematite, 52 the mineral containing analogous high-spin d5 FeO6 chromophores. We plan to re-investigate models of α-hematite in the near future. The statistical analysis of computational errors indicated that CASPT2 and NEVPT2 methods provide comparable and rather high accuracy for relative energies of ligand–field states (including spin states) of aqua complexes, with mean absolute errors of 0.15 and 0.13 eV, respectively. However, the maximum errors obtained for Fe III and Co III complexes were found much larger and well above the desired chemical accuracy. These discrepancies are consistent with the earlier reports of CASPT2 having tendency to overstabilize higher–spin states (more open-shell) with respect to lower-spin states (closed shell or with fewer unpaired electrons) by several kcal/mol 27,44,95–97 (although in most previously analyzed cases, it was not possible to compare with the experimental data as conclusively as in this work). Recently, Pierloot and co-workers suggested that these CASPT2 shortcomings may be due to erratic description of core–valence correlation, whereas the

44


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good performance of NEVPT2 (in some cases) may result from a fortuitous error cancelation. 10 The limitation of these PT2-based methods may be overcome by application of more expensive coupled cluster methods. Here, it was shown—by comparison with back-corrected experimental data and higher-level CCSDT(Q) calculations—that the CCSD(T) method is able to reach the chemical accuracy for one of the most challenging spin–state splittings in our benchmark set, the 6A

1g –

4T 1g

gap of Fe III aqua complex. Note, however, that standard single-reference coupled clus-

ter calculations can be performed only for the lowest-energy state with a given spin multiplicity and spatial symmetry, precluding applications to many other interesting excited states, for which CASPT2/NEVPT2 becomes the method of choice. Obviously, one should be careful in extrapolating the present methodology conclusions to other TM complexes, especially those with more covalent metal–ligand bonds. We are currently performing benchmark studies of selected WFT and DFT methods based on quantitative experimental data of spin–state energetics accumulated for a more representative subset of TM complexes.

Supporting Information Available Detailed and additional results, optimized Cartesian coordinates, and complete refs 63, 64, and 87. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement This research was supported in part by PLGrid Infrastructure (computations have been performed in part on resources provided by ACC Cyfronet AGH/UST); Jagiellonian University (projects no. K/DSC/003792 and 004554); and National Science Centre, Poland (grant no. 2017/26/D/ST4/00774). MR acknowledges the scholarship for outstanding young scientists from Ministry of Science and Higher Education, Poland, and participation in the COST Action CM1305 (ECOSTBio).

45



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A CASSCF/CASPT2 Analysis of

[V(H2 O)6 ]3+ and [Ni(H2 O)6 ]2+ . Inorg. Chem. 2001, 40, 2595–2601. (76) Reber, C.; Zink, J. I. Unusual Features in Absorption Spectra Arising from Coupled Potential Surfaces. Comments Inorg. Chem. 1992, 13, 177–220. (77) Relevant results from ref 75 (cf figure 6 therein): For [V(H2 O)6 ] 3+ , maxima of spectral envelope 11.4, 21.4; vertical excitation energies 11.0, 21.2. For [Ni(H2 O)6 ] 2+ , maxima of spectral envelope 7.9, 11.5, 24.7; vertical excitation energies 7.4, 11.0, 24.9 (in 103 cm−1 ). (78) Holmes, O. G.; McClure, D. S. Optical Spectra of Hydrated Ions of the Transition Metals. J. Chem. Phys. 1957, 26, 1686–1694. (79) (a) Tachikawa, H.; Ichikawa, T.; Yoshida, H. Geometrical structure and electronic states of the hydrated titanium(III) ion. An ab initio CI study. J. Am. Chem. Soc. 1990, 112, 982–987; (b) Tachikawa, H.; Ichikawa, T.; Yoshida, H. Hydration structure of titanium(III) ion. Electron spin resonance and electron spin-echo study. J. Am. Chem. Soc. 1990, 112, 977–982. (80) Maurelli, S.; Livraghi, S.; Chiesa, M.; Giamello, E.; Doorslaer, S. V.; Valentin, C. D.; Pacchioni, G. Hydration Structure of the Ti(III) Cation as Revealed by Pulse EPR and DFT Studies: New Insights into a Textbook Case. Inorg. Chem. 2011, 50, 2385–2394. 54


