Born-Oppenheimer effects

Since many of these developments reach into the molecular domain, the understanding of nano-structured functional materials equally necessitates fundamental aspects of molecular physics, chemistry, and biology. The elementary energy and charge transfer processes bear much similarity to the molecular phenomena that have been revealed in unprecedented detail by ultrafast optical spectroscopies. Indeed, these spectroscopies, which were initially developed and applied for the study of small molecular species, have already evolved into an invaluable tool to monitor ultrafast dynamics in complex biological and materials systems. The molecular-level phenomena in question are often of intrinsically quantum mechanical character, and involve tunneling, non-Born-Oppenheimer effects, and quantum-mechanical phase coherence. Many of the advances that were made over recent years in the understanding of complex molecular systems can therefore be transposed and extended to the study of... 

We confine the discussion in the remainder of this section to the treatment of electronic wave functions. This confinement to electronic wave functions is justified as long as no (sharply avoided) intersystem crossings are present or other non-Born-Oppenheimer effects such as rovibronic (rota-tional/vibrational/electronic) coupling are involved. Intersystem crossings will be discussed in connection to nonradiative transitions. 

H. Koppel, L. S. Cederbaum, W. Domcke and S. S. Shaik, Angew. Chem., 95,221 (1983) Angew. Chem., Int. Ed. Engl, 22,210 (1983), discuss Non Born/Oppenheimer effects in radical cations. See also L. S. Cederbaum, W. Domcke, J. Schirmer and W. v. Niessen, Adv. Chem. Phys., 5,115 (1986) on Correlation effects in the ionization of molecules breakdown of the molecular orbital picture . 

High accuracy of quantum chemical calculations not only requires a satisfactory treatment of electron correlation, but also relativistic and beyond-Born-Oppenheimer effects need to be considered [16, 17, 18]. These are not in the scope of the present review. We further concentrate on correlation effects on the energy (and on quantities directly derivable from potential energy surfaces) and we ignore correlation effects on properties, which is in important subject at present [19]. We further shall report more on the calculation than on the interpretation of correlation effects. 

The total energy in ab initio theory is given relative to the separated particles, i.e. bare nuclei and electrons. The experimental value for an atom is the sum of all the ionization potentials for a molecule there are in addition contributions from the molecular bonds and associated zero point energies. The experimental value for the total energy of FI2O is -76.480 au and the estimated contribution from relativistic effects is -0.045 au. Including also a mass correction of 0.0028 au (a non-Born-Oppenheimer effect that accounts for the difference between finite and infinite nuclear masses) allows the experimental non-relativistic energy to be estimated as -76.438 0.003 au. ... 

Non-Born-Oppenheimer effects. The assumption of a rigorous separation of nuclear and electronic motions is in most cases a quite good approximation and there is a good understanding of when it will fail. Methods for going beyond the Born-Oppenheimer approximation are still somewhat limited in term of generality and applicability. 

H. Koppel, L. S. Cederbaum, W. Domcke, and S. S. Shaik, Angew. Ghent., 95, 221 (1983). Symmetry-Breaking and Non-Born-Oppenheimer Effects in Radical Cations. 

Such predictions require the careful consideration of a number of contributions, in addition to the electronic part we will have to account for the vibrational, relativistic or non-Born-Oppenheimer effects, as implemented in, e.g., a number of model chemistry protocols for high accuracy thermochemical predictions. The central issue is the accurate solution of the electronic Schrodinger equation, which usually contributes the main portion to the molecular energy or property. 

NON-ADIABATIC EFFECTS IN CHEMICAL REACTIONS EXTENDED BORN-OPPENHEIMER EQUATIONS AND ITS APPLICATIONS... 

There are phenomena such as the Renner and the Jahn-Teller effects where the Bom-Oppenheimer approximation breaks down, hut for the vast majority of chemical applications the Born-Oppenheimer approximation is a vital one. It has a great conceptual importance in chemistry without it we could not speak of a molecular geometry. 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Evaluating the energy e for different values of R gives the effective potential for the nuclei in the presence of the electron. This function is called the Born-Oppenheimer potential surface or just the potential surface. In order to evaluate e(R) we have to determine HAA, HAB, and SAB. These quantities, which can be evaluated using elliptical coordinates, are given by... 

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 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... 

A disadvantage of this technique is that isotopic labeling can cause unwanted perturbations to the competition between pathways through kinetic isotope effects. Whereas the Born-Oppenheimer potential energy surfaces are not affected by isotopic substitution, rotational and vibrational levels become more closely spaced with substitution of heavier isotopes. Consequently, the rate of reaction in competing pathways will be modified somewhat compared to the unlabeled reaction. This effect scales approximately as the square root of the ratio of the isotopic masses, and will be most pronounced for deuterium or... 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

The effects of deviations from the Born-Oppenheimer approximation (BOA) due to the interaction of the electron in the sub-barrier region with the local vibrations of the donor or the acceptor were considered for electron transfer processes in Ref. 68. It was shown that these effects are of importance for long-distance electron transfer since in this case the time when the electron is in the sub-barrier region may be long as compared to the period of the local vibration.68 A similar approach has been used in Ref. 65 to treat non-adiabatic effects in the sub-barrier region in atom transfer processes. However, nonadiabatic effects in the classically attainable region may also be of importance in atom transfer processes. In the harmonic approximation, when these effects are taken into account exactly, they manifest themselves in the noncoincidence of the... 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

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