DFT and MCDS Outcome for a Comparative Analysis of NO, NO₂, SO, SO₂ and SO₃ Gas Adsorption onto a NaMgPO₄ (033) Surface

Abstract

The research purpose of this work is to examine the adsorption interaction of gaseous molecules (GMs), such as NO, NO₂, SO, SO₂, and SO₃, with the surface of sodium magnesium phosphate NaMgPO₄ (033), in a neutral medium, using two different computational methods: density functional theory (DFT) and Monte Carlo dynamic simulation (MCDS). Various quantum and dynamic descriptors, such as global and local quantum descriptors and the radial distribution function (RDF), are also evaluated and discussed. The data obtained revealed that the NO₂ molecule has a small energy gap (0.363 eV) when compared to the other molecules, which means that it is highly reactive and is liable to adsorb, or stick, to the surface of NaMgPO₄ (033). Furthermore, this NO₂ molecule exhibits good adsorption in aqueous media, returning to the lowest global hardness value (0.1815 eV). MCDS predicted adsorption energies of −874.03, −819.94, −924.81, −876.33, and −977.71 kcal/mol for NO, NO₂, SO, SO₂, and SO₃, respectively. These energies are negative, implying that adsorption occurs spontaneously. Thus, the side views indicated which SO, NO, and SO₃ molecules are adsorbed in parallel to NaMgPO₄ and the other SO₂ and NO₂ molecules are adsorbed horizontally. Eventually, the theoretical results reveal that the studied gaseous molecules interact strongly with NaMgPO₄. The result obtained by radial distribution function (RDF) analysis for all complexes below 3.5 Å confirm that the adsorption is of the chemi1cal type.

1. Introduction

Due to global climate change or air pollution, there is an immediate need to reduce exhaust gases [1]. The substantial emission of harmful gases, such as NO_x, SO_x, and CO_x, from the automotive industry is released into the atmosphere, leading to various health problems, including infectious diseases and respiratory conditions [2,3,4]. Acid rain, photochemical smog, and ozone depletion are just a few of the environmental issues that also promote the corrosion or rust of equipment and industrial instruments caused by the nitrogen oxides NO_x and sulfur oxides SO_x released by cars and coal-fired power plants [5]. Therefore, the elimination of NO_x and SO_x is imperative. The regular measurement of gas concentration is necessary to avoid the hazards of these gases. Notably, advancements in NO_x removal technology have received significant attention, resulting in a relatively rapid reduction in NO_x air pollution. Technologies such as NO oxidation have become increasingly important in NO_x removal methods, like NO_x storage and reduction [6,7], dry sorbent injection [8], absorption in wet flue gas desulfurization [9,10], and the use of catalysts, considered a common method for removing these harmful gases from automotive emissions [11,12,13]. Unfortunately, meeting international regulations for NO_x removal by the three-way catalyst in a diesel engine under operating conditions is challenging due to the abundance of oxygen in the emissions [14,15]. Considering that sulfur monoxide, dioxide, and trioxide (SO_x) release disagreeable and offensive odors into the atmosphere, several studies have focused on SO_x adsorption onto various surfaces [16,17,18,19,20]. However, the advanced technologies for the elimination of gaseous molecules entail a high energy cost. Improving the efficiency and reducing the cost of the elimination of these studied molecules entails integrating simple and efficient removal processes, like adsorption and membrane separation technologies. The adsorption process is the most efficient because of its lower environmental impact and lower cost [21,22,23,24]. Furthermore, research has revealed that phosphate adsorbents have a significant adsorption capacity for both organic and inorganic pollutants due to their natural abundance [25]. Many researchers have focused for the adsorption of various gaseous molecules on different adsorbents. For instance, Sajid et al. focused on the adsorption of gaseous molecules with Be₁₂O₁₂, Mg₁₂O₁₂, and Ca₁₂O₁₂. The results indicated that the adsorption energies of N₂O@Ca₁₂O₁₂, NO₂@Ca₁₂O₁₂, NO@Ca₁₂O₁₂, H₂S@Ca₁₂O₁₂, SO₂@Ca₁₂O₁₂, and SO₃@Ca₁₂O₁₂ were −11. 79, −46.53, −26.51, −50.26, −78.64, and −123.62 kcal/mol, respectively [26]. Similarly, Gao et al. studied the adsorption of NO, NO₂, SO₂, and SO₃ on single vacancy graphene with three doped nitrogen atoms (Ni-SVN3/GN) on graphene adsorbent. According to their findings, the adsorption energy values for NO, NO₂, SO₂, and SO₃ were 3.51 eV, 2.50 eV, 1.72 eV, and 2.37 eV, respectively [27]. Furthermore, organic aza-macrocyclic hexaazabipyH2 (HA) was utilized as an adsorbent to remove gaseous molecules, such as N₂O, NO₂, H₂S, SO₂, and SO₃. The obtained data revealed the interaction energies to be −4.80, −4.86, −7.09, −7.42, and −11.64 kcal/mol for NO₂, N₂O, H₂S, SO₂, and SO₃, respectively [28]. Subsequently, we conducted a theoretical study of NO, NO₂, SO, SO₂, and SO₃ gases on a NaMgPO₄ surface. NaMgPO₄ is a member of the ABPO₄ family with A denoting alkaline metals (A = Li, Na, and K) and B denoting alkaline earth metals (M = Mg, Ca, Sr, and Ba), which is gaining popularity. NaMgPO₄ is especially intriguing due to its luminescent properties, remarkable structural properties, and better chemical and thermal stability [29].

