DFT Calculations for Structural, Electronic, and Magnetic Properties of ZnFe₂O₄ Spinel Oxide: The Role of Exchange-Correlation Functional

Abstract

In this study, quantum-mechanical calculations in the framework of the Density Functional Theory (DFT) were performed to investigate the role of exchange-correlation functional in describing structural, electronic, and magnetic properties of ZnFe₂O₄. Herein B3LYP, PBE0, B1WC, and WC1LYP functionals implemented in the CRYSTAL17 code were considered due to the different amounts of the exact Hartree-Fock exchange fraction. In particular, the role of HF fraction on ZnFe₂O₄ properties was addressed for the first time. Indeed, structural, electronic, and magnetic properties indicate the dependence upon the exchange fraction, where WC1LYP with a 16% exact HF exchange exhibits the best performance compared to the other hybrid functionals. The obtained results reveal an excellent agreement for bandgap, local magnetic moment, long-range magnetic ordering, and unit-cell lattice parameters, overcoming previous theoretical studies based on local/semilocal exchange-correlation treatments. These results confirm the importance of hybrid HF/DFT with controlled HF term contribution to describe the essential features of strongly correlated materials.

Keywords:
ZnFe₂O₄; Antiferromagnetic; DFT; Exchange-correlation functional; WC1LYP

1. Introduction

The Zinc Ferrite (ZnFe₂O₄) is a solid oxide material with the normal spinel structure, where the Zn²⁺ occupies the tetrahedral sites, and the Fe³⁺ occupies the octahedral sites of the cubic Fd-3m structure¹. It is an important material with special characteristics, showing chemical and thermal stability² with unique magnetic properties that enable applications such as gas sensors, electronic components, magnetic materials, and medical diagnosis and drug delivery³. Besides the other characteristics, the semiconductor behavior with a bandgap of ~1.9 eV is used in the photocatalysis of residual colored water. Due to his magnetic behavior, it can be easily removed after the process⁴.

In the last decades, the first-principles calculations, mainly Density Functional Theory (DFT), have played a fundamental role in solid-state investigations, showing a reliable description for different properties^5-8. The theoretical understanding of the ZnFe₂O₄ reported until now had applied different computational methods to describe this material's structural and electronic properties in agreement with the experimental result. For instance, Yao et al.⁹ confirmed the accuracy of the revised Perdew–Burke–Ernzerhof functional (RPBE), based on the Gradient Generalized Approximation (GGA), to investigate the effect of vacancies on the crystal and electronic structure in comparison to other GGA-based functionals. Rodríguez et al.¹⁰ studied the stability, structural, electronic, and magnetic properties of the ZnFe₂O₄ surface (001) with DFT applying the Hubbard corrected PBE (PBE+U) approach. On the other hand, Fritsch¹¹ compared GGA+U and hybrid PBE0 formalism to investigate the electronic and optical properties of the material.

However, the obtained theoretical predictions for the magnetic properties of ZnFe₂O₄ did not approach the experimental results in most cases. In particular, the precise understanding of magnetic structure for spinel ferrites is exciting from the viewpoint of technological applications in several fields. The magnetic ground state definition can affect the description of cation doping, structural, and magnetic phase transitions^9,12,13. Indeed, the treatment of magnetic materials using DFT is challenging once the strong correlation generated from unpaired electrons should be balanced with the orbital localization. In commonly used Local-Density Approximation (LDA) or GGA exchange-correlation functionals, such refined treatment falls on the Self-Interaction Error (SIE) that provides a spurious interaction of an electron with itself, resulting in underestimated band gaps, erroneous localization for d–f orbitals, spin contamination, and the tendency toward magnetism^7,14. Recently, a series of studies have involved a class of functional, which incorporates a fraction of the Hartree–Fock exchange on the DFT treatment to improve the description of strongly correlated materials denominated hybrid functional. Many hybrid functionals are described in the literature for different applications^14-17. In particular, Zuo and co-workers deeply investigated the role of such exchange-correlation treatment on magnetic properties of different spinel ferrites, confirming that exact exchange fraction plays a vital role in the analysis of exchange coupling constant and magnetic properties^18-21.

