Born-Oppenheimer approximation, for

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... 

Perhaps the first evidence for the breakdown of the Born-Oppenheimer approximation for adsorbates at metal surfaces arose from the study of infrared reflection-absorption line-widths of adsorbates on metals, a topic that has been reviewed by Hoffmann.17 In the simplest case, one considers the mechanism of vibrational relaxation operative for a diatomic molecule that has absorbed an infrared photon exciting it to its first vibrationally-excited state. Although the interpretation of spectral line-broadening experiments is always fraught with problems associated with distinguishing... 

Potential energy surface for a chemical reaction can be obtained using electronic structure techniques or by solving Schrodinger equation within Born-Oppenheimer approximation. For each geometry, there is a PE value of the system. 

Fig. 2.1 The adiabatic correction to the Born-Oppenheimer approximation for H2 and HD schematic, not to scale AC = C(H2)-C(HD). In each case the uncorrected potential lies to the left, the corrected to the right...

It is mentioned in passing that the proper masses mA and mB to be used in Equation 3.3 are the atomic masses (nucleus + electrons) rather than the respective nuclear masses as might be expected from a strict Born-Oppenheimer approximation. For further discussion of this point, reference should be made to the reading lists at the end of this chapter and of Chapter 2. The combination of Equations 3.1 and 3.2 corresponds to a classical harmonic oscillator with force constant f and mass p. The harmonic oscillator frequency v is given by the well-known formula... 

Studies of the Pgl electron spectra have shown that, in spite of all the mentioned complications, many Pgl systems with molecular targets can still be well described within the theory of simple Pgl, if only the possibility of vibrational transitions is incorporated into the function r(fl). Within the Born-Oppenheimer approximation for both the projectile-target motion and the intramolecular motion, this is done in the following way. We denote by r,(rt) the width belonging to a certain final electronic state, defined as in (11.85). Then r,( ) can, at any distance R, be decomposed as... 

Chapter 3 describes radiationless transitions in the tunneling electron transfers in multi-electron systems. The following are examined within the framework of electron Green s function approach the dependence on distance, the influence of crystalline media, and the effect of intermediate particles on the tunneling transfer. It is demonstrated that the Born-Oppenheimer approximation for the wave function is invalid for longdistance tunneling. 

The usual Born-Oppenheimer approximation for molecular wave functions gives rise to the separation of the wave function (b into the nuclear and electronic components... 

The field gradient is measured at a fixed point within the molecule, the translational part of the wave-function is thus of no consequence for (qap)-The effect of molecular rotation does, however, modify (qap) but the relationship between the rotating and stationary (qap) s has already been treated in the chapter dealing with microwave spectroscopy. In the present context, we are interested in the field gradients in a vibrating molecule in a fixed coordinate system. The Born-Oppenheimer approximation for molecular wave-functions enables us to separate the nuclear and electronic motions, the electronic wave functions being calculated for the nuclei in various fixed positions. The observed (qap) s will then be average values over the vibrational motion. 

There are two reasons why so much is unknown. First, at high densities three (and even four) body forces are important. This is particularly so when chemically reactive atoms are present. Then, even for two-body forces, the strongly repulsive regime is not well understood and, in addition, close in, as one approaches the united atom limit, there is considerable promotion of molecular orbitals. This is a universal mechanism for electronic excitation which means a breakdown of the Born-Oppenheimer approximation for close collisions. 

Another interesting example of a classical chemical concept is the nuclear framework of a molecule. The usual Born-Oppenheimer approximation for molecules shows that this is not a strictly classical structure. With respect to the states and (see Fig. 3), for example, the... 

A molecular Hamiltonian usually has embedded parameters. Within the Born-Oppenheimer approximation, for instance, the nuclear positions are parameters. Without loss of generality, any parameters can be embedded in an arbitrary Hamiltonian, and these will be designated a,b,c,. . ., ... 

In the case of polynuclear systems like molecules and solids, it is common to use the standard nonrelativistic Born-Oppenheimer approximation for the separation of... 

The one electron He- H++ ion system, considered written the Hamiltonian (Born-Oppenheimer approximation) for the electronic energy in atomic units ... 

The method makes no reference to electrons, and so cannot (except by some kind of empirical algorithm) throw light on electronic properties like charge distributions or nucleophilic and electrophilic behaviour. Note that MM implicitly uses the Born-Oppenheimer approximation, for only if the nuclei experience what amounts to a static attractive force, whether from electrons or springs, does a molecule have a distinct geometry (section 2.3). 

Fig. 4 Possible structures (Born-Oppenheimer approximation), for methanium ion, 2, optimized at the MP2(full)/6-31G(d) level. 2a is the absolute minimum 2b and 2c are transition states.

Note that the wave functions for the initial and final states and include both donor and acceptor. This equation is usually simplified by making the Born-Oppenheimer approximation for the separation of nuclear and electron wave functions, resulting in equation (12), in which V is the electronic matrix element describing the coupling between the electronic state of the reactants with those of the product, and FC is the Franck-Condon factor. 

Wodtke, A.M., Tully, J.C., Auerbach, D.J. Electronically non-adiabatic interactions of molecules at metal surfaces Can we trust the Born-Oppenheimer approximation for surface chemistry , Int. Rev. Phys. Chem. 2004,23,513. 

Schwenke, D.W. Beyond the potential energy surface Ab initio corrections to the Born-Oppenheimer approximation for H2O, J. Phys. Chem. A 2001,105, 2352-60. 

For atoms, these dimensional scalings amount to nothing more than a simple change of variables. We are free to use one scaling at = 0 and another at = 1 no inconsistency results om this. In the case of molecules, however, there is an additional subtlety, due to the Born-Oppenheimer approximation. For example, the potential energy for H, in cylindrical coordinates, is... 

The potential energy curves for the motion of the nuclei for electronic states computed at the Born-Oppenheimer approximation for diatomics... 

Reactions of the type (37) are not suitable for study in electrochemical cells ( ) because they inTOlve the isotopically-mixed species HD. Howeirer, a reaction like (36) can be investigated directly in an electrochemical cell. The basic idea is to compare an experimental value of K for reaction (36) with two calculated (by Wolfsberg, et al.) values of K, one obtained invoking the Born-Oppenheimer approximation, and the other obtained without invoking the Born-Oppenheimer approximation. In this way the reliability of the Born-Oppenheimer approximation for hydrogen-isotope-exchange reaction can be tested experimentally. 

Fig. 7.6. Failure of the Born-Oppenheimer approximation for HD. The shaded circle on the left represents the deuterium nucleus, the shaded circle on the right the proton. The frozen electron distribution is indicated by an ellipse.

In the Born-Oppenheimer approximation for the proton, the scheme of the calculation is as follows if a condition similar to (45) holds... 

Born-Oppenheimer Approximation for Continuum Electronic States ... 

It is beyond the scope of this work to enter the subject of gauge theory deeply. We shall only illustrate some results relevant to the group-Born-Oppenheimer approximation. For more details we refer to Ref. 5 and references therein. 

In Chap. 2, we formulate our basic framework of the chemical theory based on the Born-Oppenheimer approximation. We briefly discuss how valid or how accurate the Born-Oppenheimer approximation for bound states is. Also a theory of electron scattering by polyatomic molecules within the Born-Oppenheimer framework (or the so-called fixed nuclei approximation) is presented. This is one of the typical theories of electron dynamics, along with the theory of molecular photoionization. 

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