First‐Principles Study of Optical Properties of Linde‐Type A Zeolite

Within the framework of density functional theory, electronic structures of Linde‐type A (LTA) zeolite membranes are calculated, and based on the electronic structures, several key optical properties of LTA zeolite membranes can be conveniently determined, with the method of the full‐potential linearized augmented plane wave under the random phase approximation for the boundaries and the generalized gradient approximation for the exchange–correlation potentials. The calculated optical properties as a function of frequency include the real and imaginary parts of the dielectric function ε(ω), the refraction index n(ω), the extinction coefficient k(ω), the reflectivity R(ω), the energy loss function L(ω), the absorption coefficient α(ω), and finally the optical conductivity σ(ω), along with specific directions of the primitive cell. It is demonstrated that the optical properties of the sodium LTA zeolite cluster appear anisotropic within the range of UV. The nature of anisotropy implies that all the functions are 3 × 3 tensors with nonzero off‐diagonal elements. Three diagonal and three off‐diagonal elements are simultaneously presented.


Introduction
[3] This unique feature grants them the properties of molecular sieves, making them highly valuable for the selective separation of gases and organic molecules.Zeolites are commonly referred to as aluminosilicates, and their physicochemical properties are greatly influenced by three main parameters: the crystalline structure, the Si/Al ratio, and the type of exchange cation present in the framework.Indeed, these factors play a crucial role in defining the characteristics and behavior displayed by zeolites, thereby influencing the wide array of practical applications that can be utilized.While zeolites can be found naturally, the number of identified types with distinct crystalline structures is relatively limited, standing at around 40. [4] Due to the significance of structure in their properties, the range of applications for these natural zeolites is constrained.Additionally, natural zeolites may contain undesired contaminants, which can obstruct their pores and pose challenges in terms of removal or may incur high costs, further limiting their practical use.As a result, there is a growing interest in the synthesis of zeolites, as it offers more control over their structure and purity, expanding their potential applications and enhancing their performance in various industries.
Among synthetic zeolites, the Linde-type A (LTA) zeolite stands out as one of the most representative.It exhibits exceptional selectivity in various processes, including pervaporation, [5] serving as an additive in laundry detergent, [6] facilitating the industrial dehydration of ethanol, [7][8][9][10] influencing bacterial adhesion, [11] and participating in the functionalization of textiles, [12] among other applications.Its versatility and effectiveness make it a valuable and widely used zeolite in numerous industrial and scientific fields.In the last years, LTA zeolite clusters have been synthesized with microwaves, and the catalysis performance of the materials has been widely studied. [13,14][15] Desired elements may be attached to specific surfaces of the clusters. [15,16]he developments enable the zeolite materials to increase catalysis efficiencies, and the properties of the zeolites may be influenced externally by light in the range of UV and visible DOI: 10.1002/pssb.202300378Within the framework of density functional theory, electronic structures of Lindetype A (LTA) zeolite membranes are calculated, and based on the electronic structures, several key optical properties of LTA zeolite membranes can be conveniently determined, with the method of the full-potential linearized augmented plane wave under the random phase approximation for the boundaries and the generalized gradient approximation for the exchange-correlation potentials.The calculated optical properties as a function of frequency include the real and imaginary parts of the dielectric function ε(ω), the refraction index n(ω), the extinction coefficient k(ω), the reflectivity R(ω), the energy loss function L(ω), the absorption coefficient α(ω), and finally the optical conductivity σ(ω), along with specific directions of the primitive cell.It is demonstrated that the optical properties of the sodium LTA zeolite cluster appear anisotropic within the range of UV.The nature of anisotropy implies that all the functions are 3 Â 3 tensors with nonzero off-diagonal elements.Three diagonal and three offdiagonal elements are simultaneously presented.
An LTA zeolite membrane may be complicated as there are usually more than 100 elements involved, and the optical properties of the clusters are particularly sensitive to the inclusion and location of key elements.The computational task of calculating the electronic structures of such huge mesoscopic clusters is usually challenging.Therefore, to understand the optical interaction of the material and also to find the application of LTA zeolite membranes in optoelectronics and catalysis, it is appealing to have a reliable ab initio study on the electronic structures as well as the optical properties of the material.The calculation should be sensitive and reactive to any changes in the optical properties due to modification of the cluster, such as the inclusion of elements, position changes of the elements, and so on.Particularly, one pays attention to the metallic or conductive elements, as the inclusion of such elements alters the plasmonic properties.It is worth mentioning that the optical properties are expected to be anisotropic for the LTA zeolite.
In the present work, within the framework of density functional theory (DFT), the electronic structures of LTA zeolite membranes are determined using the full-potential linearized augmented plane wave (FP-LAPW) method and the generalized gradient approximation (GGA) for the exchange-correlation potential. [23,24]On the base of the electronic structure, several key optical properties of LTA zeolite membranes are calculated under the random phase approximation (RPA).The calculations are performed using WIEN2K package, a widely used program for FP-LAPW. [24]The optical properties were extracted from the calculated electronic structures. [25,26]Six tensor elements are output for all functions.
In passing we note also that it is in general desirable and plausible to have a reliable numerical process to extract the optical properties of complicated huge molecules from the density of state calculated based on the DFT.The most important application is to serve as a fingerprint of the synthesized materials.This is helpful for material characterization and in turn for guiding the synthesis production.This is particularly needed for LTA zeolites.Numerical estimation of optical properties provides a simple way to analyze the materials (see, for instance, refs.[2-5]).
The present article is organized as follows.In Section 2, the theory and program are to be introduced and outlined.In Section 3, some calculated results are presented and discussed.Finally, the work is concluded in Section 4.

