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.

## INTRODUCTION

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 molecular^{3,5} and periodic^{6} 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 Table^{9} 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 SOC^{13} on the absorption spectra of complexes such as in M(bpy)_{3}^{2+} (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 O_{2} 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.

## METHOD

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 DFTB^{9} and are included here for completeness.

### Extended tight-binding method

The total energy in GFN1-xTB^{4} comprises electronic energy (*E*_{el}), atom-pairwise repulsion (*E*_{rep}), dispersion (*E*_{disp}), and halogen bonding term (*E*_{XB}), which is represented as

The electronic energy is given as

where *H*_{0} is the zero-order hamiltonian, *ψ*_{i} is the single-electron wave function of a valence molecular orbital (MO), and *n*_{i} 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. *q*_{A} is the Mulliken charge of atom *A*, and Γ_{A} is the charge derivative of the atomic Hubbard parameter. *T*_{el}*S*_{el} 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 *γ*_{AB,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 model^{13} is given as

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, $L\u0302$ is the angular momentum operator, and $S\u0302$ is the spin operator. $H\u0302\mu \nu L\u0302\u22c5S\u0302$ 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 SOC^{8} reads as

Here, *k*_{l} and *k*_{l′} are the Hückel parameters for angular momentum *l* for atom *A* and *l*′ for atom *B* and *K*_{AB} is a scaling constant. The energy levels for atom *A* and atom *B* are represented as $hAl$ and $hBl\u2032$ with *l* and *l*′ being the angular momentum, respectively. *S*_{μv} is the overlap matrix, and Δ*EN*_{AB} is the electronegativity difference of two atoms with *k*_{EN} as a proportionality coefficient. $\Pi RAB,ll\u2032$ is a distance and *l*-dependent function.

### Calculation of SOC parameters

We have calculated the SOC parameters for free atoms throughout the Periodic Table employing AMS-BAND^{30} 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 *Δ**H* in terms of *Δ**H*_{T} (Thomas precession energy) and *Δ**H*_{L} (Larmor interaction energy) can be given as

where $\gamma =1\u2212Zn\alpha 22$ is derived by using $vc=Zn\alpha $ in $\gamma =1\u2212v2c2$. 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

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.

### Computational details of the benchmark calculations

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.

### Geometries

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.

. | $Febpy32+$ . | $Rubpy32+$ . | $Osbpy32+$ . | |||
---|---|---|---|---|---|---|

Electronic MO model^{14}
. | SOC-GFN1-xTB MO model . | Electronic MO model^{14}
. | SOC-GFN1-xTB MO model . | Electronic MO model^{14}
. | 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 model^{14}
. | SOC-GFN1-xTB MO model . | Electronic MO model^{14}
. | SOC-GFN1-xTB MO model . | Electronic MO model^{14}
. | 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 |

. | $Febpy32+$ . | . | $Rubpy32+$ . | . | $Osbpy32+$ . | . | |||
---|---|---|---|---|---|---|---|---|---|

Excitations . | Electronic MO m^{14}
. | SOC-GFN1-xTB . | σ (SD) . | Electronic MO m^{14}
. | SOC-GFN1-xTB . | σ (SD) . | Electronic MO m^{14}
. | SOC-GFN1-xTB . | σ (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 |

$1A1\u2032$ | −975 | −796 | 126.6 | −1285 | −1190 | 67.18 | −2785 | −2957 | 121.6 |

$2A1\u2032$ | 855 | 701 | 108.9 | 1185 | 1089 | 67.88 | 2085 | 2153 | 48.08 |

$3A1\u2032$ | −550 | −438 | 79.19 | 10 | 76 | 46.67 | 450 | 789 | 239.7 |

$4A1\u2032$ | −1980 | −1835 | 102.5 | −1350 | −1288 | 43.84 | −750 | −384 | 258.8 |

$5A1\u2032$ | −2025 | −1914 | 78.48 | −1565 | −1529 | 25.46 | −1135 | −718 | 294.9 |

$6A1\u2032$ | −2525 | −2577 | 36.77 | −3095 | −3026 | 48.79 | −5165 | −5543 | 267.3 |

$1A2\u2032$ | −1175 | −1247 | 50.91 | −1750 | −1662 | 62.25 | −3350 | −3756 | 287.1 |

$2A2\u2032$ | −545 | −447 | 69.29 | −50 | −29 | 14.89 | 950 | 1052 | 72.12 |

$3A2\u2032$ | −550 | −438 | 79.19 | 10 | 76 | 46.67 | 450 | 789 | 239.7 |

$4A2\u2032$ | −1980 | −1835 | 102.5 | −1350 | −1288 | 43.84 | −750 | −384 | 258.8 |

