INTRODUCTION

Fe(CO)4 organometallic fragment is isolobal with \({\text{CH}}_{3}^{ + }\) organic fragment. Structure, bonding and the related compounds of Fe(CO)4 have been reported [1–3]. The interaction of numerous ligands with the Fe(CO)4 fragment has been considered [4–8]. The interesting structures of the metal–π ligand complexes, giving creative attention within organometallic chemistry [9].

Boron cluster chemistry is an attractive subject in inorganic chemistry [10–17]. Various researches have been reported recently to the [XnYn] nano-structures (X = Group III elements, Y = Group V elements) [18–33]. These nano-materials indicate noteworthy chemical and physical properties, for instance wide band-gap semiconductors. The XY bond causes [XnYn] molecules to show a reactivity pattern different from that of carbon analogue [34]. [Al12P12] and [B12P12] molecules are valuable nano-cages, and they have suitable adsorption ability, large HOMO–LUMO gap, small electron attraction, and exceptional properties [35–41]. Many researches have been reported on adsorption properties of several molecules on the surface of [X12P12] (X = Al and P) nanocages. A DFT investigation was used on the hydrogen atom interaction with boron phosphide nano-cluster [42]. The consequences illustrated a significant role of electron density of adsorbing atoms in hydrogen adsorption on the boron phosphide nano-cage. In other investigation, computational study of the effect of Ni and Pd transition metal functionalized on Interaction of mercaptopyridine with [B12N12] nano-cage has been reported [43].

DFT investigation of doping of the first row transition metals (from Sc to Zn) onto possible adsorption positions of the outside surface of [B12N12] nano-cage has been reported [44].

An experimental investigation about of tetra-carbonyl derivate of C60 has been reported by 57Fe Mossbauer spectroscopy and experimental data compared to similar organometallic complexes [45].

In the basis of our researches, interactions Fe(CO)4 and [X12Y12] (X = B or Al, Y = N or P) nano-cages have been not studied. Therefore, we interested to illustration the electronic, structural and optical properties of Fe(CO)4 [X12Y12] (X = B or Al, Y = N or P) complexes at the mPW1PW91/6-311G(d,p) level of theory. Two possible isomers of the interaction between [X12Y12] nano-cluster and Fe(CO)4 are considered. Dipole moment values, polarizability parameters and non-linear optical properties of these complexes are investigated.

COMPUTATION DETAILS

The studied molecules were optimized at the mPW1PW91/6-311G(d,p) level of theory using Gaussian software package [46]. The standard 6-311G (d, p) basis set is considered [47, 48]. the parameter hybrid functional with adapted Perdew-Wang exchange and correlation (mPW1PW91) was considered [49]. This functional is suitable for transition metal complexes in compared to B3LYP [50–53]. Identity of the studied molecules as an energy minimum was considered with a vibrational analysis.

The bonding interaction values between the Fe(CO)4 and [X12Y12] (X = B or Al, Y = N or P) nano-cages were analyzed using the energy decomposition analysis (EDA) handled in Multiwfn 3.5 package [54]. In this analysis, the interaction energy (ΔEint) between two fragments can be divided into three major components:

$$\Delta {{E}_{{{\text{int}}}}} = \Delta {{E}_{{{\text{polar}}}}}~ + \Delta {{E}_{{{\text{els}}}}} + \Delta {{E}_{{{\text{Ex}}}}},$$

where ΔEpolar is electron density polarization term (also called as induction term)

$$\Delta {{E}_{{{\text{polar}}}}} = E\left( {{\text{SCF last}}} \right)-E\left( {{\text{SCF 1st}}} \right),$$

where E (SCF last) and E (SCF 1st) are the SCF energy at the first and last cycles of SCF process, respectively; ΔEels is electrostatic interaction term, and ΔEEx is exchange repulsion term. The last two terms may be combined into a single one, which is called the steric-repulsion term (ΔEsteric = ΔEels + ΔEEx).

Bond order values are calculated by Laplacian bond order (LBO) method [55]. This method is definition of covalent bond order based on the Laplacian of electron density ∇2ρ in fuzzy overlap space. The LBO between atom A and B can be simply written as

$${{L}_{{{\text{A,B}}}}} = - 10~\,\, \times \int\limits_{{{\nabla }^{2}}{{\rho }}\,\, < \,\,0} {{{w}_{{\text{A}}}}(r){{w}_{{\text{B}}}}(r){{\nabla }^{2}}{{\rho }}(r)dr,} $$

where w is a smoothly varying weighting function proposed by Becke and represents fuzzy atomic space; hence, wA and wB correspond to fuzzy overlap space between A and B.

