Chemical bond breaking/forming as a Franck–Condon electronic process

Abstract

We describe chemical bond changes as Franck–Condon electronic processes within a new theoretical ansatz that we call ‘rigged’ Born–Oppenheimer (R-BO) approach. The notion of the separability of nuclear and electron states implied in the standard Born–Oppenheimer (BO) scheme is retained. However, in the present scheme the electronic wave functions do not depend upon the nuclear coordinate ( R-space). The new functions are obtained from an auxiliary Hamiltonian corresponding to the electronic system ( r-coordinates) submitted to a Coulomb potential generated by external sources of charges in real space ( α-coordinates) instead of massive nuclear objects. A stationary arrangement characterized by the coordinates α 0A, is determined by a particular electronic wave function, ψ( r; α 0A); it is only at this stationary point, where an electronic Schrödinger equation: H e( r, α 0A)| Ψ( r; α 0A)〉= E( α 0A)| Ψ( r; α 0A)〉 must hold. This equation permits us to use modern electronic methods based upon analytic first and second derivatives to construct model electronic wave functions and stationary geometry for external sources. If the set of wave functions { Ψ( r; α 0A)} is made orthogonal, the energy functional in α-space, E( α; α 0A)=〈 Ψ( r; α 0A)| H e( r, α 0A)| Ψ( r; α 0A)〉 is isomorphic to a potential energy function in R-space: E( R; α 0A)=〈 Ψ( r; α 0A)| H e( r, R)| Ψ( r; α 0A)〉. This functional defines, by hypothesis, a trapping convex potential in R-space and the nuclear quantum states are determined by a particular Schrödinger equation. The total wave function for the chemical species A reads as a product of our electronic wave function with the nuclear wave function ( Ξ ik ( R; α 0A)): Φ ik ( r, R)= Ψ i ( r, α 0A) Ξ ik ( R; α 0A). This approach facilitates the introduction of molecular frame without restrictions in the R-space. Two molecules (characterized with different electronic spectra) that are decomposable into the same number of particles (isomers) have the same Coulomb Hamiltonian and they are then characterized by different electronic wave functions for which no R-coordinate ‘deformation’ can possibly change its electronic structure. A bond breaking/forming process must be formally described as a spectroscopic-like electronic process. The theory provides an alternative to the adiabatic as well as the diabatic scheme for understanding molecular processes. As an illustration of the present ideas, the reaction of H 2+CO leading to formaldehyde is examined in some detail.