Figure 1 shows the monomeric, homoleptic optimized structure of tris-chromium Cr(III) and Cr(VI) carboxylate (acetate) complexes along with the ligand with numbering the atoms. The optimized molecular geometry parameters of the tris [Cr^III(AC)₃]⁰ and [Cr_VI(AC)₃]³⁺ complexes were reported in Table 1 which allied with Cr-O and C-O distances. As of tabulated data, a comparison between the bond length of free ligand and corresponding complexes of Cr(III) and Cr(VI) oxidation states were checked and found the C-O bond length increased in the presence of metal ions [31] [32]. The calculated average Cr-O bond lengths in the [Cr^III(AC)₃]⁰ and [Cr_VI(AC)₃]³⁺ complexes were 2.01 Å and 1.85 Å respectively whereas experimentally measured Cr^III-O bond length found 2.008 Å [9]. The III(AC) ₃] ⁰ and tris [Cr ^VI(AC) ₃] ³⁺ complexes. On the other hand the angles of III(AC) ₃] ⁰ and [Cr ^VI(AC) ₃] ³⁺ respectively [31] [32]. These decreases in bond lengths and bond angles were due to the ionic sizes of Cr(III) and Cr(VI) ions. These data indicates that Cr(VI) and carboxylate interaction is more stronger compare to Cr(III) metal ion. The geometry parameters are in fairly agrees well with the experimental data. The DFT optimized [Cr ^III(AC) ₃] ⁰ complex demonstrates several exciting features ( Figure 1). The three carboxylate (acetate) radiates from the CrO ₆-core to form a distorted octahedron with D₃ symmetry. The ligands carboxylate persuade a minor average bite angle (

^aRef. [5], ^bRef. [9], ^cRef. [40].

FT-IR is one of the well-established diagnostic probes of carboxylate coordination in metal complexes. The difference in wavenumbers between the antisymmetric and symmetric CO₂ stretches (Δ) distinguishes between monodentate and free ion (>200 cm⁻¹), bridging (200 - 100 cm⁻¹) and bidentate (<100 cm⁻¹) carboxylate bonding geometries [31].

The IR spectra of tris chromium acetate complex [Cr^III(AC)₃]⁰ are shown in Figure 2. The scaled wavenumbers of free carboxylate (acetate), tris-chromium acetate complexes [Cr(AC)₃]ⁿ (where n = 0, +3) of are reported in Table 2. These data demonstrates the antisymmetric ( ν asym COO ) stretch at 1526 cm⁻¹ and the symmetric ( ν sym COO ) stretch in the ~1494 cm⁻¹ region yielding a respective separation ( Δ ν a - s = ν asym COO − ν sym COO ) is ~32 cm⁻¹ of tris [Cr^III(AC)₃]⁰ complex. The presence of more than one band which may be assigned to (COO~) vibrations suggests that the three carboxylate groups are non-equivalent. These two leading infrared (IR) peaks are close in frequency to those previously calculated and experimental data observed for tris-chromium carboxylates complex Cr^III(EH)₃ at 1515 and 1455 cm⁻¹ respectively [16]. These data support carboxylate as bidentate coordination with the chromium centre [16]. This suggests the hexacoordination [15] of chromium provided one of the acetate groups is asymmetric with one Cr-O bond is normal and other significantly longer. All the ^υCO₂ stretches was assigned supported by the computational modeling. The IR peaks with high intensities arose from the large charge polarization since the positive Cr(III) ion being en-closed by three negatively charged acetate ligands. Due to charge polarization, the vibrations that result in unsymmetrical distortions convince large dipole moment deviations and large IR intensities. Table 2 reports the vibrational normal modes allied with the bidentate carboxylate groups and their distinctive absorptions reflect the dissimilarities in molecular structure of Cr-acetate complex. The assignment of the theoretically calculated frequencies is based on the experimentally observed band frequencies and intensities in the IR spectra [15]. The monomer bidentate structure [Cr^III(AC)₃]⁰ demonstrates good harmony with the experimental infrared spectra.

