#
Molecular Structure of M(N_{13}) Compounds with 12-Membered Nitrogen-Containing Cycle and Axial Nitrogen Atom (M = Mn, Fe): Quantum-Chemical Design by DFT Method

^{1}

^{2}

^{*}

## Abstract

**:**

_{13}) chemical compounds (M = Mn, Fe) that are unknown for these elements has been predicted. Data on the structural parameters, the multiplicity of the ground state, APT and NBO analysis, and standard thermodynamic parameters of formation (standard enthalpy Δ

_{f}H

^{0}, entropy S

^{0}, and Gibbs’s energy Δ

_{f}G

^{0}) for these compounds are presented.

## 1. Introduction

_{4}and KFeO

_{4}). Be that as it may, there is no information about the compounds of formula II in the literature, although a number of publications have considered two-element chemicals containing atoms of s-, p- or d-elements and nitrogen atoms (see, in particular, [11,12,13,14,15,16]). It should be noted that almost every one of these works mentioned the possible use of such compounds as potential high-energy materials. A discussion of the possibility of the existence of type II compounds for various M of 3d elements, as well as the dependence of the parameters of their molecular and electronic structure on the nature of M, is the subject of this article.

## 2. Method

_{f}H

^{0}, S

^{0}, and Δ

_{f}G

^{0}) for the M(N

_{13}) compounds under examination were calculated according to the methodology described in [33].

## 3. Results and Discussion

_{13}) type having molecular structures (II) can exist only for two 3d elements, namely for Mn and Fe, as we assumed above. The interatomic distances of M—“axial” nitrogen atoms for these M, depending on the calculation method, is in the range of 150.0–152.5 pm, which is significantly (by more than 30 pm) shorter than the bond lengths of M—“equatorial” nitrogen atoms; this may serve as an indication that the chemical bond formed between them is not single or even double, but triple. In this regard, it should be noted that in the case of 3d elements following Fe in Mendeleev’s Periodic System (i.e., Co, Ni, Cu, and Zn), on the one hand, the interatomic distances of M—“axial” nitrogen atoms (in the framework of the OPBE/TZVP method for M = Co, 155.3; M = Ni, 166.9; M = Cu, 186.1; M = Zn, 194 pm), on the other hand, the difference between these distances and the bond lengths of M—“equatorial” nitrogen atoms (which are equal to 181.1, 186.2, 188.3, and 209.4 pm, respectively) is much smaller, since the formation of a triple bond in the case of each of these four M becomes very doubtful. In view of this important circumstance, in what follows, we will only discuss two type II compounds, namely, Mn(N

_{13}) and Fe(N

_{13}). The most important geometric parameters of the molecular structures of these compounds (the lengths of chemical bonds between atoms and bond angles) obtained within the framework of each of the variants of the DFT method used by us are presented in Table 1. As follows from the data presented in it, a grouping of four nitrogen atoms bound with the M atom by single bonds, in both of the above compounds of M(N

_{13}), is strictly flat, because the sum of the angles N1N4N7, N4N7N10, N7N10N1, and N10N1N4 in each of them is 360.0°, and this takes place within each of the three variants of the method used in the work DFT. There is, however, a small nuance. In the case of Mn(N

_{13}) within the framework of the DFT B3PW91/TZVP, DFT OPBE/TZVP, and DFT M062X/Def2TZVP methods, all the above non-bonding angles are equal to each other and amount to 90°, while according to the data from DFT M06/TZVP, they are equal only in pairs, although the deviation of their values from 90° is smaller than 0.5°. However, in the case of Fe(N

_{13}), within the framework of the DFT B3PW91/TZVP and DFT OPBE/TZVP methods, all the above non-bonding angles are equal to each other and amount to 90°, while according to the data of the DFT M06/TZVP and DFT M062X/Def2TZVP methods, they are equal only in pairs, but the deviation of their values from 90° exceeds 0.5° (Table 1). Both in the case of Mn(N

