Accurate calculation of spin-state energy gaps in Fe(III) Spin-Crossover systems using Density Functional Methods

Fe(III) complexes are receiving ever-increasing attention as spin crossover (SCO) systems because they are usually air stable, as opposed to Fe(II) complexes, which are prone to oxidation. Here, we present the first systematic study exclusively devoted to assess the accuracy of several exchange-correlation functionals when it comes to predicting the energy gap between the high-spin ( S =5/2) and the low-spin ( S =1/2) states of Fe(III) complexes. Using a dataset of 24 different Fe(III) hexacoordinated complexes, it is demonstrated that the B3LYP* functional is an excellent choice not


Introduction
Switchable molecules and materials are key elements in the design of the new generation of nanodevices due to their inherent bi-stable behavior. Among them, spincrossover (SCO) systems, molecules and materials that have access to two alternative electronic states close in energy, are particularly interesting due to their flexibility in design and tunability. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] In SCO systems, the spin-state manipulation can be done using an external stimulus, usually temperature, but can be also induced by means of pressure or electromagnetic radiation. The thermal transition, which is by far the most common one, takes place when the entropic term overcomes the enthalpic one, shifting the system from the low-spin state to the high-spin one. 19,20 The temperature with equal populations of both spin-states is defined as the transition temperature (T 1/2 ), and this is a key parameter in the physical characterization of the system. Since their discovery by Cambi and co-workers in 1931, 21 SCO systems have been the focus of attention of many researchers due to their potential and practical application as molecular level switches. 4,5,7,9,14,22 Much development has been done in the field over the last decades, expanding the number of metal centers and coordination environments able to exhibit such behavior. As much experimental information as there is, the rational design of new SCO systems with tailored properties relies heavily on the experience. For that reason, the vast majority of newly reported SCO systems build upon the Fe II -hexacoordinated motif with six nitrogen donor atoms. [23][24][25][26][27][28][29][30][31] However, d 6 -Fe(II) SCO complexes are air unstable, and tend to oxidize and lose their switching behavior, which makes its application in actual devices challenging. For that reason, there is an increasing interest in the design of new SCO systems with d 5 -Fe(III) metal centers, [31][32][33][34][35] systems that exhibit a much larger stability towards oxidation and, therefore, show more potential towards its use in technological devices. In the case of Fe(III) centers, the high-spin (HS) and low-spin (LS) states involved in the spin transition are commonly a sextet spin state (S=5/2) and a doublet spin state (S=1/2), respectively. Despite the increasing interest in such systems, [31][32][33][34][35] which has been recently reviewed, there is less experimental information and, therefore, the rational design of new Fe(III)-SCO systems with tailored properties becomes challenging. In particular, designing a molecule with a given transition temperature is extremely difficult. In that sense, great progress has been done in the field of computational modelling of SCO systems, and some methodologies have been presented aiming to reproduce the behavior of such systems, and even their intersystem crossing rates. 20, In particular, much development has been done over the last years in the modelling of T 1/2 for several families of SCO systems, [63][64][65][66][67][68] including the study of spin-state energy gaps in Fe(III) systems. 45 Motivated by such results, we decided to explore the potential use of such methods to quantitatively calculate T 1/2 in Fe(III)-SCO systems. In this work, several DFT methods have been used towards a dataset of SCO Fe(III) systems to evaluate their performance in terms of accurate calculation of the T 1/2 . To the best of our knowledge, this is the first systematic work with an exclusive focus on Fe(III)-SCO systems, with the only exception of a specific study of Fe(III) quinolylsalicylaldiminate compounds, previously reported. 69 While several benchmark studies have been reported on the accuracy of different exchange-correlation functionals to predict energy differences between spin states and T 1/2 for Fe(II)-SCO systems, 52,70 no systematic study on a large dataset of Fe(III)-SCO systems has been reported yet. Given that it has already been shown that a given functional (or a given value of U in DFT+U approaches) does not result in the same accuracy when predicting spin-state energetic gaps of Fe(II) or Fe(III) systems, 45,52 it is clear that a systematic study fully devoted to Fe(III)-SCO systems is mandatory. The results of the quantitative methodology we have employed will hopefully be used in the virtual screening of new spin-crossover molecules with tailored properties, a tool that will accelerate the discovery of new members of the Fe(III) SCO family. The paper is organized as follows. First, we will present the results, followed by their discussion, and finally, the conclusions of the work.

