M-Encapsulated Be₁₂O₁₂ Nano-Cage (M = K, Mn, or Cu) for CH₂O Sensing Applications: A Theoretical Study

Abstract

DFT and TD-DFT studies of B3LYP/6–31 g(d,p) with the D2 version of Grimme’s dispersion are used to examine the adsorption of a CH₂O molecule on Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages (M = K, Mn, or Cu atom). The energy gap for Be₁₂O₁₂ was 8.210 eV, while the M encapsulation decreased its value to 0.685–1.568 eV, whereas the adsorption of the CH₂O gas decreased the E_g values for Be₁₂O₁₂ and CuBe₁₂O₁₂ to 4.983 and 0.876 eV and increased its values for KBe₁₂O₁₂ and MnBe₁₂O₁₂ to 1.286 and 1.516 eV, respectively. The M encapsulation enhanced the chemical adsorption of CH₂O gas with the surface of Be₁₂O₁₂. The UV-vis spectrum of the Be₁₂O₁₂ nano-cage was dramatically affected by the M encapsulation as well as the adsorption of the CH₂O gas. In addition, the adsorption energies and the electrical sensitivity of the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ nano-cages to CH₂O gas could be manipulated with an external electric field. Our results may be fruitful for utilizing Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages as candidate materials for removing and sensing formaldehyde gas.

1. Introduction

Recently, the problem of pollution of the air, soil, and water has attracted the attention of many scientists. There have been several attempts to reduce pollution sources, whether by capturing the pollutants or detecting the contaminated materials. Formaldehyde (CH₂O) is considered one of these pollutants. That is because of its diverse household uses in addition to its assorted uses in several industries [1,2,3,4,5,6,7,8]. CH₂O gas has a pungent smell and no color [9]. Exposure to CH₂O gas can cause voluminous hazards or even death to humans [10,11,12,13]. Accordingly, it is necessary to search for a material capable of removing the CH₂O or that can be used as a CH₂O sensor. Beryllium oxide (BeO) has some electronic features, making it a candidate material for this purpose. BeO is a semiconductor possessing an energy gap (E_g) value of 8.21 eV [14]. The Be-O bond appears to be a shared behavior of ionic and covalent bonds [15]. Several experimental and theoretical efforts have been made to investigate BeO nano-structures, which have been synthesized in various forms such as nano-fibers [16] and nano-particles [17,18]. The geometrical and electrical characteristics of diverse nano-clusters of BeO have been inspected, and this has proven that a Be₁₂O₁₂ nano-cage possesses adequate stability [19]. Moreover, Be₁₂O₁₂ is a semiconductor with E_g = 7.41–8.29 eV [20,21,22,23,24,25]. Furthermore, metal oxide gas sensors have desirable properties such as a small size, lower cost, and extended lifetime [26]. Accordingly, Be₁₂O₁₂ nano-cages are employed for various uses such as a catalyst to convert methane to organic compounds [27]; a sensor for sulfur mustard [23], tabun, mercaptopyridine, formaldehyde, sulfur hydride, and sulfur dioxide [20,22,28,29]; and a hydrogen storage material [25]. The impact of Li, Na, and K deposition and encapsulation on the electronic and non-linear optical properties of Be₁₂O₁₂ has been studied [24,28]. It was found that the deposition and encapsulation of the alkali metals sharply decreased the E_g of the nano-cage to 1.10–1.56 eV and 0.68–0.69 eV, respectively. Additionally, it proved that the non-linear optical properties of Be₁₂O₁₂ nano-cages could be modified by the alkali metals. The UV-vis spectra for the Be₁₂O₁₂ nano-cage were predicted by Fallahi et al. [22] and Jouypazadeh et al. [23]. They found that the λ_max absorbance peak for a Be₁₂O₁₂ nano-cage is located at 140 to 150 nm. Kosar et al. [30] found that the encapsulation of Be₁₂O₁₂ with Be, Mg, and Ca atoms led to a red shift that was located at 505 nm, 781 nm, and 1136 nm, respectively.

Several efforts have been made to sense the CH₂O molecules [31,32]. Diverse compounds such as MgO [33], BN [34,35,36,37], N-doped TiO₂ [38], ZnO [4,39], transition metal-doped MoS₂ [40], SnS [41], In₂O₃ [42], SnO₂ [43], NiO [44], and BeO [20] have been employed as CH₂O sensors.

According to previous studies [45,46,47,48,49,50,51,52], the existence of an external electric field (EF) impacts the geometric structure, substrate–adsorbate interaction, and electrical characteristics. Additionally, the electric field has a significant impact on key sensor properties such as the adsorption energy (E_ads), sensitivity, and recovery time (τ). Our previous study [20] focused on studying the effect of the EF on the interaction characteristics of CH₂O on a pristine Be₁₂O₁₂ nano-cage in different solvents. It was discovered that the magnitude and orientation of the EF can alter the sensing parameters of Be₁₂O₁₂ for the CH₂O molecule.

According to our knowledge, there is a deficiency in the investigation of the impact of encapsulation of 3d metal atoms on the electrical, optical, and adsorption features of Be₁₂O₁₂. Moreover, no work has been interested in the effect of the existence of an EF on the adsorption features of CH₂O molecules on metal-encapsulated Be₁₂O₁₂. Therefore, in this work, we sought to study the effect of an alkali metal (K) or a transition metal (Mn or Cu) as well as the effect of an EF on the adsorption characteristics of the CH₂O molecule on a Be₁₂O₁₂ nano-cage.

2. Methods

The BeO nano-cage was represented by 12 Be-O dimers. Then, as a trial to modify the properties of the beryllium oxide nano-cage (Be₁₂O₁₂), an M atom (M = K, Mn, or Cu) was encapsulated into the nano-cage to form MBe₁₂O₁₂ nano-cages (see Figure 1). DFT-D2 computations were performed to inspect the adsorption features of the CH₂O molecule on the Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages. To estimate the optimized energetic geometrical structures and the electronic properties, the DFT-D2 computations were carried out by utilizing the hybrid functional B3LYP and 6–31 g(d,p) basis set [53]. The exchange functional B3 refers to Becke’s three parameters, while the correlation functional LYP refers to the correlation functional of Lee, Yang, and Parr [54]. The investigated structures were fully optimized under the following cutoff conditions for the total force on an atom (0.129 eV/Å), the root mean square of the force (0.086 eV/Å), the displacement ($5.29\times {10}^{-3}\mathrm{\AA }$), and the root mean square of the displacement ($3.53\times {10}^{-3}\mathrm{\AA }$).

