Born-Oppenheimer approximation introduced

For the majority of systems the Born-Oppenheimer approximation introduces only very small errors. The diagonal Born-Oppenheimer correction (DBOC) can be evaluated relatively easy, as it is just the second derivative of the electronic wave function with respect to the nuclear coordinates, and is therefore closely related to the nuclear gradient and second derivative of the energy (Section 10.8). 

Abstract. The Born-Oppenheimer approximation, introduced in the 1927 paper On the quantum theory of molecules , provides the foundation for virtually all subsequent theoretical and computational studies of chemical binding and reactivity, as well as the justification for the universal ball and stick picture of molecules as atomic centers attached at fixecl distances by electronic glue. 

The Franck-Condon principle is essentially a restatement of the Born-Oppenheimer approximation (introduced in Chapter 2), as it assmnes that the electronic transition occurs so quickly that the nuclear coordinates remain stationary. Phonon... 

At first, we shall calculate the transition probability for fixed quantum numbers of the electrons, a and jS. The subscripts a and (3 will, however, be temporarily omitted. To separate the electronic and nuclear movements in the channels of the reaction, we shall use the Born-Oppenheimer approximation. Introducing the integral representation of the 8 function, Eq. (14) may be transformed to... 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

In this section the Born-Oppenheimer approximation will be presented in what is necessarily a very simplified form. It has already been introduced without justification in Section 6.5. It is certainly the most important - and most satisfactory - approximation in quantum mechanics, although its rigorous derivation is far beyond the level of this book. Consider, therefore, the Mowing argument... 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

Abstract The Born-Oppenheimer approximation is introduced and discussed. This approximation, which states the potential energy surface on which the molecule vibrates/rotates is independent of isotopic substitution, is of central importance in... 

Background Philosophy. Within the framework of the Born-Oppenheimer approximation (JJ ), the solutions of the Schroedin-ger equation, Hf = Ef, introduce the concept of molecular structure and, thereby, the total energy hyperspace provided that the electronic wave function varies only slowly with the nuclear coordinates, electronic energies can be calculated for sets of fixed nuclear positions. The total energies i.e. the sums of electronic energy and the energy due to the electrostatic re-... 

The terms p and ep are similar to diabatic surfaces (35,6A, 65). We have introduced them starting directly from the Born-Oppenheimer approximation. 

The Born-Oppenheimer separation19-22 of the electronic and nuclear motions in molecules is probably the most important approximation ever introduced in molecular quantum mechanics, and will implicitly or explicitly be used in all subsequent sections of this chapter. The Born-Oppenheimer approximation is crucial for modern chemistry. It allows to define in a rigorous way, within the quantum mechanics, such useful chemical concepts like the structure and geometry of molecules, the molecular dipole moment, or the interaction potential. In this approximation one assumes that the electronic motions are much faster than the nuclear... 

In both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

Phenomenological treatments which approximate the molecular potential field (Born-Oppenheimer approximation) by a series of classical energy equations and adjustable parameters. These treatments may be called classical mechanical only in the sense that harmonic force-field expressions stemming from vibrational analysis methods are often introduced, though strictly speaking one is free to select any set of functions that reproduces the experimental data whitin chosen limits of accuracy. 

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

This method of dividing up problems in the behaviour of systems of electrons and nuclei, called the Born-Oppenheimer approximation , was introduced by Max Born and J. R Oppenheimer, working together in Gottingen in 1927... 

The Born-Oppenheimer approximation (Chapter 6) that separates the motion of the nuclei from the motion of the electrons. This approximation allows us to introduce the concept of a 3-D structure of the molecule the heavy nuclear framework of the molecule kept together by "electronic glue moves (translation), and at the same time rotates in space. 

This matrix was introduced by F. T. Smith [25] for the treatment of non-adiabatic (diabatic) couplings in atomic collisions. It is now familiar also in molecular structure problems, to indicate local breakdowns of the Born-Oppenheimer approximation. Within the hyperspherical formalism, it has been introduced in the three-body Coulomb problem [20] and in chemical reactions [21-24], see also Section 3. Also, from equation (A4)... 

The Born-Oppenheimer approximation exploits the fact that the nuclear mass is very much larger than the electronic mass, and therefore the nuclear dynamics are expected to be slow in comparison to the electronic dynamics. Thus, it is convenient to introduce an electronic state, i R ), that is determined by a set of static nuclear coordinates, R. i R ) thus depends parametrically on R. ... 

In Section 2.1, the electronic problem is formulated, i.e., the problem of describing the motion of electrons in the field of fixed nuclear point charges. This is one of the central problems of quantum chemistry and our sole concern in this book. We begin with the full nonrelativistic time-independent Schrodinger equation and introduce the Born-Oppenheimer approximation. We then discuss a general statement of the Pauli exclusion principle called the antisymmetry principle, which requires that many-electron wave functions must be antisymmetric with respect to the interchange of any two electrons. 

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