The Born-Oppenheimer Principle

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, 

The total Hamiltonian for a diatomic molecule with n electrons is 

These contributions to H include the nuclear kinetic energy, the electronic kinetic energy, the electron-electron repulsions, the electron-nuclear attractions, the nuclear-nuclear repulsion, and the spin-orbit coupling. A priori, the electronic coordinates r and nuclear coordinates R appear to be inseparably mixed in i , and the electronic and nuclear coordinates are strongly coupled. The Schrodinger equation for the diatomic becomes 

The resulting electronic states fc(r R) will depend parametrically on R in that the choice of fixed nuclear positions influences the electronic states (physically, pulling the nuclei apart will naturally distort the molecule s electron cloud). The e R) are the diatomic potential energy curves—the electronic energies of the 

For the solutions M (r, R) to the full diatomic Hamiltonian, we now try expansions of the form 

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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. 

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... 

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. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47... 

Among the functions one can, at least in principle, calculate at the Schroedinger level is the Born-Oppenheimer (BO) potential surface, the potential of the forces among the nuclei assuming that at each nuclear configuration the time-independent Schroedinger equation is satisfied. We may think of this as the electron-averaged potential. Such an N-body potential Ujj often may be adequately represented as a sum of pair potentials... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Electronic and optical properties of complex systems are now accessible thanks to the impressive development of theoretical approaches and of computer power. Surfaces, nanostructures, and even biological systems can now be studied within ab-initio methods [53,54]. In principle within the Born-Oppenheimer approximation to decouple the ionic and electronic dynamics, the equation that governs the physics of all those systems is the many-body equation ... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

In the Born-Oppenheimer approximation, the relative importance of channels (la) and (lb), together with their dependence on wavelength would depend upon the matrix elements for the transition between the electronic states, the Franck-Condon factors, the Honl-London factors, and upon the probabilities for spontaneous dissociation of the excited state formed. In principle, except for the last one, these are well known quantities whose product is the transition probability for that particular absorption band of Cs. When multiplied by the last quantity, and with an adjustment of numerical constants i becomes the cross section for the photolysis of Cs into Cs + Cs. It is the measurement of this cross section that lies at the focus of this work. 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

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. 

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