Theory and Computation
In the present work, the calculations of the electronic structure were realized within the framework of DFT, with the FP-LAPW method on the platform of WIEN2K program. [24]The program makes use of the RPA for the boundaries and the GGA for the exchange-correlation potentials. [23]The program solves a set of eigen equations for a many-electron system: which is also referred to as the Kohn-Sham equation in the DFT.
In the Kohn-Sham approach of DFT, the potential v eff ðrÞ is named the Kohn-Sham potential, and the eigen energy ε i and its corresponding wave function φ i ðrÞ (spin degenerated) are called the Kohn-Sham orbital energy and the Kohn-Sham orbital, respectively.For an N electron system, the electron density can be written as a sum of individual Kohn-Sham orbital densities as and as a function of the charge density, the total energy of the system becomes where the first term denotes the Joule energy stemming from the external potential v ext , the second term is the kinetic energy, the third term is the Hartree or Coulomb energy, and the last term is the exchange-correlation energy.The effective Kohn-Sham potential can be obtained from a derivation of the total energy over the charge density as where, in the right-hand side of the equation, the second term is the Hartree potential and the last term denotes the exchangecorrelation potential.A self-consistent iterative procedure calculates the Kohn-Sham potential starting from an initial electron density.Then, through a solution of Equation ( 1), and the use of the relation in Equation ( 2), one arrives a new electron density.If the initial and new densities are identical, then the ground state density has been found, or the energy is minimized.Otherwise, one must select a new trial density through the total energy in Equation ( 3) and repeat the iterative procedure.
To effectively speed up the convergence process to minimize the energy, the WIEN2K program divides the unit cell into nonoverlapping spheres centered at atomic sites called muffin-tin (MT) spheres and the interstitial region.In the MT spheres, the Kohn-Sham orbitals are expanded as a linear product of radial functions and spherical harmonics, and as a plane wave expansion in the interstitial region.The wave functions at each MT sphere are divided into core and semicore valence subsets.The core states are treated within the spherical part of the potential only and are assumed to have a spherically symmetric charge density that is completely confined within the MT spheres.
Once the electronic structure is determined, key optical parameters are ready to be extracted from possible interband transitions between unoccupied and occupied states.Optical calculations are performed in the RPA using WIEN2K code.The imaginary part of the dielectric function is obtained by calculating momentum matrix elements between the occupied and unoccupied states: [27] Imfε where the matrix element of α and β components of momentum is calculated between ψ C n k and ψ V n k states, which are the crystal wave vector k, respectively.The interband expansion on the corresponding real part was obtained by the Kramers-Kronig transformation: The reflectivity at normal incidence is given by where n and k are real and imaginary parts of complex refractive index, which are known as refractive index and the extinction coefficient, respectively, and are given by the following relations: The absorption coefficient and the optical conductivity were calculated using the following relations: The electron energy loss function in terms of the imaginary part of the dielectric tensor ε αβ ðωÞ is given by The technical parameters we used on WIEN2K are as follows.The convergence number, that is, the smallest MT radii times the plane wave cutoff parameter, R min MT K max ¼ 6.The maximum index for the wave function l max ¼ 10.G max ¼ 12:0=Bohr.The energy separation between the valence states and the nuclei was À7.5 Ry.The number of k points used in the calculations was 14 kp (3 Â 3 Â 3 mesh) for the electric properties and 112 kp (6 Â 6 Â 6 mesh) for the optical properties, all in the irreducible Brillouin zone.The convergence criterium was chosen 0.0001e.Within the DFT approximation, the choice of the exchangecorrelation potential used to calculate the electronic properties is very important.In this study, the Tran-Blaha modified Becke-Johnson (TB-mBJ) [28] was used; with this approximation, the bandgaps can be calculated efficiently and accurately.
In this study, our focus was on the sodium LTA zeolite with a Si/Al = 1 ratio, ensuring the distribution of aluminum atoms adheres to Löwenstein's rule. [1]To fulfill this criterion, we employed the unit cell outlined by Antunez et al., [2] which is depicted in Figure 1.For this cell, the parameters are a = b = c = 32.465Å and the angles α = β = γ = 60°.The number of atoms within the unit cell is 168 (96 oxygen and 24 atoms of aluminum, 24 of silicon, and 24 of sodium).The spatial group P1 was assigned to this unit cell, based on the methodology described in refs.[29,30], and the corresponding Brillouin zone is shown in Figure 1b.The MT radii R MT were set to 1.70, 1.38, 1.60, and 1.90 for Al, O, Si, and Na atoms, respectively.Al (1s 2 2s 2 ), O (1s 2 ), Si (1s 2 2s 2 ), and Na (1s 2 ) electronic states are treated as core, while Al (2p 6 3s 2 3p 1 ), O (2s 2 2p 4 ), Si (2p 6 3s 2 3p 2 ), and Na (2s 2 2p 6 3s 1 ) electronic states are considered as valence states.We note that details of the cage described in Figure 1 can be found in ref. [2].In Figure 1, the unit cell of the cage is depicted, as well as its Brillouin zone, which is needed in the present simulations.