_{13}) and in the case of Fe(N

_{13}), the N4 group has the shape of either a square (within the DFT B3PW91/TZVP and DFT OPBE/TZVP methods) or a parallelogram (within the DFT M06/TZVP and DFT M062X/Def2TZVP methods). However, the grouping of MN4 atoms in both of these chemical compounds has a tetragonal-pyramidal structure with a very significant (more than 45°) deviation from the plane formed by four “equatorial” nitrogen atoms bonded to the M atom (Table 1). This deviation somewhat depends on the DFT method used in the calculation, but it is always more pronounced for Mn(N

_{13}), which is quite natural if we take into account the somewhat larger radius of the Mn atom compared to the radius of the Fe atom. Thus, in both M(N

_{13}) compounds, the M atom is to some extent elevated above the plane of the four “equatorial” nitrogen atoms. Within the framework of the DFT B3PW91/TZVP and DFT OPBE/TZVP methods, the lengths of the four M–N bonds in both compounds under consideration are the same; within the framework of the DFT M06/TZVP and DFT M062X/Def2TZVP methods, they are different, although not too strongly (Table 1). The 12-membered macrocycles formed by nitrogen atoms in each of these two compounds are also non-coplanar, which is clearly seen even from the images of their molecular structures, presented in Figure 3. The values of the electrical dipole moments (μ) of these compounds, as expected, are quite noticeably different from 0 and are 1.66 and 1.70 in the DFT B3PW91/TZVP method and 1.77 and 1.79 Debye units in the DFT M06/TZVP method. Calculation by the DFT OPBE/TZVP method gives significantly lower values of this parameter; moreover, interestingly, for both of these compounds, it turns out to be almost the same, namely, 1.27 Debye units.

_{13}). Characteristically, the effective charges on the M atoms in the framework of the APT analysis are positive (although very small in absolute value), while in the framework of the NBO analysis, they are negative in the case of Mn(N

_{13}) and positive in the case of Fe(N

_{13}) (Table 2). On the whole, a similar situation occurs in the case of the other three DFT methods (see the Supplementary Materials). Taking into account the important fact that the electronegativity of the nitrogen atom is much greater than the electronegativity of the iron and manganese atoms, the effective charges on the atoms presented in Table 2 look rather unusual; the question of how much they correspond to reality is still open to us.

_{13}) is a spin triplet, which is in good agreement with the values of the squared intrinsic angular momentum of the total spin <S**2>, equal to 2.0016 (in the case of the DFT B3PW91/TZVP), 2.0009 (in the case of DFT OPBE/TZVP), 2.0209 (in the case of DFT M06/TZVP), and 2.0264 (in the case of DFT M062/Def2TZVP). The Fe(N

_{13}) ground state is a spin doublet, which also corresponds to the <S**2> values for this spin multiplicity (0.7508 (DFT B3PW91/TZVP), 0.7501 (DFT OPBE/TZVP), 0.7537 (DFT M06/TZVP), and 1.4530 (M062/Def2TZVP)). As can be seen from these data, for both studied compounds, the three variants of the DFT method used by us give <S**2> values that are very close to each other, although their functionals differ quite significantly from each other. However, if in the case of Fe(N

_{13}), the multiplicity of the ground state (which corresponds to the presence of one unpaired electron) seems quite natural, since the Fe atom is bound to nitrogen atoms by seven bonds and its electronic configuration in this compound can be 3d

^{1}, then in the case of Mn(N

_{13}), the triplet ground state looks somewhat unexpected, since in this case, with the same number of metal–nitrogen bonds as in Fe(N

_{13}), its electronic configuration should be 3p

^{5}4s

^{1}(but not 3p

^{6}, which seems a priori more probable). In this regard, it is worth noting that the nearest excited state for Mn(N

_{13}), according to the data of each of these three DFT methods, is a spin singlet whose energy shows 23.3 (DFT B3PW91/TZVP), 12.8 kJ/mol (DFT OPBE/TZVP), and 39.6 (DFT M062X/Def2TZVP) more ground state energy. A similar situation also occurs in the case of Fe(N

_{13}), where, according to the data of each of these methods, the nearest excited state is a spin quartet whose energy exceeds the energy of the ground state by 160.6, 190.1, and 174.0 kJ/mol, respectively. Images of the highest occupied and lowest vacant molecular orbitals (HOMO and LUMO, respectively) for the compounds under consideration, obtained by each of these three DFT methods above, are presented in Figure 4 and Figure 5.

