# Manganese(I) and Rhenium(I) Chelate Complexes with 2-Azabutadienes (RS)2C=C(H)-N=CPh2: Topological AIM Bonding Analysis and Molecular Structure of fac-MnBr(CO)3[(iPrS)2C=C(H)-N=CPh2]

Marek Kubicki, Abderrahim Khatyr, Michael Knorr

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Marek Kubicki, Abderrahim Khatyr, Michael Knorr. Manganese(I) and Rhenium(I) Chelate Complexes with 2-Azabutadienes (RS)2C=C(H)-N=CPh2: Topological AIM Bonding Analysis and Molecular Structure of fac-MnBr(CO)3[(iPrS)2C=C(H)-N=CPh2]. Chemistry, 2025, 7 (5), pp.145. 10.3390/chemistry7050145. [hal-05250693](https://hal.science/hal-05250693v1)

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# Manganese(I) and Rhenium(I) Chelate Complexes with 2-Azabutadienes (RS)₂C=C(H)-N=CPh₂: Topological AIM Bonding Analysis and Molecular Structure of *fac*-MnBr(CO)₃[($^{1}$PrS)₂C=C(H)-N=CPh₂]

Marek M. Kubicki <fnref n="1" />,* , Abderrahim Khatyr <fnref n="2" /> and Michael Knorr <fnref n="2" />,*

## Abstract

The thioether-functionalized 2-azabutadiene (¹PrS)₂C=C(H)-N=CPh₂ **L1** ligates to Mn(CO)₅Br to form the five-membered chelate compound *fac*-MnBr(CO)₃[($^{1}$PrS)₂C=C(H)-N=CPh₂] **MnPropBr**, whose crystal structure has been determined from X-ray diffraction data. In the crystal, different secondary intermolecular interactions, such as Br···HC and π···π, give rise to a supramolecular network. The electronic properties of the metal–ligand bonds in **MnPropBr** are similar to those of complex **MnPhBr** (with R = SPh instead of ¹PrS); this also applies to a series of structurally analogous *fac*-ReX(CO)₃[(RS)₂C=C(H)-N=CPh₂] (X = Cl, Br and I; R = S¹Pr, SPh and S†Bu) rhenium complexes and are discussed on the basis of QT-AIM (Quantum Theory of Atoms in Molecules) calculations. New bond length/electron density relationships are proposed for the metal–halide bonds, including, for the first time, complexes of one given metal and all three corresponding halides. In order to obtain a set of coherent data, three manganese complexes that belong to the family *fac*-MnX(CO)₃[N∩N] (X = Cl, Br and I; N∩N is a chelating ligand with two coordinating N atoms) were included in this study.

**Keywords:** crystal structure; halotricarbonyl complexes; manganese; rhenium; 2-azabutadiene; X-ray diffraction; Quantum Theory of Atoms in Molecules; bond length–electron density relationship; thioether

## 1. Introduction

The coordination chemistry of 1,4-diazabutadienes (α-diimines) and related N∩N chelate complexes [1–7] has been the topic of hundreds of articles and reviews since this type of ligand confers catalytic and rich photophysical properties. Furthermore, their potential in the domain of bioinorganic chemistry has emerged [8]. In contrast, both 1-azabutadienes R₂C=C(H)-CH=NR' and 2-azabutadienes R₂C=C(H)-N=CR'₂ (R, R' = Hal, alkyl, aryl), featuring a π-conjugated C=C(H)-C=N or C=C(H)-N=C array, are far less commonly used as ligands in coordination chemistry [9–13], although they have versatile potential for binding to metal centers through their olefinic C=C bond and imine-type N-donor ligands. Interestingly, the simplest 2-azabutadiene, H₂C=C(H)-N=CH₂, has been the object of several theoretical studies, and its occurrence has been spectroscopically detected in the atmosphere of Titan [14].

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Academic Editor: Marius Andruh

Received: 20 July 2025

Revised: 27 August 2025

Accepted: 28 August 2025

Published: 9 September 2025

**Citation:** Kubicki, M.M.; Khatyr, A.; Knorr, M. Manganese(I) and Rhenium(I) Chelate Complexes with 2-Azabutadienes (RS)₂C=C(H)-N=CPh₂: Topological AIM Bonding Analysis and Molecular Structure of *fac*-MnBr(CO)₃[($^{1}$PrS)₂C=C(H)-N=CPh₂]. *Chemistry* **2025**, *7*, 145.
https://doi.org/10.3390/chemistry7050145

**Copyright:** © 2025 by the authors.
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<footnote n="1">¹ Institut ICMUB CNRS UMR 6302, Université Bourgogne Europe, 9, Avenue Alain Savary, 21078 Dijon, France</footnote>

<footnote n="2">² Institut UTINAM UMR 6213 CNRS, Université Marie et Louis Pasteur, 16, Route de Gray, 25030 Besançon, France</footnote>

<footnote>* Correspondence: marek.kubicki@u-bourgogne.fr (M.M.K.); michael.knorr@univ-fcomte.fr (M.K.)</footnote>
In the past, our laboratories have achieved novel synthetic access to 4,4'-dichloro-1,1-diaryl-2-azabuta-1,3-dienes $Cl_2C=C(H)-N=CAr_2$ via 1,3 dipolar cycloaddition [15] and explored the potential of these dihalogenated compounds for the activation of vinylic Cl-C=C bonds (oxidative addition) across low-valent Pd and Pt compounds, yielding organometallic $\sigma$-alkenyl and $\mu$-vinylidene complexes [16]. Reaction of $Cl_2C=C(H)-N=CAr_2$ with thiolates NaSR also allowed for the preparation of a series of alky- and aryl-thioether-functionalized 2-azbutadienes of the type $Ar_2C=N-C(H)=C(SR)_2$ [17,18]. These latter compounds are valuable chelating ligands in coordination chemistry, since they can be chelated as $S \cap N$ donors to a variety of transition metals. Examples include the octahedral complex $[(OC)_4Mo({^1PrS})_2C=C(H)-N=CPh_2]$ [19], the crystallographically characterized complexes $[CdI_2({^1PrS})_2C=C(H)-N=CPh_2]$ and $[HgI_2({^1PrS})_2C=C(H)-N=CPh_2]$ and the dinuclear Cu(I) complex $[LCu(\mu_2-I_2)CuL]$ [20]. The presence of one aryl group could even be exploited for orthometallation reactions, allowing for the preparation of several square-planar cyclometalated Pd(II) and Pt(II) complexes $[(RS)_2C=C(H)-N=C(Ph)C_6H_4]MCl$ complexes [19]. In the context of our interest in the photophysical properties of transition metal complexes, we recently prepared an extended series of octahedral *fac*-[[(OC)₃ReX{[(RS)₂C=C(H)-N=CAr₂}]] (X = Cl, Br, I, triflate) complexes [21]. For comparison, we have also communicated very recently the crystal structures of two *fac*-MnBr(CO)₃[(ArS)₂C=C(H)-N=CPh₂] complexes, S,N-chelated by 2-azabutadiene ligands bearing aromatic -SAr thioether functions [22].

