# Assessment of Double-Hybrid Density Functional Theory for Magnetic Exchange Coupling in Manganese Complexes

## Abstract

**:**

## 1. Introduction

## 2. Test Set of Dinuclear Manganese Complexes

**S**and

_{1}**S**:

_{2}**1**) represents a case of ferromagnetic coupling, with a ground state spin of S = 3 [63]. The other complexes have low-spin ground states. Complex

**2**is a weakly antiferromagnetically coupled Mn(III,III) dimer (S = 0) [64]. Complex

**3**is a mixed-valence system with asymmetric ligation [65] that facilitates spin localization. This complex has a moderate antiferromagnetic coupling resulting in a spin doublet (S = 1/2) ground state and has been the subject of a recent study [20] that evaluated the use of the density matrix renormalization group [66] in the multireference treatment of exchange coupling [18,22]. Complex

**4**is a classic example of a strongly coupled bis-μ-oxo Mn(IV,IV) system [67]. Finally, complex

**5**reaches the far limit of strong antiferromagnetic coupling [68]. The tris-μ-oxo ligation in complex

**5**brings the manganese ions in such close proximity that, in addition to ligand-mediated superexchange, direct metal–metal interaction contributes significantly in stabilizing the low-spin state [39]. This situation is common in face-sharing d

^{3}–d

^{3}systems [51,69].

## 3. Selection of Functionals

## 4. Results and Discussion

#### 4.1. Conventional Density Functionals

**1**, but improves over TPSS for all other complexes. The SCAN results suggest possible non-linearity because the results are rather poor in the weak coupling cases but improve significantly toward the strong exchange-coupling regime (complexes

**4**and

**5**).

^{−1}for complexes

**1**,

**3**, and

**4**, with the largest deviation being 25 cm

^{−1}for the strongest antiferromagnetic coupling in the test set (complex

**5**). The good performance of TPSSh documented here is in agreement with previous studies on synthetic manganese complexes and bioinorganic model systems [34,35,36,38,40,41,42,43,80,81,82,83,84,85]. The increase in the percentage of HF exchange to 20% in the B3LYP functional leads to a slight overestimation of the stability of high-spin states and larger deviations from experiment. Only complex

**2**appears to be better described by B3LYP compared with TPSSh. Further increase of HF exchange to 25% in PBE0 results in exaggerated ferromagnetic coupling for complex

**1**, too small antiferromagnetic coupling for

**3**–

**5**, and qualitatively incorrect reversal of the ground spin state for

**2**from low to high spin. The effect of additional diffuse functions [86] in the basis set was tested and found to be negligible (variation of less than 0.2 cm

^{−1}in the computed exchange coupling constants) because they do not have a differential effect on the energies of the high-spin and broken-symmetry solutions. The conductor-like polarizable continuum model (CPCM) [87] was additionally tested with an infinite dielectric in order to investigate possible effects on the computed exchange coupling constants. Compared to the gas-phase results, the CPCM calculations show variations in the J values of less than 2 cm

^{−1}for the antiferromagnetically coupled dimers and up to 9 cm

^{−1}for complex

**1**. Given that these values were obtained under the extreme assumption of a perfect conductor, it is concluded that the continuum model has only a limited effect on the computed values as it does not strongly favor any particular solution. In conclusion, the various technical aspects of the calculations appear to be converged. In terms of the performance of individual functionals, even though exceptions exist at the quantitative level for specific complexes, TPSSh offers the most balanced performance.

#### 4.2. Double-Hybrid Density Functionals

**2**, and all DHDFs overestimate the stability of the high-spin state.

**1**, but a qualitatively different result for complex

**2**, a similar error albeit with opposite sign than TPSSh for complex

**3**, and significantly greater errors than TPSSh for the more strongly coupled complexes

**4**and

**5**. mPW2-PLYP tracks closely the B2-PLYP results, but with uniformly increased errors for all complexes. The reparametrized versions of B2-PLYP, i.e., the general-purpose B2GP-PLYP and the other two functionals (B2K-PLYP and B2T-PLYP) that were optimized for specific applications similarly show no improvement. Complex

**4**yields an outlier for B2K-PLYP, which overestimates the strength of the antiferromagnetic coupling.

**4**, yielding unrealistically large antiferromagnetic exchange coupling constants, particularly in the case of DSD-PBEP86. This is presumably related to the coefficients employed for the SCS-MP2 correction, to the overall MP2 contribution, or both. Moreover, both DSD functionals fail to predict the absolute and relative sign of the coupling for complexes

