Reaction mechanisms perturbation theory calculations

In calculating the transition probability for the nonadiabatic reactions, it is sufficient to use the lowest order of quantum mechanical perturbation theory in the operator V d. For the adiabatic reactions, we must perform the summation of the whole series of the perturbation theory.5 (It is insufficient to retain only the first term of the series that appeared in the quantum mechanical perturbation theory.) Correct calculations in both adiabatic and diabatic approaches lead to the same results, which is evidence of the equivalence of the two approaches. 

First we shall consider entirely nonadiabatic reactions for which formulas of the quantum mechanical perturbation theory in the first order in the interaction, leading to reaction may be used for the calculation of the... 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

Equation 4.9 has been extensively applied to study the mechanisms of electrophilic (e.g., protonation) reactions, drug-nucleic acid interactions, receptor-site selectivities of pain blockers as well as various other kinds of biological activities of molecules in relation to their structure. Indeed, the ESP has been hailed as the most significant discovery in quantum biochemistry in the last three decades. The ESP also occurs in density-based theories of electronic structure and dynamics of atoms, molecules, and solids. Note, however, that Equation 4.9 appears to imply that p(r) of the system remains unchanged due to the approach of a unit positive charge in this sense, the interaction energy calculated from V(r) is correct only to first order in perturbation theory. However, this is not a serious limitation since using the correct p(r) in Equation 4.9 will improve the results. 

Since the PE surfaces are anharmonic in this limit, the Franck-Condon contributions can be difficult to treat. The approaches to this limit can be separated into two classes (a) perturbation theory corrections of the weak-coupling limit and (b) quantum-mechanical calculations of the reaction coordinate. The latter tend to be done on a reaction-by-reaction basis this makes it difficult to generalize. The perturbation theory approaches have the advantage that they make use of the parameters used in the weak-coupling limit, and this can provide usefiil insights into general trends and patterns. 

Quantum mechanical approaches for describing electron transfer processes were first applied by Levich [4] and Dogonadze, and later also in conjunction with Kuznetsov [5]. They assumed the overlap of the electronic orbitals of the two reactants to be so weak that perturbation theory, briefly introduced in the previous section, could be used to calculate the transfer rate for reactions in homogeneous solutions or at electrodes. The polar solvent was here described by using the continuum theory. The most important step is the calculation of the Hamiltonians of the system. In general terms the latter are given for an electron transfer between two ions in solution by... 

To aid the modelers in developing improved reaction mechanisms as well as to aid the experimentalists in their interpretation of the data, we have calculated the energetics of molecular intermediates and products arising from these reactions. Our approach was to use the highly accurate fourth order M ller-Plesset perturbation theory (24) with bond-additivity corrections. (2 )... 

However, the data of direct quantum chemical calculations on the PES of this reaction indicate strong steric repulsion in the case of the supra-antara approach I, which makes this mechanism energetically unfavored [2, 3]. But the route II of the parallel approach with the synchronous formation of two bonds C—C is equally energetically unrealizable. The MINDO/3 calculations with precise localization of the transition state by minimization of the gradient norm lead to the structure III. A three-center Cj—C2—C3 interaction, predictable from the perturbation theory [5], takes place in this structure. The form of the transition vector III reflects the character of the carbon atom shifts which determine the reaction path where the processes of breaking and making of the C—C bonds are sharply asynchronous. The ab initio calculations [3, 6] lead to the same conclusion, they indicate a transition state with the structure IV in which two CC bonds lying in parallel planes are spaced 2.237 A apart. 

An important conclusion throwing some light on the reasons for the discrepancy between the semiempirical and the ab initio calculations of the mechanism of [4 4-2]-cycloaddition reactions was recently arrived at in Ref. [26] where the effect the correlation corrections have on the PES of this reaction was studied by means of the M ller-Plesset perturbation theory (see Sect. 2.2.4). The authors used the minimal STO-3G basis set and restricted themselves to the calculation of the section of the PES assuming the sum of the lengths of the forming 5-bonds to be equal to 4.4 A. Figure 10.2 shows... 

The principal notions and conceptual systems of theoretical organic chemistry have been evolved from generalizations and rationalizations of the results of research into reaction mechanisms. In the sixties the data from quantum mechanical calculations began to be widely invoked to account for and predict the reactivity of organic compounds. In addition to and in place of the notions derived on the basis of the resonance and mesomerism theories that earlier had been treated semiquantitatively by means of correlation equations, novel research tools came to be employed such as reactivity indices, perturbation MO theory, or the Woodward-Hoffmann rules. It is very characteristic of these approaches, which have now taken firm root in the field of theoretical chemistry, that they, on the whole, imply an a priori assumption of the mechanism and probable structures of the transition states of reactions. 

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