In this context, the focus of the study is to use DFT and MCDS to examine the mechanism of interaction behavior of NO, NO₂, SO, SO₂, and SO₃ molecules on the surface of NaMgPO₄ (033) in aqueous media. The adsorption of the studied molecules onto the surface has significant applications in many industrials processes. Our results show that SO, NO, SO₂, NO₂, and SO₃ molecules are adsorbed in parallel/horizontally on the NaMgPO₄ surface. GM adsorption on the NaMgPO₄ (033) face is chemical in nature, indicating a strong interaction.

2. Materials and Methods

2.1. Quantum Chemical Calculations

In this study, the Gaussian 09 package was implemented to conduct full DFT calculations. The molecule’s geometry was optimized using the B3LYP functional, which is commonly used in these calculations because it provides a good balance of accuracy and computational efficiency [30]. Additionally, the LanL2DZ basis was applied in the calculations [31]. It was selected as the most precise basis set from the available options. The computation was conducted with the minimum energy and with water as the solvent. This calculation is typically used to investigate a variety of characteristics, including the electronic properties of gaseous molecules, the impact of the energies of the lowest and highest occupied molecular orbitals (LUMOs and HOMOs), and the distinction between both of them. As a result, the DFT approach has grown in popularity in recently [32]. Quantum reactivity descriptors are a set of molecular properties that are calculated from the output of quantum chemical calculations and are used to assess the chemical reactivity of a molecule. The most important molecular descriptors extracted directly from the output files are those related to molecule reactivity, which include the energy of the lowest unoccupied molecular orbital (E_LUMO), the energy of the highest occupied molecular orbital (E_HOMO), electronic affinity (AE = −E_LUMO), ionization potential (IE = −E_HOMO), energy gap (Eg = E_LUMO − E_HOMO), hardness (ƞ), absolute electronegativity (χ), global softness (S), global electrophilicity index (ω), back donation energy (E_b–d), the number of transferred electrons (ΔN), and chemical potential (µ). These descriptors are used to understand the preferred adsorption sites of compounds as well as to predict their reactivity and predict the outcomes of chemical reactions. The following mathematical expressions were used to calculate the above parameters [33,34,35,36].

$${\mathrm{E}}_{\mathrm{g}}={\mathrm{E}}_{\mathrm{LUMO}}-{\mathrm{E}}_{\mathrm{HOMO}}$$

(1)

$${\mathsf{\mu }}_{\mathrm{CP}}=-\mathsf{\chi }=-\frac{\left(\mathrm{IP}+\mathrm{EA}\right)}{2}$$

(2)

$$\mathsf{\eta }=\frac{(\mathrm{IP}-\mathrm{EA})}{2}$$

(3)

$$\frac{1}{\mathrm{S}}=2\mathsf{\eta }={\left(\frac{\partial \mathsf{\mu }}{\partial \mathrm{N}}\right)}_{\mathrm{v}\left(\mathrm{r}\right)}=\left(\frac{{\partial }^2\mathrm{E}}{{\partial }^2\mathrm{N}}\right)$$

(4)

$$\mathsf{\omega }=\frac{{\mathsf{\chi }}^2}{2\mathsf{\eta }}=\frac{{\mathsf{\mu }}^2}{2\mathsf{\eta }}$$

(5)

$${\mathrm{E}}_{\mathrm{b}-\mathrm{d}}=-\frac{\mathsf{\eta }}{4}$$

(6)

$$\mathsf{\Delta }\mathsf{{\rm N}}=\frac{\mathsf{\chi }}{2\mathsf{\eta }}$$

(7)

