Born-Oppenheimer principle

These terms have very different values and can vary independently. This is called the Born-Oppenheimer principle. 

Fluorescence is defined simply as the electric dipole tranation from an excited electronic state to a lower state, usually the ground state, of the same multiplicity. Mathematically, the probability of an electric-dipole induced electronic transition between specific vibronic levels is proportional to R f where Rjf, the transition moment integral between initial state i and final state f is given by Eq. (1), where represents the electronic wavefunction, the vibrational wavefunctions, M is the electronic dipole moment operator, and where the Born-Oppenheimer principle of parability of electronic and vibrational wavefunctions has been invoked. The first integral involves only the electronic wavefunctions of the stem, and the second term, when squared, is the familiar Franck-Condon factor. 

Equation [73] has the same form as the equations of motion for molecules with constrained internal coordinates, and we already know that such equations can be solved effectively using the SHAKE algorithm4 ° Equations [72] and [73] play a key role in the Car-Parrinello method and enable one to run the dynamics for both ionic and electronic degrees of freedom in parallel. With carefully chosen effective mass p and a small time step, the electronic state adjusts itself instanteously to the nuclear configuration (Born-Oppenheimer principle), and, therefore, the atomic dynamics is computed along the system s Born-Oppenheimer surface. Note that there is no need to carry out the costly matrix-diagonalization procedure for performing electronic structure calculations. 

The values of these energies are very different and according to the Born— Oppenheimer principle they can vary independently of each other. 

In the elementary theory of H2, it is considered as a simple system in which vibrational, electronic and rotational motions can be separated (the Born-Oppenheimer principle) and fully analytic solutions exist (uniquely for a molecule) which show that the molecule is stable. This, however, is not the complete story. In fact, as is separated into H and H+, one encounters an additional shallow minimum near the dissociation limit, at much larger internuclear distances than its equilibrium separation. This second minimum, which arises from a dipole in the neutral fragment induced by the presence of the charged fragment, is capable of supporting... 

The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable. A formal definition of the Born-Oppenheimer principle can be made by considering the time-independent Schrodinger equation of a molecule, which is of the form... 

The Born-Oppenheimer principle is a cornerstone of molecular spectroscopy, an organizing principle that vastly simplifies the assignment of different spectral features to different types of molecular motion. Without it, electronic and nuclear motions would be scrambled in complicated molecular Hamiltonians,... 

In our discussion of the Born-Oppenheimer principle (Section 3.1) we pointed out that eigenfunctions k(r R)> of the electronic Hamiltonian... 

Eight or nine of the eleven chapters in this book can be comfortably accommodated within a one-semester course. The underlying time-dependent perturbation theory for molecule-radiation interactions is emphasized early, revealing the hierarchies of multipole and multiphoton transitions that can occur. Several of the chapters are introduced using illustrative spectra from the literature. This technique, extensively used by Herzberg in his classic series of monographs, avoids excessive abstraction before spectroscopic applications are reached. Diatomic rotations and vibrations are introduced explicitly in the context of the Born-Oppenheimer principle. Electronic band spectra are examined with careful attention to electronic structure, angular momentum... 

The calculations are based on the Born-Oppenheimer principle which separates the electronic motion from the vibrational and rotational motion of the nuclei by considering that the light electrons move much faster in a molecule than the heavy nuclei, i.e. 

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