This latter investigation was also driven by the paucity of structural information on $[(OC)_3MnBrl(S \cap N)]$ complexes prepared by other research groups. A search of the CSD database (CSD release 2025 1.1) [23] reveals only three entries for related $[(OC)_3MnBr[(S \cap N)]$ complexes with a thioether function, namely bromo-tricarbonyl-(2-(phenylsulfanyl)-N-[2-(phenylsulfanyl)ethyl]ethan-1-amine)manganese A (CSD refcode DIXLEE), tricarbonyl-bromo-(2-(phenylthiomethyl)pyridine-N,S)manganese B (CSD ref code IHIMIV) and bromo-tris(carbonyl)-[2-(ethylsulfanyl)-N-[(quinolin-4-yl)methyl]ethan-1-amine]manganese C (CSD refcode DOKXAG) (Scheme 1) [24–26]. We found no entries for similar chloro- or iodo-manganese complexes. Interestingly, compounds A and C are also biologically active. Quite recently, the crystal structure of the chelate complex *fac*-[Mn(CO)₃Br{[(4-CH₃)C₆H₃N-2-NH(P(=S)Ph₂)]}] D, with diphenylphosphinesulfide coordination, was also reported [27]. For comparison, also the $\alpha$-diime complex bromo-tricarbonyl-(N,N'-dimesitylbutane-2,3-diimine)manganese E (CSD refcode MOZZOS) is shown in Scheme 1 [2].

<figure><img src="image_2.png" /><figcaption>**Scheme 1.** Examples of structurally characterized chelate complexes $[MnBr(CO)_3S \cap N]$ and $[(OC)_3MnBr[N \cap N]]$.</figcaption></figure>
In this article, we report the crystal structure of a Mn(I) complex bearing stronger electron-donating SPr<sup>i</sup> groups instead of SAr, namely *fac*-MnBr(CO)<sub>3</sub>[(iPrS)<sub>2</sub>C=C(H)-N=CPh<sub>2</sub>], thus complementing the family of Mn and Re *fac*-halotricarbonyl adducts with thioether-functionalized 2-azabutadienes. We describe therein its synthesis, crystal structure and the electronic characteristics of the Mn-ligand bonds. The Quantum Theory of Atoms in Molecules is employed for this purpose.

## 2. Experimental

Ligand L was prepared according to the procedure described in the literature [19]. *Preparation of the complex MnPropBr*: To a degassed solution of MnBr(CO)<sub>5</sub> (155 mg, 0.5 mmol) in CHCl<sub>3</sub> (5 mL), ligand L1 (0.55 mmol) was added. The resulting mixture was gently refluxed for 24 h, and the reaction was followed by IR spectroscopy. The volume of red solution was reduced to ca. 2 mL under reduced pressure. Layering with hexane yielded red needles after several days at 5 °C. Yield: 61%. Anal. Calc. for C<sub>24</sub>H<sub>25</sub>BrMnNO<sub>3</sub>S<sub>2</sub> (M.W. = 574.44 g.mol<sup>-1</sup>): C, 50.18; H, 4.38; N, 2.43; S, 11.16%. Found: C, 50.49; H, 4.10; N, 2.19, S, 10.99%. <sup>1</sup>H-NMR (CDCl<sub>3</sub>) at 298 K: δ 7.28-7.84 (m, CH Ar and vinylic H), 3.88 ppm (m, broadened, 1H, CH(CH<sub>3</sub>)<sub>2</sub>), 3.47 ppm (m, broadened, 1H, CH(CH<sub>3</sub>)<sub>2</sub>), 1.32-1.61 (overlapped signals, 12H, 2 CH(CH<sub>3</sub>)<sub>2</sub>) ppm. IR (CH<sub>2</sub>Cl<sub>2</sub>): ν(CO): 2032vs, 1956s, 1916s cm<sup>-1</sup>.

### 2.1. Apparatus

The <sup>1</sup>H spectra were recorded on a Bruker AC 300 (Bruker, Wissembourg, France) at 300 MHz, using CDCl<sub>3</sub> as a solvent. The infrared spectrum was recorded on a NICOLET Avatar 320 in solution. The UV-visible absorption spectrum was measured on a VARIAN-Cary 100 spectrophotometer.

### 2.2. X-Ray Diffraction

The X-ray data collection for **MnPropBr** was performed on a Enraf Nonius KappaCCD with the KappaCCD supergui [28] program. These data were reduced with the Denzo and Scalpack programs [29]. No absorption correction was applied to the collected data. The structure was solved with SHELXS97 [30]. The structure was refined with SHELXL [31]. All non-hydrogen atoms were refined with anisotropic temperature factors. The hydrogen atoms were placed in calculated positions and refined in an isotropic riding model. The intermolecular interactions were explored with the MERCURY program [32], and the molecular structure was plotted with ORTEP3 [33]. Crystal data and refinement details are presented in Table 1.

<table><caption>**Table 1.** Crystal data, data collection and structure refinement for **MnPropBr**.</caption><thead><tr><th>Formula</th><th>C<sub>24</sub>H<sub>25</sub>BrMnNO<sub>3</sub>S<sub>2</sub></th></tr></thead><tbody><tr><td>Formula weight</td><td>574.41</td></tr><tr><td>Temperature/K</td><td>120(2)</td></tr><tr><td>Wavelength/Å</td><td>0.71073</td></tr><tr><td>Crystal system</td><td>orthorhombic</td></tr><tr><td>Space group</td><td>Pbca</td></tr><tr><td>a/Å</td><td>13.6907(3)</td></tr><tr><td>b/Å</td><td>18.7839(4)</td></tr><tr><td>c/Å</td><td>19.6170(4)</td></tr><tr><td>a/°</td><td>90.0</td></tr></tbody></table>
<table><caption>**Table 1. Cont.**</caption><thead><tr><th>Formula</th><th>C<sub>24</sub>H<sub>25</sub>BrMnNO<sub>3</sub>S<sub>2</sub></th></tr></thead><tbody><tr><td>β/°</td><td>90(0)</td></tr><tr><td>γ/°</td><td>90.0</td></tr><tr><td>Volume/Å<sup>3</sup></td><td>5044.80(19)</td></tr><tr><td>Z</td><td>8</td></tr><tr><td>ρ calc. g/cm<sup>3</sup></td><td>1.515</td></tr><tr><td>μ/mm<sup>-1</sup></td><td>2.298</td></tr><tr><td>F(000)</td><td>2336</td></tr><tr><td>Crystal size/mm</td><td>0.10 × 0.10 × 0.07</td></tr><tr><td>θ range for data collection/°</td><td>2.08 to 27.47</td></tr><tr><td>Index ranges</td><td>-17 ≤ h ≤ 17,<br>-24 ≤ k ≤ 24,<br>-25 ≤ l ≤ 25</td></tr><tr><td>Reflections collected</td><td>10808</td></tr><tr><td>Independent reflections</td><td>5750 [R(int) = 0.0596]</td></tr><tr><td>Refl. greater [I > 2σ(I)]</td><td>3922</td></tr><tr><td>Absorption correction</td><td>none</td></tr><tr><td>Transmission max</td><td>0.999</td></tr><tr><td>Transmission min</td><td>0.704</td></tr><tr><td>Refinement method</td><td>Full-matrix least squares on F<sup>2</sup></td></tr><tr><td>Data/restraints/parameters</td><td>5750/0/349</td></tr><tr><td>Goodness-of-fit on F<sup>2</sup></td><td>1.017</td></tr><tr><td>Final R indexes [I > 2σ(I)]</td><td>R1 = 0.0427,<br>wR2 = 0.0726</td></tr><tr><td>R indexes (all data)</td><td>R1 = 0.0819,<br>wR2 = 0.0833</td></tr><tr><td>Largest diff. peak<br>and hole/e.Å<sup>-3</sup></td><td>0.437 and -0.560</td></tr></tbody></table>

The crystallographic data have been deposited at the Cambridge Crystallographic Data Centre: deposition number CCDC 941427 contains detailed crystallographic data for this publication. These data may be obtained free of charge from the Cambridge Crystallographic Data Center through www.ccdc.cam.ac.uk/data_request/cif, accessed on 27 August 2025.