Results and Discussions
Figure 1a shows the optimized unit cell used to describe the sodium LTA zeolite.From this cell, the sodalite cage was extracted along with its six double four-membered ring (D4R) cages as shown in Figure 1c.In the latter figure, three sodium atoms (green spheres) orbit around each D4R, while only one (yellow sphere) is located at the center of a six-membered ring (6MR).Previous investigations [3,4] focusing on sodalite zeolite (SOD) have unveiled that the equilibrium position of Na cations lies around the 6MR plane.Therefore, although the crystal structures of SOD and LTA zeolites markedly differ, the distinctive arrangement of Na cations in these two zeolites implies disparate mechanisms for ionic exchange as well as dissimilar processes for ionic diffusion.
Figure 2a shows the band structure obtained for the sodium LTA zeolite along the different trajectories described by the different special points. [29]It shows a large bandgap at 6.5 eV and within the forbidden zone, two distinct bands at 4.5 and 5.2 eV.The total density of states (TDOS) shown in Figure 2b shows that the contribution of the bands around 4.5 and 5.2 eV is small (indicated by dashed arrows).Figure 2c depicts the partial density of states (PDOS) for the distinct chemical elements constituting the zeolite.A direct comparison among these figures reveals that the states emerging within the forbidden band arise from the direct interaction between Na and O atoms.This phenomenon is akin to observations made in recent studies focusing on the lamellar form of ZSM-5 zeolite.In that context, the investigation underscores that the peculiar nature of the Na─O ionic bond prevents complete saturation of the O valence orbital.This, in turn, results in a conduction phenomenon between sodium and oxygen, thereby leading to the emergence of states within the forbidden band.However, a closer analysis of Figure 2b,c shows that despite the reduction in the bandgap, the zeolite continues to exhibit insulating behavior.At the same time, the density of states introduced remains modest.Consequently, although electron conduction between sodium and oxygen atoms occurs, it is very low compared to that between O with Al and Si atoms at a higher bandgap.These calculations were performed using the TB-mBJ potential.Having obtained the band structure, one continues to calculate the optical properties.As the material presents a semiconductor behavior, the intraband transitions are not considered in the present work; only the interband transitions were taken into account.As already mentioned, the optical calculations  were made in a 6 Â 6 Â 6 k-point mesh and the TB-mBJ approximation.
It is worth mentioning that it is expected that the optical properties are anisotropic for the clusters.The nature of anisotropy implies that all the functions are 3 Â 3 tensors with nonzero offdiagonal elements.There should be nine elements in the tensors, but the tensors are symmetric.Therefore, the theoretical process should be able to output six elements, i.e., three diagonal elements and three off-diagonal elements.The six elements are schematically shown in Equation ( 13 One of the most important optical parameters of a material is the complex dielectric function ε ¼ ε 1 þ iε 2 , especially the imaginary part ε 2 .This imaginary part of the function contains energy loss information as well as the absorption properties of the material.However, the real part ε 1 provides information on the polarization when an electric field is applied. In Figure 3, both the calculated real ε 1 and imaginary parts ε 2 of the dielectric function are, respectively, presented.One observes from this figure ε 2 that there is significant optical excitation for energies energy less than 10 eV above 8 eV with two peaks at %10 and %12 eV.A detailed analysis between 0 and 5 eV does not show absorption peaks.From the calculated PDOS (Figure 2), we can infer that the response at %7.5 eV and above came from electronic transitions between oxygen valence electronic p states and conduction states of silica, aluminum, and sodium.The anisotropy is apparent in the components xx, yy, and zz of the imaginary part.In Figure 3, the real part of the dielectric function peaks with the maximum value of 2.2 for diagonal tensor elements xx, yy, and zz between 9 and 11 eV.On the other hand, the dielectric function becomes insignificant for off-diagonal xy, xz, and yz tensor elements.The maximum value for the real part of the dielectric function occurs at 10 eV.Both the real and imaginary parts of the dielectric function appear anisotropic as indicated by the nonzero values of the off-diagonal elements.Besides, we observe that the real part ε 1 only takes positive values for energies between 0 and 20 eV, with two peaks at %9 and %11 eV.The static dielectric constant ε 0 is given by the low energy limit of ε 1 function, i.e., ε 1 (0), with a value of %1.5 for the three components of the main diagonal.
As the peaks in the dielectric function stem from the transits between valence and conduction bands, the significant values for the dielectric function indicate considerable energy absorption, which will be shown below by other optical parameters.
In Figure 4, it is shown the refraction index and the extinction coefficient, respectively.As expected, the refraction index peaks with the maximum value of 1.45 for diagonal tensor elements xx, yy, and zz between 9 and 11 eV.On the other hand, the dielectric function becomes insignificant for offdiagonal xy, xz, and yz tensor elements.In Figure 4b, the extinction coefficient peaks between 11 and 20 eV, and becomes insignificant for off-diagonal xy, xz, and yz tensor elements.
Based on the results presented in Figure 3 and 4, respectively, for the dielectric function and the refraction index, some important parameters, namely, the energy loss function, the optical reflectivity, and the optical absorption coefficient, can be readily extracted.The relevant mathematical formulae are already outlined and explained in the previous paragraph.
The energy loss function is presented as a function of photon energy in Figure 5.A resonance energy loss peak is found near 2 eV for the component x-y.Otherwise, the rest of the area and the other tensor elements show only insignificant losses.
The relative optical reflectivity is shown in Figure 6 as a function of photon energy.It confirms that the optical activities occur mainly below 20 eV.Up to full reflection takes place mainly in the region between 5 and 20 eV for the three off-diagonal tensor elements, and the molecules are not quite anisotropic, whereas the three diagonal tensor elements show insignificant optical reflection.
In Figure 7, the absorption coefficient is presented as a function of photon energy for six tensor elements.The absorption correlates well with the total possible optical interband transitions and the density of state distribution presented in Figure 2. The absorption appears anisotropic, as the three off-diagonal tensor elements are insignificant.
In Figure 8, the conductivity function is presented as a function of photon energy for six tensor elements.The absorption correlates well with the total possible optical interband transitions and the density of state distribution presented in Figure 2. The conductivity appears anisotropic, as the three off-diagonal tensor elements are insignificant.