_{f}H

^{0}, S

^{0}, and Δ

_{f}G

^{0}) are presented in Table 3. As can be seen from it, each of these parameters is positive. According to canons of thermodynamics, none of them can be obtained from the simple substances formed by chemical elements in their compositions (i.e., N and corresponding M). Nevertheless, according to the data obtained as a result of the quantum-chemical calculation carried out by us, the molecular structures of the given compounds and the full totality of their geometric parameters can be realized as a single whole. Thus, it can be argued that they are capable of existence, at least in the gas phase as individual molecules. It should be noted that, according to data of each of the DFT methods indicated above, Δ

_{f}H

^{0}and Δ

_{f}G

^{0}values for Fe(N

_{13}) are greater than for Mn(N

_{13}), whereas, for S

^{0}values, an inverse ratio takes place (Table 2). It should be noted that the calculation of Δ

_{f}H

^{0}, S

^{0}, and Δ

_{f}G

^{0}parameters using DFT M062/Def2TZVP was considered inappropriate by us, because M062X is a global hybrid functional with 54% HF exchange, and it is the top performer within the 06 functionals for main group thermochemistry, kinetics, and non-covalent interactions; however, it cannot be used for cases where multi-reference species are or might be involved, such as in transition metal thermochemistry and organometallics [34]).

_{13}) and Fe(N

_{13}) compounds studied by us in this work with similar data of the Mn(N

_{12}) and Fe(N

_{12}) compounds close to them in composition, which were presented in our work [1] and its Supplementary Materials and also calculated with DFT B3PW91/TZVP, DFT M06/TZVP, and DFT OPBE/TZVP. A comparison of these data shows that the key fragment of the M(N

_{13}) molecular structure, namely, the N12 macrocycle, as a whole, does not undergo noticeable changes compared to that in the M(N

_{12}) molecular structure. What has just been said also applies to the degree of deviation of the MN4 groupings from coplanarity, which in M(N

_{13}), for any of the two M we considered, and in the framework of any of the three variants of the DFT method used in the calculation, is only slightly greater than in the corresponding compound M(N

_{12}) (in particular, within the framework of the DFT B3PW91/TZVP method, in the case of Fe(N

_{13}), it is 49.2°, while in the case of Fe(N

_{12}), it is 47.1°). This slight change can be attributed to the fact that the formation of a rather short triple bond by the M atom with the “axial” nitrogen atom should additionally “raise” the M atom above the plane of the four N atoms that form chemical bonds with it.

## 4. Conclusions

_{13}) and Fe(N

_{13}), containing a cyclic group of twelve nitrogen atoms and an “axial” N atom bonded to the Mn or Fe atom via a triple bond. At the same time, each of these three methods testifies to the impossibility of the existence of compounds of the M(N

_{13}) type for all other 3d elements. The results of calculating the molecular structure of both these compounds, obtained by the above variants of the DFT method, are in good agreement with each other not only qualitatively, but also quantitatively. Both of these compounds have a tetragonal-pyramidal structure of the MN4 group (M = Mn, Fe) with a very significant (more than 45°) deviation from coplanarity; however, the grouping of four N4 nitrogen atoms bonded to the M atom in any of these compounds is strictly planar. However, the 12-membered macrocycles formed by nitrogen atoms in both Mn(N

_{13}) and Fe(N

_{13}) are non-coplanar, with very significant deviations from coplanarity. Comparison of the calculation data for the parameters of molecular structures and the standard thermodynamic parameters of the M(N

_{13}) (M = Mn, Fe) compounds we considered with the analogous parameters of the M(N

_{12}) compounds of the same 3d elements, characterized in our previous article [1], allows us to note quite a significant similarity between them.

_{f}H

^{0}and Δ

_{f}H

^{0}values, as a rule, are 1500 kJ/mol or more, depending on the DFT method used), and if successful in obtaining them, they will undoubtedly find some practical application, at least in the capacity indicated.

