A DFT and TD‐DFT Approach to the Understanding of Statistical Kinetics in Substitution Reactions of M₃Q₄ (M=Mo, W; Q=S, Se) Cuboidal Clusters

Abstract

For many years it has been known that the nine water molecules in [M(3)Q(4)(H(2)O)(9)](4+) cuboidal clusters (M = Mo, W; Q = S, Se) can be replaced by entering ligands, such as chloride or thiocyanate, and kinetic studies carried out mainly on the substitution of the first water molecule at each metal centre reveal that the reaction at the three metal centres occurs with statistical kinetics; that is, a single exponential with a rate constant corresponding to the reaction at the third centre is observed instead of the expected three-exponential kinetic trace. Such simplification of the kinetic equations requires the simultaneous fulfilment of two conditions: first that the three consecutive rate constants are in statistical ratio, and second that the metal centres behave as independent chromophores. The validity of those simplifications has been checked for the case of the reaction of [Mo(3)S(4)(H(2)O)(9)](4+) with Cl(-) by using DFT and TD-DFT theoretical calculations. The results of those calculations are in agreement with the available experimental information, which indicates that the H(2)O ligands trans to the μ-S undergo substitution much faster than those trans to the μ(3)-S. Moreover, the energy barriers for the substitution of the first water molecule at the three metal centres are close to each other, the differences being compatible with the small changes in the numerical values of the rate constants required for observation of statistical kinetics. TD-DFT calculations lead to calculated electronic spectra, which are in reasonable agreement with those experimentally measured, but the calculations do not indicate that the three metal centres behave as independent chromophores, although the mathematical conditions required for simplification of the kinetic traces to a single exponential are reasonably well fulfilled at certain wavelengths. A re-examination of the kinetics of the reaction by using global fitting procedures yields results, which are compatible with statistical kinetics, although an alternative interpretation in which substitution only occurs at a single metal centre under reversible conditions is also possible.

Equilibrium and kinetics studies on bibrachial lariat aza-crown/Cu(II) systems reveal different behavior associated with small changes in the structure

“…4 indicate that the molar absorptivities of the binuclear species double the values for the mononuclear species. These facts can be considered strong evidence on the operation of statistical kinetics in the decomposition of the dinuclear complexes, a phenomenon previously reported for other macrocyclic complexes and clusters [44][45][46][47]. In that case the dinuclear complex will decompose in two steps which involve the sequential dissociation of both Cu 2+ ions with rate constants in the statistical 2:1 ratio, but because of mathematical simplifications only the value of the rate constant for the exit of the second Cu 2+ can be measured and so, the values are similar to those found for the decomposition of the mononuclear complex.…”

“…In that case the dinuclear complex will decompose in two steps which involve the sequential dissociation of both Cu 2+ ions with rate constants in the statistical 2:1 ratio, but because of mathematical simplifications only the value of the rate constant for the exit of the second Cu 2+ can be measured and so, the values are similar to those found for the decomposition of the mononuclear complex. The conditions required for this mathematical simplification have been summarized recently and essentially consist in the independent behavior of the metal centres in the binuclear species both from the kinetic and spectral points of view [45].…”

Reaction Schemes That Are Easily Confused with a Reversible First-Order Reaction

“…These coincidences may seem rare at first sight, but they are quite common (and also valid in the entire useful wavelength range) in practice when several equivalent and independent reaction centers are involved, and there are also a number of equivalent absorbing moieties. Experimental examples of this phenomenon can be found both in the recent and older chemical literature .…”

“…It should be emphasized that, in the classic cases of statistical kinetics , there are no approximations or experimental limitations involved; it is the exact mathematical solution of the scheme that may be misleading in experimental studies because of the coincidences. However, there is another and arguably much more common reason why a kinetic curve could be misidentified as a simpler one: The usual experimental errors in measuring the absorbance or other features of the reactive system may make two mathematically distinct types of kinetic curves indistinguishable.…”

Benchmarking of DFT methods using experimental free energies and volumes of activation for the cycloaddition of alkynes to cuboidal Mo3S4 clusters

“…Moreover, each metal center features a distorted octahedral geometry if the MoMo bonds are ignored, and therefore, three additional coordination sites are normally occupied by mono‐ and/or polydentate ligands. Besides the reactivity associated with the Mo(IV) centers and its outer coordination environment, [ 5 ] the three bridging (μ‐S) ligands provide important reactivity paths. In fact, two main processes are known to occur at these sulfur sites: (a) incorporation of a second metal (M') to form heterometallic [Mo 3 M'S 4 ] 4+ clusters [ 6 ] and (b) alkyne insertion to produce dithiolene adducts, as shown in Scheme 1.…”