Born-Oppenheimer models

B3.1.1.1 THE UNDERLYING THEORETICAL BASIS—THE BORN-OPPENHEIMER MODEL... 

In the Car-Parrinello method [6] (and see, e.g., [24, 25, 16, 4]), the adiabatic time-dependent Born-Oppenheimer model is approximated by a fictitious Newtonian dynamics in which the electrons, represented by a set of... 

Strictly speaking, a chiral species cannot correspond to a true stationary state of the time-dependent Schrodinger equation H

time scale for such spontaneous racemization is extremely long. The wavefunction of practical interest to the (finite-lived) laboratory chemist is the non-stationary [Pg.42]

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

In summary, the adiabatic approximation defined within the Born-Oppenheimer model leads to the equation. 

The full derivation of the Born-Oppenheimer model is rather more involved and can be found in Dynamical Theory of Crystal Lattices by M. Born and K. Huang (OUP, 1954). 

Fig. 5.22. An experimental scan over two lines corresponding to 2vz In <— Ooo and 2i2 loi transitions in D2H+ ion. Full and open arrows indicate theoretical predictions based on semi-empirical and ah initio non-Born-Oppenheimer model treatments, respectively.

Here E and E are the exact energies of the two individual molecules A and B when they are isolated, while E" is the exact energy of the supersystem (molecular complex, for example). Theoretically, these quantities can be obtained from the exact solution of the Schrodinger equation for the corresponding systems. (We remain within the nonrelativistic Born-Oppenheimer model.) This requires the definition of the Hamiltonians H", H and H" , and one feels challenged to handle these Hamiltonians in a common (e.g., perturbational) scheme. This point is not at all trivial especially if approximate model Hamiltonians are used. In what follows we shall consider this issue emphasizing the points where the second quantized approach can help to clarify the situation. 

No such problem is encountered with the partition of nuclei expressed by the symbols a G A, as far as one works within the Born-Oppenheimer model. When considering the electronic Schrodinger equation, nuclei are not treated as quantum particles, their role is merely to give rise an external potential for electrons. 

Here, 1 > and 2 > are the diabatic electronic states (exciton and CT states, respectively), J their electronic coupling parameter, uj the vibrational frequency, 6 and h the boson creation and annihilation operators, and g and g2 the equilibrium position shifts in the excited states 1 > and 2 >, and AE the zeroth order splitting between the two electronic states. The zero energy is set to ( AE - Eg (jS) where ft = 1. and Eg is the ground state electronic energy of a Born-Oppenheimer model. 

The second kind of evidence concerned the explanatory applications of quantum mechanics to molecules. These explanations, 1 argue, fail to display the direction of explanation that physicalism requires. Remember Woolley s point that Born-Oppenheimer models assume but do not explain molecular structures. It is natural to read the attribution of such structures as the direct attribution of a state to the molecule as a whole, a state that is not further explained in terms of the more fundamental force laws governing pairwise interactions between the constituent electrons and nuclei. Civen that this state constrains the quantised motions of the functional groups appearing in the spectroscopic explanation, the direction... 

From the preceding sections, it seems evident that a real description of ion specificities in solutions can only be done if the geometry and the properties of water molecules are explicitly taken into account. Such models are called non-primitive or Born-Oppenheimer models. In the 1970s and 1980s, they were developed in two different directions. In particular, integral equation theories, such as the hypernetted chain (HNC) approach, were extended to include angle-dependent interaction potentials. The site-site Ornstein-Zernike equation with a HNC-like closure and the molecular Ornstein-Zernike equation are examples. For more information, see Ref. 17. 

The theoretical approach based on the HNC integral equation is described in the context of ionic specificity. Two levels of description of the water medium are considered. Within the Primitive Model (continuous solvent), ionic specificity is introduced via effective, solvent-averaged, dispersion forces. The agreement with experimental data in bulk or at air-water interfaces is only partial and illustrates the limits of that approach. Within the Born-Oppenheimer model, the molecular HNC equation is solved with an explicit description of the solvent molecules (SPC water). Ionic and solvent profiles in bulk and at interfaces are enriched by short-range osdUated structures. The ionic polaris-ability is introduced via the self-consistent mean-field theory, the polarisable ions carrying an effective, fixed dipole moment. The study of the air-water interface reveals the limits of the conventional HNC approach and the needs for improved integral equations. 

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