Quantum theory of atoms in molecules (QTAIM) analysis, Independent Gradient Model (IGM) analysis based on promolecular density [56, 57] and Laplacian bond order (LBO) values were provided by Multiwfn 3.5 package [58, 59].

The total static first hyperpolarizability (βtot) was computed from the following equation:

$${{\beta }_{{{\text{tot}}}}} = \sqrt {\beta _{x}^{2} + \beta _{y}^{2} + \beta _{z}^{2}} ,$$

where

$${{\beta }_{i}} = {{\beta }_{{iii}}} + \frac{1}{3}\mathop \sum \limits_{i \ne j} ({{\beta }_{{ijj}}} + {{\beta }_{{jij}}} + {{\beta }_{{jji}}}).$$

The Kleinman symmetry show that [60]:

$${{\beta }_{{xyy}}} = {{\beta }_{{yxy}}} = {{\beta }_{{yyx}}};\,\,\,\,{{\beta }_{{yyz}}} = {{\beta }_{{yzy}}} = {{\beta }_{{zyy}}}.$$

Therefore

$${{\beta }_{{{\text{tot}}}}} = \sqrt {{{{({{\beta }_{{xxx~}}} + {{\beta }_{{xyy~}}} + {{\beta }_{{xzz~}}})}}^{2}} + {{{({{\beta }_{{yyy~}}} + {{\beta }_{{yzz~}}} + {{\beta }_{{yxx~}}})}}^{2}} + {{{({{\beta }_{{zzz~}}}zxx + {{\beta }_{{zyy~}}})}}^{2}}} $$

RESULTS AND DISCUSSION

Energetics Aspects

Two modes coordination of Fe(CO)4 fragment to [X12Y12] (X = B or Al, Y = N or P) nano-cages are presented in Fig. 1. In isomer I, Fe(CO)4 fragment interacts with common X–Y bond between six-membered rings. In isomer II, this fragment interacts with common X–Y bond between six-membered and four-membered of cage. Energy and relative energy values of the investigated structures are gathered in Table 1. It can be observed, isomer II has more stability than isomer I. On the other hand, the relative energy values decrease with increasing of atomic numbers of X and Y.

Fig. 1.
figure 1

(a) Front view and (b) 3D structures of two coordination modes of the Fe(CO)4 fragment to [X12Y12] (X = B or Al, Y = N or P) nano-cages.

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Table 1.   Energy (E, a.u.), relative energy (ΔE, kcal/mol), dipole moment (Debye, μ), and isotropic (au, αiso) values of the two modes coordination of Fe(CO)4 fragment to [X12Y12] (X = B or Al, Y = N or P) nano-cages
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Bond Distances

The Fe–C, Fe–X and Fe–Y bonding distances of the studied complexes are listed in Table 2. These values reveal Fe–X and Fe–Y bonds are longer in isomer I than isomer II. On the other hand, Fe–Coax bond distances are longer than average of Fe–COeq bond distances. The Fe–CO bond distance is 178.1 pm in free Fe(CO)4. It can be found, Fe–COax bond distances are longer than free Fe(CO)4. Therefore, electron density of Fe–COax is smaller than Fe–COax fragment. On the other hand, Fe–CO bonds are longer in trans position to X than cis position to Y.

Table 2.   Fe–C, Fe–X, and Fe–Y bonding distances (in pm) of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes
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Energy Decomposition Analysis (EDA)

Energy decomposition analysis (EDA) was useful to explain interactions of Fe(CO)4 fragment to [X12Y12] (X = B or Al, Y = N or P) nano-cages. EDA results calculations are listed in Table 3. It can be observed, interactions are stronger in Fe(CO)4···[B12P12] complexes than Fe(CO)4···[B12N12] complexes. On the other hand, it can be found stronger interactions for Fe(CO)4···[Al12N12] complexes than Fe(CO)4···[Al12P12] complexes. EDA consequences specify the most negative ΔEint value in the Fe(CO)4···[Al12N12] molecule.