Computational Details
All density functional calculations (DFT) have been carried out with Gaussian 16 (revision B0.1) 71 electronic structure package with a 10 −8 convergence criterion for the density matrix elements, using the latest triple-ζ basis set with polarization functions for all elements (def2-TZVP) by Ahlrichs and co-workers. 72,73 The corresponding vibrational analysis was done for all optimized structures to ensure that they were minimums along the potential energy surface. The transition temperatures (T 1/2 ) were estimated by means of the following expression, which holds under the condition of thermodynamical equilibrium: In this equation, ∆ !"!!" is the enthalpy difference between the HS and LS states (which is assumed to be a independent of temperature) and ∆ !"!!" is the entropy difference between the HS and LS states. Both ∆ !"!!" and ∆ !"!!" have an electronic and a vibrational contribution: is the adiabatic energy difference between the HS and LS states. To obtain ∆ !"!# , the electronic entropies for each state need to be evaluated, which can be done through Eq. 4: The vibrational enthalpy and entropy for each state, which are needed to evaluate ∆ !"# and ∆ !"# , can be obtained by means of the frequencies of the vibrational normal modes ( ! ) and the standard harmonic-oscillator approximations employed in statistical thermodynamics:

Results
To evaluate the accuracy of DFT calculations towards spin-state energy gaps in Fe III spin-crossover systems, we assembled a benchmark dataset of 18 molecules, as illustrated in Figure 1.
Results from Table 1 can be properly visualized by plotting the average value as well as the standard deviation for the computed dataset. These results are shown in Figure   2. As can be seen in the figure, only B3LYP* and TPSSh are able to correctly predict the ground state for most systems in our dataset. This is consistent with previous benchmarks done for the TPSSh functional. 59,70 However, the most remarkable result is that B3LYP* provides with ΔE HS-LS values that fit in the energy window that usually is associated with SCO to occur, this is, ΔE HS-LS between 2 and 8 kcal/mol.    Table 2: Enthalpy and entropy change for the 18 systems studied in this work using B3LYP* and TPSSh functionals, as well as the corresponding computed and experimental values for the transition temperature (T 1/2 ). Enthalpies in kcal·mol -1 , entropies in cal·K -1 ·mol -1 and temperatures in K.
[b] The experimental T 1/2 has been obtained from the ∆H and ∆S measured by the Evans 1 H-NMR method (see Table S5 of ESI). [c] Average T 1/2 value for different counterions and co-crystallizing solvent molecules (see Table S4 of ESI) As can be seen from Table 2, even though both functionals perform correctly in terms of predicting the ground state for all the studied systems, the computed range in energies using TPSSh is larger than with B3LYP*. This translates in much larger computed T 1/2 values when using the TPSSh functional, while B3LYP* provides with much closer values towards the experimental data.

Discussion
The accurate calculation of transition temperatures in SCO systems is a hard task for any computational method. However, Table 2 shows that both, TPSSh and, in a more quantitative way B3LYP* can be used to compute T 1/2 in Fe III -based SCO systems.
One may assume that the amount of exact exchange Hartree-Fock mixed in the functionals (10% for TPSSh and 15% for B3LYP*) is responsible for their different accuracy towards T 1/2 , given that this quantity modulates the relative stability of the different spin-states. For that reason, we explored the possibility of adjusting the amount of Hartree-Fock in the B3LYP functional towards the calculation of the spinstate energy gap in Fe III systems. Using S4, S6 and S11 as test cases (all of them have a sharp single step transition without hysteresis), we computed the spin state energy gap adjusting the amount of Hartree-Fock exchange mixed in the BLYP functional from 10 to 20%. Results are shown below for S6 (see ESI for S4 and S11 systems), both for the spin-state energy gap as well as the computed T 1/2 . As can be seen in Figure 3, there is a linear correlation between the amount of Hartree-Fock mixed and the spin-state energy gap. Thus, one can envision a specific ad-hoc functional for Fe III systems. The problem is that the optimal amount of Hartree-Fock exchange mixed to reproduce the experimental temperature is different for each system, being 14%, 16% and 17% for systems S4, S6 and S11 respectively (see ESI). Thus, it seems that in order to quantitatively match the experimental values, individual reparameterizations are required, with values oscillating around the optimal 15% adjusted for B3LYP*.  Table 3.    A key difference between B3LYP* and TPSSh in terms of computing T 1/2 for Fe(III) systems is the energy window that the first one gives for SCO to occur (Figure 2).
Thus, we wanted to validate the functional towards other Fe(III) systems that are either low-spin or high-spin, but do not exhibit SCO. The results are summarized in  values with great accuracy, but also we can predict the corresponding T 1/2 for other members of the same family and explain the trend on the basis of the relevant d-based molecular orbitals. Obviously, the lack of inclusion of crystal packing effects in our calculations, and the use of the lowest energy harmonic frequencies to compute the thermochemical quantities makes the quantitative agreement between computed and experimental T 1/2 a much more challenging problem. The B3LYP* functional is also able to discriminate between Fe(III) systems that do not exhibit SCO, and classify them as either high-spin or low-spin, in excellent agreement with the experimental data. The presented work thus enables the use of the B3LYP* functional to scan for new spin-crossover systems and predict, to some extent, their transition temperatures using computational tools.