## Supplementary Materials

_{13}) calculation by B3PW91/TZVP method; Mn(N

_{13}) calculation by M06/TZVP method; Mn(N

_{13}) calculation by OPBE/TZVP method; Mn(N

_{13}) calculation by M062X/Def2TZVP method; Fe(N

_{13}) calculation by B3PW91/TZVP method; Fe(N

_{13}) calculation by M06/TZVP method; Fe(N

_{13}) calculation by OPBE/TZVP method; Fe(N

_{13}) calculation by M062X/Def2TZVP method.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mikhailov, O.V.; Chachkov, D.V. Twelve-Nitrogen-Atom Cyclic Structure Stabilized by 3d-Element Atoms: Quantum Chemical Modeling. Int. J. Mol. Sci.
**2022**, 23, 6560. [Google Scholar] [CrossRef] [PubMed] - Klapötke, T.M.; Harcourt, R.D. The interconversion of N
_{12}to N_{8}and two equivalents of N_{2}. J. Mol. Struct. (Theochem)**2001**, 541, 237–242. [Google Scholar] [CrossRef] - Olah, G.A.; Prakash, G.K.S.; Rasul, G. N
_{62}^{+}and N_{42}^{+}Dications and Their N_{12}and N_{10}Azido Derivatives: DFT/GIAO-MP2 Theoretical Studies. J. Am. Chem. Soc.**2001**, 123, 3308–3310. [Google Scholar] [CrossRef] [PubMed] - Li, Q.S.; Zhao, J.F. Theoretical Study of Potential Energy Surfaces for N
_{12}Clusters. J. Phys. Chem. A**2002**, 106, 5367–5372. [Google Scholar] [CrossRef] - Bruney, L.Y.; Bledson, T.M.; Strout, D.L. What Makes an N
_{12}Cage Stable? Inorg. Chem.**2003**, 42, 8117–8120. [Google Scholar] [CrossRef] - Samartzis, P.C.; Woodtke, A.M. All-nitrogen chemistry: How far are we from N
_{60}? Intern. Revs. Phys. Chem.**2006**, 25, 1952–2005. [Google Scholar] [CrossRef] - Greschner, M.J.; Zhang, M.; Majumdar, A.; Liu, H.; Peng, F.; Tse, J.S.; Yao, Y. A New Allotrope of Nitrogen as High-Energy Density Material. J. Phys. Chem. A
**2016**, 120, 2920–2925. [Google Scholar] [CrossRef] - Mikhailov, O.V.; Chachkov, D.V. Molecular structures and thermodynamics of stable N
_{4}, N_{6}and N_{8}neutral polynitrogens according to data of QCISD(T)/TZVP method. Chem. Phys. Lett.**2020**, 753, 137594. [Google Scholar] [CrossRef] - Chachkov, D.V.; Mikhailov, O.V. Tetra-, hexa-, and octanitrogen molecules: A quantum chemical design and thermodynamic properties. Russ. Chem. Bull.
**2020**, 69, 2067–2072. [Google Scholar] [CrossRef] - Mikhailov, O.V. Molecular and Electronic Structures of Neutral Polynitrogens: Review on the Theory and Experiment in 21st Century. Int. J. Mol. Sci.
**2022**, 23, 2841. [Google Scholar] [CrossRef] - Straka, M. N
_{6}ring as a planar hexagonal ligand in novel M(η6-N6) species. Chem. Phys. Lett.**2002**, 358, 531–536. [Google Scholar] [CrossRef] - Choi, C.; Yoo, H.-W.; Goh, E.M.; Cho, S.G.; Jung, Y.S. Ti(N
_{5})_{4}as a Potential Nitrogen-Rich Stable High-Energy Density Material. J. Phys. Chem. A**2016**, 120, 4249–4255. [Google Scholar] [CrossRef] [PubMed] - Brathwaite, A.D.; Abbott-Lyon, H.L.; Duncan, M.A. Distinctive Coordination of CO vs N
_{2}to Rhodium Cations: An Infrared and Computational Study. J. Phys. Chem. A**2016**, 120, 7659–7670. [Google Scholar] [CrossRef] - Ding, K.; Xu, H.; Yang, Y.; Li, T.; Chen, Z.; Ge, Z.; Zhu, W.; Zheng, W. Mass Spectrometry and Theoretical Investigation of VN
_{n}+ (n = 8, 9, and 10) Clusters. J. Phys. Chem. A**2018**, 122, 4687–4695. [Google Scholar] [CrossRef] [PubMed] - Bykov, M.; Bykova, E.; Koemets, E.; Fedotenko, T.; Aprilis, G.; Glazyrin, K.; Liermann, H.-P.; Ponomareva, A.V.; Tidholm, J.; Tasnadi, F.; et al. High-pressure synthesis of a nitrogen-rich inclusion compound ReN8 xN2 with conjugated polymeric nitrogen chains. Angew. Chem. Int. Ed.
