Spin–orbit coupling (SOC) is crucial for correct electronic structure analysis in molecules and materials, for example, in large molecular systems such as superatoms, for understanding the role of transition metals in enzymes, and when investigating the energy transfer processes in metal–organic frameworks. We extend the GFN-xTB method, popular to treat extended systems, by including SOC into the hamiltonian operator. We followed the same approach as previously reported for the density–functional tight-binding method and provide and validate the necessary parameters for all elements throughout the Periodic Table. The parameters have been obtained consistently from atomic SOC calculations using the density–functional theory. We tested them for reference structures where SOC is decisive, as in the transition metal containing heme moiety, chromophores in metal–organic frameworks, and in superatoms. Our parameterization paves the path for incorporation of SOC in the GFN-xTB based electronic structure calculations of computationally expensive molecular systems.

Density–functional theory (DFT) is a computationally feasible first-principles method known for its chemical accuracy and computational scalability.1,2 However, most of the fascinating chemistry occurring at the biochemical and material level incorporates thousands of atoms. Metallo-proteins, metal–organic frameworks (MOFs), superatoms, and nanoclusters involve thousands of atoms. GFN-xTB,3,4 an advanced semi-empirical quantum mechanical method, has proven its usefulness in describing the chemistry and physics of such large molecular3,5 and periodic6 systems. To date, all extensions of GFN-xTB are formulated in a spin-restricted way and only consider scalar relativistic effects in the form of effective core potentials but do not account for spin–orbit coupling (SOC).7 Here, we incorporate SOC into the GFN1-xTB method following the same approach as in density–functional based tight-binding (DFTB).8 Our approach for incorporation of SOC to GFN1-xTB is transferrable to other extensions of GFN-xTB, such as GFN2-xTB, and possible future versions. We provide parameters covering the elements throughout the Periodic Table9 and validate them with typical molecular benchmark systems.

SOC is a relativistic effect originating from the Dirac equation, a relativistic analog of the Schrödinger equation (SE). SOC can be added to the hamiltonian operator through approximate decoupling of a fully relativistic Dirac equation in two-components and, then, splitting scalar and SOC parts. This can be performed, for example, by using the exact two-component (X2C) method, the zeroth order regular approximation (ZORA),10,11 or the Douglas–Kroll–Hess (DKH) approximation.12 Semi-empirical quantum mechanical methods solve the SE, and relativistic corrections can be added to it in two parts. First, the inertial mass of the electrons is corrected via pseudopotentials or by employing ZORA. One important fact to be mentioned here is that the mass–velocity correction for the s electrons gets counteracted by the Darwin term. Second, by coupling of the spin of the electron with the magnetic field in the reference frame of the electron, known as SOC.8 

SOC is crucial for electronic structure analysis of many molecular systems. SOC perturbs the electronic structure of molecules and accounts for various interesting properties. One example is MOFs, an emerging material class with applications, among others, in energy applications including the photochemical conversion of solar energy. Studies concerning the effect of SOC13 on the absorption spectra of complexes such as in M(bpy)32+ (M = Fe, Ru, Os; bpy = 2,2′-bipyridine) have facilitated the research on understanding of energy transfer processes in MOFs.14,15 Similarly, Chakraborty et al. emphasized the effect of SOC in dipole–dipole energy transfer in Ru(II), Ir(III), and Os(II) polypyridyl complexes incorporated into the backbone of the MOF UiO-67.16 

Superatoms are an exciting class of clusters with free atom-like properties and, thus, can serve as building blocks for advanced nanomaterials.17,18 Assemblies of ligated magnetic superatoms can serve as better molecular electronic devices, as weak fields can control the coupling;19 SOC can affect the electron affinity of superatoms to a great extent, as for example in WAu12 with an electron affinity difference of ∼2 eV,20 which affects the charge transfer process.

