Born-Oppenheimer approximation potential curve

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

A good basis for the qualitative understanding of the Pgl process and its theoretical description is the potential curve model of Pgl, 21 which was developed and applied6-14 prior to the theoretical formulation of Pgl (see Fig. 1). The spontaneous ionization occurring with probability F(Rt)/h at some distances R, is the vertical transition V+(RI)—>V+(RI), as indicated in the diagram. This vertical condition is a consequence of the Born-Oppenheimer approximation and has nothing to do with the approxima-... 

This is so because no coupling between electronic and nuclear motion is assumed within the Born-Oppenheimer approximation, which in the classical limit leads to separate conservation of the instantaneous heavy-particle motion. Denoting by EA(R,) and (/ ,) the instantaneous kinetic energy at the moment of transition in the upper- and lower potential curve,... 

Consider a diatomic molecule such as H Cl. Within the Born-Oppenheimer approximation, we focus attention on the electronic wavefunction and calculate enough data points to give a potential energy curve. Such a curve shows the variation of the electronic energy with intemuclear separation. The nuclei vibrate in this potential. 

Indeed, Dunham s energy-level formula [Eq. (2.1.1)] is based both on the concept of a potential energy curve, which rests on the separability of electronic and nuclear motions, and on the neglect of certain couplings between the angular momenta associated with nuclear rotation, electron spin, and electron orbital motion. The utility of the potential curve concept is related to the validity of the Born-Oppenheimer approximation, which is discussed in Section 3.1. 

The potential energy curves for the motion of the nuclei for electronic states computed at the Born-Oppenheimer approximation for diatomics... 

Coming finally to genuine chemistry,we may accept the Born-Oppenheimer approximation ( ) and replace the potential curve (for each many-electron state) of a diatomic species,in a system with N nuclei having (3N-6) mutually independent ipternuclear distances. 

The equilibrium structure of a molecule is conceived as the hypothetical vibrationless state, i.e. the state in which all intramolecular modes of vibration are imagined as frozen at the minima of their potential energy curves. This concept, and indeed the entire concept of a potential energy surface which arises from the electronic structure of a molecule, depends on the Born-Oppenheimer approximation which is virtually always made in studies of molecular structure. 

Separation of Electronic and Nuclear Motion. Because, in general, electrons move with much greater velocities than nuclei, to a first approximation electron and nuclear motions can be separated (Born-Oppenheimer theorem [3]). The validity of this separation of electronic and nuclear motions provides the only real justification for the idea of a potential-energy curve of a molecule. The eigenfunction Y for the entire system of nuclei and electrons can be expressed as a product of two functions F< and T , where is an eigenfunction of the electronic coordinates found by solving Schrodinger s equation with the assumption that the nuclei are held fixed in space and Yn involves only the coordinates of the nuclei [4]. 

Once the potential energy curve E (R ) vs R of a diatomic molecule has been determined from the Schrodinger equation of the electronic problem in the fixed-nucleus approximation (Born-Oppenheimer), there are various methods to determine the force constants and vibrational frequencies non-empirically. These methods will now be described below. 

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