Conclusion and Perspective
In conclusion, within the framework of DFT, electronic structures of LTA zeolite membranes are calculated, and based on    the electric structures, several key optical properties of LTA zeolite membranes can be conveniently determined, with the method of the full-potential linearized augmented plane wave under the RPA for the boundaries and the GGA for the exchange-correlation potentials.The calculated optical properties as a function of frequency include the real and imaginary parts of the dielectric function ε(ω), the refraction index n(ω), the extinction coefficient k(ω), the reflectivity R(ω), the energy loss function L(ω), the absorption coefficient α(ω), and finally the optical conductivity σ(ω), along with specific directions of the primitive.It is demonstrated that the optical properties of the sodium LTA zeolite cluster appear anisotropic within the range of UV.The nature of anisotropy implies that all the functions are 3 Â 3 tensors with nonzero off-diagonal elements.Three diagonal and three off-diagonal elements are simultaneously presented.
In terminating, we emphasize that the calculated example presented in the present work shows the computational feasibility of implementing the density theory in asserting electrical structures and optical properties for LTA zeolite clusters.Of course, similar molecules can be modeled for other purposes.Continued work will be twofold, computational and experimental.In computational work, specific elements of interest can be relocated and the changes in electrical and optical properties would be immediately observed.On the other hand, experimental work can be carried out to represent the changes in elements.Both computational and experimental results will be in our forthcoming articles.