Enzymes have manifested themselves as vital elements of the biosphere and have contributed to the advent of life in its current form. Enzymatic processes, as in binding of oxygen to hemoglobin and myoglobin, are supposed to be of low yield, as for example the reaction between the singlet O2 and quintet heme moiety is spin-forbidden. The presence of transition metals in the enzymes lifts the spin prohibition,21 as SOC results in mixing of the states. Hence, SOC is crucial for correctly describing the high yield in one of the most important biological processes, binding of oxygen to hemoglobin. Similarly, SOC affects the yield of final products in various chemical processes, such as spin catalysis.22–24 SOC also facilitates the singlet to triplet intersystem crossing in systems containing heavy compounds.23 Thus, a proper account of SOC in chemical reactions is crucial for the correct prediction of the yield of end products.

As SOC is a physical effect impacting various electronic and transport properties of extended molecules and molecular framework materials, it makes a very useful and timely addition to the GFN-xTB approaches. One can extend the GFN-xTB hamiltonian to include a SOC correction based on an atom-dependent parameterization with similar computational cost as of a non-collinear spin-polarized calculation. One limit of such implementations is the availability of accurate SO splitting parameters throughout the Periodic Table, which was overcome in our previous work.9 

In the present study, we have implemented the SOC extension within the LS coupling model to GFN1-xTB, using exactly the same approach as earlier reported for DFTB.8,9 We validated the approach and parameters on a variety of reference structures, including chromophores in MOFs, superatoms, and transition metal containing heme moieties. We calculated the spin–orbit splitting of valence molecular orbitals and compared with the reference values calculated at the DFT level with SOC-ZORA relativistic corrections. We observed excellent agreement between both methods, and the error bar of our SOC correction is typically lower than that expected for the molecular orbitals.

In this section, we give the SOC extension to the GFN-xTB method and the calculation of parameters that will be included into the model. While we have implemented it here for both GFN1-xTB and GFN2-xTB, we concentrated us in the tests in the more wide-spread GFN1-xTB variant. Extension and parameterization are identical to our previous work on DFTB9 and are included here for completeness.

The total energy in GFN1-xTB4 comprises electronic energy (Eel), atom-pairwise repulsion (Erep), dispersion (Edisp), and halogen bonding term (EXB), which is represented as

EGFN−xTB=Eel+Erep+Edisp+EXB.
(1)

The electronic energy is given as

Eel=iocc⋅niψi|H0|ψi+12A,BlAlBplAplBγAB,ll+13AΓAqA3TelSel,
(2)

where H0 is the zero-order hamiltonian, ψi is the single-electron wave function of a valence molecular orbital (MO), and ni is the occupation number of MO of index i. The second and third terms comprise the self-consistent charge (SCC) contributions to the electronic energy. qA is the Mulliken charge of atom A, and ΓA is the charge derivative of the atomic Hubbard parameter. TelSel is the electronic free energy of the system. A and B are two distinct atoms of the system, l and l′ are the orbital angular momentums of the atomic shells of atoms A and B, respectively. plA is the charge distributed over the atomic shell with orbital angular momentum number l at atom A. The distance dependence of the Coulomb interaction is given as γAB,ll following the generalized Mataga–Nishimoto–Ohno–Klopman formalism.25–28 A detailed description of the GFN1-xTB formalism is provided in the supplementary material.

SOC incorporation to the GFN1-xTB hamiltonian in the LS coupling model13 is given as

ĤμνL̂Ŝ=12SμνϵμL̂ZL̂L̂+L̂Z+ϵνL̂ZL̂L̂+L̂ZwithμlA,vlB,
(3)

where μ and v are the atomic shell labels for corresponding AOs with angular momentum l at atom A and l′ at atom B, respectively. Here, ɛ is the SOC parameter, L̂ is the angular momentum operator, and Ŝ is the spin operator. ĤμνL̂Ŝ is the hamiltonian matrix for dual SOC, where dual stands for considering the off-site corrections in addition to on-site corrections.29 

The full GFN1-xTB hamiltonian matrix with consideration of spin polarization and SOC8 reads as