**1**and

**2**.

**2**.

#### 4.3. Energetic Contributions to Exchange Coupling from Double-Hybrid Density Functionals

_{DFT}in Table 4). This is due to the high HF admixture in all DHDFs. Looking at these results alone, the usual and expected correlation between the percentage of HF exchange and the stabilization of the high-spin state that was discussed for the conventional hybrid functionals (Table 2) becomes immediately apparent. B2-PLYP, mPW2-PLYP, and PWPB95 appear practically identical here because the results are primarily defined by the fact that all three functionals have almost the same ${\alpha}_{\mathrm{X}}$ factor (0.50–0.55). Reparametrized functionals with higher ${\alpha}_{\mathrm{X}}$ factors yield even greater stabilization of the high-spin state. In terms of the methodological utility of these DHDFs in the prediction of exchange coupling constants, the question is, to what extent can the perturbational contribution correct the flawed J

_{DFT}picture.

_{PT2}values in Table 4 are illuminating in this respect. Almost without exception the perturbational component correctly stabilizes the low-spin state, thereby enhancing antiferromagnetic coupling. For the functionals that were termed “well-behaved” above, the perturbational corrections are all very similar. What can be concluded on the basis of these results is that the perturbational correction for these functionals is well-controlled but never sufficient to fully recover the experimentally determined strength of antiferromagnetic coupling. Interestingly, the SOS-MP2 used in PWPB95 gives practically the same corrections as the MP2 component of B2-PLYP and mPW2-PLYP.

**4**. The decomposition of the energetic contributions in Table 4 reveals clearly the origin of the weakness of the DSD functionals for the present application. Their weakness stems in large part from the SCS-MP2 approach used in these functionals, or more precisely from the associated ${c}_{\mathrm{C}}$, ${c}_{\mathrm{O}}$, and ${c}_{\mathrm{S}}$ parameters that lead to very large and hence unreliable perturbational terms. These two families of DHDFs vividly demonstrate the pitfalls of pursuing property-specific functional parametrizations.

## 5. Computational Methods

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Structures of the manganese complexes included in this study. Hydrogen atoms bound to carbons are omitted for clarity (Mn: purple; C: grey; N: blue; O: red; B: yellow; Cl: green).

**Table 1.**Dinuclear manganese complexes considered in this study, with their crystallographic identifiers, Mn oxidation states, Mn···Mn distance R (in Å), and exchange coupling constant J (in cm

^{−1}).

Compound ^{a} | Refcode | Ox. States | R | J | Ref. | |
---|---|---|---|---|---|---|

1 | [Mn_{2}O(O_{2}BPh)_{2}(Me_{3}tacn)_{2}](PF_{6})_{2} | TIPFAZ | IV, IV | 3.185 | +10 | [63] |

2 | [Mn_{2}O(OAc)_{2}(H_{2}O)_{2}(bpy)_{2}](PF_{6})_{2}‚ 1.75H_{2}O | GEFKAD | III, III | 3.131 | −3.4 | [64] |

3 | [Mn_{2}O_{2}(OAc)(Me_{3}tacn)(OAc)_{2}] | KUVPEW | III, IV | 2.665 | −90 | [65] |

4 | [Mn_{2}O_{2}Cl_{2}(bpea)_{2}](ClO_{4})_{2} | ZEQGOR | IV, IV | 2.756 | −147 | [67] |

5 | [Mn_{2}O_{3}(Me_{3}tacn)_{2}](PF_{6})_{2}‚ H_{2}O | VADDAF | IV, IV | 2.297 | −390 | [68] |

^{a}Definition of ligand abbreviations: Me

_{3}tacn = 1,4,7-trimethyl-1,4,7-triazacyclononane; bpy = bipyridine; bpea = N,N-bis(2-pyridylmethyl)ethylamine.

**Table 2.**Exchange coupling constants J (in cm

^{−1}) computed with selected conventional density functionals for the five manganese complexes studied in this work, compared with experimentally fitted values. Mean absolute deviations (MAD) in cm

^{−1}.