Table 3.   Energy decomposition analysis (EDA) and thermochemical analysis results (kcal/mol) of interaction of Fe(CO)4 fragment to [X12Y12] (X = B or Al, Y = N or P) nano-cages
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The negative polarization energy values stabilize these molecules. The computed most negative ΔEpolar values is observed for Fe(CO)4···[B12P12] molecule. Moreover, positive steric values energy destabilized these molecules. The calculated most positive ΔEsteric values is revealed for Fe(CO)4···[B12P12] molecule.

Dipole Moment

Dipole moment values of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes are calculated (Table 1). The calculated dipole moment values show that polarity of the studied complexes depend on the nano-cage character. It can be deduced, this property decreases as: [B12P12] > [Al12P12] > [B12N12] > [Al12N12]. On the other hand, polarity of isomer II is larger than isomer I in the presence of [B12N12], [Al12N12], and [Al12P12] nano-cages. This property is larger for isomer I than isomer II in the presence of [B12P12] nano-cage. The most polarity is attributed to isomer II of Fe(CO)4[B12P12] complex.

Polarizability

Isotropic polarizability values of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes are calculated (Table 1). Isotropic polarizability ⟨αiso⟩ was calculated using following equation and considering only diagonal elements:

$${{\alpha }_{{iso}}} = \frac{{{{\alpha }_{{xx}}} + {{\alpha }_{{yy}}} + {{\alpha }_{{zz}}}}}{3}.$$

It can be observed these values show that polarity of the studied complexes depend on the nano-cage character. This property decreases as: [Al12P12] > [B12P12] > [Al12N12] > [B12N12]. It can be observed, the most polarizability is attributed to Fe(CO)4[Al12P12] complexes. Least electronegativity and largest size values of Al and P than other atoms cause the electron could is most easily distorted in Fe(CO)4[Al12P12] complexes.

Vibrational Analysis

Stretching vibrational modes of carbonyl ligands are indicated in Fig. 2. Frequencies values of these vibrations (ω) are listed in Table 4. It can be found that the values of ω1 and ω2 (symmetric and asymmetric vibrational frequencies values of COaxial bonds) are larger than ω3 and ω4 values (those values of COequatorial bonds). On the other hand, asymmetric stretching vibrational frequencies (ω1 and ω3) are smaller than symmetric vibrational frequencies (ω2 and ω4).

Fig. 2.
figure 2

Stretching vibrational modes of carbonyl ligands in the investigated complexes.

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Table 4.   Vibrational frequencies values of the stretching vibrational modes of Fe-CO bonds vibrations in the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes (ω, cm–1). Scale factor 0.9567 is considered from Computational Chemistry Comparison and Benchmark DataBase (https://cccbdb.nist.gov/vsf2x.asp)
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Thermochemical Analysis

Thermodynamics parameters of the interactions of Fe(CO)4 fragment and [X12Y12] nano-cages are calculated by the following reaction equation:

$$\begin{gathered} {\text{F}}{{{\text{e}}}_{{\text{2}}}}{{\left( {{\text{CO}}} \right)}_{9}} + 2\left[ {{{{\text{X}}}_{{{\text{12}}}}}{{{\text{Y}}}_{{{\text{12}}}}}} \right] \to \\ \to \,\,\left[ {{{{\text{X}}}_{{{\text{12}}}}}{{{\text{Y}}}_{{{\text{12}}}}}} \right] \cdot \cdot \cdot {\text{Fe}}{{\left( {{\text{CO}}} \right)}_{4}} + 2{\text{CO}};\,\,\,\,\Delta X = G,H. \\ \end{gathered} $$

Free Fe(CO)4 fragment is not stable, therefore Fe2(CO)9 complex is considered as a reference level for illustration of thermodynamics of the [X12Y12]···Fe(CO)4 formations. Free energy change (ΔG) and enthalpy change (ΔH) values of these reaction are gathered in Table 3 at 298 K and 1 atm. It can be concluded, these reactions are spontaneous and exothermic in the presence of [Al12N12], [B12P12], and [Al12P12] nano-cages. In addition, ΔG values show that isomer II formation reactions are more spontaneous than isomer I. However, the reactions are non-spontaneous and endothermic in the presence [B12N12].