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(81) Jørgensen, C. K.; Hansen-Nygaard, L.; Pedersen, C.; Hartiala, K.; Veige, S.; Diczfalusy, E. Ligand Field Spectrum of Vanadium(II) Hexaquo Ions. Acta Chem. Scand. 1958, 12, 1537– 1538. (82) Figgis, B. N. Introduction to ligand fields; Interscience Publishers, 1966. (83) Lempicki, A.; Andrews, L.; Nettel, S. J.; McCollum, B. C.; Solomon, E. I. Spectroscopy of Cr3+ in Glasses: Fano Antiresonances and Vibronic “Lamb Shift”. Phys. Rev. Lett. 1980, 44, 1234–1237. (84) Brawer, S. A.; White, W. B. Optical properties of trivalent chromium in silicate glasses: A study of energy levels in the crossing region. J. Chem. Phys. 1977, 67, 2043. (85) Tregenna-Piggott, P. L. W.; Weihe, H.; Barra, A.-L. High-Field, Multifrequency EPR Study of the [Mn(OH2 )6 ]3+ Cation: Influence of π-Bonding on the Ground State Zero-Field-Splitting Parameters. Inorg. Chem. 2003, 42, 8504–8508. (86) Taylor, P.R., Can sixty-plus years of crystal-field theory be wrong? Some experience with d– d spectra of transition-metal complexes. Electronic Structure Theory for Strongly Correlated Systems conference, Palermo, Italy, 2012. (87) MRCC, a quantum chemical program suite by M. Kállay and co-workers. See Rolik, Z. et al. An efficient linear-scaling CCSD(T) method based on local natural orbitals. J. Chem. Phys. 2013, 139, 094105 and www.mrcc.hu (accessed Dec 2017). (88) Cotton, F. A.; Meyers, M. D. Magnetic and Spectral Properties of the Spin-Free 3d6 Systems Iron(II) and Cobalt(III) in Cobalt(III) Hexafluoride Ion: Probable Observation of Dynamic Jahn-Teller Effects. J. Am. Chem. Soc. 1960, 82, 5023–5026. (89) We note in passing that some of the interpretations made in ref 51 are suspicious, like the assignment of one of the dynamic-JT maxima for the 5 Eg state to the triplet state.

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(90) Jørgensen, C. K. Studies of Absorption Spectra. III. Absorption Bands as Gaussian Error Curves. Acta Chem. Scand. 1954, 8, 1495. (91) Ballhausen, C. J.; Jørgensen, C. K. Studies of Absorption Spectra. IX. The Spectra of Cobalt(II) Complexes. Acta Chem. Scand. 1955, 9, 397–404. (92) Solomon, E. I.; Ballhausen, C. Identification of the structure of the 3 T1g (I) ← 3 A2g band in the Ni(H2 O)++ 6 complex. Mol. Phys. 1975, 29, 279–299. (93) (a) Domingo, A.; Rodríguez-Fortea, A.; de Graaf, C. The Absorption Spectrum of Cytosine Tautomers: Beyond the Static Approach. J. Chem. Theory Comput. 2012, 8, 235–244; (b) Eilmes, A. Spiropyran to Merocyanine Conversion: Explicit versus Implicit Solvent Modeling. J. Phys. Chem. A 2013, 117, 2629–2635. (94) Radoń, M. Revisiting the role of exact exchange in the DFT spin state energetics. Phys. Chem. Chem. Phys. 2014, 16, 14479–14488. (95) Kepenekian, M.; Robert, V.; Le Guennic, B. What zeroth-order Hamiltonian for CASPT2 adiabatic energetics of Fe(II)N6 architectures? J. Chem. Phys. 2009, 131, 114702. (96) Radoń, M.; Broclawik, E.; Pierloot, K. Electronic Structure of Selected {FeNO}7 Complexes in Heme and Non-Heme Architectures: A Density Functional and Multireference ab Initio Study. J. Phys. Chem. B 2010, 114, 1518–1528. (97) Vancoillie, S.; Zhao, H.; Tran, V. T.; Hendrickx, M. F. A.; Pierloot, K. Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems. J. Chem. Theory Comput. 2011, 7, 3961–3977.

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´ Gabriela Drabik Spin States and Other Ligand–Field Mariusz Radon, States of Aqua Complexes Revisited with Multireference Ab Initio Calculations Including Solvation Effects.

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Spin States and Other LigandâField States of Aqua Complexes

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