Kinetic energy operator Born-Oppenheimer approximation

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

The Born-Oppenheimer approximation consists neglecting the kinetic energy operator T in the full Hamiltonian (la.I), which means solving the wave equation (2.1) at fixed nuclear coordinates by representing the wave function for a given electronic state by the product... 

We will start by reviewing the Born-Oppenheimer approximation in more detail. The total (non-relativistic) Hamiltonian operator can be written as kinetic and potential energies of the nuclei and electrons. 

Where T and Tei are the kinetic energy operators for the nuclei and electrons, and Vmi, Vne and Vee are the electrostatic potential energies arising from intemuclear, nucleus-electron, and interelectronic interactions. At this point, the Born-Oppenheimer approximation is invoked, i.e., the assumption that the movement of electrons is much faster than that of the nuclei and therefore the two can be decoupled from one another. This is mathematically represented by a separation of variables in the wavefunction, Y(R,r)= (r(R)) %(R), where <1> and % are the electronic and nuclear wavefunctions, respectively, and O is a function of r parameterized by R. Thus, with some rearrangement, Equation (21) becomes... 

The function irk(r, R) represents an eigenfunction of the electronic Hamiltonian Ho(Ry, i.e., the Hamiltonian H, in which the kinetic energy operator for the nuclei is assumed to be zero (the clamped nuclei Hamiltonian) The eigenvalue of the clamped nuclei Hamiltonian depends on positions of the nuclei and in the Born-Oppenheimer approximation, it is mass-independent. This energy as a function of the configuration of the nuclei represents the potential energy for the motion of the nuclei (Potential Energy Surface, or PES). 

Vibronic coupling through the nuclear kinetic energy operator f(Q) rather than through Q-dependence in the electronic transition moment can be treated in a manner that parallels our discussion of the Born-Oppenheimer approximation in diatomics (Section 3.1). The Born-Oppenheimer theory of vibronic coupling predicts that the induced transition moment will be... 

Effects due to the geometric phase (GP) have been reported by Joubert-Doriol, Ryabinkin and Izmaylov.In particular they report on symmetry breaking and spatial localisation, and on GP effects studied with the multi-dimensional LVQ model. In the first study by Ryabinkin and Izmaylov the ground state dynamics is considered of a two-state system approximated by (a) a Hamiltonian of a two-state Cl model, (b) the Born-Oppenheimer (BO) model and (c) a BO model augmented with an explicit GP dependence in the kinetic energy operator. It is demonstrated that... 

The masses of the nuclei, m, are at least three orders of magnitude larger than the mass of an electron. We can therefore assume that the electrons will instantaneously adjust to a change in the positions of the nuclei and that we can find a wavefunction for the electrons for each arrangement of nuclei. In the Born-Oppenheimer approximation the total molecular Hamilton operator Hnuc,e from Eq. (2.1) is thus partitioned in the kinetic energy operator of the nuclei, field free electronic Hamiltonian defined as... 

An approximation that can be made is to realize that the motion of the nuclei is sluggish relative to the motion of the electrons due to the large differences in mass. Due to the great difference in motion between the nuclei and the electrons, the electrons are capable of instantaneously adjusting to any change in position of the nuclei. Hence, the electron motion is determined for a fixed position of the nuclei making the distances Rab in Equation 9-1 now a constant. This approximation is called the Born-Oppenheimer approximation. The Bom-Oppenheimer approximation removes the kinetic energy operators for the nuclear motion in Equation 9-1. 

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