Assessment of the SCAN Functional for Spin-State Energies in Spin-Crossover Systems

The Strongly-Constrained and Appropriately Normed (SCAN) functional has been tested towards the calculation of spin-state energy differences in a dataset of 20 spin-crossover (SCO) systems, ranging from d 4 to d 7 . The results shown that SCAN functional is able to correctly predict the low-spin state as the ground state for all systems, and the energy window provided by the calculations falls in the approximately range of energies that will allow for SCO to occur. Moreover, because SCAN is a pure meta-GGA functional, one can use such method in periodic calculations, accounting for the effect of collective crystal vibrations and counterions in the thermochemistry of the spin-transition. Our results validate this functional as a potential method for in silico screening of new SCO systems at both, molecular and crystal packed levels.


Introduction
Spin-crossover molecules are very appealing systems either experimentally, due to their switching abilities that can be harnessed for electronic devices, or from the theoretically point of view, because of the computational challenge that accurate calculation of spin- to calculate the low-high-spin state electronic energy difference. Surprisingly, highlyaccurate post Hartree-Fock methods, such CASPT2 or NEVPT2 do not provide much better results than DFT approaches.5-8 Taking into account these points, the most common theoretical approach is to study qualitatively the spin-crossover systems by performing molecular calculations.5, [9][10][11][12][13][14][15][16] If the low-spin state is only below 10-12 kcal/mol more stable than the high spin state, such difference can be compensated with the entropic contribution and thus, the system is suitable to exhibit spin-crossover behavior. However, the quantitative calculation of the transition temperatures, as mentioned above, present many drawbacks. In order to solve some of these problems, DFT methods seem to be an appropriate approach because it is possible to perform molecular or periodic systems calculations, analytic gradient and frequencies in molecular calculations but numerically in the PBC approaches, and relatively accurate low-high spin state energy differences. Regarding this last point, recently, we have established a benchmark set of 20 spin-crossover systems including examples of all the first-row transition metal showing such property (see Fig. 1) and moreover, they present a sharp low-high spin transition to have a well determined transition temperature.
Among the high number of exchange-correlation functionals tested, only the hybrid meta-GGA TPSSh functional provides the right description of the ground state for the whole set of molecules, and relatively small low-high energy differences. The goal of this study is to apply the meta-GGA Strongly-Constrained and Appropriately Normed (SCAN) functional to verify is this non-hybrid functional can achieve similar accuracy to determine low-high electronic energy differences. This will facilitate its application in PBC approaches to handle the whole unit cell where the calculation of exact-type exchange is problematic, and it requires huge amounts of computer time. SCAN functional was developed to satisfy the all 17 exact constrains of a meta-GGA functional and consequently, is not empirical and usually, it provides results closer to the hybrid functionals than other pure meta-GGA ones. Due to its nature, the SCAN functional has attracted a lot of attention and has been tested for several applications already.17-21

Results and Discussion
A total of 20 spin-crossover systems were computed using the TPSSh and SCAN functionals. For each system, the electronic energy difference (i.e, E = EHS-ELS) was computed. In Table 1 we present the results for both functionals. The results for the TPSSh functional have been also used to validate the computational methodology and transferability of the results (see SI).  scheme implementation over the electronic energy differences have been also evaluated for the SCAN functional (Table 2). Similarly, to previously reported results, ZPE corrected energy gaps are smaller, while the inclusion of dispersion effects increases the electronic energy differences between the two spin-states. Furthermore, it is also worth noting with the SCAN functional are much better than those provided by the pure meta GGA TPSS (see Table S2).42  Although the electronic energy differences with the SCAN functional tend to be a bit larger than the ones using TPSSh, and therefore the corresponding T1/2 are going to be much larger than the experimental ones, the fact that the SCAN functional is a pure functional allows us to use it in the study of SCO systems in periodic conditions. This has been done for the [Fe(bpp)2][BF4]2 system (S13 with a relatively large experimental T1/2 of 256 K, Figure 2), for which the unit cells containing two molecules were used in the calculation of the electronic energy differences and also, as a reference system to check the influence of the grid size, The periodic calculation will allow as well to determine the influence of the intermolecular dispersion, the intramolecular dispersion stabilize the low-spin state 2.72 kcal/mol for the S13 system per unit cell. The calculation of the thermodynamic magnitudes for the periodic model has been performed by using the post processing Phonopy code with a k-sampling using a 48x48x24 mesh in combination with the SCAN energies obtained with Turbomole code.
The calculation of the phonon modes due to the lack of analytic gradients using a numerical approach results in 6N single point calculations being N the total number of atoms in the unit cell (132 atoms for the S13 system). The calculated acoustic modes that must be strictly zero in  point are negative and have absolute values below 0.3 cm-1. The calculated entropy difference between high-and low-spin states using the optic phonons is 11.3 cal/K·mol per molecule while a value of 12.8 cal/K·mol was obtained for one isolated molecule with the SCAN functional using the Q-Chem code.
Thus, the difference between both values at room temperature is only around 0.4 kcal/mol. Consequently, the use of discrete molecule to estimate the thermodynamic magnitudes could be justified taking into account the complexity of the phonon calculation in periodic systems.

Concluding Remarks
In summary, The SCAN functional has been employed to calculate the energy differences between high-and low-spin for a test case of 20 molecules. Such energy differences are very challenging from the computational point of view because most of the theoretical methods fail not only in a quantitative approach also in the prediction of the right ground state and reasonable high-and low-spin energy differences. Previously, we shown that hybrid meta-GGA TPSSh functional provided a good estimation of such energy differences with the right ground state for the whole set of calculated systems.
The non-hybrid SCAN functional also gives the right ground state for the whole set of test cases being the unique pure DFT functional to provide good results for such challenging test (see Figure 3). It is worth noting the well-known large dependence of the results with the grid size using SCAN functional. In comparison with the TPSSh functional, the calculated energy differences are slightly larger. Hence, in few cases with SCAN values are out the range to have spin transitions below room-temperature (entropy can compensate energy differences below 12 kcal/mol) but larger grids increase the number of failures Furthermore, the periodic and molecular calculations were performed were done using the riper module using less-diffuse basis sets (pob-TZVP keyword) to avoid linear dependence problems and also, an auxiliary basis sets for the resolve identity approach (dhf-TZVP keyword). Dispersion effect were included using the Becke-Johnson D3 method. The grid employed in such calculations was defined through the keywords gridsize=5 and radsize=50 that it is the recommended in this code for the SCAN functional. The density of radial points also is variable for each element but is much larger than those employed with the Q-Chem code, for instance the usual reference for first-row atoms (Li-Ne) is 2475 radial points. The calculations were performed for a kpoint mesh (4,4,2) including 32 k-points, these is reduction of 2.0 kcal/mol in comparison with the (2,2,1) mesh with only 4 k-points. NWChem calculations were employed to calculate the energies using the Laplacian version of SCAN (SCAN-L) using the same basis set and the grid was defined with fine (70 radial 590 angular) and xfine (100 radial 1202 angular) grids.