## 2.3. TD-DFT Computing and AIM Analysis

The wave functions for the AIM topological analyses were calculated using the Gaussian 09 package [34] with the hybrid B3LYP functional. The 6-311G(d,p) standard basis set was applied for all atoms except the manganese and rhenium atoms. Since the AIM theory should preferentially use the all-electron and not the ECP (effective core potential) basis sets, the double zeta polarized DZP-DKH basis set of Jörge was applied for the Mn and Re atoms [35]. Because we were interested in a comparative interpretation of metric parameters found in the X-ray structures, these calculations were performed on the molecular geometries really present in the crystal structures. Topological parameters were calculated using the AIMALL packages [36]. GausSum [37] was used for analysis of the TD-DFT files. The IR spectrum of MnPropBr was calculated using a triple dzeta def2-TZP basis set. The
full TD-DFT method without TDA was used for the calculation of the UV/vis spectrum of MnPropBr. The corresponding data are given in Table S1. The calculations were performed either at the Centre of Calculations of the University Burgundy Europe or on a local PC computer with GW09-W [34].

**3. Results and Discussion**

**3.1. Synthesis and Characterization**

The title complex *fac*-MnBr(CO)₃[($^{1}$PrS)₂C=C(H)-N=CPh₂] **1** (further abbreviated as **MnPropBr**) was obtained in a carbonyl substitution reaction via treatment of [Mn(CO)₅Br] with an equimolar amount of the azabutadiene ligand **L1** in refluxing chloroform solution. Upon cooling, small red crystals of air-stable **MnPropBr** were isolated (Scheme 2). For comparison, Scheme 2 also illustrates the preparation of derivatives **MnPhBr** and **MnTolBr** and that of the Re(I) analog **RePropBr**, whose synthesis and crystal structures were reported previously [21,22].

<figure><img src="image_3.png" /><figcaption>**Scheme 2.** Synthesis of the *S*,*N*-chelate complex **MnPropBr**, its *S*-aryl derivatives **MnPhBr** and **MnTolBr** and that of its Re(I) analog **RePropBr**.</figcaption></figure>

There are three intense bands corresponding to ν(CO) stretching vibrations (2032, 1956 and 1916 cm⁻¹) in the experimental IR spectrum of the **MnPropBr** complex. The calculated IR spectrum (gas phase) in the corresponding region is shown in Figure 1a. The presence of three strong bands (2020, 1968 and 1920 cm⁻¹) is fully compatible with trivial symmetry (C₁ point group) of this and of other *fac*-MX(CO)₃(*S*,*N*-chelate) complexes and matches with the experimental IR spectrum recorded in CH₂Cl₂, (Figure S1).

The calculated UV-vis spectrum (Figure 1b) exhibits two bands at 310 and 383 and a weak broad band near 530 nm. This last band explains the red color of the complex and originates mainly from two transitions: one at 539 nm occurs from HOMO-1 (62%) and HOMO (34%) into the LUMO and the second one at 533 nm from HOMO-1 (35%) and HOMO (59%) also into the LUMO. All twenty calculated transitions are described in Table S1 of the Supplementary Materials. Both occupied MOs are localized roughly in the Br-Mn-CO part of the molecule, whereas the LUMO is localized in the main chain of the
azabutadiene ligand (Figure 2). Moreover, its shape is very close to that of the free ligand L ($^{1}$PrS)₂C=C(H)-N=CPh₂.

<figure><img src="image_4.png" /><figcaption>**Figure 1.** (a) Calculated IR spectrum of **MnPropBr** scaled by a factor of 0.96; (b) calculated UV-vis spectrum of **MnPropBr**.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Molecular orbitals of **MnPropBr** implicated in electronic transition responsible of broad band near 530 nm. (a) HOMO-1, (b) HOMO, (c) LUMO and (d) LUMO of the free ($^{1}$PrS)₂C=C(H)-N=CPh₂ ligand molecule.</figcaption></figure>

## 3.2. X-Ray Structural Studies

### 3.2.1. Molecular Structure

The crystal structure of **MnPropBr** was determined by X-ray diffraction. The complex molecule is shown in Figure 3. Like its S-aryl analogs **MnPhBr** and **MnTolBr** [22], **MnPropBr** exhibits a *fac*-octahedral geometry in which three *facial* carbonyl groups, one bidentate S,N-azabutadiene and one bromide form the coordination sphere around the central Mn(I) atom.

The metric parameters found in **MnPropBr** are very close to those reported for its thio-aryl analogs **MnPhBr** and **MnTolBr** [22]. They are also very close to other similar complexes bearing the S,N-chelating ligands: bromo-tricarbonyl-(2-phenylsulfanyl)-N-[2-(phenylsulfanyl)-ethane-1-amine]manganese(I) DIXLEE (Mn-Br 2.5288(5) Å, Mn-S 2.3747(6) Å, Mn-N 2.125(2) Å, S-Mn-N 85.06(5)°) [24] and bromo-tricarbonyl-[2-(phenylthiomethyl)pyridine-N,S]manganese(I) IHIMIV (Mn-Br 2.5381(5) and 2.5369(5) Å, Mn-S 2.3449(7) and 2.3468(7) Å, Mn-N 2.083(2) and 2.082(2) Å, S-Mn-N 83.25(5)° and 83.39(5)°) [26]. This indicates high rigidity of the metal coordination sphere in this family of complexes. The five-membered metallacycle M–S–C–C–N is almost planar, with the highest deviation from the mean plane observed for the N atom (0.10 Å).
<figure><img src="image_6.png" /><figcaption>**Figure 3.** ORTEP (50% probability level) of the molecular structure of the complex **MnPropBr**. Selected bond lengths (Å) and angles (°): Mn–Br 2.5348(6), Mn–S 2.3201(9), Mn–N 2.137(2), Mn–C17 1.796(3), Mn–C16 1.813(3), Mn–C18 1.801(3), N–Mn–S1 82.95(7).</figcaption></figure>

### 3.2.2. Supramolecular Features

Several intermolecular non-covalent interactions are present in the crystal structure of **MnPropBr**. The Br···H hydrogen bonds form supramolecular ribbons (Figure 4) whereas the van der Waals (vdW) H···H interactions connect the molecules from adjacent ribbons (Figure 5).