D2 refers to Grimme’s dispersion [55,56], which takes into account the van der Waals interactions. The optimal multiplicity M (M = 2S + 1) was examined for the considered nano-cages, where S is the total spin. The dynamic stability for the considered nano-cages was checked by utilizing the vibrational frequencies, which were estimated by using the FT-IR spectra as well as a molecular dynamic simulation using the ADMP model. The UV-vis spectra were estimated for the considered structures by using the time-dependent DFT (TD-DFT) method. In TD-DFT calculations, an adequate number of excited states (n = 15) is estimated to cover all the probable transitions in the appropriate range (0–2000 nm).

The binding energy per atom (E_b) for the investigated structures was assessed using Equation (1) [57].

$${\mathrm{E}}_{\mathrm{b}}=\frac{1}{\mathrm{n}}\left({\mathrm{E}}_{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}-\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{E}}_{\mathrm{i}}\right)$$

(1)

Here, ${\mathrm{E}}_{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}$ is the optimized nano-cage total energy, ${\mathrm{E}}_{\mathrm{i}}$ is the single atom energy, and n is the number of the cage atoms.

The ionization potentials (IPs) for the considered cages were evaluated as shown below [5,58]:

$$\mathrm{I}\mathrm{P}={\mathrm{E}}_{{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}^{+}}-{\mathrm{E}}_{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}$$

(2)

where ${\mathrm{E}}_{{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}^{+}}$ is the energy of the cage after removing an electron while keeping the same geometry of the neutral cage. The Fermi level (E_F), hardness (η), and electrophilicity (ω) were evaluated as shown in Equations (3)–(5), respectively [59,60]:

$${\mathrm{E}}_{\mathrm{F}}=\frac{1}{2}\left({\mathrm{E}}_{\mathrm{H}\mathrm{O}\mathrm{M}\mathrm{O}}+{\mathrm{E}}_{\mathrm{L}\mathrm{U}\mathrm{M}\mathrm{O}}\right)$$

(3)

$$\mathsf{\eta }=\frac{1}{2}\left({\mathrm{E}}_{\mathrm{L}\mathrm{U}\mathrm{M}\mathrm{O}}-{\mathrm{E}}_{\mathrm{H}\mathrm{O}\mathrm{M}\mathrm{O}}\right)$$

(4)

$$\omega \approx {\mathrm{E}}_{\mathrm{F}}^2/2\eta$$

(5)

where ${\mathrm{E}}_{\mathrm{H}\mathrm{O}\mathrm{M}\mathrm{O}}$ ${\mathrm{E}}_{\mathrm{L}\mathrm{U}\mathrm{M}\mathrm{O}}$ are the energies of the highest occupied and the lowest unoccupied molecular orbitals, respectively. The adsorption energy (E_ads) for CH₂O on a nano-cage was calculated by Equation (6):

$${\mathrm{E}}_{\mathrm{a}\mathrm{d}\mathrm{s}}={\mathrm{E}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}/\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}-\left({\mathrm{E}}_{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}+{\mathrm{E}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}\right)$$

(6)

where ${\mathrm{E}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}/\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}$, ${\mathrm{E}}_{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}$and ${\mathrm{E}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}$ are the energies for the $\mathrm{C}{\mathrm{H}}_2\mathrm{O}/\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}$ complex, the nano-cage and the CH₂O molecule, respectively.

For the adsorption of multiple n molecules of CH₂O gas, the adsorption energy per molecule (${\overline{\mathrm{E}}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$) was calculated by Equation (7):

$${\overline{\mathrm{E}}}_{\mathrm{a}\mathrm{d}\mathrm{s}}=\frac{1}{\mathrm{n}}\left({\mathrm{E}}_{\mathrm{n}\mathrm{C}{\mathrm{H}}_2\mathrm{O}/\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}-\left({\mathrm{E}}_{\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}+\mathrm{n}{\mathrm{E}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}\right)\right)$$

(7)

where ${\mathrm{E}}_{\mathrm{n}\mathrm{C}{\mathrm{H}}_2\mathrm{O}/\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}}$is the energy for the $\mathrm{n}\mathrm{C}{\mathrm{H}}_2\mathrm{O}/\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e}$ complex. A negative sign for the E_ads and ${\overline{\mathrm{E}}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ indicates an exothermic reaction and more stable products.

The Gaussian 09 program was used to achieve all the calculations, whereas the Gauss View 5 was used for the visualization of the results [61]. The GaussSum 3.0 program estimated the partial density of states (PDOS) of the adsorbates–substrates [62]. The NBO version 3.1 measured the charges of atoms [63]. The Multiwfn 3.7 software package was employed to achieve the Quantum Theory of Atoms in Molecules (QTAIM) analysis [64].

3. Results and Discussion

3.1. Effect of Metal Encapsulation in Be₁₂O₁₂

In this section, the impact of doping on the geometrical, electrical, optical, and magnetic characteristics of the Be₁₂O₁₂ nano-cage was scrutinized. Therefore, full geometrical optimization was performed for the pristine and M-encapsulated Be₁₂O₁₂ nano-cage (M = K, Mn, or Cu). The optimized structures for Be₁₂O₁₂ as well as MBe₁₂O₁₂ are shown in Figure 1. The Be₁₂O₁₂ nano-cage comprises eight hexagonal and six tetragonal rings. Furthermore, the Be-O bonds are distinguished into two types. The first type, denoted by d₁, is shared between hexagonal and tetragonal rings. The second type, denoted by d₂, is shared between two hexagonal rings. It is found that d₁ and d₂ have bond lengths of 1.58 and 1.52 Å, respectively, for the Be₁₂O₁₂ nano-cage in good agreement with previous works [20,21,29,65]. Whereas the metal encapsulation into Be₁₂O₁₂ elongates d₁ and d₂ bonds to be 1.62 and 1.57 Å for KBe₁₂O₁₂, 1.60 and 1.55 Å for MnBe₁₂O₁₂, and 1.61 and 1.55 Å for CuBe₁₂O₁₂.

To investigate the stability of Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages, molecular dynamic (MD) simulations by the ADMP model as implemented in Gaussian 09 were performed. The MD simulations were performed at room temperature (300 K) for 500 fs. Figure S1, in the Supplementary Materials, illustrates the potential energy (PE) fluctuation against time besides the structure of Be₁₂O₁₂ as well as MBe₁₂O₁₂ at the end of the time. One can notice that the PE irrelevantly fluctuates, and unimportant distortion occurs for the cages. Therefore, the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages have stable structures. Additionally, in Figure S2, in the Supplementary Materials, the disappearance of the imaginary frequencies for all investigated nano-cages declares that the optimized nano-cages are true minima on the potential energy surfaces [66,67,68,69]. This is in agreement with our previous work on the Be₁₂O₁₂ nano-cage [29].