Electrophilicity-Based Charge Transfer (ECT)

The electrophilicity-based charge transfer (ECT) values of Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) isomers are evaluated. The difference between ΔNmax values of interacting molecules is defined as ECT [61]:

$${\text{ECT }} = \Delta {{N}_{{{\text{max}}}}}\left( {{\text{Fe}}{{{\left( {{\text{CO}}} \right)}}_{4}}} \right)-\Delta {{N}_{{{\text{max}}}}}\left( {{{{\text{X}}}_{{{\text{12}}}}}{{{\text{Y}}}_{{{\text{12}}}}}} \right),$$

where ΔNmax is defined as:

$${{(\Delta {{N}_{{{\text{max}}}}})}_{i}} = \frac{{{{\mu }_{i}}}}{{{{\eta }_{i}}}}.$$

In this equation, η and μ are global hardness and chemical potential, respectively. They are provided on the basis of Koopman’s theorem [62] and known as global reactivity descriptors [63–66]. These values for Fe(CO)4, [X12Y12] molecules are computed by the subsequent equations and results are mentioned in Table 5.

$$\eta = ~\frac{{E\left( {{\text{LUMO}}} \right) - E\left( {{\text{HOMO}}} \right)}}{2},$$
$$\mu = \frac{{E\left( {{\text{HOMO}}} \right) + E\left( {{\text{LUMO}}} \right)}}{2}.$$
Table 5.   Frontier orbital energy, hardness (η), and chemical potential (μ) of Fe(CO)4 and [X12Y12] (X = B or Al, Y = N or P) nano-cages (eV)
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The calculated ΔNmax value are –1.54, –0.75, –0.20, and 0.25 for interaction of Fe(CO)4 with [B12N12], [Al12N12], [B12P12], and [Al12P12], respectively. The positive value of ECT reveals charge flow from [Al12P12] to Fe(CO)4. On the other hand, the negative values of ECT show charge flow from Fe(CO)4 to [B12N12], [Al12N12], and [B12P12].

Hyperpolarizability

The first hyperpolarizabilities values of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes are calculated. The βtot, βx, βy, and βz values of these complexes are presented in Table 6. The calculated βtot values show that NLO properties of the studied complexes depend on the nano-cage character. It can be deduced; this property increases as: [B12N12] < [Al12N12] < [B12P12] < [Al12P12]. On the other hand, NLO property of isomer I is larger than isomer II in the presence of [B12N12], [Al12N12], and [B12P12] nano-cages. This property is larger for isomer II than isomer I in the presence of [Al12P12] nano-cage. The most activity of NLO is attributed to isomer II of Fe(CO)4[Al12P12] complex.

Table 6.   Components and total first hyperpolarizability values (esu) of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes
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QTAIM

Results of QTAIM computations at the bond critical points (BCP) of Fe–C, Fe–X, and Fe–Y bonds of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes are listed in Table 7. Larger ρBCP(Fe–X) and ρBCP(Fe–X) values are well-matched with longer Fe–X and Fe–Y (Table 2). There is good linear correlation between ρBCP(Fe–X) and r(Fe–X) values:

$${{\rho }_{{{\text{BCP}}}}} = -0.0009r\left( {{\text{Fe}}-{\text{X}}} \right) + 0.263;\,\,\,\,{{R}^{2}} = 0.9936.$$
Table 7.   QTAIM results at the bond critical points of Fe–C, Fe–X, and Fe–Y bonds of the Fe(CO)4[X12Y12] (X = B or Al, Y = N or P) complexes
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Maximum electron density in bond critical points is belonged to Fe–It can be seen, ρBCP(Fe–Cax) values are smaller than ρBCP(Fe–Ceq) in the studied complexes. These results are compatible with longer Fe–Cax bonds than Fe–Ceq bonds.

Positive Laplacian of electron density (∇2ρ) values are compatible with closed–shell interactions for the Fe–C, Fe–X and Fe–Y bonds.