Method | 1 | 2 | 3 | 4 | 5 | MAD |
---|---|---|---|---|---|---|

exp. | +10 | −3.4 | −90 | −147 | −390 | - |

BLYP | −26.6 | −71.6 | −180.8 | −261.4 | −618.1 | −107.6 |

TPSS | −13.4 | −48.7 | −147.7 | −216.7 | −549.7 | −71.2 |

SCAN | −20.4 | −29.2 | −113.9 | −155.6 | −402.4 | −20.2 |

TPSSh | +13.5 | −19.3 | −95.1 | −140.9 | −415.0 | −7.3 |

B3LYP | +26.2 | −11.4 | −77.8 | −115.2 | −360.7 | +16.3 |

PBE0 | +40.0 | +1.3 | −57.8 | −89.6 | −327.2 | +37.4 |

**Table 3.**Exchange coupling constants J (in cm

^{−1}) computed with selected double-hybrid density functionals for the five manganese complexes studied in this work, compared with experimentally fitted values. Mean absolute deviations (MAD) in cm

^{−1}.

Method | 1 | 2 | 3 | 4 | 5 | MAD |
---|---|---|---|---|---|---|

exp. | +10 | −3.4 | −90 | −147 | −390 | - |

B2-PLYP | +13.1 | +4.6 | −83.8 | −109.9 | −326.8 | +23.5 |

mPW2-PLYP | +19.6 | +6.0 | −72.5 | −101.8 | −317.7 | +30.8 |

B2GP-PLYP | +13.3 | +11.7 | −79.8 | −135.1 | −332.6 | +19.6 |

B2K-PLYP | +11.2 | +14.4 | −96.4 | −318.7 | −351.5 | −24.1 |

B2T-PLYP | +15.6 | +9.2 | −73.6 | −114.6 | −323.2 | +26.8 |

DSD-PBEP86 | −17.4 | +16.0 | −107.5 | −861.1 | −402.5 | −150.4 |

DSD-PBEB95 | −15.5 | +11.2 | −97.6 | −230.6 | −394.1 | −21.2 |

PWPB95 | +8.4 | −0.5 | −78.2 | −108.3 | −318.9 | +24.6 |

**Table 4.**Density functional theory (DFT)-only exchange coupling constants J

_{DFT}(in cm

^{−1}) obtained from the Kohn–Sham orbitals of the double hybrid functionals by excluding the perturbational energy component, and the corresponding perturbational contribution (ΔJ

_{PT2}) that leads to the final results of Table 3.

J_{DFT} | ΔJ_{PT2} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | |

B2-PLYP | +66.5 | +12.5 | −18.7 | −43.7 | −277.8 | −53.4 | −7.9 | −65.1 | −66.2 | −49.0 |

mPW2-PLYP | +67.0 | +12.2 | −17.7 | −42.7 | −277.7 | −47.5 | −6.2 | −54.8 | −59.1 | −40.0 |

B2GP-PLYP | +77.2 | +13.4 | −6.4 | −28.6 | −269.2 | −63.9 | −1.7 | −73.4 | −106.5 | −63.3 |

B2K-PLYP | +83.2 | +13.2 | +3.3 | −13.1 | −261.1 | −71.9 | +1.2 | −99.7 | −305.6 | −90.4 |

B2T-PLYP | +72.5 | +13.2 | −11.9 | −36.0 | −273.7 | −56.9 | −4.0 | −61.7 | −78.6 | −49.6 |

DSD-PBEP86 | +94.0 | +16.1 | +12.7 | +9.9 | −244.8 | −111.4 | 0.0 | −120.2 | −871.1 | −157.7 |

DSD-PBEB95 | +88.7 | +15.1 | +2.2 | −11.5 | −249.2 | −104.2 | −3.9 | −99.8 | −219.1 | −144.9 |

PWPB95 | +67.1 | +12.9 | −22.8 | −50.3 | −278.4 | −58.7 | −13.3 | −55.4 | −58.0 | −40.4 |

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Pantazis, D.A.
Assessment of Double-Hybrid Density Functional Theory for Magnetic Exchange Coupling in Manganese Complexes. *Inorganics* **2019**, *7*, 57.
https://doi.org/10.3390/inorganics7050057

**AMA Style**

Pantazis DA.
Assessment of Double-Hybrid Density Functional Theory for Magnetic Exchange Coupling in Manganese Complexes. *Inorganics*. 2019; 7(5):57.
https://doi.org/10.3390/inorganics7050057

**Chicago/Turabian Style**

Pantazis, Dimitrios A.
2019. "Assessment of Double-Hybrid Density Functional Theory for Magnetic Exchange Coupling in Manganese Complexes" *Inorganics* 7, no. 5: 57.
https://doi.org/10.3390/inorganics7050057