One can judge the relative stability as well as relative reactivity of Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages in terms of the following: E_b, HOMO–LUMO energy gap (E_g), IP, E_F, η, and ω. Their values are listed in

Table 1. Electronic properties of Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages (M = K, Mn, or Cu). HOMO and LUMO energy levels (eV), HOMO–LUMO gap (E_g, eV), average binding energy per atom (E_b, eV), NBO charges (Q, e), ionization potential (IP, eV), Fermi level (E_F, eV), hardness (η, eV), electrophilicity (ω, eV), and dipole moment (D, Debye).

	Be12O12	KBe12O12	MnBe12O12	CuBe12O12
HOMO (α)	−8.463	−2.128	−2.281	−3.544
LUMO (α)	−0.253	−0.973	−0.547	−0.373
HOMO (β)	−8.463	−8.607	−3.803	−7.446
LUMO (β)	−0.253	−1.443	−1.224	−1.977
Eg	8.210	0.685	1.057	1.568
Eb	−5.630	−5.137	−5.201	−5.405
QBe	1.210	1.241	1.199	1.185
QO	−1.210	−1.249	−1.211	−1.194
QM	-	0.096	0.237	0.108
IP	10.135	3.298	3.639	5.216
EF	−4.358	−1.786	−1.753	−2.760
η	4.105	0.343	0.529	0.784
ω	2.314	4.655	2.905	4.860
D	0.001	0.003	0.001	0.105

. The E_b is calculated via Equation (1). The E_b value for Be₁₂O₁₂ is −5.630 eV, which is in agreement with Sajid et al. [21], while the M encapsulation decreases its value by 8.77%, 7.63%, and 4.01% for KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂, respectively. This indicates that the MBe₁₂O₁₂ nano-cages are lower in stability and consequently higher in reactivity than the Be₁₂O₁₂ nano-cage. Moreover, the E_g value for Be₁₂O₁₂ is 8.210 eV and it decreased to 0.685, 1.057, and 1.568 eV for KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂, respectively, which is in good agreement with previous studies [20,21,65].

Furthermore, the IP, E_F, η, and ω values for Be₁₂O₁₂ are 10.135, −4.358, 4.105, and 2.314 eV, respectively. While the IP values decrease by 67.46%, 64.10%, and 48.54%, the E_F values lower to −1.786, −1.753, and −2.760 eV, and the η values decrease by 91.65%, 87.12%, and 80.90%, whereas the ω values rise by 101.18%, 25.56%, and 110.07% for KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂, respectively. It was stated [69,70,71,72] that the clusters distinguished by lower E_g, IP, and η values and larger ω values are chemically less stable and more reactive. Thus, our results assert that the M-encapsulated Be₁₂O₁₂ nano-cages are chemically higher in reactivity than the Be₁₂O₁₂ nano-cage.

For more intuition, the NBO atomic charges were estimated. As listed in Table 1, for MBe₁₂O₁₂ nano-cages, the K, Mn, and Cu atoms own positive charges of 0.096, 0.237, and 0.108 e.

This indicates the occurrence of charge transfer from the M atoms to the rest of the beryllium oxide nano-cage. Despite the charge transfer that occurred, the charge distribution on the beryllium and oxygen atoms is still symmetric. Therefore, no obvious change in the magnitudes of the dipole moment values of the MBe₁₂O₁₂ nano-cages was observed. Figure 2 represents the charge density difference ($\mathsf{\Delta }\rho$) for MBe₁₂O₁₂ nano-cages.

Obviously, the M atom is enclosed by positive and negative values of $\mathsf{\Delta }\rho$, which indicates the occurrence of a donation-back donation of charges between the metal atom and the rest of the nano-cage atoms. In other words, there is a charge transfer from the M atom to the nano-cage and vice versa. Additionally, the electronic configuration was calculated for M atoms in a free state as well as in the MBe₁₂O₁₂ nano-cages and listed in

Table 2. The electronic configurations for M atom in a free state and MBe₁₂O₁₂ nano-cages.

Structure	4s	3d	4p
K	1.00	-	-
KBe12O12	0.46	0.23	0.41
Mn	2.00	5.00	-
MnBe12O12	0.63	5.76	0.41
Cu	1.00	10.0	-
CuBe12O12	0.76	9.67	0.48

. It is obvious that because of the interaction of the K, Mn, and Cu atoms with the rest of the KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂ nano-cages atoms, the 4s sublevel loses a charge of 0.54, 0.37, and 0.24 e, and the 4p sublevel gains 0.41, 0.41, and 0.48 e, respectively. Meanwhile, the 3d sublevel gains 0.23 and 0.76 e for KBe₁₂O₁₂ and MnBe₁₂O₁₂ nano-cages, respectively, and it loses 0.33 e for CuBe₁₂O₁₂ nano-cage. This emphasizes the donation-back donation mechanism of charges between the M atom and the rest of the nano-cage atoms.

In Figure 3, the molecular electrostatic potential (MESP) for Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages was displayed. Figure 3a reveals that positive and negative electrostatic potentials surround the Be sites and the O sites, respectively, which is in good agreement with our previous work [20]. Figure 3b–d show that the K, Mn, or Cu encapsulation in the nano-cage has no obvious effect on the MESP distribution around the nano-cage. Therefore, the Be and O sites for Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages are expected to perform as electrophilic and nucleophilic sites, respectively.

Furthermore, the PDOS as well as the surfaces of the frontier orbitals (HOMO and LUMO) for Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages are depicted in Figure 4. As seen in Figure 4a the HOMO and LUMO of the Be₁₂O₁₂ nano-cage are located at −8.463 and −0.253 eV, respectively. Therefore, Be₁₂O₁₂ nano-cage is a wide-band gap semiconductor possessing an E_g value of 8.21 eV [14].