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Fragment of the chains formed through intermolecular Br···H interactions in the structure of **MnPropBr**. Br···H19 2.999 (−0.051)Å, Br···H22 2.864 (−0.186)Å. Shortening with respect to the sum of vdW radii. Symmetry operation #: −1/2+x, 1/2-y, 1-z.</figcaption></figure>

There are four phenyl groups in the central part of Figure 5; two of them are parallel. A different view (Figure S2) strongly suggests the presence of supplementary interactions between the π-electron clouds of two neighboring parallel phenyl rings (Figure 6). The shortest carbon–carbon distance of 3.494 Å is longer than the sum of “standard” Bondi’s van der Waals radii of carbon atoms (rC<sub>vdW</sub> = 1.7 Å) [38], but it is shorter than that of the value more recently proposed by Alvarez (1.77 Å) [39]. The question of its presence or absence will be subjected to an AIM analysis of the electron density in the concerned region and discussed later.
<figure><img src="image_8.png" /><figcaption>**Figure 5.** The H···H interactions interconnecting the molecules from neighbor ribbons. H2···H2O2# = 2.267 (−0.133)Å.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Possible π···π interaction between two neighboring molecules in the structure of MnPropBr. Dimeric couple over the symmetry center at 1,0,1/2 in the *Pbca* space group.</figcaption></figure>

### 3.3. Electronic Features and AIM Bonding Analysis

In order to obtain information about the nature of bonding in complexes **MnPropBr** and **MnPhBr**, we calculated the wave functions on molecular geometries present in their X-ray structures. They were further used in QT-AIM analysis (Quantum Theory of Atoms in Molecules), as developed by Bader [40,41]. Topological analysis of electron density is a powerful tool for the study of different weak interatomic interactions (hydrogen bonding, [42,43] Van der Waals, σ–π and π–π [44–46]), as well for studying the nature of metal–metal [47,48] and metal–ligand [49,50] interactions, in addition to the classical case of covalent bonds. Various topological properties at *Bond Critical Points* (BCPs defined with (rank, signature) code (3, −1)) are used for these purposes. The most frequently used topological parameters are the electron charge density (ρ, e⁻/a₀³) and the Laplacian of electron density (∇²ρ, e⁻/a₀⁵). Deeper insight into the electronic structure is obtained from the values of local kinetic, potential and total energy densities (G, V and H, Ha/a₀³), as well as from derived quantities like the dimensionless ratio |V|/G and the bond degree
(H/ρ, Ha/e⁻) at the BCP. The potential energy density V (always negative) is related to the covalent contribution to the bond, whereas the kinetic G (always positive) one is related to its non-covalent part. Thus, the total energy density H (sum of potential and kinetic contributions), resulting from competition between V and G, is negative for predominantly covalent bonds, while the positive value of H indicates a non-covalent nature of the bond. The integrations of electron density over atomic basins allow, among other properties, the calculations of atomic charges and of delocalization indexes (DI<sub>A-B</sub>), which indicate the number of electron pairs localized between atoms A and B and are thus roughly related to the classical bond orders. The chemical interacting systems are classified within the QT-AIM frame into three main types depending on the values of the parameters mentioned above [42,51]. These are as follows:

(1) Shared-shared (SS), covalent, $∇²ρ < 0$, $H < 0$ and $|V|/G > 2$;

(2) Shared-closed (SC), transit or intermediate, $∇²ρ > 0$, $H < 0$ and $1 < |V|/G < 2$;

(3) Closed-closed (CC), hydrogen bonds, $∇²ρ > 0$, $H > 0$ and $|V|/G < 1$.

The molecular graphs built on bond paths for **MnPropBr** and **MnPhBr** are shown in Figures 7 and 8. The molecular graph of complex **MnPropBr** (Figure 7) shows the presence of six expected bond paths around the Mn atom with Br, S, N and three C atoms of carbonyl groups. Two intramolecular C–H···Br hydrogen bonds, as well as two C–H···π (C17 atom of CO group) and intramolecular π···π (carbonyl C16–phenyl C4) interactions, are also present therein.

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Molecular graph of the complex molecule in the crystal structure of **MnPropBr**. Colors of critical points: bond (BCP 3, −1)—red; ring (RCP 3, +1)—orange. Interatomic distances for non-covalent interactions (Å): Br···H11 3.021, Br···H243 3.309, C16···C4 2.966, C17···H5 2.759, C17···H203 2.563.</figcaption></figure>

The molecular graph of complex **MnPhBr** (Figure 8) exhibits the six Mn–ligand bond paths, but only one C–H···Br and one C–H···π interaction is retained here. Those involving the S<sup>i</sup>Pr groups in **MnPropBr** are absent for the SPh groups in **MnPhBr**. The intramolecular π···π (carbonyl C4–phenyl C38) is also present. It is worth noting that the C–H···X hydrogen bonds between the halide and one phenyl group of the imine part of the ligand are also present in all Re complexes (with the exception of **ReButBr**) (Figures S3–S7 in the Supporting Materials). On the other hand, and without exception the C–H···π(CO) and C···C π–π(CO···Ph) interactions are present in all Mn and Re complexes.
<figure><img src="image_11.png" /><figcaption>**Figure 8.** Molecular graph of the complex molecule in the crystal structure of **MnPhBr**. Colors of critical points (CP): bond (BCP 3, −1)—red; ring (RCP 3, +1)—orange. Interatomic distances for non-covalent interactions (Å): Br···H33 3.004, C38···C4 2.936, C5···H43 2.661.</figcaption></figure>

The topological parameters calculated on the geometries found in the crystal structures of **MnPropBr** and **MnPhBr** are gathered in Table 2. We mentioned in the Introduction that a series of *fac*-rhenium complexes with *S*,*N*-azabutadiene chelating ligands has been recently described by us [21]. We have carried out the wave function and QT-AIM calculations on these complexes. The obtained parameters are also included in Table 2. The molecular graphs of the rhenium complexes are given in Figures S2–S7 of the Supplementary Materials. To establish a correct correlation between the bond lengths and electron densities for the Mn–X bonds (*vide infra*), there was a need to introduce some new complexes with *N*,*N*-chelating ligands. Their topological parameters are included in Table 2, and their molecular graphs are included in the Supplementary Materials.