^a,bexperimentally and theoretically calculated frequencies by sydaro et al. [16].

The binding energies (ΔE) are correlated with the stability of the corresponding complexes. This elucidates that the stability of the complexes in gas phase escalations with the increasing absolute binding energy. The binding energies (ΔE), enthalpies (ΔH) and Gibbs free energies (ΔG) of tris chromium acetate complexes with different oxidation states and spin states were calculated and summarized in Table 3. Electronic energies only included for the calculation of above ΔE’s binding energies. Because of the covalent nature of the metal ligand bindings, the vibrational, thermal and entropic contributions to ΔE turned out to be insignificant. According to Table 3, the calculated binding energies trend at differentoxidation states are as follows: Cr(VI) > Cr(V) > Cr(IV) > Cr(III) > Cr(II) > Cr(I). This trend shows that carboxylate (AC) has a higher probability of selective complexation with Cr(VI). But the other oxidation states like Cr(V), Cr(IV), Cr(II) and Cr(I) complexes are not stable. These complexes with these oxidation states are limited in the environment. The two major oxidation states Cr(VI) and Cr(III) predominant in the environment and Cr(VI) reduced to Cr(III) under the acidic conditions.

In case of Cr (III) complexes, the relative energy of low-spin state is higher than high-spin state by 385.79 kcal·mol⁻¹ (see Table 3). Likewise, for Cr (VI) complex the relative energy at low spin state is larger than high spin states by 768.69 kcal·mol⁻¹ respectively. Based on the relative energies of Cr(III) and Cr(VI) tris-acetate complexes, it was viewed that the stability of the complexes intensification in the order of HS > LS. Comparing all the estimated binding energy values, it was concluded that Cr(VI) at high spin state was the most preferable to form the tris chromium-acetate complex. According to Boys-Bernandi counterpoise (CP) [34] corrected method we have checked the basis set superposition error (BSSE) for thetris [Cr(AC)₃]⁰ and [Cr(AC)₃]³⁺ complexes of high spin state only. The BSSE-corrected energy E complex C P ( − 1730.06   au ) is compared with the uncorrected energy E_complex (−1730.06 au). The relative deviation defined as | ( E complex − E complex C P ) / E complex | × 100 . The calculated BSSE were within 0.001% of the complexes energy which was in the range of computational error.

By employing the conductor-like polarizable continuum model (CPCM), [35] [36] [37] we also examined the effects of solvent for the Cr(III) and Cr(VI) tris acetate complexes. The estimated energy of each complex was almost unaffected

^aRef. [5].

in water solvent, only fluctuating by less than 0.008%. Due to the presence of the solvent, the structural change of each complex was not considered in this calculation. We however guess that the near-octahedron geometry around the central metal of each compound rests approximately unbroken with the introduction of the solvent. Thus, the binding energies of metal-ligand apparently will not fluctuate far from those in the solvent.

Due to the upturn of the amount of the precise interchange energy, high spin state with higher number of unpaired electrons is strongly stabilized compared to the low spin states. Thus energy investigation displays that the smaller ionic radius with more charge like Cr(VI), and the shorter bond distances M-O(Cr-O, 1.85 Å) allows the metal ion to withdraw more electron density from the carboxylates. These outcomes lead to the increase the charge transfer among the acetate (AC) and chromium metal ions. Furthermore, binding enthalpy (ΔH_bind) of coordination shows that the interactions between carboxylate and chromium ions are affected by the thermal correction in water. It is noticed that complexes with the same ligand acetate, Cr(VI) is more stable compare to the Cr(III). Thus a correlation is noticed between the complexation capacity and size of the metal ions, because the metal-ligand fascination proves to be the opposite of the ionic radius. Therefore, these results are reliable with the difference in size among the ions, 0.76 [38] and 0.44˚A [39] for Cr(III) and Cr(VI), respectively.