In particular, in the present work, quantum-mechanical calculations performed for different hybrid (B3LYP, B1WC, PBE0, and WC1LYP) functionals were used to study the structural, electronic, and magnetic properties of ZnFe₂O_4, aiming to provide a general picture of the role of hybrid exchange-correlation treatment for this material. Moreover, the theoretical results reported here can guide the choice for exchange-correlation treatment of spinel ferrite materials, providing a reliable computational method to precisely predict and describe the structural, electronic, and mainly magnetic properties of such strongly correlated materials. The benchmarking analysis was based on the comparison between this paper (theoretical data) and previously reported experimental studies in the literature.

2. Computational Methods

Density functional theory (DFT) calculations using the B3LYP²², B1WC²³, PBE0²⁴, and WC1LYP²⁵ exchange-correlation functional were carried out within CRYSTAL17 code²⁶ to investigate the structural, electronic, and magnetic properties for the ZnFe₂O₄ material. This method has been successfully employed for electronic, structural, photoluminescence, surface, morphology, and magnetic properties of several materials classes^27-31. These exchange-correlation treatments were selected to consider the additional role of the exact Hartree-Fock exchange term in the description of correlated spins.

An atom-centered all-electron basis set described the Zn, Fe, and O atomic centers as 86-411d31G, 86-411d41G, and 8-411d1, respectively, according to the CRYSTAL basis set library. The XCrySDen program was used to analyze the ZnFe₂O₄ periodic model³².

Here, ZnFe₂O₄ was simulated in a normal spinel configuration considering a conventional (56-atom) unit cell of the Fd-3m cubic symmetry, as illustrated in Figure 1a. In this normal structure, the Zn atoms are located at tetrahedral A-sites (Wyckoff position 8a (1/8, 1/8, 1/8)), whereas the Fe atoms (which carry a magnetic moment due to the partially filled 3d shell) occupy the octahedral B-sites (16d, (1/2, 1/2, 1/2)). The O atoms are at 32e (u,u,u) positions of the f.c.c. structure (u= 0.258³³ for the -3m origin). This methodology is in agreement with other theoretical studies published in the literature³⁴.

Figure 1
a) Schematic representation of conventional unit cell for crystalline structure for ZnFe₂O₄ in the normal spinel structure. The gray and orange polyhedral represent the [ZnO₄] and [FeO₆] clusters, respectively. b) Schematic representation of FEM and AFM magnetic models considering only open-shell Fe-sublattices.

The crystalline structure was fully relaxed (in terms of atomic positions and cell lattice parameters) as a function of the total energy. The electronic integration over the Brillouin zone was performed using a 6×6×6 Monkhorst−Pack-mesh³⁵. Five thresholds were chosen to control the accuracy of the Coulomb and exchange integral calculations (8, 8, 8, 8, and 14). The root-mean-square (RMS) gradient, RMS displacement, maximum gradient, and maximum displacement were set to 3×10⁻⁴, 1.2×10⁻³, 4.5×10⁻⁴, and 1.8×10⁻³ a.u., respectively.

The optimized crystalline structures were used to investigate the band structures and density of states (DOS) to explore the electronic structure of ZnFe₂O_4. The band structures were calculated for 30 k-points along the Γ (0; 0; 0), X (½, 0, ½), L (½, ½, ½), and Γ (0; 0; 0) high-symmetry points of the Brillouin zone.

The magnetic properties of ZnFe₂O₄ material were investigated by considering two collinear magnetic configurations (Figure 1b), defined as ferromagnetic (FEM) and antiferromagnetic (AFM), in agreement with other theoretical and experimental studies^34,36. In this case, the magnetic models consider the interactions up to third neighbors, helping to describe the long-range magnetic ordering. The magnetic ground-state definition was based on the energy difference criteria between the investigated models.

Abbreviations are presented in the Supplementary Information.

3. Results and Discussion

3.1. Structural properties

The quantum-mechanical calculations in the framework of the Density Functional Theory (DFT optimized lattice parameters (a = b = c) of ZnFe₂O₄ agree with experimental results published in the literature as summarized in Table 1. The lowest lattice parameter was obtained using B1WC, and the highest values correspond to the B3LYP level of theory. In comparison to experimental data, all employed methods reasonably describe the crystalline structure of ZnFe₂O₄.

Table 1
Results for lattice parameter (Å), oxygen parameter (u), and bond distances (d_Zn-O and d_Fe-O) in (Å) for the B3LYP, B1WC, PBE0, and WC1LYP hybrid functionals. Additional sources of experimental and theoretical data were considered for comparative purposes.