<table><caption>**Table 2.** Topological properties of bond critical points ((3, −1) BCP) in MX(CO)₃AzBuR complexes: *d* bond distance; *ρ* electron density; ∇²*ρ* Laplacian of *ρ*; V, G and H potential, kinetic and total energy densities; H/*ρ* pressure of total energy per electron (“bond degree”); DI delocalization index. All but *d* (interatomic distances) data are given in atomic units.</caption><thead><tr><th>BCP</th><th>d, Å</th><th>ρ, e<sup>-</sup>/a<sub>0</sub><sup>3</sup></th><th>∇<sup>2</sup>ρ, e<sup>-</sup>/a<sub>0</sub><sup>5</sup></th><th>|V|/G</th><th>H, Ha/a<sub>0</sub><sup>3</sup></th><th>|H|/ρ, Ha/e<sup>-</sup></th><th>DI</th></tr></thead><tbody><tr><td><strong>M-X</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MnPropBr</td><td>2.5348</td><td>0.0503</td><td>0.133</td><td>1.277</td><td>-0.01273</td><td>0.253</td><td>0.508</td></tr><tr><td>MnPhBr</td><td>2.5326</td><td>0.0507</td><td>0.132</td><td>1.280</td><td>-0.01278</td><td>0.252</td><td>0.509</td></tr><tr><td>RePropBr</td><td>2.6293</td><td>0.0605</td><td>0.152</td><td>1.195</td><td>-0.00921</td><td>0.152</td><td>0.605</td></tr><tr><td>RePhCl</td><td>2.4951</td><td>0.0667</td><td>0.199</td><td>1.144</td><td>-0.00840</td><td>0.126</td><td>0.593</td></tr><tr><td>RePhBr</td><td>2.6295</td><td>0.0605</td><td>0.152</td><td>1.195</td><td>-0.00921</td><td>0.152</td><td>0.608</td></tr><tr><td>RePhI</td><td>2.8174</td><td>0.0531</td><td>0.104</td><td>1.262</td><td>-0.00918</td><td>0.173</td><td>0.637</td></tr><tr><td>ReButBr</td><td>2.6208</td><td>0.0619</td><td>0.152</td><td>1.203</td><td>-0.00970</td><td>0.157</td><td>0.621</td></tr><tr><td>MnN-NCl</td><td>2.3742</td><td>0.0565</td><td>0.213</td><td>1.113</td><td>-0.00679</td><td>0.12</td><td>0.501</td></tr><tr><td>MnN-NBr</td><td>2.5339</td><td>0.0503</td><td>0.135</td><td>1.27</td><td>-0.01246</td><td>0.248</td><td>0.505</td></tr><tr><td>MnN-NI</td><td>2.7569</td><td>0.0431</td><td>0.078</td><td>1.365</td><td>-0.01115</td><td>0.259</td><td>0.529</td></tr></tbody></table>
<table><caption>Table 2. Cont.</caption><thead><tr><th>BCP</th><th>d, Å</th><th>ρ, e<sup>-</sup>/a<sub>0</sub><sup>3</sup></th><th>∇<sup>2</sup>ρ, e<sup>-</sup>/a<sub>0</sub><sup>5</sup></th><th>|V|/G</th><th>H, Ha/a<sub>0</sub><sup>3</sup></th><th>|H|/ρ, Ha/e<sup>-</sup></th><th>DI</th></tr></thead><tbody><tr><td colspan="8"><strong>M-S</strong></td></tr><tr><td>MnPropBr</td><td>2.3401</td><td>0.0664</td><td>0.212</td><td>1.250</td><td>-0.01766</td><td>0.266</td><td>0.509</td></tr><tr><td>MnPhBr</td><td>2.3454</td><td>0.0647</td><td>0.222</td><td>1.229</td><td>-0.01645</td><td>0.254</td><td>0.488</td></tr><tr><td>RePropBr</td><td>2.4585</td><td>0.0763</td><td>0.212</td><td>1.194</td><td>-0.01274</td><td>0.167</td><td>0.588</td></tr><tr><td>RePhCl</td><td>2.4598</td><td>0.0757</td><td>0.214</td><td>1.144</td><td>-0.00840</td><td>0.111</td><td>0.574</td></tr><tr><td>RePhBr</td><td>2.4538</td><td>0.0767</td><td>0.214</td><td>1.195</td><td>-0.01293</td><td>0.169</td><td>0.592</td></tr><tr><td>RePhI</td><td>2.46</td><td>0.0763</td><td>0.212</td><td>1.195</td><td>-0.01288</td><td>0.169</td><td>0.591</td></tr><tr><td>ReButBr</td><td>2.4763</td><td>0.0738</td><td>0.204</td><td>1.189</td><td>-0.01192</td><td>0.162</td><td>0.576</td></tr><tr><td colspan="8"><strong>M-N</strong></td></tr><tr><td>MnPropBr</td><td>2.137</td><td>0.0627</td><td>0.337</td><td>1.105</td><td>-0.00882</td><td>0.149</td><td>0.397</td></tr><tr><td>MnPhBr</td><td>2.114</td><td>0.066</td><td>0.328</td><td>1.105</td><td>-0.00965</td><td>0.146</td><td>0.41</td></tr><tr><td>RePropBr</td><td>2.249</td><td>0.0775</td><td>0.302</td><td>1.097</td><td>-0.00815</td><td>0.105</td><td>0.471</td></tr><tr><td>RePhCl</td><td>2.247</td><td>0.0775</td><td>0.306</td><td>1.095</td><td>-0.00808</td><td>0.104</td><td>0.473</td></tr><tr><td>RePhBr</td><td>2.254</td><td>0.0761</td><td>0.303</td><td>1.091</td><td>-0.00754</td><td>0.099</td><td>0.47</td></tr><tr><td>RePhI</td><td>2.26</td><td>0.0755</td><td>0.299</td><td>1.091</td><td>-0.00747</td><td>0.099</td><td>0.473</td></tr><tr><td>ReButBr</td><td>2.26</td><td>0.0748</td><td>0.298</td><td>1.088</td><td>-0.00722</td><td>0.097</td><td>0.458</td></tr><tr><td colspan="8"><strong>M-C<sup>X</sup></strong></td></tr><tr><td>MnPropBr</td><td>1.796</td><td>0.1421</td><td>0.655</td><td>1.255</td><td>-0.05907</td><td>0.416</td><td>1.174</td></tr><tr><td>C<sup>X</sup>-O</td><td>1.149</td><td>0.4639</td><td>0.422</td><td>1.885</td><td>-0.80938</td><td>1.745</td><td>1.448</td></tr><tr><td>MnPhBr</td><td>1.801</td><td>0.14</td><td>0.664</td><td>1.243</td><td>-0.05320</td><td>0.380</td><td>1.149</td></tr><tr><td>CX-O</td><td>1.144</td><td>0.4686</td><td>0.469</td><td>1.875</td><td>-0.82023</td><td>1.750</td><td>1.545</td></tr><tr><td>RePropBr</td><td>1.901</td><td>0.1674</td><td>0.475</td><td>1.384</td><td>-0.07417</td><td>0.443</td><td>1.273</td></tr><tr><td>CX-O</td><td>1.155</td><td>0.4562</td><td>0.411</td><td>1.885</td><td>-0.89061</td><td>1.727</td><td>1.544</td></tr><tr><td>RePhCl</td><td>1.890</td><td>0.