Additionally, the HOMO is commonly attributed to the 2p orbitals of the O atoms, whereas the LUMO is commonly attributed to the 2p orbitals of the Be atoms. Thus, the HOMO and LUMO are chiefly localized on the O and the Be sites, respectively. Figure 4b–d show that the presence of the K, Mn, and Cu atoms, respectively, causes noticeable changes in the states attributed to the Be₁₂O₁₂ nano-cage. Moreover, for KBe₁₂O₁₂, new donor and acceptor localized states appear at −2.14 and −1.45 eV, respectively. While for MnBe₁₂O₁₂, new donor localized states appear at −6.74, −6.62, −5.88, −3.84, and −2.28 eV, and new acceptor localized states appear at −0.98 and −0.38 eV. Whereas for CuBe₁₂O₁₂, new donor localized states appear at −7.58 and −3.54 and new acceptor localized states appear at −1.97, −0.36, and −0.02 eV. Consequently, the E_g values for KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂ are narrowed to 0.69, 1.30, and 1.57 eV, respectively. Furthermore, for MBe₁₂O₁₂ nano-cages, an overlap is observed between the occupied states of the K, Mn, and Cu atoms and the occupied states of the rest of the nano-cage atoms, which indicates an interaction among them has occurred. The electrical conductivity (σ) is governed by the E_g value as follows [73,74,75,76,77,78].

$$\mathsf{\sigma }\propto {\mathrm{T}}^{3/2}\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\mathrm{E}}_{\mathrm{g}}}{2\mathrm{k}\mathrm{T}})$$

(8)

where k is Boltzmann’s constant, and T is the temperature. Subsequently, the metal doping increases the $\mathsf{\sigma }$ value of the Be₁₂O₁₂ nano-cage.

To scrutinize the influence of metal encapsulation on the optical properties of the Be₁₂O₁₂ nano-cage, the UV-vis spectra for the Be₁₂O₁₂, KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂ nano-cages were estimated via TD-DFT calculations and graphed in Figure 5. It is obvious the Be₁₂O₁₂ nano-cage has a λ_max absorbance peak in the UV region at 169 nm, which is consistent with previous studies [22,23]. Moreover, the KBe₁₂O₁₂ nano-cage has two absorbance peaks; the first peak is located in the visible region at 424 nm, and the second peak is located in the IR region at 1527 nm, while the MnBe₁₂O₁₂ nano-cage has a λ_max absorbance peak in the IR region at 889 nm. However, the CuBe₁₂O₁₂ nano-cage has a λ_max absorbance peak in the visible region at 431 nm. In other words, the metal encapsulation obviously affects the optical activity of the Be₁₂O₁₂ nano-cage. The Be₁₂O₁₂ nano-cage is an ultraviolet active compound, while the MBe₁₂O₁₂ nano-cages are active compounds in the visible and IR regions.

3.2. Adsorption of CH₂O on M-Doped Be₁₂O₁₂

To examine the interaction characteristics of the CH₂O molecule with the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages, an optimization was carried out for the free CH₂O, the bare nano-cages (Be₁₂O₁₂ and MBe₁₂O₁₂), and the CH₂O/nano-cages (CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂) complexes. Our calculations show that the bond lengths of the C-O and C-H bonds for the CH₂O molecule are 1.21 and 1.11 Å, respectively. The geometrical parameters for the bare Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages were discussed in the previous section. Additionally, in our previous work [20], it was found that negative and positive electrostatic potentials surround the O atom and CH₂ group, respectively, of the CH₂O molecule. While Figure 3 shows that positive and negative electrostatic potentials surround the Be and O sites, respectively, of the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages. Therefore, the interaction of CH₂O with the nano-cage is investigated in different orientations, as shown in Figure S3 in the Supplementary Materials. The optimization process for the suggested orientations shows that the CH₂O molecule always interacts via its O head with the Be site of the nano-cage, as depicted in Figure 6.

One can see that the distance between the CH₂O and the Be₁₂O₁₂ nano-cage (d_Be1-O1 = 1.76 Å) is longer than that between the CH₂O and the MBe₁₂O₁₂ nano-cage (d_Be1-O1 in the range 1.49–1.50 Å). Moreover, the C-O bond length of the CH₂O molecule is elongated by 0.83%, 10.74%, 11.57%, and 11.57% for the CH₂O/Be₁₂O₁₂, CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂, respectively. Furthermore, the CH₂O adsorption elongates the bond lengths between the adsorbing site (Be1) and the neighboring oxygen sites (O2, O3, and O4); this elongation is higher for the CH₂O/MBe₁₂O₁₂ complexes than that for the CH₂O/Be₁₂O₁₂ complex.

Table 3. Adsorption properties of CH₂O on MBe₁₂O₁₂ nano-cages. Adsorption energies (E_ads, eV), HOMO and LUMO energy levels (eV), HOMO–LUMO gap (E_g, eV), NBO charges (Q, e), and dipole moment (D, Debye).

	CH2O/Be12O12	CH2O/KBe12O12	CH2O/MnBe12O12	CH2O/CuBe12O12
Eads	−0.963	−2.518	−2.551	−1.541
HOMO	−7.869	−2.665	−3.007	−2.900
LUMO	−2.886	−1.380	−1.491	−2.024
Eg	4.983	1.286	1.516	0.876
QBe1	1.134	1.230	1.208	1.191
QO2	−1.221	−1.233	−1.181	−1.177
QO3	−1.210	−1.232	−1.191	−1.182
QO4	−1.210	−1.229	−1.174	−1.181
QM	-	0.443	0.570	0.578
QO1	−0.516	−0.858	−0.853	−0.849
QH1	0.194	0.134	0.134	0.134
QH2	0.180	0.148	0.153	0.151
QC	0.290	−0.128	−0.124	−0.126
${\mathrm{Q}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}$	0.148	−0.703	−0.690	−0.690
D	5.387	8.494	6.804	6.962

lists the adsorption features of CH₂O on MBe₁₂O₁₂ nano-cages.

