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2017 | OriginalPaper | Chapter

5. Adiabatic, Born-Oppenheimer, and Non-adiabatic Approaches

Author : Monika Stanke

Published in: Handbook of Computational Chemistry

Publisher: Springer International Publishing

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Abstract

A detailed derivation of the adiabatic approximation and the Born-Oppenheimer approximation is presented, the difference between these two approximations is discussed and the circumstances under which the adiabatic approximation collapses are discussed. It is shown that the solution of the Schrödinger equation in the adiabatic approximation can be divided into one representing the motion of electrons in the field of fixed nuclei and another one representing the motion of nuclei in the potential generated by the presence of the electrons. The shapes of the potential energy curves generated by the electrons and the motion of the nuclei in these potentials are also analyzed. Finally, the state-of-the-art highly accurate calculations for diatomic molecules performed without the use of the Born-Oppenheimer approximation is presented.

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Precisely, the BO and adiabatic approximations are based on the difference of the time scales of movements of the nuclei and the electrons. For example, in some weakly bound molecular anions, the excess electron is very weakly bound (e.g., in the water anion). Such an electron moves at a speed comparable to the oscillating motion of atomic nuclei. A complete description of the dynamics of such an anion in the framework of the adiabatic approximation is doomed to failure. The failure can be attributed to the fact that the electron may be unbound for some large areas of the configuration space of the nuclei. In these areas the electronic wave function is not square integrable and the adiabatic corrections are divergent.

We use the most common definition of the new coordinates. Internal coordinates may be defined in many different ways, see, e.g., Piela (2007).

“Frolov’s calculations showed that when one mass increased without limit (the atomic case), any discrete spectrum persisted, but when two masses were allowed to increase without limit (the molecular cause), the Hamiltonian ceased to be well defined, and this failure led to what he called adiabatic divergence in attempts to compute discrete eigenstates of (21). This divergence is discussed in some mathematical detail in the Appendix to Frolov (1999). It does not arise from the choice of a translationally invariant form for the electronic Hamiltonian; rather it is due to the lack of any kinetic energy term to dominate the Coulomb potential. Thus it really is essential to characterize the spectrum of H _elec to see whether the traditional approach can be validated.” (Sutcliffe 2003)

For the sake of simplicity, we denote ψ _k ^el(r; R) ≡ ψ _k ^el and χ _k(R) ≡ χ _k.

As a result of simple transformations:

More inventive faculty trying to explain to students the BO approximation takes an example of a cow (symbolizes the nucleus) and flies flying around it (electrons). Flies almost immediately adapt to the current position of the cow, just because they are lighter and move faster. Therefore, the cow only sees a cloud of flies, while the flies only see a static cow.

In fact, there exist rovibrational states, e.g., in the helium dimer, which cannot be properly described by the model of a harmonic oscillator.

This model works well for molecules like NH₃, CH₃Cl, C₆H₆. In a general case the asymmetric top model should be used (see Haken and Wolf 2004 for details).

Crossing the states of the same symmetry is possible if you work within the adiabatic approximation.

From now on the symbol “tot” will be used to denote the sum of the total nonrelativistic energy and the corrections up to the certain order in terms of the hyperfine constant α.

Optimal values of these parameters in the basis functions (126) and (127) are determined using the gradient method.

Unfortunately in the results of Wolniewicz, the values for the v = 22 are missing.

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Title: Adiabatic, Born-Oppenheimer, and Non-adiabatic Approaches
Author: Monika Stanke
Publisher: Springer International Publishing
Book: Handbook of Computational Chemistry
Print ISBN: 978-3-319-27281-8

Electronic ISBN: 978-3-319-27282-5

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-27282-5_41

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