Hμν=KAB12kl+kl12hAl+hBlSμν1+kENΔENAB2ΠRAB,ll1001+12SμνClγAC,ll+γBC,llplC1001+12SμvqA2ΓA+qB2ΓB1001±Sμv2lAWAllpAlzpAlypAlxpAly+pAlxpAlz×lBWBllpBlzpBlypBlxpBly+pBlxpBlz+Sμν2εμL̂ZL̂L̂+L̂Z+ενL̂ZL̂L̂+L̂ZμlA,vlB.
(4)

Here, kl and kl are the Hückel parameters for angular momentum l for atom A and l′ for atom B and KAB is a scaling constant. The energy levels for atom A and atom B are represented as hAl and hBl with l and l′ being the angular momentum, respectively. Sμv is the overlap matrix, and ΔENAB is the electronegativity difference of two atoms with kEN as a proportionality coefficient. ΠRAB,ll is a distance and l-dependent function.

We have calculated the SOC parameters for free atoms throughout the Periodic Table employing AMS-BAND30 software with two-component relativistic corrections at the SOC-ZORA level and TZ2P basis set with the all electron approach and then renormalized, as reported earlier.9 We will, for completeness, briefly outline the approach below. All elements are in their ground state atomic configuration.

Spin–orbit potential ΔH in terms of ΔHT (Thomas precession energy) and ΔHL (Larmor interaction energy) can be given as

ΔH=ΔHT+ΔHL=22γ3γ+1μBmeec21rUrrLS,
(5)

where γ=1Znα22 is derived by using vc=Znα in γ=1v2c2. Here, Z is the atomic number of the atom, n is the principal quantum number, γ is the Lorentz factor, and α is the fine structure constant or Sommerfeld constant. The renormalized spin–orbit coupling parameter expression is given as

ε=2Δl1γ3γ+1.
(6)

Here, Δ is the SO splitting from atomic calculations with SOC relativistic corrections at the ZORA level, two-component relativistic approximation to the Dirac equation, l is the angular quantum number for respective shells, and ɛ is the renormalized spin–orbit coupling constant. Table S1 (supplementary material) contains the calculated SOC parameters throughout the Periodic Table and a detailed derivation is provided in the supplementary material.

FIG. 1.

Optimized geometric parameters—bond distances and bond angles at DFT and GFN1-xTB levels, respectively, for heme moiety of hemoglobin.

FIG. 1.

Optimized geometric parameters—bond distances and bond angles at DFT and GFN1-xTB levels, respectively, for heme moiety of hemoglobin.

Close modal

Geometries were optimized using the ANCopt method in the GFN1-xTB framework at an optimization level tight as implemented in the xTB package. We also optimized the geometries at the DFT level with ZORA scalar relativistic corrections for a consistent benchmarking reference. We used the all electron approach with TZ2P quality of the basis set in conjugation with the GGA-PBE exchange–correlational functional.

Very small differences were found between the optimized geometries at GFN1-xTB and DFT levels. There are small differences in the bond distances of ∼0.01 to 0.03 Å and in the bond angles ∼2° to 5° going from DFT to GFN1-xTB (see the supplementary material, Sec. 3, for details). As an example, Fig. 1 shows these differences in bond distances and bond angles for heme moiety of hemoglobin. As expected, incorporation of SOC has only a marginal effect on geometries of the structures with the change in bond distances and bond angles of ∼10−3 and ∼0.2° to 0.5°, respectively. Therefore, in the following, single-point SOC calculations in GFN1-xTB, termed as SOC-GFN1-xTB, were performed in the LS coupling model through our parameterization using the SOC parameters, as given in Table I. For the validation of results from GFN1-xTB, we performed single-point calculations at the DFT level employing AMS-ADF software. We used the GGA-PBE exchange–correlation functional with SOC-ZORA relativistic corrections. The all electron approach with the basis set of TZ2P quality was used.

TABLE I.

Parameters for the calculation of absorption spectra. All values are in cm−1.