One can notice that the CH₂O adsorption on the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ nano-cages is a chemisorption. Meanwhile, the E_ads values for the CH₂O/MBe₁₂O₁₂ complexes are more negative than the E_ads value for the CH₂O/Be₁₂O₁₂ complex. In other words, the presence of the M atom enhances the CH₂O adsorption on the nano-cage. Thus, the Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages can be utilized as removal materials for formaldehyde gas, whereas the MBe₁₂O₁₂ nano-cages are more efficient than the pristine Be₁₂O₁₂ nano-cage for this purpose. Furthermore, it was found in previous studies that the E_ads of CH₂O on the B₃O₃ monolayer [79], Ti-functionalized porphyrin-like C70 fullerenes [80], carbon nano-tube (CNT) [81], Pd-loaded CNT [81], BeO nano-tube [82], Zn₁₂O₁₂ nano-cage [4], and Al-deposited Zn₁₂O₁₂ nano-cage [4] are −0.402, −1.862, −0.106, −1.299, −1.088, −1.27, and −2.23 eV, respectively. Therefore, the present work shows that KBe₁₂O₁₂ and MnBe₁₂O₁₂ nano-cages as CH2O sorbent materials are more efficient than those mentioned in the previous studies. In addition, the CH₂O adsorption decreases the E_g value for Be₁₂O₁₂ and CuBe₁₂O₁₂ by 39.31% and 44.13%, respectively, and increases its value for KBe₁₂O₁₂ and MnBe₁₂O₁₂ by 87.74% and 43.42%, respectively. For a more detailed explanation of the results, NBO atomic charge analysis, charge density difference ($\mathsf{\Delta }\mathsf{\rho }$) analysis, QTAIM analysis, and PDOS analysis were performed. The NBO atomic charge analysis shows that for the CH₂O/Be₁₂O₁₂ complex, the CH₂O molecule earns a positive charge of 0.148 e due to a charge transfer from the CH₂O molecule to the Be₁₂O₁₂ nano-cage. On the other side, for CH₂O/MBe₁₂O₁₂ complexes, the CH₂O molecule acquires negative charges of 0.703, 0.960, and 0.960 e. Furthermore, the positive charges for K, Mn, and Cu rose by 0.347, 0.333, and 0.470 e for CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂, respectively; i.e., a charge transfer has occurred from the MBe₁₂O₁₂ nano-cages, mainly from the metal atom to the CH₂O molecule. This may be owing to the lower IP value and consequently higher ability to donate electrons (higher basicity) for the MBe₁₂O₁₂ nano-cages than the Be₁₂O₁₂ nano-cages. Additionally, the electronic configurations of 2s and 2p orbitals for the O and C atoms of CH₂O were estimated for the free CH₂O, CH₂O/Be₁₂O_12, and CH₂O/MBe₁₂O₁₂ complexes (see

Table 4. The electronic configurations for 2s and 2p orbitals for oxygen and carbon atoms of CH₂O for free CH₂O, CH₂O/Be₁₂O_12, and CH₂O/MBe₁₂O₁₂.

Structure	O		C
	2s	2p	2s	2p
CH2O	1.72	4.76	1.05	2.70
CH2O/Be12O12	1.66	4.84	1.07	2.62
CH2O/KBe12O12	1.66	5.18	1.07	3.04
CH2O/MnBe12O12	1.66	5.18	1.07	3.03
CH2O/CuBe12O12	1.66	5.18	1.07	3.03

).

It is clear that the interaction of the CH₂O molecule with the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ nano-cages is accompanied by losing electrons from the 2s and gaining electrons to the 2p orbitals of the oxygen atom. Whereas for the carbon atom, the 2s orbital has an insignificant change in its electronic configuration, while the 2p orbital loses electrons for the CH₂O/Be₁₂O₁₂ complex and gains electrons for the CH₂O/MBe₁₂O₁₂ complexes. This means a donation-back donation mechanism has happened between the CH₂O molecule and the nano-cages. Figure 7 demonstrates the charge density difference ($\mathsf{\Delta }\mathsf{\rho }$) for CH₂O/Be₁₂O₁₂ as well as the CH₂O/MBe₁₂O₁₂ complexes. One can see that the CH₂O molecule for all the investigated complexes is surrounded by positive (blue color) and negative (red color) $\mathsf{\Delta }\mathsf{\rho }$ values. This means the CH₂O molecule loses and gains charges, emphasizing the donation-back donation mechanism between the CH₂O molecule and the nano-cages. It is worth noticing that the high dipole moment values for the CH₂O/Be₁₂O₁₂ and the CH₂O/MBe₁₂O₁₂ complexes in Table 3 refer to the charge redistribution between the CH₂O molecule and the nano-cages.

For more insight into the nature of the CH₂O adsorption on the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ nano-cage, the QTAIM analysis was accomplished. Figure S4, in the Supplementary Materials, shows the bond critical points (BCP) of type (3,−1) for the CH₂O/Be₁₂O₁₂ and the CH₂O/MBe₁₂O₁₂ complexes. The topological parameters are stated in

Table 5. The estimated topological parameters. Electron densities ($\mathsf{\rho }$), Laplacian of charge density (${\nabla }^2\mathsf{\rho }$), kinetic electron density (G(r)), potential energy density (V(r)), and energy density (H(r)). All units are in au.

Complex	BCP *	$\mathsf{\rho }$	${\nabla }^2\mathsf{\rho }$	G(r)	V(r)	H(r)	−G(r)/V(r)
CH2O/Be12O12	Be1-O1	0.048	0.327	0.077	−0.072	0.005	1.069
	O2-H1	0.014	0.054	0.012	−0.010	0.002	1.200
CH2O/KBe12O12	Be1-O1	0.110	0.858	0.212	−0.211	−0.002	1.005
CH2O/MnBe12O12	Be1-O1	0.113	0.879	0.219	−0.219	−0.001	1.000
CH2O/CuBe12O12	Be1-O1	0.112	0.867	0.216	−0.215	−0.001	1.005

* The atom numbering shown in the Figure 6 is used.

.

The kind of bond can be distinguished by the BCP parameters [83,84,85]. As reported, the ionic bond, weak hydrogen bond, and van der Waals interaction are characterized by ${\nabla }^2\mathsf{\rho }$ > 0, H(r) > 0, −G(r)/V(r) > 1. Furthermore, the strong interaction is categorized by ${\nabla }^2\mathsf{\rho }$ >10⁻¹ au, and the weak interaction is categorized by ${\nabla }^2\mathsf{\rho }$ < 10⁻¹ au. Additionally, the partly covalent interaction is categorized by ${\nabla }^2\mathsf{\rho }$ > 0 and H(r) < 0. Our results show that for the CH₂O/Be₁₂O₁₂ complex, there are two BCPs; the first BCP is Be1-O1, which has a ${\nabla }^2\mathsf{\rho }$ of 0.327 au, H(r) of 0.005 au, and −G(r)/V(r) ratio of 1.069, while the other BCP is O2-H1, which has a ${\nabla }^2\mathsf{\rho }$ of 0.054 au, H(r) of 0.002 au, and −G(r)/V(r) ratio of 1.200. This indicates the formation of a pure ionic bond between the O1 atom of the CH₂O molecule and the Be1 site of the Be₁₂O₁₂ nano-cage. Meanwhile, a weak hydrogen bond is found between the H1 atom of the CH₂O molecule and the O2 site of the Be₁₂O₁₂ nano-cage. For CH₂O/MBe₁₂O₁₂ complexes, only one BCP exists between the O1 atom of the CH₂O molecule and the Be1 site of the MBe₁₂O₁₂ nano-cage. This BCP has ${\nabla }^2\mathsf{\rho }$ values of 0.858, 0.879, and 0.867, −G(r)/V(r) ratios of 1.005, 1.000, and 1.005, and H(r) values of −0.002, −0.001, and −0.001 for CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂ complexes, respectively. These results categorize the interaction as an ionic interaction with a partially covalent character. Therefore, one can say that the presence of the metal atom has an obvious role in altering the character of the interaction between the CH₂O molecule and the nano-cages. Furthermore, the higher $\mathsf{\rho }$ values [80] for the CH₂O/MBe₁₂O₁₂ complexes explain the higher adsorption of the CH₂O molecule than the CH₂O/Be₁₂O₁₂ complex.