$5A2\u2032$ | −2025 | −1914 | 78.48 | −1565 | −1529 | 25.46 | −1135 | −718 | 294.9 |

$6A2\u2032$ | −2525 | −2577 | 36.77 | −3095 | −3026 | 48.79 | −5165 | −5543 | 267.3 |

. | $Febpy32+$ . | . | $Rubpy32+$ . | . | $Osbpy32+$ . | . | |||
---|---|---|---|---|---|---|---|---|---|

Excitations . | Electronic MO m^{14}
. | SOC-GFN1-xTB . | σ (SD) . | Electronic MO m^{14}
. | SOC-GFN1-xTB . | σ (SD) . | Electronic MO m^{14}
. | SOC-GFN1-xTB . | σ (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 |

$1A1\u2032$ | −975 | −796 | 126.6 | −1285 | −1190 | 67.18 | −2785 | −2957 | 121.6 |

$2A1\u2032$ | 855 | 701 | 108.9 | 1185 | 1089 | 67.88 | 2085 | 2153 | 48.08 |

$3A1\u2032$ | −550 | −438 | 79.19 | 10 | 76 | 46.67 | 450 | 789 | 239.7 |

$4A1\u2032$ | −1980 | −1835 | 102.5 | −1350 | −1288 | 43.84 | −750 | −384 | 258.8 |

$5A1\u2032$ | −2025 | −1914 | 78.48 | −1565 | −1529 | 25.46 | −1135 | −718 | 294.9 |

$6A1\u2032$ | −2525 | −2577 | 36.77 | −3095 | −3026 | 48.79 | −5165 | −5543 | 267.3 |

$1A2\u2032$ | −1175 | −1247 | 50.91 | −1750 | −1662 | 62.25 | −3350 | −3756 | 287.1 |

$2A2\u2032$ | −545 | −447 | 69.29 | −50 | −29 | 14.89 | 950 | 1052 | 72.12 |

$3A2\u2032$ | −550 | −438 | 79.19 | 10 | 76 | 46.67 | 450 | 789 | 239.7 |

$4A2\u2032$ | −1980 | −1835 | 102.5 | −1350 | −1288 | 43.84 | −750 | −384 | 258.8 |

$5A2\u2032$ | −2025 | −1914 | 78.48 | −1565 | −1529 | 25.46 | −1135 | −718 | 294.9 |

$6A2\u2032$ | −2525 | −2577 | 36.77 | −3095 | −3026 | 48.79 | −5165 | −5543 | 267.3 |

### Effect of SOC on charge transport properties of chromophores in MOFs

We reproduced the results of the effect of SOC on absorption spectra of the ions M(bpy)_{3}^{2+} (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 *dπ* → *π** electronic transitions from *dπ* orbitals (with T_{2g} symmetry) of the metal center to $\pi *1$ 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 *dπ*_{E} and $d\pi A1$ orbitals generated after lifting the degeneracy of the *dπ* orbitals. Similarly, Γ is the energy difference between the $\pi 1E*$ and $\pi 1A2*$ orbitals generated after lifting the degeneracy of the $\pi 1*$ 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 d^{5} electron results in 12 orbitals with E, six orbitals with A_{1}, and six orbitals with the A_{2} 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 A_{1} symmetry to states of the E symmetry are XY polarized, and the transitions to A_{2} 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.

### Superatoms

$WAu12$ 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 s^{2},1p^{6},1d^{10}.

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 (Δ*EA*) compared to DFT calculations with SOC-ZORA. The electron affinity difference (Δ*EA*) is given as

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 Δ*EA* of the cluster in our model is estimated at 2.122 eV and matches very well with the reported theoretical value of 2.090 eV^{20} and the experimental value of 2.020 eV.^{20} Overall, we observe excellent agreement between experiments, DFT and SOC-GFN1-xTB.

### Effect of SOC on binding of O_{2} on ferrous deoxyheme

Binding of O_{2}, 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 O_{2} 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 ∼10^{11} 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 (Fe^{3+}O_{2}^{−}) 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^{−1}^{21,31} for ferrous deoxyheme in hemoglobin and myoglobin and a theoretical value of 0.96 kJ mol^{−1}^{33} 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).

## CONCLUSIONS

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+.

## SUPPLEMENTARY MATERIAL

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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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).

## DATA AVAILABILITY

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).