The negative total electron energy density (H) values are considered as an indicator of covalency. There is the following equation between H and its components:

$$H = G + V.$$

In this equation, V and G are virial energy density and Lagrangian kinetic energy, respectively. The positive ∇2ρ values and negative H values of Fe–C, Fe–X and Fe–Y bonds are compatible with similar systems [67, 68]. These values are compatible with a combination of the shared and closed–shell interactions for the Fe–L bonds.

|V|/G ratio is useful for description of bond characterization. |V(r)|/G(r) < 1 and V(r)|/G(r) > 2 are characteristics of a typical ionic interaction and “classical” covalent interactions, respectively [69]. |V|/G rations of Fe–C, Fe–X and Fe–Y bonds are listed in Table 7. These values are between 1 and 2. Therefore, it can be observed mixture of the shared and closed–shell interactions for the Fe–L bonds.

Laplacian Bond Order (LBO)

Bond orders are valuable values to illustrate of the Fe–C, Fe–X, and Fe–Y bonds. An appropriated method of covalent bond is Laplacian bond order (LBO) [55]. In this method bond orders are computed in the basis of the Laplacian of electron density ∇2ρ in fuzzy overlap space.

The calculated LBO values for Fe–C, Fe–X, and Fe–Y bonds in the studied complexes are listed in Table 8. These values show that, LBO values of Fe–C bond are close to 1, therefore, the Fe–C bonds are the covalent single bond. On the other hand, LBO values of Fe–X and Fe–Y bonds are smaller than 1 reveal that the weaker interactions in compared to the classic covalent single bond. There are good linear correlations between average of LBO and bond distances values of Fe–X and Fe–Y bonds as:

$$\begin{gathered} {\text{Fe}}{{\left( {{\text{CO}}} \right)}_{4}} \cdot {\kern 1pt} \cdot {\kern 1pt} \cdot \left[ {{{{\text{X}}}_{{{\text{12}}}}}{{{\text{N}}}_{{{\text{12}}}}}} \right]\left( {{\text{X = B and Al}}} \right){\kern 1pt} :\left\langle {{\text{LBO}}} \right\rangle {\text{ }} \\ {\text{ = }}\,\,-239.16\left\langle r \right\rangle + 241.62;\,\,\,\,{{R}^{2}} = 0.9949; \\ {\text{Fe}}{{\left( {{\text{CO}}} \right)}_{4}} \cdot {\kern 1pt} \cdot {\kern 1pt} \cdot \left[ {{{{\text{X}}}_{{{\text{12}}}}}{{{\text{P}}}_{{{\text{12}}}}}} \right]\left( {{\text{X = B and Al}}} \right){\kern 1pt} :\left\langle {{\text{LBO}}} \right\rangle \\ = \,\,-195.65\left\langle r \right\rangle + 271.76;\,\,\,\,{{R}^{2}} = 0.9502. \\ \end{gathered} $$
Table 8.   Calculated LBO values for Fe–C, Fe–X, and Fe–Y bonds in the studied complexes
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In addition, LBO values of Fe–CO bonds are smaller in trans position to X than cis position to Y. This result explains longer Fe–CO bonds are in trans position to X than cis position to Y.

Independent Gradient Model (IGM)

Independent Gradient Model (IGM) is a significance tool for illustration of interfragment and intrafragment interactions [56, 57]. The provided isosurface graphs of the studied systems are presented in Fig. 3. The major van der Waals interaction region is exhibited as green isosurface for each of molecules.

Fig. 3.
figure 3

IGM analysis results of the studied complexes.

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CONCLUSIONS

In this work, electronic, structural, and optical properties of Fe(CO)4 [X12Y12] (X = B or Al, Y = N or P) complexes were investigated at the mPW1PW91/6-311G(d,p) level of theory. According to the computations, isomer II was more stable than isomer I. EDA consequences identified the most interaction in the Fe(CO)4···[Al12N12] molecule. The positive value of ECT revealed charge flow from [Al12P12] to Fe(CO)4. On the other hand, the negative values of ECT indicated charge flow from Fe(CO)4 to [B12N12], [Al12N12], and [B12P12]. LBO values of Fe–C bond were close to 1 indicate that the Fe–C bonds were the covalent single bond. The negative H and positive ∇2ρ values of Fe–C, Fe–X, and Fe–Y bonds revealed mixture of the shared and closed–shell interactions for the Fe–L bonds. On the other hand, LBO <1 values of Fe–X and Fe–Y bonds indicated the weaker interactions than the covalent single bond. As future outlook of this study, solvent effect on the calculated parameters can be useful. Dielectric constant dependency of the solvent on these parameters will give useful information.