ZnFe2O4	a = b = c	u	dZn-O	dFe-O	Reference
B3LYP	8.521	0.261	2.003	2.042	This work
B1WC	8.398	0.260	1.969	2.016	This work
PBE0	8.449	0.261	1.987	2.024	This work
WC1LYP	8.470	0.260	1.986	2.033	This work
Experimental (nanoparticles)	8.430	0.259	-	-	Bock et al.37
Experimental (nanoparticles)	8.523	-	-	-	Oliveira et al.38
Experimental (Thin Films)	8.451	-	-	-	Jin et al.39
Experimental (nanoparticles)	8.434	-	-	-	Andhare et al.40
GGA	8.390	-	-	-	Jin et al.39
GGA	8.380	0.261	-	-	Quintero et al.41
GGA+U	8.457	0.260	-	-	Quintero et al.41
GGA+U	8.541	0.261	-	-	Fritsch11
PBE0	8.446	0.261	-	-	Fritsch11

On the other hand, interesting trends were verified when comparing the hybrid functional results with previous theoretical calculations. For instance, previous theoretical calculations developed with semilocal GGA functional showed smaller lattice parameters when compared with hybrid functional or GGA+U approaches. The exception was found for B1WC, where the calculated results are very close to GGA. Moreover, the PBE0-CRYSTAL results reported here are closer to previous calculations with plane-wave basis set within VASP⁴¹, indicating good reproducibility of such hybrid exchange-correlation functional. Further, theoretical results obtained with B3LYP, PBE0, and WC1LYP were close to previous GGA+U calculations (U = 5.0^{41 and U} = 5.25 eV¹¹). To select an adequate hybrid functional to describe ZnFe₂O₄ results, additional results concerning bandgap, magnetic ground-state, and local magnetic moment should be analyzed, as described in other sections.

3.2. Electronic properties

Electronic properties were investigated using Band structure with primary atomic orbitals (AOs), Mulliken Analysis Populations, Charge Density Maps, and partial projected DOS. Herein, the experimental AFM ground state was considered for the electronic structure analysis.

Figure 2a-d presents the calculated band structure for ZnFe₂O₄ obtained with different hybrid functional. The energy is in eV, and the Fermi Level was set at the maximum of the valence band (VBM). The dashed line represents the location of a Fermi level set as zero. The analyses of the principal atomic orbital (AOs) components of selected bands were performed with the ANBD option of the CRYSTAL17 code by using a threshold of 0.07 au for the important eigenvector coefficients following previous theoretical studies⁴².

Figure 2
Representation of the ZnFe₂O₄ band structure with main atomic orbitals in VB and CB.

The E_gap value was obtained for all band structures as the difference between the valence band maximum (VBM) and the bottom of the conduction band (CBM). Once the VBM and CBM were located at Γ point for all investigated models, the minimal E_gap values correspond to a direct transition calculated as 2.61, 3.21, 4.06, and 2.53 eV for B1WC, B3LYP, PBE0, and WC1LYP hybrid functionals, respectively. Experimental bandgap results for ZnFe₂O₄ can be divided into two groups: (i) 1.78–2.01 eV obtained for mesoporous powder or nanoparticles and (ii) 2.61–3.25 eV reported for thin films and smaller nanoparticles, as reported by Ulpe et al.⁴³ and the references therein. However, comparing the hybrid DFT calculations reported here with previous theoretical data obtained with GGA+U (1.68 - 2.2 eV)^11,41,44,45 G₀W₀ (2.02 eV)⁴⁶ and PBE0 (3.68 eV)¹¹, it was observed that B1WC and WC1LYP hybrid functionals exhibit closer results to GGA+U and experimental data, while B3LYP and PBE0 overestimate the bandgap value as a function of the Hartree-Fock fraction.

For all band structures, the VB is formed by O-2p states (2p_x, 2p_y, and 2p_z) atomic orbitals, with minor contributions from Fe-3d and Zn-3d atomic orbitals. The relative contributions of atomic orbitals for occupied and virtual bands depend upon the exchange-correlation treatment. In the VB, the Fe-3d_z² levels shift down to the O-2p_x and O-2p_z levels with a change of hybrid functional. In the B3LYP level, the top of VB has largely O-2p atomic orbitals, while for WC1LYP, PBE0, and BW1C, the Fe-3d_x²_-y² and Fe3d_z² contributions become larger close to VBM. In CB, the main contribution for all approaches came from Fe-3d_xz with a minor contribution of 3d-Zn atomic orbitals. These shifts are a consequence of the self-interaction correction provided by each hybrid functional.