**2018**, 57, 9048–9053. [Google Scholar] [CrossRef][Green Version] - Ding, K.; Chen, H.; Xu, H.; Yang, B.; Ge, Z.; Lu, C.; Zheng, W. Identification of octahedral coordinated ZrN12+ cationic clusters by mass spectrometry and structure searches. Dalton Trans.
**2021**, 50, 10187–10192. [Google Scholar] [CrossRef] - Zhao, Y.; Truhlar, D.G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: Two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor. Chem. Acc.
**2008**, 120, 215–241. [Google Scholar] [CrossRef][Green Version] - Becke, A.D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Revs. A
**1988**, 38, 3098–3100. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Wang, Y. Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys. Revs. B
**1996**, 54, 16533–16539. [Google Scholar] [CrossRef][Green Version] - Medvedev, M.G.; Bushmarinov, I.S.; Sun, J.; Perdew, J.P.; Lyssenko, K.A. Density functional theory is straying from the path toward the exact functional. Science
**2017**, 355, 49–52. [Google Scholar] [CrossRef] - Mikhailov, O.V.; Chachkov, D.V. DFT Quantum-Chemical Modeling Molecular Structures of Cobalt Macrocyclic Complexes with Porphyrazine or Its Benzo-Derivatives and Two Oxygen Acido Ligands. Int. J. Mol. Sci.
**2020**, 21, 9085. [Google Scholar] [CrossRef] - Hoe, W.M.; Cohen, A.; Handy, N.C. Assessment of a new local exchange functional OPTX. Chem. Phys. Lett.
**2001**, 341, 319–328. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed][Green Version] - Paulsen, H.; Duelund, L.; Winkler, H.; Toftlund, H.; Trautwein, A.X. Free Energy of Spin-Crossover Complexes Calculated with Density Functional Methods. Inorg. Chem.
**2001**, 40, 2201–2203. [Google Scholar] [CrossRef] - Swart, M.; Groenhof, A.R.; Ehlers, A.W.; Lammertsma, K. Validation of Exchange—Correlation Functionals for Spin States of Iron Complexes. J. Phys. Chem. A
**2004**, 108, 5479–5483. [Google Scholar] [CrossRef] - Swart, M.; Ehlers, A.W.; Lammertsma, K. Performance of the OPBE exchange-correlation functional. Mol. Phys.
**2004**, 102, 2467–2474. [Google Scholar] [CrossRef] - Swart, M. Metal–ligand bonding in metallocenes: Differentiation between spin state, electrostatic and covalent bonding. Inorg. Chim. Acta
**2007**, 360, 179–189. [Google Scholar] [CrossRef] - Møller, C.; Plesset, M.S. Note on an approximation treatment for many-electron systems. Phys. Rev.
**1934**, 46, 618–622. [Google Scholar] [CrossRef][Green Version] - Head-Gordon, M.; Pople, J.A.; Frisch, M.J. MP2 energy evaluation by direct methods. Chem. Phys. Lett.
**1988**, 153, 503–506. [Google Scholar] [CrossRef] - Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G.A. Gaussian 09, Revision A.01; Gaussian, Inc.: Wallingford, UK, 2009; Available online: https://www.scienceopen.com/document?vid=7625a2b3-85a4-4746-8a93-fb3335021944 (accessed on 1 January 2023).
- Cioslowski, J. A New Population Analysis Based on Atomic Polar Tensors. J. Am. Chem. Soc.
**1989**, 111, 8333–8336. [Google Scholar] [CrossRef] - Weinhold, F.; Landis, C.R.; Glendening, E.D. What is NBO analysis and how is it useful? Int. Rev. Phys. Chem.
**2016**, 35, 399–440. [Google Scholar] [CrossRef] - Ochterski, J.W. Thermochemistry in Gaussian; Gaussian, Inc.: Wallingford, CT, USA, 2000. [Google Scholar]
- Mardirossian, N.; Head-Gordon, M. Thirty years of density functional theory in computational chemistry: An overview and extensive assessment of 200 density functionals. Mol. Phys.
**2017**, 115, 2315–2372. [Google Scholar] [CrossRef][Green Version]

**Figure 3.**Molecular structures of the Mn(N

_{13}) and Fe(N

_{13}) compounds obtained as a result of DFT B3PW91/TZVP quantum-chemical calculation.