2.2. Monte Carlo Dynamic Simulation Details

To conduct our investigation of the adsorption characteristics of NO, NO₂, SO, SO₂, and SO₃ compounds on the NaMgPO₄ (033) surface, MCD computations were conducted using the Material Studio 8.0 Software (Accelrys; BIOVIA, Dassault Systems, San Diego, CA 92121, USA) [37,38]. In a simulation box (20 Å)³, the interaction of a single molecule (NO, NO₂, SO, SO₂, or SO₃) with the NaMgPO₄ (033) surface was simulated. As shown in Figure 1, a 20 Å thickness vacuum slab was built above the NaMgPO₄ (033) face, which was then enlarged into a (4 × 4 × 4) supercell. The (033) plane was used for all molecular systems (M: NO, NO₂, SO, SO₂, and SO₃/500 H₂O/NaMgPO₄ (033)). A liquid solution of 500 H₂O molecules was appended to predict the solvent effect, which can influence the adsorption process. This simulation was set to run for a total simulation time of 500 ps at a 1.0 bar of pressure. The universal force field was used to compute the energy values and search for equilibrium configurations during the over-all simulation procedure with the charges for the used current. Atom-based and Ewald & Group summation methods were performed to obtain the potential energy of the complex in the simulation procedure. The fine quality choice was adopted to assure preciseness in the analysis of electrostatic interaction contributions. The purpose of this calculation study was to understand the interaction between GMs and the NaMgPO₄ (033) face and to discover the relationship between the reactivity of the molecules and their ability to adsorb onto the surface. By calculating the energy adsorption (E_ads) of each of the molecules, it is possible to determine which molecules have a stronger affinity for the surface and to identify any small energy adsorption centers that may be present. Thus, the E_ads emitted when the expanded adsorbate constituent is simultaneously deposited on the adsorbent was calculated using the following expression [39].

$${\mathrm{E}}_{\mathrm{ads}}={\mathrm{E}}_{\mathrm{surface}/\mathrm{molecule}}-{(\mathrm{E}}_{\mathrm{surface}}+{\mathrm{E}}_{\mathrm{molecule}})$$

(8)

where E_{molecule/surface} reflects the total energy of the GM and NaMgPO₄ (033) face system, E_surface represents the total energy of the isolated NaMgPO₄ (033), and E_Molecule represents the total energy of the isolated GMs.

The RDF analysis enabled us to comprehend the nature of the interactions between the adsorbent and the adsorbate, such as whether they are physisorbed (held by weak van der Waals forces) or chemisorbed (held by chemical bonds) [40,41]. It is often used to study the structure and properties of materials, such as adsorbents and adsorbates, in order to understand their behavior during the adsorption process. Thus, RDF or g(r) represents the probability of finding a particle at a distance r from a reference particle within a system, rather than specifically an atom from another atom. In this case, “r” represents the distance between two particles. The g (r) to perform the distance analysis between two atoms, and α and β are determined using the following expression.

$${\mathrm{g}}_{\mathsf{\alpha }\mathsf{\beta }}\left(\mathrm{r}\right)=\frac{1}{{〈{\mathsf{\rho }}_{\mathsf{\beta }}〉}_{\mathrm{local}}}\times \frac{1}{{\mathrm{N}}_{\mathsf{\alpha }}}\left({\displaystyle \sum _{\mathrm{i}\in \mathsf{\alpha }}^{{\mathrm{N}}_{\mathsf{\alpha }}}{\displaystyle \sum _{\mathrm{i}\in \mathsf{\beta }}^{{\mathrm{N}}_{\mathsf{\beta }}}\frac{\mathsf{\delta }\left({\mathrm{r}}_{\mathrm{ij}}-\mathrm{r}\right)}{{4\mathsf{\pi } \mathrm{r}}^2}}}\right)$$

(9)

where 〈ρ_β〉 local represents the particle density of β averaged over all shells around particle α. N_α is the number of particles and r_ij is the positions of particles i and j.