Figure 8 depicts the HOMO and LUMO surfaces and the PDOS for the CH₂O molecule, CH₂O/Be₁₂O₁₂, and CH₂O/MBe₁₂O₁₂ complexes. Figure 8a shows four occupied states of the CH₂O molecule located at −7.30, −10.86, −12.24, and −13.44 eV. Looking at Figure 8b–e, obvious changes in the occupied states of the CH₂O molecule indicate a strong reaction between the CH₂O and the nano-cages. Regarding Figure 8b, one can see the HOMO of Be₁₂O₁₂ rises to −7.87 eV and an acceptor state is created at −2.89 eV; thus, the E_g value decreases from 8.21 to 4.98 eV. Looking at Figure 4b–d, it is obvious that there are occupied states located at −2.13, −2.28, and −3.54 eV, which are attributed to the metal atoms for KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂, respectively. As seen in Figure 8c–e, these states disappear for the complexes CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂.

This confirms that the adsorption of the CH₂O leads to a charge loss from the metal atom. Additionally, new occupied states assigned to CH₂O appear at −2.67, −3.01, and −2.90 eV for CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂, respectively. This confirms that the adsorption of the CH₂O leads to a charge transfer from the MBe₁₂O₁₂ to the CH₂O molecule. Furthermore, the adsorption of the CH₂O molecule causes a shift for LUMO states from −1.44, −1.22, and −1.98 eV to −1.38, −1.49, and −2.02 eV for KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂, respectively. Accordingly, the E_g values increase from 0.69 and 1.06 eV to 1.29 and 1.52 eV for KBe₁₂O₁₂ and MnBe₁₂O₁₂ nano-cages, respectively, while E_g value decreases from 1.57 to 0.88 eV for CuBe₁₂O₁₂ nano-cage. Consequently, due to the CH₂O adsorption and Equation (8), the electrical conductivity is increased for the Be₁₂O₁₂ and CuBe₁₂O₁₂, whereas it is decreased for the KBe₁₂O₁₂ and MnBe₁₂O₁₂. Therefore, the Be₁₂O₁₂ and CuBe₁₂O₁₂ nano-cages can be utilized as electrical sensors for the CH₂O molecule. In addition, the essential sensing factor, the recovery time (τ), depends on the E_ads as in the following equation [20,86,87].

$$\tau ={\nu }_{\mathsf{o}}^{-1}\exp (-\frac{{\mathrm{E}}_{\mathrm{a}\mathrm{d}\mathrm{s}}}{\mathrm{k}\mathrm{T}})$$

(9)

where ${\nu }_{\mathsf{o}}$ is the attempt frequency, k is Boltzmann’s constant, and T is the temperature. Therefore, the τ value is in the following trend: MnBe₁₂O₁₂ > KBe₁₂O₁₂ > CuBe₁₂O₁₂ >Be₁₂O₁₂.

3.3. Effect of EF

In the present section, the influence of EF on the electronic properties of the CH₂O molecule, Be₁₂O₁₂ as well as MBe₁₂O₁₂ nano-cages, and the CH₂O/Be₁₂O₁₂ as well as CH₂O/MBe₁₂O₁₂ complexes was examined. The EF was considered in the range of −514 to +514 kV/mm with a step of 102.8 kV/mm. All the investigated structures are fully optimized for each EF value. The direction of the EF relative to the investigated structures is sketched in Figure 9. The influence of the EF on the dipole moment for the CH₂O molecule and the Be₁₂O₁₂ nano-cage, as well as the MBe₁₂O₁₂ nano-cages, was investigated. Because the EF was applied in the X-axis direction, the change in the dipole moment was only evident in its x-component.

Therefore, the x-component only of the dipole moment was considered. Figure 10a shows the dipole moment for the CH₂O molecule versus EF. It is clear that with the EF varying from −514 to +514 kV/mm, the dipole moment of the CH₂O molecule decreases. The dipole moments versus EF for the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages are illustrated in Figure 10b.

It is seen that with the varying of the EF from −514 to +514 kV/mm, the dipole moment of the investigated nano-cages decreases from 0.311, 2.041, 0.901, and 0.420 Debye to −0.311, −2.032, −0.903, and −0.427 Debye for Be₁₂O₁₂, KBe₁₂O₁₂, MnBe₁₂O₁₂, and CuBe₁₂O₁₂, respectively. It is worth noticing that the polarity of the dipole moment is inverted as the EF direction is inverted. Additionally, the rate of change in the dipole moment with the EF is in this trend: KBe₁₂O₁₂ > MnBe₁₂O₁₂> CuBe₁₂O₁₂> Be₁₂O₁₂. These results manifest that the EF affects the charge distribution for the CH₂O molecule, Be₁₂O₁₂, and MBe₁₂O₁₂ nano-cages. This is emphasized by Figure 10b–f, which demonstrates the charge density difference ($\mathsf{\Delta }\mathsf{\rho }$) for the CH₂O molecule, Be₁₂O₁₂, and MBe₁₂O₁₂ nano-cages for EF values of −514 and +514 kV/mm. The dipole moment of a molecule plays an important role in its reactivity with the surrounding medium [88]. Therefore, it is expected that the EF would have an obvious impact on the CH₂O adsorption on the Be₁₂O₁₂ as well as MBe₁₂O₁₂nano-cages.

The manipulation of the EF on the CH₂O adsorption on the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages has been completed under the same criteria mentioned above. The E_ads values for the CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂ complexes are estimated and plotted against the EF in Figure 11.