Figure 2 .
Figure 2. Calculated a) band structure (the different colors only differentiate the different bands); b) TDOS and c) PDOS for zeolite LTA.

Figure 1 .
Figure 1.Schematic view of a) the optimized unit cell of zeolite LTA with a triclinic structure; red, blue, dark blue, and yellow spheres correspond to O, Al, Si, and Na atomic positions, respectively; b) the Brillouin zone and the special points; and c) a wireframe representation of sodalite cage connected with six D4R cages; both green and yellow spheres represent sodium atoms.
) with a 3 Â 3 tensor a ↔ , with three diagonal elements a xx , a yy , and a zz and three off-diagonal elements (the other three elements are symmetric) a xy = a yx , a xz = a xy , and a xz = a xx .a ↔ ¼ a xx a xy a xz a yx a yy a yz a zx a zy a zz

Figure 5 .
Figure 5.The energy loss as a function of photon energy L(ω), with six tensor elements.

Figure 6 .
Figure 6.The reflectivity as a function of photon energy R(ω), with six tensor elements.

Figure 7 .
Figure 7.The absorption coefficient as a function of photon energy α(ω), with six tensor elements.

Figure 4 .
Figure 4. a) The refraction index n(ω) and b) the extinction coefficient k(ω) as a function of photon energy, with six tensor elements.

Figure 8 .
Figure 8.The conductivity function as a function of photon energy σ(ω), with six tensor elements.