3. Results

3.1. Frontier Molecular Orbitals and MEP

By plotting the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) densities and the MEP (molecular electrostatic potential) of the GMs under study in aqueous media, determined by the B3LYP functional with a LanL2DZ basis, as shown in Figure 2, it was possible to gain insights into the reactivity and stability of the GMs [42]. When the HOMO and LUMO densities are concentrated in different regions of the molecule, it means a higher reactivity of the molecules. Similarly, A positive MEP in one region of the molecule and negative in another indicates that the molecule is unstable and prone to reacting with other molecules. As illustrated in Figure 2, in SO₂ and SO₃ molecules, the HOMOs are usually localized on the oxygen atoms. Conversely, the HOMO and LUMO electron densities for the SO, NO, and NO₂ molecules are centered on the entirety of the three molecules. Furthermore, the HOMO electron density provides information about the sites of the molecule that are more likely to donate electrons to an acceptor molecule’s appropriate orbital. This can lead to a chemical reaction or change in the electronic properties of the adsorbent. In contrast, the LUMO density is frequently employed to describe the reactivity of GMs, since it is the orbital that is the most available to accept an electron from the NaMgPO₄ (033) face. Thus, Figure 2 indicates that the LUMO is located mostly on the O, S, and N atoms of the investigated molecules, except for the SO₃ molecule, where it is present on the oxygen atom. This suggests that these atoms are the most likely sites for electron transfer to occur during the adsorption process [43]. It can also be observed that the adsorption process of GMs on the NaMgPO₄ (033) face is likely mediated by the interaction of the nitrogen, oxygen, and sulfur atoms with the surface. These findings were confirmed by MEP, which represents the molecule’s electronegativity and electropositivity. As shown in Figure 2, the electrostatic potentials on the MEP map are represented by various hues, with the electrostatic potential value increasing steadily from red to blue. The nucleophilic regions are represented in blue and light blue, while the electrophilic regions are represented in red. Furthermore, the green color represents the neutral charge. The NO, NO₂, SO, SO₂, and SO₃ molecules demonstrate that the N and S atoms have positive electrostatic potentials, ranging between 0.0083 and 0.0323 a.u. for the N atom and varying between 0.0425 and 0.0586 a.u. for the S atom, whereas the oxygen atoms have negative electrostatic potentials raging between −0.0459 and −0.0143 a.u. Most electron-rich areas (electrophilic sites) are mainly found near oxygen atoms, and electron-deficient areas are found near the N atoms and S atoms of the NO, NO₂, SO, SO₂, and SO₃ molecules. This indicates that negatively charged heteroatoms can interact with the adsorbent surface via an electron donor–acceptor reaction [44]. This type of interaction can be important in various chemical reactions and processes, such as adsorption.

3.2. Global Quantum Descriptors

To comprehend the reactivity of the studied molecules,

Table 1. Quantum chemical parameters (in eV) of the GMs in aqueous media.

Parameter (eV)
Molecule	EHOMO	ELUMO	Eg	S	χ	η	µ	ω	Eb–d	ΔN
NO	−9.434	−3.236	6.198	0.323	6.335	3.099	−6.335	−1.584	−0.775	1.022
NO2	−9.766	−9.403	0.363	5.510	9.584	0.181	−9.584	−2.396	−0.045	26.403
SO	−8.123	−5.297	2.825	0.708	6.710	1.412	−6.710	−1.677	−0.353	2.375
SO2	−9.100	−6.461	2.639	0.758	7.781	1.320	−7.781	−1.945	−0.330	2.948
SO3	−12.193	−10.318	1.874	1.067	11.255	0.937	−11.255	−2.814	−0.234	6.004

lists the calculated parameters disregarding their structural or electronic characteristics in aqueous media. The E_LUMO is a critical factor in determining a molecule’s ability to accept electrons. In general, a molecule with a low E_LUMO is more inclined to accept electrons from the adsorbent surface, while a molecule with a high E_LUMO is less inclined to do so. Furthermore, the E_HOMO also expresses a compound’s capacity to provide electrons. Molecules with a high HOMO energy are more likely to donate electrons to other molecules, making them more reactive. On the other hand, molecules with a low HOMO energy are less inclined to provide electrons and may be less reactive as a result. Nonetheless, the E_HOMO can be an important factor when predicting and designing molecules for chemisorption applications. There are, however, other important descriptors for understanding many chemical reactions. Table 1 displays the obtained global parameters for the reactivity of NO, NO₂, SO, SO₂, and SO₃ species.