Febpy32+Rubpy32+Osbpy32+
Electronic MO model14 SOC-GFN1-xTB MO modelElectronic MO model14 SOC-GFN1-xTB MO modelElectronic MO model14 SOC-GFN1-xTB MO model
Δ 100 185 500 585 800 888 
Γ −1500 −1500 −1600 −1600 −2100 −2100 
K 800 800 850 850 850 850 
λ 440 561 1200 1154 3000 3555 
Febpy32+Rubpy32+Osbpy32+
Electronic MO model14 SOC-GFN1-xTB MO modelElectronic MO model14 SOC-GFN1-xTB MO modelElectronic MO model14 SOC-GFN1-xTB MO model
Δ 100 185 500 585 800 888 
Γ −1500 −1500 −1600 −1600 −2100 −2100 
K 800 800 850 850 850 850 
λ 440 561 1200 1154 3000 3555 
FIG. 2.

Absorption spectrum of Mbpy32+M=Fe,Ru,Os;bpy=2,2bipyridine complexes calculated at the reference MO model (blue dashed lines) and the SOC-GFN1-xTB MO model (red lines) with system specific Δ, Γ, λ, and K values (Table II). XY and Z polarization are the polarization of π* electronic excitations.

FIG. 2.

Absorption spectrum of Mbpy32+M=Fe,Ru,Os;bpy=2,2bipyridine complexes calculated at the reference MO model (blue dashed lines) and the SOC-GFN1-xTB MO model (red lines) with system specific Δ, Γ, λ, and K values (Table II). XY and Z polarization are the polarization of π* electronic excitations.

Close modal
TABLE II.

Relative energy of π* electronic transitions calculated with the SOC-GFN1-xTB MO model and the reference MO model for ions Mbpy32+M=Fe,Ru,Os;bpy=2,2bipyridine with parameters from Table I. Double prime symbols indicate excited states. σ is the standard deviation of peak values at the SOC-GFN1-xTB model with respect to the reference MO model.