The theoretical absorption spectrum of tris [Cr^III(AC)₃]⁰ complex was calculated using Gaussian16 program package at DFT level using the optimized lowest energy structure. Figure 3 illustrates the UV-vis absorption spectrum of the tris [Cr^III(AC)₃]⁰ complex. The calculated wavelength, oscillatory strength and excitation energy of tris [Cr^III(AC)₃]⁰ complex was tabulated in Table 4. Two major strong peaks were witnessed at 420 nm and 528 nm. The absorption maximum was placed at 528 nm, supportive by the experimental wavelength (600 nm) of the leading peak originate from tris [Cr^III(AC)₃]⁰. The calculated wavelength of the studied complex are supportive by the experimental wavelengths at 448 nm and 600 nm originated from tris(methacrolato) chromium(III) [16]. The calculated wavelengths are also supportive the experiment done by Valencia-Centeno et al. at 416 nm and 576 nm [40]. Therefore, this results are in agreement with the established transitions, indicating that in Cr(O₂CCH₃)₃ the chromium ion exhibits an oxidation state of III and an octahedral coordination sphere. To illustrate the electronic transitions, six frontier molecular orbitals (MOs) were shown, extending from the highest occupied MO-2(HOMO-2) to the lowest unoccupied MO+1 (LUMO+1) (Figure 4). The absorption maximum at 528 nm arose mostly from the HOMO to LUMO transition. The electron density in the HOMO is mostly disseminated over the three acetate ligands. The formation of

^aRef. [16], ^bRef. [40].

HOMO orbital is primarily owing to 62% contribution from the chromium iron. In the LUMO presented in Figure 4, however, the electron density is mostly moved out of the acetates into the chromium ion. Thus, the strongest peak of the tris acetate complex in the UV-vis spectrum created obviously from ligand-to-metal (L→M) charge transfer. The peak at 420 nm mainly (66%) involved the transition from HOMO to LUMO+1. The orbital illustration of LUMO+1 has likewise the similar type of configuration leading the major involvement of oxygen has 62.89 % contribution for the construction of LUMO+1 orbital. Again, this peak originated from the ligand-to-metal charge transfer; the electron density in HOMO is dispersed mostly over acetate ligands. The contribution of electron density from metal ion has only 7.1%. The peaks located at 420 and 528 nm arose mostly from the HOMO to LUMO+1 and HOMO to LUMO transitions correspondingly. The HOMO-1 is alike to the HOMO-2 in that the electron density is extended over acetate ligands. The LUMO+1 is alike to the LUMO in that electron density is contained round the metal−oxygen interaction area. Another is that LUMO+1 does not have π character of the LUMO orbital.

The atomic charges on the metal ions were checked and reported in Table 5. Six different schemes were applied to estimate the charges: the schemes are natural population analysis (NPA), [41] Merz-Singh-Kollman(MK), [42] CHelpG, [43] CHelp [44] methods to fit the electrostatic potential, and the charge fitting method, HLYGAt [45] and NBO [18] [19]. The atomic charges varied from different charge schemes. Such as, the charge at NPA is 1.203 for Cr, whereas the charge was 1.502 if the MK scheme was applied. Generally, the atomic charges

^aNatural Population Analysis (NPA) [41]; ^bMK (Merz-Singh-Kollman) [42]; ^cCHelpG (CH-arges from Electrostatic Potentials using a Grid based method) [43]; ^dCHelp methods to fit the electrostatic potential method [44].

decrease with enlarging nuclear charges. The calculated results indicate the more reactivity and stability of tris [Cr^VI(AC)₃]³⁺ complex compare to [Cr^III(AC)₃]⁰. The charges on Cr(VI) in tris [Cr^VI(AC)₃]³⁺ complex are smaller than Cr(III) ion in [Cr^III(AC)₃]⁰.