The obtained results show that the variation in E_LUMO values is in the following order in the aqueous phase: SO₃ < NO₂ < SO₂ < SO < NO. On the other hand, E_HOMO follows the sequence SO₃ < NO₂ < SO₂ < NO < SO. Notably, the SO molecule has a higher E_HOMO value than the NO₂, NO, SO₂, and SO₃ molecules, indicating that it is more likely to donate electrons. On the other hand, the SO₃ molecule has a lower E_LUMO value than the NO₂, SO, SO₂, and NO molecules, indicating that it is more likely to accept electrons [45]. This means that SO is more likely to act as an electron donor in a chemical reaction, while SO₃ is more likely to act as an electron acceptor. The energy gap (E_g) explains the adsorbate molecule’s reactivity (such as a GM) for adsorption on the adsorbent surface (such as NaMgPO₄). The lower value of E_g of the NO₂ molecule indicates that it is easier to remove an electron from the adsorbate molecule or to add an electron to it, making it more reactive and more likely to adsorb onto the surface of NaMgPO₄ [46]. As shown in Table 1, the quantum chemistry calculations show that the E_g of the NO₂ molecule is the smallest compared to that of the other molecules, which show a decrease in E_g values for the NO, SO, SO₂, SO₃, and NO₂ molecules of 6.1979 > 2.8251 > 2.6395 > 1.8745 > 0.363 eV, respectively. In addition, the species’ total number of transported electrons or ΔN was estimated for each species. The outcomes are presented in Table 1. It has been reported that ΔN reveals a molecule’s propensity to donate electrons [47,48]. In fact, for a molecule’s inclination to donate electrons to electron-deficient species to be high, it is necessary that the value of ΔN is positive and high. In the case of the studied molecules, a higher ΔN was noticed for the NO₂ molecule, which involves a stronger propensity to interact with the NaMgPO₄ (033) face (i.e., a higher inclination to adsorb onto the adsorbent surface), suggesting a strong adsorption capacity on the adsorbent surface. The studied molecules’ ΔN values were in the following order 26.4033 > 6.0045 > 2.9478 > 2.3751 > 1.0221 eV for NO₂, SO₃, SO₂, SO, and NO respectively. Additionally, back donation energy, or E_b–d, is a measure of the energy required for an electron to transfer from a GM to a NaMgPO₄ surface. The results show that E_b–d is negative for all molecules, suggesting that the molecule investigated has an energetically favorable back donation to the NaMgPO₄ surface. Moreover, hardness (ƞ) and softness (S) are two other properties that can be employed to analyze a molecule’s stability and reactivity. In general, molecules that are harder are more stable and less reactive, while molecules that are softer are less stable and more reactive. The NO₂ molecule has a low hardness and high softness value, calculated at 0.181 eV and 5.510 eV, respectively, than the other NO, SO, SO₂, and SO₃ molecules, which means they are more reactive and less stable. Furthermore, the results of E_b–d, ΔN, S, and ƞ are in agreement. All studied NO, NO₂, SO, SO₂, and SO₃ molecules ha a negative chemical potential calculated at −6.335, −9.584, −6.710, −7.781, and −11.255 eV, respectively. This means that GMs may be able to form strong interactions with other species in the system, resulting in the formation of new chemical bonds. Alternatively, GMs may be capable of stabilizing the system by filling empty sites.

3.3. Mulliken Charge Distribution

The analysis of the Mulliken population typically provides details about a molecule’s active sites. The active sites are the nitrogen (N), sulfur (S), and oxygen (O) atoms, which are critical components in the reactivity of the compounds under investigation. The calculated Mulliken charges of these three atoms in the NO, NO₂, SO, SO₂, and SO₃ molecules are listed in

Table 2. Mulliken atomic charges obtained for each molecule.

	N	O	S
NO	0.7797	−0.7797
NO2	0.2779	−0.1389
SO		−0.5456	0.5456
SO2		−0.5216	1.0432
SO3		−0.4687 −0.4535 −0.4531	1.3753

. The outcomes show that the oxygen atoms have a higher negative charge in all investigated molecules, suggesting that these atoms are the likely active sites for attacking the adsorbent atoms. Consequently, these active sites will facilitate the adsorption process of the selected molecules onto the NaMgPO₄ surface by increasing the adsorption energy. On the other hand, the nitrogen and sulfur atoms in these molecules have a positive Mulliken population, indicating that they are electron-deficient and may be more likely to accept electrons from other atoms The oxygen Mulliken population values of the studied molecules were as follows: −0.7797, −0.1389, −0.5456, −0.5216, −0.4687, −0.4535, and −0.4531 for NO, NO₂, SO, SO₂, and SO₃, respectively.

As it can be seen from this result, oxygen atoms are capable of conferring electrons to NaMgPO₄ atoms to create a coordination bond. This type of bond is often seen in compounds containing transition metals, where the metal atom acts as the central atom and the ligand atoms (such as oxygen) donate their electrons to form the bond.

3.4. Fukui Function Calculations

The Fukui function (f) measures the electron density at a particular site in a molecule. It can be utilized to identify the most reactive sites in a molecule and to predict the reactivity of a molecule towards different types of reagents. The different parts of the molecule can be distinguished based on their chemical characteristics using the condensed Fukui function and local reactivity indices [49]. These indices can be used to predict which parts of a molecule are more likely to react with other molecules, and can be useful for understanding the behavior of a molecule in chemical reactions. The Fukui indices are descriptive terms that identify the kind of attack (radical, electrophile, or nucleophile, or any combination of the three). The possibility of the adsorption of the adsorbate on the adsorbent is recalled by the reactive sites with high negative charge densities. A nucleophile is a species that is attracted to electron-poor regions and tends to attack atoms or functional groups with a high f⁺ value. On the other hand, an electrophile is attracted to electron-rich regions and tends to attack atoms or functional groups with a high f⁻ value. As it is well-known, Δf < 0 represents a suitable region for the electrophilic attack. Moreover, Δf > 0 is the most expected region of a nucleophilic attack. The Fukui indices can be calculated by employing the following formulae for three different situations [47]:

For a reaction with nucleophiles:

$$f_i{\left(\overrightarrow{r}\right)}^{+}={\mathrm{q}}_{\mathrm{i}}\left(\mathrm{N}+1\right)-{\mathrm{q}}_{\mathrm{i}}\left(\mathrm{N}\right)$$

(10)

For a reaction with electrophiles:

$$f_i{\left(\overrightarrow{r}\right)}^{-}={\mathrm{q}}_{\mathrm{i}}\left(\mathrm{N}\right)-{\mathrm{q}}_{\mathrm{i}}\left(\mathrm{N}-1\right)$$

(11)

where q_N, q_N+1, and q_N−1 are the atomic charges of the systems with N, N + 1, and N − 1 electrons, respectively, at a particular point r in the molecule.