Febpy32+Rubpy32+Osbpy32+
ExcitationsElectronic MO m14 SOC-GFN1-xTBσ (SD)Electronic MO m14 SOC-GFN1-xTBσ (SD)Electronic MO m14 SOC-GFN1-xTBσ (SD)
1E″ 950 1060 77.78 1610 1676 46.67 2650 2989 239.7 
2E″ −525 −518 4.950 35 70 24.75 1165 1381 152.7 
3E″ −480 −335 102.5 250 312 43.84 1450 1816 258.8 
4E″ −1025 −1153 90.51 −1495 −1426 48.79 −2965 −3343 267.3 
5E″ −550 −439 78.48 10 76 46.67 450 789 239.7 
6E″ −2025 −1914 78.48 −1565 −1529 25.46 −1135 −818 224.2 
7E″ −2675 −2747 50.91 −3350 −3262 62.23 −5550 −5946 280.0 
8E″ −2475 −2329 103.2 −2885 −2302 412.2 −4985 −3824 820.9 
9E″ −1980 −1835 102.5 −1350 −1288 43.84 −750 −384 258.8 
10E″ −645 −589 39.60 −415 −320 67.18 −115 −284 119.5 
11E″ −2045 −1948 68.59 −1650 −1629 14.85 −1250 −842 288.5 
12E″ −2525 −2577 36.77 −3095 −3026 48.79 −5165 −5543 267.3 
1A1 −975 −796 126.6 −1285 −1190 67.18 −2785 −2957 121.6 
2A1 855 701 108.9 1185 1089 67.88 2085 2153 48.08 
3A1 −550 −438 79.19 10 76 46.67 450 789 239.7 
4A1 −1980 −1835 102.5 −1350 −1288 43.84 −750 −384 258.8 
5A1 −2025 −1914 78.48 −1565 −1529 25.46 −1135 −718 294.9 
6A1 −2525 −2577 36.77 −3095 −3026 48.79 −5165 −5543 267.3 
1A2 −1175 −1247 50.91 −1750 −1662 62.25 −3350 −3756 287.1 
2A2 −545 −447 69.29 −50 −29 14.89 950 1052 72.12 
3A2 −550 −438 79.19 10 76 46.67 450 789 239.7 
4A2 −1980 −1835 102.5 −1350 −1288 43.84 −750 −384 258.8 
5A2 −2025 −1914 78.48 −1565 −1529 25.46 −1135 −718 294.9 
6A2 −2525 −2577 36.77 −3095 −3026 48.79 −5165 −5543 267.3 
Febpy32+Rubpy32+Osbpy32+
ExcitationsElectronic MO m14 SOC-GFN1-xTBσ (SD)Electronic MO m14 SOC-GFN1-xTBσ (SD)Electronic MO m14 SOC-GFN1-xTBσ (SD)
1E″ 950 1060 77.78 1610 1676 46.67 2650 2989 239.7 
2E″ −525 −518 4.950 35 70 24.75 1165 1381 152.7 
3E″ −480 −335 102.5 250 312 43.84 1450 1816 258.8 
4E″ −1025 −1153 90.51 −1495 −1426 48.79 −2965 −3343 267.3 
5E″ −550 −439 78.48 10 76 46.67 450 789 239.7 
6E″ −2025 −1914 78.48 −1565 −1529 25.46 −1135 −818 224.2 
7E″ −2675 −2747 50.91 −3350 −3262 62.23 −5550 −5946 280.0 
8E″ −2475 −2329 103.2 −2885 −2302 412.2 −4985 −3824 820.9 
9E″ −1980 −1835 102.5 −1350 −1288 43.84 −750 −384 258.8 
10E″ −645 −589 39.60 −415 −320 67.18 −115 −284 119.5 
11E″ −2045 −1948 68.59 −1650 −1629 14.85 −1250 −842 288.5 
12E″ −2525 −2577 36.77 −3095 −3026 48.79 −5165 −5543 267.3 
1A1 −975 −796 126.6 −1285 −1190 67.18 −2785 −2957 121.6 
2A1 855 701 108.9 1185 1089 67.88 2085 2153 48.08 
3A1 −550 −438 79.19 10 76 46.67 450 789 239.7 
4A1 −1980 −1835 102.5 −1350 −1288 43.84 −750 −384 258.8 
5A1 −2025 −1914 78.48 −1565 −1529 25.46 −1135 −718 294.9 
6A1 −2525 −2577 36.77 −3095 −3026 48.79 −5165 −5543 267.3 
1A2 −1175 −1247 50.91 −1750 −1662 62.25 −3350 −3756 287.1 
2A2 −545 −447 69.29 −50 −29 14.89 950 1052 72.12 
3A2 −550 −438 79.19 10 76 46.67 450 789 239.7 
4A2 −1980 −1835 102.5 −1350 −1288 43.84 −750 −384 258.8 
5A2 −2025 −1914 78.48 −1565 −1529 25.46 −1135 −718 294.9 
6A2 −2525 −2577 36.77 −3095 −3026 48.79 −5165 −5543 267.3 

We reproduced the results of the effect of SOC on absorption spectra of the ions M(bpy)32+ (M = Fe, Ru, Os; bpy = 2,2′-bipyridine), given their importance in the dipole–dipole energy transfer in MOFs as reported by Chakraborty et al.16 In this reference work, the SO splitting parameter λ of Ru(III), Fe(III), and Os(III) was reported as 1200, 440, and 3000 cm−1, respectively, and matches well with the SOC-GFN1-xTB results, which yield values for λ = 1154, 561, and 3555 cm−1, respectively. Following the same MO model as reported in the literature, we calculated the absorption spectra of the above complexes. The absorption spectra at the SOC-GFN1-xTB level, in combination with the MO model from the literature (SOC-GFN1-xTB MO model), matches very well with the reference spectra (Fig. 2).