**Figure 4.**The pictures of HOMO and LUMO in the Mn(N

_{13}) obtained according to the DFT D3PW91/TZVP, DFT OPBE/TZVP, DFT M06/TZVP, and DFT M062/Def2TZVP methods. The energy values of the given MOs (in brackets) are expressed in eV.

**Figure 5.**The pictures of HOMO and LUMO in the Fe(N

_{13}) obtained according to the DFT D3PW91/TZVP, DFT OPBE/TZVP, DFT M06/TZVP, and DFT M062/Def2TZVP method. The energy values of the given MOs (in brackets) are expressed in eV.

**Table 1.**Key parameters of molecular structures of Mn(N

_{13}) and Fe(N

_{13}) compounds calculated by DFT B3PW91/TZVP, OPBE/TZVP, M06/TZVP, and M062X/Def2TZVP levels.

Mn(N_{13}) | Fe(N_{13}) | |||||||
---|---|---|---|---|---|---|---|---|

Structural Parameter | B3PW91/ TZVP | OPBE/ TZVP | M06/ TZVP | M062X/ Def2TZVP | B3PW91/ TZVP | OPBE/ TZVP | M06/ TZVP | M062X/ Def2TZVP |

M–N bond lengths in the MN_{4} chelate node, pm | ||||||||

M1N1 | 188.4 | 188.5 | 190.7 | 191.1 | 183.0 | 182.6 | 184.5 | 184.9 |

M1N4 | 188.4 | 188.5 | 189.3 | 191.1 | 183.0 | 182.6 | 183.1 | 188.8 |

M1N7 | 188.4 | 188.5 | 189.3 | 191.1 | 183.0 | 182.6 | 183.1 | 188.8 |

M1N10 | 188.4 | 188.5 | 190.7 | 191.1 | 183.0 | 182.6 | 184.5 | 184.9 |

M–N bond lengths between M and nitride N atom, pm | ||||||||

M1N13 | 151.0 | 152.5 | 150.9 | 147.3 | 150.9 | 151.5 | 150.0 | 154.4 |

Nitrogen-nitrogen bond lengths in macrocycle, pm | ||||||||

N1N2 | 134.9 | 134.9 | 138.4 | 142.1 | 134.5 | 134.9 | 138.8 | 136.2 |

N2N3 | 127.2 | 128.2 | 125.9 | 123.4 | 127.5 | 128.2 | 125.8 | 123.8 |

N3N4 | 134.9 | 134.9 | 134.0 | 142.1 | 134.5 | 134.9 | 134.0 | 142.9 |

N4N5 | 134.9 | 134.9 | 137.6 | 129.9 | 134.5 | 134.9 | 137.6 | 130.0 |

N5N6 | 127.2 | 128.2 | 124.6 | 132.3 | 127.5 | 128.2 | 124.6 | 131.3 |

N6N7 | 134.9 | 134.9 | 137.6 | 130.0 | 134.5 | 134.9 | 137.6 | 130.0 |

N7N8 | 134.9 | 134.9 | 134.0 | 142.1 | 134.5 | 134.9 | 134.0 | 142.9 |

N8N9 | 127.2 | 128.2 | 125.9 | 123.4 | 127.5 | 128.2 | 125.8 | 123.8 |

N9N10 | 134.9 | 134.9 | 138.4 | 142.1 | 134.5 | 134.9 | 138.8 | 136.2 |

N10N11 | 134.9 | 134.9 | 132.0 | 129.9 | 134.5 | 134.9 | 131.4 | 137.9 |

N11N12 | 127.2 | 128.2 | 129.4 | 132.3 | 127.5 | 128.2 | 129.9 | 124.2 |

N12N1 | 134.9 | 134.9 | 132.0 | 130.0 | 134.5 | 134.9 | 131.4 | 137.9 |

Bond angles in the MN_{4} grouping, deg | ||||||||

N1M1N4 | 76.3 | 76.0 | 76.0 | 74.6 | 77.7 | 77.7 | 77.4 | 76.2 |

N4M1N7 | 76.3 | 76.0 | 75.7 | 75.6 | 77.7 | 77.7 | 77.1 | 78.0 |

N7M1N10 | 76.3 | 76.0 | 76.0 | 74.6 | 77.7 | 77.7 | 77.4 | 76.2 |

N10M1N1 | 76.3 | 76.0 | 76.3 | 75.6 | 77.7 | 77.7 | 78.0 | 76.0 |

Bond angles sum (BAS), deg | 305.2 | 304.0 | 304.0 | 300.4 | 310.8 | 310.8 | 309.9 | 306.4 |

Deviation from coplanarity, deg | 54.8 | 56.0 | 56.0 | 59.6 | 49.2 | 49.2 | 50.1 | 53.6 |

Non-bond angles in the MN_{4} grouping, deg | ||||||||

N1N4N7 | 90.0 | 90.0 | 90.4 | 90.0 | 90.0 | 90.0 | 90.5 | 88.8 |

N4N7N10 | 90.0 | 90.0 | 90.4 | 90.0 | 90.0 | 90.0 | 90.5 | 88.8 |

N7N10N1 | 90.0 | 90.0 | 89.6 | 90.0 | 90.0 | 90.0 | 89.5 | 91.2 |

N10N1N4 | 90.0 | 90.0 | 89.6 | 90.0 | 90.0 | 90.0 | 89.5 | 91.2 |

Non-bond angles sum (NBAS), deg | 360.0 | 360.0 | 360.0 | 360.0 | 360.0 | 360.0 | 360.0 | 360.0 |

Deviation from coplanarity, deg | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Bond angles in 5-membered cycles, deg | ||||||||