1719</td><td>0.477</td><td>1.400</td><td>-0.07900</td><td>0.460</td><td>1.317</td></tr><tr><td>CX-O</td><td>1.162</td><td>0.4489</td><td>0.35</td><td>1.898</td><td>-0.77094</td><td>1.718</td><td>1.550</td></tr><tr><td>RePhBr</td><td>1.916</td><td>0.1612</td><td>0.475</td><td>1.362</td><td>-0.06745</td><td>0.418</td><td>1.250</td></tr><tr><td>CX-O</td><td>1.140</td><td>0.4719</td><td>0.561</td><td>1.855</td><td>-0.82403</td><td>1.746</td><td>1.555</td></tr><tr><td>RePhI</td><td>1.921</td><td>0.1596</td><td>0.48</td><td>1.353</td><td>-0.06556</td><td>0.411</td><td>1.198</td></tr><tr><td>CX-O</td><td>1.100</td><td>0.5209</td><td>1.018</td><td>1.787</td><td>-0.941015</td><td>1.806</td><td>1.549</td></tr><tr><td>ReButBr</td><td>1.920</td><td>0.1601</td><td>0.471</td><td>1.361</td><td>-0.06648</td><td>0.415</td><td>1.249</td></tr><tr><td>CX-O</td><td>1.148</td><td>0.4627</td><td>0.475</td><td>1.871</td><td>-0.80294</td><td>1.735</td><td>1.553</td></tr><tr><td colspan="8"><strong>M-C<sup>S</sup></strong></td></tr><tr><td>MnPropBr</td><td>1.813</td><td>0.1378</td><td>0.624</td><td>1.252</td><td>-0.05242</td><td>0.380</td><td>1.045</td></tr><tr><td>C<sup>S</sup>-O</td><td>1.148</td><td>0.465</td><td>0.420</td><td>1.886</td><td>-0.81267</td><td>1.748</td><td>1.567</td></tr><tr><td>MnPhBr</td><td>1.816</td><td>0.1373</td><td>0.604</td><td>1.258</td><td>-0.05264</td><td>0.383</td><td>1.049</td></tr><tr><td>C<sup>S</sup>-O</td><td>1.150</td><td>0.4636</td><td>0.409</td><td>1.888</td><td>-0.80921</td><td>1.746</td><td>1.568</td></tr><tr><td>RePropBr</td><td>1.924</td><td>0.1605</td><td>0.451</td><td>1.373</td><td>-0.06699</td><td>0.417</td><td>1.151</td></tr><tr><td>C<sup>S</sup>-O</td><td>1.146</td><td>0.4658</td><td>0.492</td><td>1.869</td><td>-0.81113</td><td>1.741</td><td>1.587</td></tr><tr><td>RePhCl</td><td>1.939</td><td>0.1548</td><td>0.449</td><td>1.353</td><td>-0.06131</td><td>0.396</td><td>1.127</td></tr><tr><td>C<sup>S</sup>-O</td><td>1.130</td><td>0.4828</td><td>0.653</td><td>1.839</td><td>-0.85044</td><td>1.761</td><td>1.586</td></tr><tr><td>RePhBr</td><td>1.946</td><td>0.1530</td><td>0.442</td><td>1.351</td><td>-0.06970</td><td>0.390</td><td>1.122</td></tr><tr><td>C<sup>S</sup>-O</td><td>1.137</td><td>0.4754</td><td>0.588</td><td>1.850</td><td>-0.83316</td><td>1.752</td><td>1.585</td></tr><tr><td>RePhI</td><td>1.941</td><td>0.1540</td><td>0.451</td><td>1.350</td><td>-0.06067</td><td>0.394</td><td>1.155</td></tr><tr><td>C<sup>S</sup>-O</td><td>1.134</td><td>0.4782</td><td>0.610</td><td>1.846</td><td>-0.83964</td><td>1.756</td><td>1.584</td></tr><tr><td>ReButBr</td><td>1.944</td><td>0.1543</td><td>0.436</td><td>1.357</td><td>-0.06095</td><td>0.395</td><td>1.113</td></tr><tr><td>CS-O</td><td>1.138</td><td>0.4745</td><td>0.577</td><td>1.852</td><td>-0.83116</td><td>1.752</td><td>1.585</td></tr><tr><td colspan="8"><strong>M-C<sup>N</sup></strong></td></tr><tr><td>MnPropBr</td><td>1.801</td><td>0.1423</td><td>0.624</td><td>1.265</td><td>-0.05629</td><td>0.396</td><td>1.095</td></tr><tr><td>C<sup>N</sup>-O</td><td>1.146</td><td>0.4680</td><td>0.430</td><td>1.882</td><td>-0.82013</td><td>1.7530</td><td>1.570</td></tr><tr><td>MnPhBr</td><td>1.808</td><td>0.1384</td><td>0.604</td><td>1.242</td><td>-0.05234</td><td>0.378</td><td>1.080</td></tr><tr><td>C<sup>N</sup>-O</td><td>1.142</td><td>0.4721</td><td>0.476</td><td>1.875</td><td>-0.829786</td><td>1.758</td><td>1.574</td></tr><tr><td>RePropBr</td><td>1.908</td><td>0.1680</td><td>0.451</td><td>1.409</td><td>-0.07499</td><td>0.446</td><td>1.223</td></tr></tbody></table>
<table><caption>**Table 2. Cont.**</caption><thead><tr><th>BCP</th><th>d, Å</th><th>ρ, e<sup>-</sup>/a<sub>0</sub><sup>3</sup></th><th>∇<sup>2</sup>ρ, e<sup>-</sup>/a<sub>0</sub><sup>5</sup></th><th>|V|/G</th><th>H, Ha/a<sub>0</sub><sup>3</sup></th><th>|H|/ρ, Ha/e<sup>-</sup></th><th>DI</th></tr></thead><tbody><tr><td>C<sup>N</sup>-O</td><td>1.144</td><td>0.4679</td><td>0.506</td><td>1.867</td><td>-0.81685</td><td>1.746</td><td>1.586</td></tr><tr><td>RePhCl</td><td>1.933</td><td>0.1578</td><td>0.449</td><td>1.371</td><td>-0.06471</td><td>0.418</td><td>1.184</td></tr><tr><td>C<sup>N</sup>-O</td><td>1.141</td><td>0.4718</td><td>0.542</td><td>1.859</td><td>-0.82499</td><td>1.749</td><td>1.597</td></tr><tr><td>RePhBr</td><td>1.916</td><td>0.1643</td><td>0.442</td><td>1.397</td><td>-0.07120</td><td>0.433</td><td>1.225</td></tr><tr><td>C<sup>N</sup>-O</td><td>1.153</td><td>0.4589</td><td>0.424</td><td>1.882</td><td>-0.79575</td><td>1.734</td><td>1.594</td></tr><tr><td>RePhI</td><td>1.927</td><td>0.1604</td><td>0.451</td><td>1.383</td><td>-0.06724</td><td>0.419</td><td>1.209</td></tr><tr><td>C<sup>N</sup>-O</td><td>1.144</td><td>0.4682</td><td>0.512</td><td>1.865</td><td>-0.81745</td><td>1.746</td><td>1.591</td></tr><tr><td>ReButBr</td><td>1.900</td><td>0.1696</td><td>0.436</td><td>1.409</td><td>-0.07686</td><td>0.453</td><td>1.268</td></tr><tr><td>CN-O</td><td>1.156</td><td>0.4559</td><td>0.393</td><td>1.889</td><td>-0.78877</td><td>1.730</td><td>1.571</td></tr></tbody></table>