The total and atom-resolved DOS projections, as shown in Figure 3, were analyzed to in-depth understand the role of atomic orbitals on the symmetry-adapted crystalline orbitals (SACO’s). In this case, DOS analysis considered the VB SACO’s located between 0 and -2 eV. The conduction band (CB) was analyzed with different energy ranges according to the employed hybrid functional. For BW1C, the CB was observed between 2.61 and 5 eV, B3LYP between 3.21 and 5 eV, PBE0 between 4.06 and 5 eV, and WC1YLP between 2.53 and 5 eV.

Figure 3
Total and Partial Density of States (DOS) for ZnFe₂O₄, periodic models BW1C, B3LYP, PBE0, and WC1LYP hybrid functional.

In the DOS analysis illustrated in Figure 3, the main contribution to VB, in all cases, was associated with O 2p (px, py, pz) orbitals with a small content of Zn and Fe orbitals. Concerning the conduction band (CB) composition, the obtained results indicate a dominant contribution of Fe 3d atomic orbitals (3d_xz, 3d_xy, 3d_yz, 3d_z², 3d_x²_-y²) with a small content of Zn 3d atomic orbitals hybridized with oxygen atomic orbitals. Thus, it is possible to assume that an electron transfer inside the bandgap region should occur between 2p orbitals of the oxygen atom and 3d orbitals of the Fe and Zn in octal- and tetrahedral configuration, respectively, represented by [ZnO₄], and [FeO₆] clusters.

The electron density partition and its analogs (net charge and bonding population) correspond to a common characteristic used to analyze the nature of chemical bonds in ZnFe₂O₄. The Mulliken population analysis⁴⁷ was employed for both atomic charge and bonding overlap, as shown in Table 2 and Table 3, considering different hybrid exchange-correlation functionals.

Table 2
Theoretical results of the Mulliken population analysis for atomic charge net (in |e|) performed on ZnFe₂O₄ structure.

ZnFe2O4	B1WC	B3LYP	PBE0	WC1LYP
Atoms	Atomic charge (|e|)
Zn	1.310	1.321	1.351	1.295
Fe	2.253	2.260	2.321	2.227
O	-1.454	-1.460	-1.498	-1.437

Table 3
Overlap population (in m|e|) of the Zn-O and Fe-O bonds in the AFM structure.

ZnFe2O4 bonds	B1WC	B3LYP	PBE0	WC1LYP
Zn-O	0.107	0.109	0.104	0.110
Fe-O	0.023	0.024	0.021	0.025

From Table 2, it can be seen positive charges on the Fe and Zn cations and negative charges on O anions. The O-atoms act as electron acceptors, whereas Zn and Fe atoms act as electron donors. Nevertheless, such theoretical results depend on the atomic basis set; then, the discussion about net charge will be performed through a qualitative viewpoint. The calculated charge for the O atoms was calculated following the formal charge O^-2, and the remaining values for Zn and Fe in the tetra- and the octahedral site did not approach the formal values of +2 and +3, respectively.

The Mulliken bond population analysis is listed in Table 3. The positive values for Zn-O and Fe-O indicate the dominant covalent character due to the effective overlap of the atomic orbitals characterizing an accumulation of electronic density in the internuclear region. The PBE0 hybrid functional makes the materials more ionic due to the less value of the overlap population than other hybrid functional. On the other hand, the WC1LYP hybrid functional indicates a more covalent picture for all Zn-O and Fe-O bonds due to the larger overlap values.