It is seen that by increasing the negative EF, the value of E_ads is gradually enhanced for the CH₂O/Be₁₂O₁₂ complex up to 3.4% at EF = −514 kV/mm with respect to their value at zero EF. Meanwhile, the values of E_ads are gradually inhibited for CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂ complexes up to 4.8%, 3.6%, and 6.2%, respectively, at EF = −514 kV/mm with respect to its value at zero EF. On the other side, by increasing the positive EF, the value of E_ads is gradually inhibited for the CH₂O/Be₁₂O₁₂ complex up to 3.1% at EF = +514 kV/mm with respect to its value at zero EF. Meanwhile, the values of E_ads are gradually enhanced for CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂ complexes up to 4.2%, 3.5%, and 6.4%, respectively, at EF = +514 kV/mm with respect to their value at zero EF. Therefore, the E_ads values for CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂ complexes are dominated by either the value or direction of the EF. These results could be explained in terms of the mechanism of CH₂O interaction with the nano-cage. For the CH₂O/Be₁₂O₁₂ complex, as mentioned before, there is a charge transfer from the CH₂O molecule to the Be₁₂O₁₂ nano-cage. Looking at Figure 9b,c, one can see that the negative EF induces a negative $\mathsf{\Delta }\mathsf{\rho }$ value on the oxygen head of the CH₂O molecule and a positive $\mathsf{\Delta }\mathsf{\rho }$ value on the side of the Be₁₂O₁₂ nano-cage facing the CH₂O molecule. This in turn encourages the charge transfer and, consequently, enhances the E_ads value. In contrast, the positive electric field induces a positive $\mathsf{\Delta }\mathsf{\rho }$ value on the oxygen head of the CH₂O molecule and a negative $\mathsf{\Delta }\mathsf{\rho }$ value on the side of the Be₁₂O₁₂ nano-cage facing the CH₂O molecule. This discourages the charge transfer and, consequently, inhibits the E_ads value. This could be confirmed by Figure 12, which shows the NBO charges of the CH₂O molecule (${\mathrm{Q}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}$) vs. electric field. It is clear that the positive ${\mathrm{Q}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}$ for the CH₂O/Be₁₂O₁₂ complex increases as the negative EF increases and decreases as the positive EF increases.

Thus, the E_ads value is enhanced by increasing the negative EF and inhibited by increasing the positive EF (see Figure 11). On the other side, for the CH₂O/MBe₁₂O₁₂ complexes, the interaction has occurred due to the charge transfer from the MBe₁₂O₁₂ nano-cage to the CH₂O molecule. Looking at Figure 9d–f, it is clear that for negative EF values, the induced positive $\mathsf{\Delta }\mathsf{\rho }$ value on the side of the MBe₁₂O₁₂ nano-cage facing the CH₂O molecule inhibits this charge transfer and consequently inhibits the E_ads values. Whereas for the positive EF values, the induced negative $\mathsf{\Delta }\mathsf{\rho }$ value on the side of the MBe₁₂O₁₂ nano-cage facing the CH₂O molecule encourages the charge transfer and thus enhances the E_ads values. This can be confirmed by Figure 12, where the negative ${\mathrm{Q}}_{\mathrm{C}{\mathrm{H}}_2\mathrm{O}}$ for CH₂O/MBe₁₂O₁₂ complexes decreases as the negative EF increases and decreases as the positive EF increases. Thus, the E_ads value is inhibited as the negative EF increases and enhanced as the positive EF increases (see Figure 11).

Figure 13 represents the impact of the EF on the E_g for the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages as well as the CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂ complexes. It is clear that the EF has a negligible impact on the E_g values of the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages. Figure 13a declares that the E_g value for the CH₂O/Be₁₂O₁₂ complex increases as the negative EF increases and decreases as the positive EF increases. On the other side, Figure 13b–d show that the E_g values for the CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, CH₂O/CuBe₁₂O₁₂ complexes increase as the negative EF decreases and the positive EF increases. Therefore, according to Equation (8), the electric conductivity (σ) for the CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂ complexes can be controlled by the EF. Here, σ for the CH₂O/Be₁₂O₁₂ complexes will increase by increasing the positive EF, while for the CH₂O/MBe₁₂O₁₂ complexes, σ will increase by increasing the negative EF.

To investigate the electrical sensitivity dependence on the EF, the percentage of the variance of E_g (ΔE_g) by the adsorption for CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂ complexes versus the EF is estimated by Equation (10) and represented in Figure 14.

$$\mathsf{\Delta }{\mathrm{E}}_{\mathrm{g}}=\frac{{\mathrm{E}}_{\mathrm{g}}({\mathrm{C}\mathrm{H}}_2\mathrm{O}/\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{o}-\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e})-{\mathrm{E}}_{\mathrm{g}}(\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{o}-\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e})}{{\mathrm{E}}_{\mathrm{g}}(\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{o}-\mathrm{c}\mathrm{a}\mathrm{g}\mathrm{e})}\times 100$$

(10)

ΔE_g values at EF = 0 were −41.2%, 87.8%, 38.9%, and −44.1% for CH₂O/Be₁₂O₁₂, CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂ complexes, respectively. It is clear that the variance in value and the direction of the EF has a negligible effect on ΔE_g for the CH₂O/Be₁₂O₁₂ complex. Whereas for the CH₂O/KBe₁₂O₁₂, CH₂O/MnBe₁₂O₁₂, and CH₂O/CuBe₁₂O₁₂ complexes, as the negative EF increases, the ΔE_g values are lowered, reaching 58.2%, 24.4%, and −55.0%, while as the positive EF increases, the ΔE_g values are rising, reaching 114.2%, 53.4%, and −34.4%, respectively. Therefore, the electrical sensitivity of Be₁₂O₁₂ to the adsorption of CH₂O molecules is independent of the EF, while the electrical sensitivity of MBe₁₂O₁₂ depends on the EF.