CBr is the carbon atom trans to Br; CS is the carbon atom trans to S; CN is the carbon atom trans to N. ReButCl is the virtual optimized complex.

The data gathered in Table 2 provide a significant amount of information about the nature of the metal-ligand bonds. The first observation is that in all complexes the Laplacians are positive and the total energy densities are negative. Thus, these bonds are the shared/closed C/S transient coordination bonds. Furthermore, it can be seen for the Mn-Br bonds that their topological parameters are very similar, indicating that the nature of the SR substituent has no or very small influence on the electronic nature of this metal-halide bond. The fact that the bromide participates in two hydrogen bonds in MnPropBr, but in only one in MnPhBr does not change the electronic properties of this bond. A similar observation was made for Re-Br bonds in analogous RePropBr and RePhBr complexes. However, the properties of Re-Br bonds in the t-butyl derivative ReButBr are slightly different. This is probably due to a different structure of the ligand with the S<sup>t</sup>Bu substituent. Both SR substituents are in a geminal position on terminal carbon atom of the vinyl part of the ligand with S<sup>t</sup>Pr and SPh substituents, whereas for the S<sup>t</sup>Bu ligand, they are vicinal and distributed on both carbon atoms of the vinyl part for steric reasons [18,21]. The topological parameters of Re-X bonds vary with the nature of the halide, as expected. On the other hand, those of M-S, M-N and M-CO bonds vary very little and thus confirm a structural rigidity of this class of complexes. There are, however, some tendencies in the evolution of parameters found for the metal-ligand bonds.

*Electron density ρ*. The electron densities ρ at BCPs are always higher for bonds around Re than for those around Mn. This is true for bonds in both the “coordination” (wernerian) and the “organometallic” parts of the molecules. They are higher for M-S and M-N bonds than for M-X ones. The highest values of ρ are calculated in the “organometallic” part of the molecules for M-CO bonds. They decrease in the order M-C<sup>X</sup> > M-C<sup>N</sup> > M-CS. For the halides, the order is ρ(Re-C<sup>Cl</sup>) > ρ(Re-C<sup>Br</sup>) > ρ(Re-C<sup>I</sup>). This is a consequence of increases in bond lengths in the order d(Re-C<sup>Cl</sup>) < d(Re-C<sup>Br</sup>) < d(Re-C<sup>I</sup>). If we assume that the *trans* influence operates in our complexes and that the chloride is a better electron-withdrawing ligand than Br and I, an inverse trend should be observed. But instead of a metal-carbon bond, we should rather consider the whole *trans* metal/carbonyl unit. The electron densities ρ retained in the C-O bonds *trans* to X ligands decrease from X = I (0.521 e<sup>-</sup>/a<sub>0</sub><sup>3</sup>) through X = Br (0.472 e<sup>-</sup>/a<sub>0</sub><sup>3</sup>) to X = Cl (0.449 e<sup>-</sup>/a<sub>0</sub><sup>3</sup>), an order compatible with the increase in *d*(C-O) bond lengths from X = I to X = Cl. The overall electron densities on *trans* to X Re-C-O units (ρ(Re-C<sup>X</sup>) + ρ(C<sup>X</sup>-O)) are equal to 0.681 (I) > 0.633 (Br) > 0.621 e<sup>-</sup>/a<sub>0</sub><sup>3</sup> (Cl), which is the expected order. The correlations between the electron densities and the lengths of the metal-halide bonds will be discussed later.

*|V|/G dimensionless parameter.* The |V|/G values in the “coordination” part of the metal coordination sphere are greater for the three Mn-Br, M-S and M-N bonds than
those for Re analogous bonds. These values decrease in the order M–Br > M–S > M–N for the Mn complexes, but in the Re series, the values for Re–Br and Re–S are similar, both being greater than for Re–N bonds. The M–N bonds are thus the less covalent ones. Recall that a higher value of |V|/G is an indicator of higher covalency. The |V|/G parameters of Re–X bonds increase in the order **RePhCl** < **RePhBr** < **RePhI**, which is the expected order of their covalent nature. This order is also observed in a series of complexes with N,N–chelating ligands **MnN-NX**. Contrary to the “coordination” (“wernerian”) bonds, the values of the |V|/G parameters for Re–C(carbonyl) bonds in the “organometallic” part are higher than those for Mn–C(carbonyl) ones. Thus, the Re–C bonds are more covalent than are the Mn–C bonds.

*Total energy densities H.* Because the total energy density H corresponds to the sum of potential and kinetic densities, the H values follow the same trends as |V|/G. Total energies are more negative for “coordination” Mn–ligand than for Re–ligand bonds, whereas the H values are more negative for Re–C than for Mn–C bonds.

|H|/ρ bond degree and delocalization index (DI). Because the absolute values of total energy densities |H| are greater for Mn “wernerian” bonds than for Re, while the corresponding electron densities ρ are lower, the |H|/ρ parameters are greater in the case of Mn than that of Re, conferring them a more covalent nature. However, this does not mean that they are stronger or of a higher order because the delocalization indexes (DIs) are greater for the Re–ligand bonds. In the case of M–C bonds, the bond degrees are higher for Re–C than for Mn–C, indicating that they are more covalent. The delocalization indexes are also higher (Re > Mn), confirming their higher covalencies and strengths.

It has been suggested in the supramolecular part that a π···π interaction between the parallel phenyl rings of neighbor molecules is present in the crystal structure of **MnPropBr**. We calculated the wave function in the relevant region on a model in which only two azabutadiene ligands from neighbor molecules are retained. An AIM analysis was further carried out, and the resulting graph is depicted in Figure 9. The bond critical point BCP between two carbon atoms C12 and C12# can be observed, confirming the presence of a very weak π···π interaction.

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Non-covalent C···C and H···H bond paths (dashed lines) in the structure of **MnPropBr**. Colors of critical points (CP): bond (BCP 3, −1)—red; ring (RCP 3, +1)—orange. Symmetry operation # = 2 − x, −y, 1 − z.</figcaption></figure>

We were also interested in correlations between the electron density ρ at the bond critical point of the Mn–X and Re–X bonds and their bond lengths R. The crystallographic so-called bond length–bond strength or bond valence curves were developed in the 1970s
by Brown [52,53]. The mostly used one has exponential expression (Equation (1)), where R<sub>ij</sub> is the length of the bond between atoms i and j, S<sub>ij</sub> is its experimental bond valence and R<sub>0</sub> and B are the parameters chosen for the best matching of valence sum rule, i.e., the sum of all valences S<sub>ij</sub> around each ion (central atom of bonding polyhedron) is equal to its valence V<sub>i</sub> (Equation (2)). The first correlation between bond order and its length was established by Pauling (Equation (3)), in which c is a constant and R<sub>0</sub> is the bond length of order 1 [54].

$$S_{ij} = \exp[(R_0 - R_{ij})/B] \qquad (1)$$

$$V_i = \sum S_{ij} \qquad (2)$$

$$R_i - R_0 = -c \log n_i \qquad (3)$$

Bader proposed that the electron density ρ, calculated via his QT-AIM approach at the bond critical points of carbon-carbon bonds, is correlated with the bond order through exponential expression (Equation (4)) analogous to (Equation (1)), in which A and B are the constants [40].

$$n = \exp[A(\rho - B)] \qquad (4)$$

ρ in this last equation corresponds to R, and n corresponds to S of Equation (1). Thus, ρ and R are also correlated. This kind of correlation has been well established, in particular for covalent carbon-carbon [55,56], sulfur-sulfur [57] and hydrogen bonds [42,58]. For the covalent C-C bonds, Alkorta suggested [59] that the linear correlation between the length R and the electron density has a coefficient of determination of R² = 0.995. On the other hand, the team of Boyd pointed out that the simplest functional relationship between ρ and R, consistent with the vanishing of ρ when R → ∞ is a power low. However, he stated that the problem is not without ambiguity since some linear correlations fit even better than the power ones. He also proposed a linear correlation for non-covalent CN···H hydrogen bonds [60,61]. Note, however, that for the whole range of H/F distances from covalent F-H to non-covalent F···H, the exponential or logarithmic function operates [23,48]. It has also been stated that for sulfur-carbon bonds, neither a linear (R² = 0.914) nor an exponential (R² = 0.891) relationship is satisfactory [62].