An atom’s electrophilic and nucleophilic attacks in a molecule are defined as follows:

$$\mathsf{\Delta }{f}_i=f_i{\left(\overrightarrow{r}\right)}^{+}-f_i{\left(\overrightarrow{r}\right)}^{-}$$

(12)

The f_k+ and f_k⁻ parameters’ absolute values increase with the increase in the tendency of accepting and donating electrons, respectively.

The Fukui indices of the GMs were calculated to predict the center of nucleophilic attack (f_i⁺) and electrophilic attack (f_i⁻), which are depicted in

Table 3. Fukui indices calculated using the B3LYP/LanL2DZ method.

Molecule	Atom	qi(N)	qi(N + 1)	qi(N − 1)	fi+	fi−	Δf
NO	O1	−0.202	0.214	−0.565	0.416	0.363	0.053
	N2	0.202	0.786	−0.435	0.584	0.637	−0.053
NO2	N1	0.498	0.194	0.837	−0.304	−0.339	0.035
	O2	−0.249	0.081	−0.597	0.33	0.348	−0.018
	O3	−0.249	0.081	−0.597	0.33	0.348	−0.018
SO	S1	0.584	1.374	−0.228	0.79	0.812	−0.022
	O2	−0.584	−0.374	−0.772	0.21	0.188	0.022
SO2	S1	1.423	1.684	0.84	0.261	0.583	−0.322
	O2	−0.711	−0.342	−0.92	0.369	0.209	0.16
	O3	−0.711	−0.342	−0.92	0.369	0.209	0.16
SO3	S1	1.985	2.025	1.427	0.04	0.558	−0.518
	O2	−0.662	−0.339	−0.809	0.323	0.147	0.176
	O3	−0.662	−0.341	−0.809	0.321	0.147	0.174
	O4	−0.661	−0.345	−0.809	0.316	0.148	0.168

. NO has the highest f_i⁺ and f_i⁻ values, observed on N₂ with values of 0.584 and 0.637, respectively, which is the most likely nucleophilic and electrophilic attack site. For NO₂, the highest values for f_i⁺ and f_i⁻ are found O1 and O2, with values of 0.33 and 0.348 (nucleophilic and electrophilic attacks), respectively. Also, for SO, the most probable nucleophilic attack (f_i⁺) and electrophilic attack (f_i⁻) are S1, with values of 0.79 and 0.812, respectively. For SO₂, the highest value for f_i⁺ is found on O₂ and O₃, with a value of 0.369 (nucleophilic attack), and the highest value for f_i⁻ is found on S1, with a value of 0.583 (electrophilic attack). Thus, SO₃ has the highest value for f_k⁺ on O₂, with a value of 0.323 (nucleophilic attack center), and f_i⁻ on S1, which represents the most probable electrophilic attack center with a value of 0.558. The higher the absolute value of the Fukui indices f_i⁺ and f_i⁻, the greater the susceptibility to accept and donate electrons. As a result, the studied molecules have the most active sites for the electron donation/acceptance type of interaction, implying an ease of adsorption onto the adsorbent surface.

3.5. Monte Carlo Dynamic Simulation Study

The MCDS was utilized to understand and explain the interactions of gaseous molecules with the NaMgPO₄ (033) surface in aqueous media. Furthermore, the adsorbent and adsorbate molecules’ combined potential and interaction energies were simulated by MCDS in a simulation box (45× 45 × 45 ų) with periodic boundary conditions. Figure 3 depicts the electrostatic, intermolecular, van der Waals, total, and total average energies for the adsorption of gaseous molecules on the adsorbent surface in aqueous media. As shown in Figure 3, the intramolecular energy is positive and stagnant. In fact, the complex SO₃/NaMgPO₄ is superior to the other molecules with a value of 530 kcal/mol, meaning that it is likely to be more stable. The order of intramolecular energies for the other complexes suggests that the stability of the complexes decreases as the intramolecular energy decreases. In the same positive stage, the intramolecular energy obtained from the studied complexes has a following order: SO₃/NaMgPO₄ > SO/NaMgPO₄ > SO₂/NaMgPO₄ > NO₂/NaMgPO₄ > NO/NaMgPO₄ (530, 442, 401, 386, and 375 kcal/mol, respectively). Nevertheless, the electrostatic energy for all studied complexes is zero, which means that the distribution of electric charge within the complexes is balanced and there are no net forces between the atoms. However, the average total energy for each complex is positive, with values for NO/NaMgPO₄ ranging from 0 to 176 kcal/mol, for NO₂/NaMgPO₄ from 129 to 232 kcal/mol, for SO/NaMgPO₄ from 183 to 442 kcal/mol, for SO₂/NaMgPO₄ from 125 to 307 kcal/mol, and for SO₃/NaMgPO₄ from 237 to 463 kcal/mol.