The natural bond orbital (NBO) analysis [18] [19] was carried out using NBO 3.1 program implemented in the Gaussian 16 to calculate the energy eigenvalues of the frontier molecular orbital (ΔE_LUMO-ΔEHOMO) energies. The NBO analyses provide information approximately the electronic structure of a complex. The interactions between the occupied and vacant orbitals signify the deviation of the molecules from the Lewis structure, and the relevant energies can be used as a measure of structure stability.

The donor-acceptor interactions strength, E i j ( 2 ) are deduced from the second-order perturbation theory analysis of the Fock-Matrix [46]. For each donor (i) and (j) acceptor in the complexes, the stabilization energy or second order perturbation energy, associated with the delocalization from was estimated as follows (2).

Δ E i j ( 2 ) = − q ( F ^ i j ) 2 ε j − ε i (2)

where q refers to the donor orbital occupancy, ε i and ε j are diagonal elements (orbital energies) and F ^ i j is the off-diagonal NBO Fock-matrix element. The higher the value of E i j ( 2 ) , the more the inter-molecular orbital interactions and consequently greater the charge transfer between the donors electron and acceptors which leads to complex stability and stronger metal-binding.

Second-order perturbation energies of interaction E i j ( 2 ) were estimated and reported the main interactions in Table 6. The most significant energies are associated with the interactions among the lone pair electrons (oxygen atom) of the ligand (LP_O) and anti-bonding orbital of the chromium metal ion ( LP M ∗ ). A greater donor-acceptor interaction indicates a higher corresponding values and the more efficient charge transfer from the ligand (oxygen atom) to the metal ion (acceptor). According to Table 6, the calculated value for the [Cr^III(AC)₃]⁰

complex (36.52 kcal·mol⁻¹) is less significant than that of hexavalent tris chromium acetate [Cr^VI(AC)₃]³⁺ complex (96.95 kcal·mol⁻¹). This supports the stronger interaction of the carboxylate and Cr(VI), that is according to the energy analysis. Comparing the calculated values of ΔE_bind in Table 3 and in Table 6 shows a correlation. In additional, an increase in values is followed by the increment of ΔE_bind values.

To know the chemical stability of a complex, it needs the information of quantum chemistry reactivity indices like a high HOMO-LUMO energy gap, ionization potential, electron affinity, chemical hardness etc. Molecules are classified as hard and soft based on the calculated HOMO-LUMO energy gap value. Soft molecules have lower energy gap comparing to hard one and are more polarizable due to their small energy of excitation. Based on calculated HOMO and LUMO energies, the ionization potential (I) and electron affinity (A) are calculated using the subsequent equations I = − E HOMO and A = − E LUMO . Thus, the electronegativity and chemical hardness are calculated applying χ = ( I + A ) / 2 and η = ( I − A ) / 2 . The chemical potential and chemical softness are estimated as the negative of electronegativity ( μ = − χ ) and inverse of hardness ( S = 1 / η ) , respectively. Using the electronic chemical potential, μ and chemical hardness, η the global electrophilicity index ω was calculated by the equation (Equation (3))

ω = μ 2 2 η (3)

The calculated quantum chemistry reactivity indices are reported in Table 7. In these calculations tris [Cr^III(AC)₃]⁰ complex considered as hard as compared to tris [Cr^VI(AC)₃]³⁺ complex. During the complex formation of tris [Cr^VI(AC)₃]³⁺ the electronic chemical hardness ( η ) and electronic chemical potential ( μ ) decreases while electronegativity increases. The chemical hardness decrease will lead

to an increase in the chemical stability of the system. The obtained result is also in agreement with the trend of ΔE_HOMO–LUMO gaps which is also an indicator of chemical stability, i.e., the greater ΔE_HOMO–LUMO gap provides less chance for electron excitation and makes the system less reactive. A decrease in the electronic chemical potential of the [Cr^VI(AC)₃]³⁺ complex in comparison to the [Cr^III(AC)₃]⁰ specifies an increase in stability.