M1N1N2 | 117.8 | 118.2 | 117.4 | 117.7 | 117.8 | 118.1 | 117.4 | 119.1 |

N1N2N3 | 113.1 | 112.6 | 111.3 | 112.4 | 112.3 | 112.0 | 110.4 | 114.9 |

N2N3N4 | 113.0 | 112.6 | 115.4 | 112.4 | 112.3 | 112.0 | 114.5 | 110.1 |

N3N4M1 | 117.8 | 118.2 | 118.0 | 117.7 | 117.8 | 118.1 | 118.2 | 116.9 |

M1N4N5 | 117.8 | 118.2 | 118.0 | 119.1 | 117.8 | 118.1 | 118.3 | 116.9 |

N4N5N6 | 113.0 | 112.6 | 113.0 | 113.1 | 112.3 | 112.0 | 112.1 | 114.1 |

N5N6N7 | 113.1 | 112.6 | 113.0 | 113.1 | 112.3 | 112.0 | 112.1 | 114.1 |

N6N7M1 | 117.8 | 118.2 | 118.0 | 119.1 | 117.8 | 118.1 | 118.3 | 116.9 |

M1N7N8 | 117.8 | 118.2 | 118.0 | 117.7 | 117.8 | 118.1 | 118.2 | 116.8 |

N7N8N9 | 113.1 | 112.6 | 115.4 | 112.4 | 112.3 | 112.0 | 114.5 | 110.1 |

N8N9N10 | 113.0 | 112.6 | 111.3 | 112.4 | 112.3 | 112.0 | 110.4 | 114.9 |

N9N10M1 | 117.8 | 118.2 | 117.4 | 117.7 | 117.8 | 118.1 | 117.4 | 119.1 |

M1N10N11 | 117.8 | 118.2 | 117.7 | 119.1 | 117.8 | 118.1 | 117.7 | 119.1 |

N10N11N12 | 113.0 | 112.6 | 113.7 | 113.1 | 112.3 | 112.0 | 112.9 | 112.0 |

N11N12N1 | 113.1 | 112.6 | 113.7 | 113.1 | 112.3 | 112.0 | 112.9 | 112.0 |

N12N1M1 | 117.8 | 118.2 | 117.7 | 119.1 | 117.8 | 118.1 | 117.7 | 119.1 |

N–M–N bond lengths between N donor atom, M, and nitride N atom, pm | ||||||||

N1M1N13 | 119.1 | 119.5 | 120.4 | 120.5 | 117.4 | 117.4 | 117.3 | 124.0 |

N4M1N13 | 119.1 | 119.5 | 118.5 | 120.5 | 117.4 | 117.4 | 118.1 | 113.5 |

N7M1N13 | 119.1 | 119.5 | 118.5 | 120.5 | 117.4 | 117.4 | 118.1 | 113.5 |

N10M1N13 | 119.1 | 119.5 | 120.4 | 120.5 | 117.4 | 117.4 | 117.3 | 124.0 |

APT Analysis Data | ||||||||

M | Effective charge on an atom, units electron charge ē | |||||||

M1 | N1 (N10) | N2 (N9) | N5 (N6) | N4 (N7) | N3 (N8) | N11(N12) | N13 | |

Mn | +0.377 | −0.151 (−0.151) | +0.079 (+0.079) | +0.025 (+0.024) | −0.151 (−0.151) | +0.079 (+0.079) | +0.025 (+0.024) | −0.185 |

Fe | +0.115 | −0.209 (−0.209) | +0.131 (+0.131) | −0.014 (−0.014) | −0.050 (−0.050) | −0.050 (−0.050) | +0.029 (+0.029) | +0.211 |