Then, the chemical complexes SO/NaMgPO₄ and SO₃/NaMgPO₄ have positive total energy values ranging from 49 to 546 kcal/mol and 144 to 620 kcal/mol, respectively, suggesting that the complexes are reactive and are likely to break apart or react with other molecules under certain conditions. NO/NaMgPO₄, NO₂/NaMgPO₄, and SO₂/NaMgPO₄ have both positive and negative total energy simultaneously, meaning that the complexes can absorb or release energy through various interactions or processes. Lastly, the van der Waals bonds have a negative and positive value for each of the complexes that were studied, with values that are no greater than −448 kcal/mol and 135 kcal/mol, respectively, indicating that the attractive and repulsive forces between the molecules in the complexes are not constant and may change under different conditions.

Figure 4 shows the lateral views of the least energy adsorption model for single gaseous molecules, including NO, NO₂, SO, SO₂, and SO₃, on the surface of NaMgPO₄ (033) in an aqueous medium. On the other hand, the NO, NO₂, and SO₂ molecules are oriented in parallel on the surface of NaMgPO₄, while the other molecules SO₃ and SO adsorb horizontally. The side views of the most stable adsorption configurations for all chemical complexes demonstrate that there is an interaction between the sorbates and the sorbent that involves electron donation and acceptance. To reinforce the ratio, all of the molecules adsorb at different distances on the adsorbent’s surface.

The evaluation of outputs for energies determined by the MCDS are summarized in

Table 4. Monte Carlo simulation outputs of the GMs on NaMgPO₄ (kcal/mol).

Molecule	Etotal	Eads	RAE	Edef	dEads/dNiH2O	dEads/dNi
NO	−472.031	−874.033	−472.563	−374.469	−0.764	−2.218
NO2	−433.835	−819.943	−434.982	−384.961	−0.781	−13.714
SO	−481.969	−924.810	−482.146	−442.664	−0.766	−72.933
SO2	−474.448	−876.333	−475.027	−401.306	−0.769	−31.010
SO3	−447.572	−977.714	−453.463	−524.251	−0.804	−155.016

, including the total (E_Tot), adsorption (E_ads), rigid adsorption (RAE), and deformation (E_def) energies. Thus, Eads is the energy required to adsorb a substance onto a solid surface. Then, the RAE is the energy required to adsorb a substance onto a surface without allowing for any deformation of the adsorbate or the surface. Therefore, the Edef is the energy required to deform the adsorbate or the surface during the adsorption process. The relaxation of adsorbates on the surface refers to the way in which the adsorbates rearrange themselves on the surface in order to minimize their total energy. Furthermore, the relaxation of the adsorbates on the NaMgPO₄ surface (033) is due to a low strain energy, and (d_Eads/dN_i) is the differential adsorption energy required to detach or remove a single molecule from the studied adsorbent surface.

The outcomes demonstrate that all adsorption energy values were generally negative, with the GM-NaMgPO₄ complexes often having a greater absolute value. As shown in Table 4, the GMs are firmly adsorbed onto the NaMgPO₄ (033) surface due to the higher negative values of adsorption energy. Obviously, the values obtained show that the SO₃-NaMgPO₄ complex has the highest negative adsorption energy of all studied gaseous molecules, calculated at −977.714 kcal/mol, which means that the SO₃ molecule is more adsorbed onto the NaMgPO₄ (033) surface than the other NO, NO₂, SO, and SO₂ molecules (−874.033, −819.943, −924.81, and −876.333 in kcal/mol, respectively). As a result, the adsorption energy and relative stability of all complexes are as follows: SO₃/500 H₂O/NaMgPO₄ (033), SO/500 H₂O/NaMgPO₄ (033), SO₂/500 H₂O/NaMgPO₄ (033), NO/500 H₂O/NaMgPO₄ (033), and NO₂/500 H₂O/NaMgPO₄ (033). Thus, their negative E_ads values characterize all systems with the strongest and most spontaneous adsorption [50]. The high absolute Eads values for all complexes GM-NaMgPO₄ indicate that all gaseous molecules in an aqueous solution are significantly adsorbed onto the NaMgPO₄ (033) face, possibly via a chemical bond. As shown in Table 4, all adsorption, deformation energies, dE_ads/dNi_MG, and dEads/dNi_H2O values obtained for SO₃ are higher than those obtained for the NO, NO₂, SO, and SO₂ molecules. This study demonstrates that, in an aqueous medium, SO3 may have chemical characteristics that make it stable and interact strongly with the NaMgPO₄ surface. The returning absolute dE_ads/dNi value for the SO₃ molecule (−155.016 kcal/mol) is higher than that of the NO, NO₂, SO, and SO₂ molecules (−2.218, −13.714, −72.933, and −31.010 kcal/mol, respectively), which reveals that SO₃ adsorption occurs with ease on the NaMgPO₄ surface in aqueous media. The higher absolute dE_ads/dNi_H2O value for the SO₃ molecule (−0.804 kcal/mol), compared with those of the NO, NO₂, SO, and SO₂ molecules (−0.764, −0.781, −0.766, and −0.769 kcal/mol, respectively), suggests that the SO₃ molecule is more strongly adsorbed onto the solid surface and that it forms more hydrogen bonds with the solid surface.