The Zn-O and Fe-O bond character and charge distribution for the ZnFe₂O₄ periodic model were investigated using electronic density isolines in [001] direction (Figure 4). These results were obtained from the optimized wave function. An isoline is a set of equal-value points in domain data showing the electron density around the atomic nucleus. Colors can perform the charge density maps analysis plot in the plane to represent the charge density distribution. These electronic density maps were described along [001] direction containing Zn, O, and Fe atoms. This direction was chosen because it provides relevant information concerning the charge connection between [ZnO₄] and [FeO₆] tetra-octahedral clusters. The deviation from the ideal ionic charge density is more extensive for Zn than for Fe, suggesting a higher degree of covalent in the Zn-O than in the Fe-O interaction. The homogeneous distribution of contour lines represents the strongly covalent character in the interaction of the Zn and Fe cations atoms with oxygen anion on the [001] analyzed plane. The observed behavior occurs because of the hybridization between the 2p O atomic orbitals with 3d (Zn and Fe) atomic orbitals.

Figure 4
Electronic charge density for ZnFe₂O₄ in [001] direction using WC1LYP.

3.3. Magnetic properties

In this section, the magnetic properties of ZnFe₂O₄ were investigated by considering different hybrid exchange-correlation treatments on first-principles calculations. For this purpose, the local magnetic moment (S_z) was calculated for Fe-site and the energy difference of AFM and FEM models (ΔE_AFM-FEM), as presented in Table 4.

Table 4
Fe magnetic moment (S_z - µ_B) and ΔE_AFM-FEM (meV/f.u.) for ZnFe₂O₄ as a function of exchange-correlation treatment.

Exchange-Correlation Functional	$S_z^{Fe}$	ΔEAFM-FEM	Reference
B1WC	4.277	-25.94	This Work
B3LYP	4.297	18.24	This Work
PBE0	4.361	27.20	This Work
WC1LYP	4.254	-14.68	This Work
Experimental	4.220	AFM	Schiessl et al.48
PBE0	4.1 – 4.3	30.00	Fritsch11
GGA+U	4.1 – 4.2	AFM	Cheng34
GGA	3.7 – 3.9	AFM	Cheng34

The obtained results indicate that WC1LYP performs the best for both Fe-site magnetic moment and ground-state magnetic ordering representation. It is important to note that commonly employed B3LYP and PBE0 hybrid functional do not correctly describe the experimental AFM magnetic ordering for ZnFe₂O₄. On the other hand, the B1WC hybrid functional successfully described the AFM ground state with a larger magnetic moment than WC1LYP. Moreover, the WC1LYP overcomes GGA and GGA+U approaches once the computed magnetic moments showed a remarkable agreement with the experimental data and reported for bandgap analysis in the previous discussion.

Therefore, combining all analyses, it was possible to note that ZnFe₂O₄ properties depend upon the exchange-correlation treatment. The hybrid exchange-correlation formalism takes into account the mixing between the exact nonlocal Fock exchange and LDA/GGA exchange-correlation terms through the adiabatic connection (ACM) between non-interacting Kohn-Sham reference system (λ = 0) to the fully interacting real system (λ = 1). In this context, the dependence can be analyzed in terms of the Fock exchange fraction once the investigated functional contains different amounts of such term combined with different DFT formalisms.

For instance, the PBE0 functional contains the correlation energy fully described by the Perdew–Burke-Ernzerhof (PBE) formalism, while the exchange energy is obtained by combining 25% of Hartree–Fock exchange energy and 75% of PBE exchange functional²⁴, being described by the following expression:

$E_{XC}^{PBE0}=\frac{1}{4}{E}_X^{HF}+\frac{3}{4}{E}_X^{PBE}+E_C^{PBE}$ (1)

On the other hand, commonly employed B3LYP uses the VWN local-density approximation and LYP functional for the correlation term, while the exchange term is described as combining Becke88 (B88), LSDA exchange functional with a 20% (a = 0.2) contribution of the exact Hartree-Fock exchange⁴⁹. The such scheme takes into account three empirical parameters to rule the mixing of HF exchange and DFT exchange and correlation:

$\begin{array}{l}{E}_{XC}^{B3LYP}=\left(1-a\right)E_X^{LSDA}+a{E}_X^{HF}+\\ b\Delta E_X^{B88}+c\Delta E_C^{LYP}+\left(1-c\right)E_C^{VWN}\end{array}$ (2)

Moreover, B1WC mixes 16% (a = 0.16) of exact exchange energy with 84% of Wu-Cohen GGA exchange, with a PW91 correlation term²³. The WC1LYP hybrid functional maintains the exchange description of B1WC, with the LYP functional for the correlation term²⁵. In these cases, the one-coefficient representation is sufficient to rule the HF/DF exchange ratio according to:

$E_{XC}^{B1WC}=\left(1-a\right)E_X^{WC-GGA}+a{E}_X^{HF}+E_C^{PW91}$ (3)

$E_{XC}^{WC1LYP}=\left(1-a\right)E_X^{WC-GGA}+a{E}_X^{HF}+E_C^{LYP}$ (4)

Hence, it was observed that by reducing the exact Hartree-Fock exchange contribution, the structural, electronic, and magnetic properties of ZnFe₂O₄ materials become closer to the experimental results published in the literature. This fact can be associated with the localization of unpaired electrons and the Fe 3d orbitals resulting in a strong correlation between the particles. Moreover, between B1WC and WC1LYP hybrid schemes, the latter becomes a more reliable choice to adequately describe the ZnFe₂O₄, mainly due to the contribution of LYP functional for energy levels distribution resulting in closer bandgap values for this treatment.

The role of the HF exchange term was confirmed by previous studies involving Fe-based materials and strongly correlated materials^50-52. In particular, for Fe³⁺ 3d orbital description, the exact exchange fraction seems to play a fundamental role in analyzing exchange coupling constant and magnetic properties, highlighting the importance of benchmark analysis to describe the magnetic properties and electronic structure adequately^17-20. Herein, it was confirmed that hybrid functional with reduced HF exchange term performs the best compared to previous studies based on LSDA+U and GGA+U schemes. Despite the worry about the computational cost involving hybrid functionals, CRYSTAL17 code can successfully perform the analysis of structural, electronic, and magnetic properties with an increased precision against the experimental with a reduced computational effort. Recently, the role of the exact exchange term was addressed as the inverse of the dielectric constant of solids, where the value can either be taken from experiments or estimated from first-principles calculations to follow the material dependency of the exchange-correlation functional^53,54. This fact can be investigated in further studies comparing different exchange-correlation treatments to verify the compromise between the exchange fraction and the correlation treatment.

Those findings aim to fulfill the critical and lacking information about the proper benchmarking analysis of employed exchange-correlation treatments. Although the Hubbard corrections commonly employed along with GGA and LSDA treatments can give quite good results for the electronic properties of solid-state materials, the local magnetic moment, ground-state definitions, and related properties of strongly correlated cases remains challenging.

4. Conclusion

Hybrid Density Functional Theory calculations employing B3LYP, PBE0, B1WC, and WC1LYP exchange-correlation functional were carried out to investigate the structural, electronic, and magnetic properties of ZnFe₂O₄. The obtained results indicate the dependence of structural, electronic, and magnetic properties on the exact Hartree-Fock exchange fraction, being the WC1LYP the functional that best performs the investigated properties. In particular, the obtained theoretical results for this approach reveal an excellent agreement for bandgap, local magnetic moment, long-range magnetic ordering, and unit-cell lattice parameters concerning the experimental results published in the literature. Therefore, we can argue that hybrid HF/DFT formalism with controlled HF term contribution is the most suitable choice to investigate the strongly correlated ZnFe₂O₄, contributing to further studies involving such technological material. It is worth noting that the role of the HF-exchange term was addressed for the first time for ZnFe₂O₄, confirming that benchmarking analysis can overcome previous theoretical studies and provide a guide for further studies of strongly correlated materials.

5. Acknowledgments

The authors would like to thank the following Brazilian research financing institutions for financial support: the National Council for Scientific and Technological Development – CNPQ, the Coordination for the Improvement of Higher Education Personnel – CAPES, the Goiás Research Foundation – FAPEG, the São Paulo Research Foundation – FAPESP (2013/07296-2 and 2021/01651-1). The authors also thank the National Laboratory for Scientific Computing (LNCC), National Center for High-Performance Processing, and the High-Performance Computing Center (NACAD) at the Federal University of Rio de Janeiro (COPPE-UFRJ) and University of Campinas (CENAPAD-SP) for providing the computational resources. And to thank the Federal University of Catalão – UFCAT including the Institute of Chemistry - UFCAT, and PPGQ UFGD/UEG/UFCAT for general support.

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Publication Dates

• Publication in this collection
  28 Oct 2022
• Date of issue
  2022

History

• Received
  29 Apr 2022
• Reviewed
  18 Aug 2022
• Accepted
  04 Oct 2022