In the literature, the π* electronic transitions from orbitals (with T2g symmetry) of the metal center to π*1 orbital of the bipyridine ligands were described with the MO theory. In this scheme, relative energies of the excited states and ground states were reported in terms of parameters as in Δ, Γ, K, and λ (Table I). Here, Δ is the energy difference between the E and dπA1 orbitals generated after lifting the degeneracy of the orbitals. Similarly, Γ is the energy difference between the π1E* and π1A2* orbitals generated after lifting the degeneracy of the π1* orbitals. K is the destabilizing energy of the metal–ligand coupled electronic state relative to the energy of the uncoupled state, and λ is the SOC constant. It is, in principle, possible to calculate these parameters within GFN1-xTB. For illustration, we calculated the Γ and K values from the GFN1-xTB method. While we found a deviation of less than 40 cm−1 in K values in comparison to the experimental data, the Γ values show large deviations of ∼400 to 500 cm−1 for Fe and Os complexes. We attribute this deviation to the fact that GFN1-xTB is not fitted for yielding excellent structures. For some peaks, even deviations of 1000 cm−1 are observed. Therefore, we recommend the reference MO model, where the electronic structure parameter is fitted to the experimental data.

The coupling between the promoted electrons in a ligand localized orbital and metal center localized d5 electron results in 12 orbitals with E, six orbitals with A1, and six orbitals with the A2 symmetry. The MO scheme given in the literature then gives the matrices for the effect of SOC in basis of these coupled states. The transitions from the ground state of the A1 symmetry to states of the E symmetry are XY polarized, and the transitions to A2 symmetry states are Z polarized. Table II gives the values of the electronic transitions for the complexes. The mean absolute deviations (MADs) were also calculated for each complex in the SOC-GFN1-xTB approach and for the reference MO model. The reference MO model shows MAD of 898.75, 1089.17, and 2024.72, while the SOC-GFN1-xTB model gives MAD of 890.125, 1063.85, and 2226.00 for Febpy32+, Rubpy32+, and Osbpy32+ complex, respectively. In Kober and Meier’s approach, the peak intensity of Z polarized transitions could not be calculated. We followed the same procedure for reproducing the results and for an effective comparison. Figure 2 shows the spin-allowed electronic transitions for the complexes.

Excellent agreement was observed between the reference spectra and the spectra obtained at the SOC-GFN1-xTB MO level. Figure 2 depicts the resemblances between the reproduced literature and our work.

WAu12 is an icosahedral 18-valence electron (VE) superatom (Fig. 3) with significant SO splitting of the electronic levels. Superatoms have an electronic configuration sequence quite different from the isolated atom but resemble atom-like electronic and chemical behaviors. Here, the electronic configuration of the reference superatom is 1 s2,1p6,1d10.

FIG. 3.

Optimized icosahedral (Ih) WAu12 gold-based superatom.

FIG. 3.

Optimized icosahedral (Ih) WAu12 gold-based superatom.

Close modal

Figure 4 depicts the SOC-imposed orbital splitting for DFT and GFN1-xTB. As most of the exciting chemistry revolves around the valence orbitals, here, we compare the HOMO d-orbitals of the clusters. SOC-GFN1-xTB gives an excellent match with the SO splitting and electron affinity difference (ΔEA) compared to DFT calculations with SOC-ZORA. The electron affinity difference (ΔEA) is given as

ΔEASOC−GFN1−xTB=EASOC−GFN1−xTBEAGFN1−xTB,
ΔEASOC−DFT=EASOC−DFTEADFT.
FIG. 4.

Spin–orbit splitting for icosahedral (Ih) WAu12 calculated at the SOC-GFN1-xTB level and at the DFT-SOC level with ZORA relativistic corrections. All values are in eV.

FIG. 4.

Spin–orbit splitting for icosahedral (Ih) WAu12 calculated at the SOC-GFN1-xTB level and at the DFT-SOC level with ZORA relativistic corrections. All values are in eV.

Close modal
FIG. 5.

Optimized geometry of ferrous oxyheme at the GFN1-xTB level and calculated SOC energy contribution to the spin flip of the triplet (Fe3+O2) to singlet (Fe3+O2) radical at the SOC-GFN1-xTB level.

FIG. 5.

Optimized geometry of ferrous oxyheme at the GFN1-xTB level and calculated SOC energy contribution to the spin flip of the triplet (Fe3+O2) to singlet (Fe3+O2) radical at the SOC-GFN1-xTB level.