3.6. Radial Distribution Function

The radial distribution function (RDF) is important to define the physisorption or chemi-sorption process that takes place on the adsorbent surface during adsorption [51]. Furthermore, the specific distance at which the peaks occur can also provide additional information about the nature of the interaction between the adsorbate and the adsorbent. In general, the chemisorption process is typically simplified when the peaks are present between 1 and 3.5 Å, while the appearance of peaks at distances greater than 3.5 Å suggests as an indication of the physisorption process. The RDF peak values of the oxygen, sulfur, and nitrogen atoms for NO, NO₂, SO, SO₂, and SO₃ and the NaMgPO₄ (033) interface are shown in Figure 5.

As shown in Figure 5, a strong bond was formed between the chemisorbed GMs and the NaMgPO₄ face, as evidenced by the fact that all the lowest link distances between the GMs and NaMgPO₄ (033) face were less than 3.5 Å. Moreover, the NO, NO₂, SO, SO₂, and SO₃ molecules appear to interact strongly with the surface of NaMgPO₄ (033). These data indicate that the GMs are firmly adsorbed on the NaMgPO₄ surface in an aqueous phase via covalent bonding. The GMs and NaMgPO₄ displayed the lowest bond distances of 0.99, 1.55, and 1.59 Å. These distance values suggest that a chemical link (within the chemisorption region) was formed between the GMs and the surface of NaMgPO₄ (033). In this study, a variety of nucleophilic and electrophilic attack centers (especially heteroatoms) were used to theoretically show that both chemical compounds under investigation are virtually parallel to the adsorbent face. Additionally, the heteroatoms lead to an electron exchange with the open orbitals of the adsorbent [52]. This caused a connection to form between the chosen molecules and NaMgPO₄. When certain gaseous molecules come into contact with an adsorbent, they adhere to the substrate to form an adsorbed layer. This material is typically referred to as an adsorbent or substrate, whereas the resultant of the adsorbed molecules is named adsorbate. We can distinguish between physisorption and chemisorption due to the natural forces at play. The excess electrons on the substrate can be transferred from it to the active sites of the investigated molecules (back-donation) [52].

4. Conclusions

In the present study, the adsorption behavior of the NO, NO₂, SO, SO₂, and SO₃ gaseous molecules on the NaMgPO₄ adsorbent was predicted using density functional theory at B3LYP with a LANL2DZ basis and Monte Carlo dynamic simulation. The reactivity indices, frontier molecular orbitals (FMO), and MEP maps were performed. The analysis of the calculated HOMO and LUMO energies revealed the evident charge transfer within the molecules. The MEP graph indicated that electrophilic sites were mostly found near oxygen atoms, while nucleophilic sites were located near nitrogen atoms and sulfur atoms. This shows that gaseous molecules can interact with the adsorbent surface through an electron donor–acceptor reaction. The Monte Carlo dynamic simulation indicated that all GM-NaMgPO₄ (with GMs = NO, NO₂, SO, SO₂, and SO₃) complexes in an aqueous medium exhibited a negative adsorption energy, with values of −819.943, −874.033, −876.333, −924.810, and −977.714 kcal/mol, respectively. These negative E_ads values indicate that GM adsorption is strong and spontaneous. Also, the MCD simulation suggested that the SO, NO, and SO₃ molecules were adsorbed in a parallel manner on the NaMgPO₄ surface, while the SO₂ and NO₂ molecules were adsorbed horizontally. Furthermore, due to the high absolute E_ads value, the GM adsorption on the NaMgPO₄ (033) surface was of the chemical type and suggests that a strong interaction took place. These results were confirmed by RDF analyses. Moreover, the SO₃-NaMgPO₄ complex had a higher ΔE_ads/dNi, with a value of −155.016 kcal/mol, compared to the other complexes, indicating that SO₃ is more adsorbed on the NaMgPO₄ (033) surface.