All up-to-date studies of ρ vs. R correlations focus on a pair of atoms, A and B. In this paper, we look at how ρ and R may correlate between a given metal and not a chosen halide, but with a series of halides. There are only two Mn(I) but five Re(I) complexes of the type *fac*-MX(CO)<sub>3</sub>(S,N-azabutadiene) studied in this paper, whose parameters are given in Table 2. Some supplementary data are required for the Mn series. Because there are no entries for mononuclear *fac*-MnX(CO)<sub>3</sub> (X = Cl, I) with S,N-chelating ligands in CCDC (version 5.44 (2023), we looked for structurally similar series with N-N chelating ligands. Three complexes were chosen: (Biacetyl-bis(phenylimine)-N,N')chlorotricarbonyl-manganese (**MnN-NCl**) [63] for the Mn-Cl bond (CSD refcode GIMMUK), bromo-tricarbonyl-(N1,N2-dimethylcyclohexane-1,2-diamine)manganese(I) (**MnN-NBr**) for the Mn-Br bond (CSD refcode RONHOU) [64] and tricarbonyl-iodo-(2-amino-2-methyl-3-(2,2-dimethylaziridino)propane)manganese(I) (**MnN-NI**) for the Mn-I bond (CSD refcode SICFEQ) [65]. The molecular graphs of these three N,N-chelates are depicted in Figures S8-S10 of the Supplementary Materials. First, the bromide complex RONHOU (**MnN-NBr**) was checked for comparison with our Br complexes **MnPropBr** and **MnPhBr**. The calculated value of ρ for **MnN-NBr** was in very good agreement, and consequently, two other complexes with Mn-Cl and Mn-I bonds were added to our study. The corresponding data are included in Table 2. The plots ρ vs. R are presented in Figure 10.
<figure><img src="image_13.png" /><figcaption>Figure 10. Plots of electron densities ρ (e⁻/a₀³) vs. bond lengths R (Å) for M-X bonds.</figcaption></figure>

A linear correlation may be proposed for the Mn (Equation (5)) and the Re (Equation (6)) series. However, a slightly better match (based on R² values) was obtained with power Equation (7) for Mn and (8) for Re complexes.

$$ \rho = 0.0348 R + 0.1387 \qquad R^2 = 0.996 \qquad (5) $$

$$ \rho = 0.0422 R + 0.1718 \qquad R^2 = 0.992 \qquad (6) $$

$$ \rho = 0.3027 \exp(-0.707 R) \qquad R^2 = 0.999 \qquad (7) $$

$$ \rho = 0.3945 \exp(-0.711 R) \qquad R^2 = 0.993 \qquad (8) $$

The most important information obtained from the established correlations is that they apply to a given metal bound to different halide ligands. Probably, the fact that the *fac*-MX(CO)₃ molecules are geometrically and electronically “rigid” contributes to these unprecedented correlations. It is also worth noting that the correlations apply even to structurally similar but a priori electronically different donor S,N vs. N,N chelating ligands.

## 4. Conclusions

In this article, we have described the structure of a new complex of the family *fac*-MnBr(CO)₃[(RS)₂C=C(H)-N=CPh₂] (R = iPr, Ph, p-Tol) and noticed that the change in the substituent R of the SR group has little influence on the metric parameters within the coordination sphere of the metal. This observation agrees with our previous results published for analogous rhenium compounds. Topological analyses of electron density distribution, carried out on Mn and Re complexes, also revealed that the electronic parameters no longer significantly change with the change in R. Both of these observations suggest that our complexes are geometrically and electronically “rigid”. AIM-derived electronic parameters, |V|/G in particular, allowed us to obtain information about the relative covalent nature of metal-ligand bonds. Particularly interesting is the observation that the Mn- “wernerian” (X, S and N) bonds are more covalent than their analogous Re bonds. In contrast, the “organometallic” Re-C(CO) bonds are more covalent in character than the Mn-C(CO) bonds. For the first time, a r/R electron density-bond length relationship has been established for a metal bound to different specific halides in structurally similar complexes.
**Supplementary Materials:** The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/chemistry7050145/s1: Figure S1: Experimental IR spectrum of MnPropBr; Figure S2: Parallel arrangement of the phenyl rings from neighbor molecules in the structure of MnPropBr; Figure S3: Molecular graph of *fac*-ReBr(CO)₃[($^{1}$PrS)₂C=C(H)-N=CPh₂] Re-PropBr; Figure S4: Molecular graph of *fac*-ReBr(CO)₃[($^{1}$BuS)₂C=C(H)-N=CPh₂] ReButBr; Figure S5: Molecular graph of *fac*-ReCl(CO)₃[(PhS)₂C=C(H)-N=CPh₂]; Figure S6: Molecular graph of *fac*-ReBr(CO)₃[Ph)₂C=C(H)-N=CPh₂]; Figure S7: Molecular graph of *fac*-ReI(CO)₃[(PhS)₂C=C(H)-N=CPh₂] RePhI; Figure S8: Molecular graph of *fac*-MnCl(CO)₃[(N,N-chelating ligand)] MnNNCl; Figure S9: Molecular graph of *fac*-MnBr(CO)₃[(N,N-chelating ligand)] MnNNBr; Figure S10: Molecular graph of *fac*-MnBr(CO)₃[(N,N-chelating ligand)] MnNNI; Table S1: Data for the electronic transitions in the calculated UV/vis spectrum of MnPropBr.

**Author Contributions:** M.K. and A.K. prepared and analyzed the complexes; M.M.K. collected the X-ray data and solved the structure; M.M.K. performed the quantum (molecular orbitals and AIM) calculations; M.M.K. and M.K. designed the study, analyzed the data and wrote the paper; A.K., M.M.K. and M.K. conceptualized the study. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work received no external funding.

**Data Availability Statement:** This information is already presented in the X-ray diffraction section. The crystallographic data have been deposited with the Cambridge Crystallographic Data Centre: deposition num-ber CCDC 941427 contains detailed crystallographic data for this publication. These data may be obtained free of charge from the Cambridge Crystallographic Data Center through www.ccdc.cam.ac.uk/data_request/cif, accessed on 27 August 2025.

**Acknowledgments:** Quantum calculations were performed using HPC resources from DNUM CCUB (Centre de Calcul de l'Université de Bourgogne). MMK thanks P. Fleurat-Lessard from University Bourgogne Europe for calculation of the IR spectrum.

**Conflicts of Interest:** The authors declare no conflicts of interest.

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