CHAPTER 1 Introduction In quantum physics, the evolution of a molecule is described by the initial-value Schr¨ odinger system, (1.1) i∂tϕ = ϕ |t=0 = ϕ0, where ϕ0 is the initial state of the molecule and H stands for the molecular Hamil- tonian involving all the interactions between the particles constituting the molecule (electron and nuclei). In case the molecule is imbedded in an electromagnetic field, the corresponding potentials enter the expression of H, too. Typically, the interac- tion between two particles of positions z and z , respectively, is of Coulomb type, that is, of the form α|z z |−1 with α IR constant. In the case of a free molecule, a first approach for studying the system (1.1) consists in considering bounded initial states only, that is, initial states that are eigenfunctions of the Hamiltonian after removal of the center of mass motion. More precisely, one can split the Hamiltonian into, H = HCM + HRel, where the two operators HCM (corresponding to the kinetic energy of the center of mass) and HRel (corresponding to the relative motion of electrons and nuclei) commute. As a consequence, the quantum evolution factorizes into, e−itH = e−itHCM e−itHRel, where the (free) evolution e−itHCM of the center of mass can be explicitly computed (mainly because HCM has constant coefficients), while the relative motion e−itHRel still contains all the interactions (and thus, all the difficulties of the problem). Then, taking ϕ0 of the form, (1.2) ϕ0 = α0 ψj where α0 depends on the position of the center of mass only, and ψj is an eigen- function of HRel with eigenvalue Ej, the solution of (1.1) is clearly given by, ϕ(t) = e−itEj (e−itHCMα0) ψj. Therefore, in this case, the only real problem is to know sufficiently well the eigenele- ments of HRel, in order to be able to produce initial states of the form (1.2). In 1927, M. Born and R. Oppenheimer [BoOp] proposed a formal method for constructing such an approximation of eigenvalues and eigenfunctions of HRel. This method was based on the fact that, since the nuclei are much heavier than the electrons, their motion is slower and allows the electrons to adapt almost instanta- neously to it. As a consequence, the motion of the electrons is not really perceived 1
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