NBO Analysis Data | ||||||||

M | Effective charge on an atom, units electron charge ē | |||||||

M1 | N1 (N10) | N2 (N9) | N5 (N6) | N4 (N7) | N3 (N8) | N11(N12) | N13 | |

Mn | −0.065 | −0.113 (−0.113) | +0.057 (+0.057) | +0.012 (+0.012) | −0.113 (−0.113) | +0.057 (+0.057) | +0.012 (+0.012) | +0.239 |

Fe | +0.056 | −0.167 (−0.167) | +0.075 (+0.075) | +0.023 (+0.023) | −0.040 (−0.040) | +0.004 (+0.004) | +0.021 (+0.021) | +0.111 |

**Table 3.**Standard thermodynamic parameters of Mn(N

_{13}) and Fe(N

_{13}) calculated by various methods.

Compound | Calculation Method | Δ_{f}H^{0}, kJ/mol | S^{0}, J/mol∙K | Δ_{f}G^{0}, kJ/mol |
---|---|---|---|---|

Mn(N_{13}) | DFT B3PW91/TZVP | 1704.9 | 424.4 | 1956.4 |

DFT OPBE/TZVP | 1411.2 | 429.5 | 1661.2 | |

DFT M06/TZVP | 1834.5 | 425.7 | 2085.6 | |

Fe(N_{13}) | DFT B3PW91/TZVP | 1821.2 | 422.7 | 2071.8 |

DFT OPBE/TZVP | 1486.3 | 416.8 | 1738.6 | |

DFT M06/TZVP | 1980.1 | 419.8 | 2231.6 |

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## Share and Cite

**MDPI and ACS Style**

Mikhailov, O.V.; Chachkov, D.V.
Molecular Structure of M(N_{13}) Compounds with 12-Membered Nitrogen-Containing Cycle and Axial Nitrogen Atom (M = Mn, Fe): Quantum-Chemical Design by DFT Method. *Quantum Rep.* **2023**, *5*, 282-293.
https://doi.org/10.3390/quantum5010019

**AMA Style**

Mikhailov OV, Chachkov DV.
Molecular Structure of M(N_{13}) Compounds with 12-Membered Nitrogen-Containing Cycle and Axial Nitrogen Atom (M = Mn, Fe): Quantum-Chemical Design by DFT Method. *Quantum Reports*. 2023; 5(1):282-293.
https://doi.org/10.3390/quantum5010019

**Chicago/Turabian Style**

Mikhailov, Oleg V., and Denis V. Chachkov.
2023. "Molecular Structure of M(N_{13}) Compounds with 12-Membered Nitrogen-Containing Cycle and Axial Nitrogen Atom (M = Mn, Fe): Quantum-Chemical Design by DFT Method" *Quantum Reports* 5, no. 1: 282-293.
https://doi.org/10.3390/quantum5010019