3.4. UV-vis Spectra Investigation

Herein, the effect of the adsorption of the CH₂O molecule as well as the EF on the UV-vis spectra of the examined nano-cages was considered. The UV-vis spectra for the CH₂O/Be₁₂O₁₂ and CH₂O/MBe₁₂O₁₂ complexes are predicted for EF values of −514, 0, and +514 kV/mm, and these are displayed in Figure 15a–c, respectively. It is worth noting that the effect of the EF on the UV-vis spectra of bare nano-cages is studied, and it is found that the EF has no noticeable effect. It is clear that at EF values of −514, 0, and +514 kV/mm, the CH₂O/Be₁₂O₁₂ complex has a λ_max absorbance peak in the UV region at 224, 252, and 260 nm, respectively. Comparing these results with Figure 5, one can observe that the adsorption of CH₂O causes a red shift for the UV-vis spectrum of the Be₁₂O₁₂. The negative EF value decreases the red shift value of λ_max, while the positive EF increases it. Furthermore, at zero EF, the CH₂O/KBe₁₂O₁₂ exhibits three peaks located at 324 nm, 509 nm, and 810 nm in the UV, visible, and IR regions of the spectra, respectively. The presence of the EF shifts these peaks to 328, 536, and 896 nm at negative EF and to 316, 484, and 736 nm at positive EF. When compared with Figure 5, the Kbe₁₂O₁₂ nano-cage has a peak in the visible region at 424 nm, whereas the adsorption of CH₂O induces a red shift in this peak to 536, 509, and 484 for EF values of −514, 0, and +514 kV/mm, respectively. Moreover, at EF values of −514, 0, and +514 kV/mm, the CH₂O/MnBe₁₂O₁₂ exhibits one prominent peak in the visible region of the spectra, which is located at 460, 463, and 448 nm, respectively. In comparison with Figure 5, it is evident that the MnBe₁₂O₁₂ nano-cage has one peak in the IR region (889 nm), while the adsorption of CH₂O causes a blue shift for λ_max to the visible region. Additionally, Figure 5 shows that the CuBe₁₂O₁₂ nano-cage is optically active in the visible region with a λ_max value of 431 nm. Meanwhile, the CH₂O/CuBe₁₂O₁₂ has peaks in the visible region at 452 and 596 nm at an EF value of −514 kV/mm, 332, 429, and 563 nm at an EF value of 0 kV/mm, and 328, 408 and 528 nm at an EF value of +514 kV/mm. From the above discussion, one can summarize the following: (i) the adsorption of a CH₂O molecule affects the UV-vis spectra of Be₁₂O₁₂ as well as the MBe₁₂O₁₂. (ii) The EF has no effect on the UV-vis spectra of Be₁₂O₁₂ as well as the MBe₁₂O₁₂. (iii) The EF has an obvious effect on the UV-vis spectra of CH₂O/Be₁₂O₁₂ as well as CH₂O/MBe₁₂O₁₂. In other words, the adsorption of the CH₂O molecule causes changes in the colors of the MBe₁₂O₁₂ nano-cages; consequently, they could be employed as bare-eye sensors for the CH₂O molecule.

3.5. Effect of Concentration

This section concerns the impact of CH₂O concentration on the adsorption characteristics. The adsorption of n molecules (n = 1–6) on the surface of the MBe₁₂O₁₂ was investigated to form nCH₂O/MBe₁₂O₁₂ complexes. The nCH₂O/MBe₁₂O₁₂ complexes were fully optimized. The adsorption energies per molecule (${\overline{\mathrm{E}}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$) were estimated by Equation (7) and graphed in Figure 16a. It is seen that as n increases, the negative value of the ${\overline{\mathrm{E}}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ decreases. Nevertheless, the ${\overline{\mathrm{E}}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ values remain within the bounds of chemical adsorption for all n values.

In Figure 16b, the relationship between E_g and n is illustrated. It is observed that for KBe₁₂O₁₂ and MnBe₁₂O₁₂, as n increases up to n = 3 and 2, respectively, the E_g decreases after which there is no significant change. Conversely, for CuBe₁₂O₁₂, the E_g decreases as n increases, reaching a minimum at n = 4, and then it rises again. According to Equations (8) and (10), it is noted that for all the investigated nano-cages, the electrical conductivity (σ) and the electrical sensitivity at all values of n > 1 are higher than their values at n = 1. Furthermore, the highest electrical conductivity (σ) and electrical sensitivity are recorded for CuBe₁₂O₁₂ at n = 4.

4. Conclusions

This work is a DFT and TD-DFT study that investigates the capability of the M atom-encapsulated Be₁₂O₁₂ nano-cage to capture or sense CH₂O gas; M = K, Mn, or Cu. The molecular dynamic simulations and the frequency calculations assert that the Be₁₂O₁₂ and MBe₁₂O₁₂ nano-cages are stable structures. In contrast, the values of the binding energies per atom (E_b), the ionization potential (IP), the hardness (η), and the electrophilicity (ω) prove that the M-encapsulated Be₁₂O₁₂ is chemically more reactive than the Be₁₂O₁₂ cage. In addition, the encapsulation of the M atom into the Be₁₂O₁₂ nano-cage increases its basicity, narrows its energy gap, and alerts its optical activity from an ultraviolet active compound into active compounds in the visible and IR regions.

Moreover, the calculated adsorption energies confirm that the CH₂O adsorption on the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ is chemisorption, while the presence of the M atom improves the adsorption of the CH₂O molecule on the nano-cage. Moreover, the QTAIM analysis confirms that the presence of the metal atom plays an obvious role in altering the character of CH₂O interaction with the nano-cages from pure ionic interaction into an ionic interaction with a partially covalent character. Additionally, the CH₂O adsorption decreases the E_g value for Be₁₂O₁₂ and CuBe₁₂O₁₂ by 39.31% and 44.13%, respectively, and increases its value for KBe₁₂O₁₂ and MnBe₁₂O₁₂ by 87.74% and 43.42%, respectively.

It is found that the existence of an external static electric field (EF) can enhance or inhibit the adsorption energies of CH₂O molecules on the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ nano-cages, depending on the value and the orientation of the EF. Furthermore, the EF has an obvious influence on the E_g of CH₂O/MBe₁₂O₁₂ complexes and, consequently, on their electrical conductivity. Thus, the electrical sensitivity of MBe₁₂O₁₂ nano-cages to CH₂O gas can be controlled via EF. Moreover, the CH₂O adsorption makes a red shift for the UV-vis spectrum of the Be₁₂O₁₂ and causes obvious changes in the absorption peaks of the MBe₁₂O₁₂ nano-cages in the visible region. Additionally, the EF has an obvious effect on the UV-vis spectra of CH₂O/Be₁₂O₁₂ as well as the CH₂O/MBe₁₂O₁₂ complexes.

Based on these results, the Be₁₂O₁₂ as well as the MBe₁₂O₁₂ nano-cages are candidate materials for removing and sensing the formaldehyde gas. The MBe₁₂O₁₂ nano-cages may be utilized as electrochemical and naked-eye sensors for CH₂O gas.