Close modal

The calculated SO splitting of HOMO d orbitals at the DFT-SOC level is 28.0 meV, while SOC-GFN1-xTB predicts the value of 29.5 meV. The ΔEA of the cluster in our model is estimated at 2.122 eV and matches very well with the reported theoretical value of 2.090 eV20 and the experimental value of 2.020 eV.20 Overall, we observe excellent agreement between experiments, DFT and SOC-GFN1-xTB.

Binding of O2, a triplet state molecule, to ferrous deoxyheme, a quintet state molecule, is a vital process for life in its present form. Nature has used the complexes of transition metals to bind, carry, and unload O2 to the cells. There are several factors that prove transition metals as the best candidate for the binding, and one crucial factor is SOC. Despite the small SOC in ferrous deoxyheme, the reaction proceeds ∼1011 faster than in the non-biological Fe–O+ system. Even though the major contribution to the binding process is based on the ligation of heme to the Fe (II) center, which facilitates the low energy interval among different spin states as its quintet ground state and the triplet state of the heme moiety differ in energy by about 10 kJ mol−1 at an Fe–O distance of 2.5 Å as in oxyheme, the SOC contribution is significant. It induces a spin flip from the quintet ground state of deoxyheme to the triplet state of deoxyheme. This lifts the spin restriction on the reaction, as now deoxyheme and dioxygen are both in triplet states. The starting oxyheme radical pair (Fe3+O2) is in a charge transfer state with triplet spin, which flips its spin state to singlet through SOC.31–33 Understanding the effect of SOC on this binding process would assist in mimicking the natural processes. Here, we have reported the values for SOC in both deoxyheme and oxyheme. We have calculated the SOC in our reference structure at the SOC-GFN1-xTB level and compared it with the results reported in the literature. Mössbauer spectroscopy estimates the SOC as ∼0.8 kJ mol−121,31 for ferrous deoxyheme in hemoglobin and myoglobin and a theoretical value of 0.96 kJ mol−133 for ferric oxyheme; our calculated values are 1.6 kJ mol−1 and 1.2 kJ mol−1, respectively, which match reasonably well with the reported Mössbauer and theoretically predicted value (Fig. 5).

We have discussed the parameterization and implementation formalism of SOC for the framework of GFN1-xTB and calculated parameters for the elements throughout the Periodic Table. We tested the SOC-GFN1-xTB formalism on the reference structures as in superatoms, transition metal containing heme moieties, and transition metal containing bipyridine complexes as chromophores in energy transfer processes in MOFs. We have used none of these systems for the calculation of the SOC parameters. The resulting SO splittings are in close accordance with DFT-based reference calculations and experiments, with the deviations smaller than those that are expected for GFN1-xTB-obtained molecular orbitals. This shows excellent transferability and assures that these parameters will be very useful for a wide range of applications where SOC is important. Examples include studying energy transfer processes, designing novel magnetic nanoclusters, and understanding the role of transition metal in spin catalysis and mimicking the natural catalysis processes.

With this work, it is now possible to incorporate SOC in all GFN-xTB calculations with far less computational costs compared to the SOC-ZORA DFT. This work also extends the availability of parameters throughout the Periodic Table which was limited to pre-calculation of the parameters for the literature specific systems.

As GFN-xTB is a well-defined approximation to DFT, extensions to the hamiltonian are system-independent and transferable within different implementations. Therefore, the presented parameters and implementation work well for various GFN-xTB extensions and implementations as in standalone GFN-xTB, AMS-GFN-xTB, and GFN-xTB in dftb+.

See the supplementary material for the detailed information on the spin–orbit coupling parameters and geometric parameters of all crystal structures considered in this work.

We thank the Center for Information Services and High-Performance Computing (ZIH) at TU Dresden for computational resources and CRC 1415 as well as SPP 2244 for financial support.

The authors have no conflicts to disclose.

Gautam Jha: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (supporting). Thomas Heine: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (equal); Resources (lead); Supervision (lead); Validation (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (lead).

Our parameters are available at GitHub for the incorporation of spin–orbit coupling for GFN-xTB implementations beyond the presented model (https://github.com/gajh494c/SOC-DFTB).

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