Born- Oppenheimer approximation, and

As ab initio MD for all valence electrons [27] is not feasible for very large systems, QM calculations of an embedded quantum subsystem axe required. Since reviews of the various approaches that rely on the Born-Oppenheimer approximation and that are now in use or in development, are available (see Field [87], Merz ]88], Aqvist and Warshel [89], and Bakowies and Thiel [90] and references therein), only some summarizing opinions will be given here. 

The equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

It should also be mentioned that a theoretical model using an empirical LEPS potential energy surface has successfully been used to reproduce the vibrational population distribution of the products of this surface reaction.40 This approach confines itself to the assumptions of the Born-Oppenheimer approximation and underscores one of the major questions remaining in this field do we just need better Born Oppenheimer potential surfaces or do we need a different theoretical approach ... 

Below we will restrict ourselves to the Born-Oppenheimer approximation and, unlike Refs. 62, 64, and 65, we will take into account the contribution from the excited vibrational states of the tunneling particle and consider the role played by the transverse quantum vibrations of the tunneling particle itself in the preparation of the potential barrier.48... 

The approximation involved in Eq. (B.17) is known as the Born-Oppenheimer approximation and this equation is called the Born-Oppenheimer equation. 

One way to simplify the Schrodinger equation for molecular systems is to assume that the nuclei do not move. Of course, nuclei do move, but their motion is slow compared to the speed at which electrons move (the speed of light). This is called the Born-Oppenheimer approximation, and leads to an electronic Schrodinger equation. 

III. Born-Oppenheimer Approximations and Point-Group Symmetry. 8... 

III. BORN-OPPENHEIMER APPROXIMATIONS AND POINT-GROUP SYMMETRY... 

A good basis for the qualitative understanding of the Pgl process and its theoretical description is the potential curve model of Pgl, 21 which was developed and applied6-14 prior to the theoretical formulation of Pgl (see Fig. 1). The spontaneous ionization occurring with probability F(Rt)/h at some distances R, is the vertical transition V+(RI)—>V+(RI), as indicated in the diagram. This vertical condition is a consequence of the Born-Oppenheimer approximation and has nothing to do with the approxima-... 

The relative motion (slow degree of freedom) is analogous to nuclear motion in the Born-Oppenheimer approximation and the fast vibrational motion is analogous to electronic motion. The nuclear wavefunction ij> (p,q) can be written as a product... 

If the wave functions of individual species are separable (Born-Oppenheimer approximation), and if there is a weak energy coupling between the system of long-lived, chemically reacting particles with other degrees of freedom, the total probability distribution function p(t, r, v, e) of the system can be separated ... 

The Smith group has also developed the methodology for making high precision calculations for small systems without invoking the Born-Oppenheimer approximation and have made calculations for two-electron atomic ions, small muonic molecules, and potentials of the screened Coulomb form. Their method for determining nonlinear parameters is now referred to as random tempering.169... 

The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born-Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as ... 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

So far, this discussion of selection rules has considered only the electronic component of the transition. For molecular species, vibrational and rotational structure is possible in the spectrum, although for complex molecules, especially in condensed phases where collisional line broadening is important, the rotational lines, and sometimes the vibrational bands, may be too close to be resolved. Where the structure exists, however, certain transitions may be allowed or forbidden by vibrational or rotational selection rules. Such rules once again use the Born-Oppenheimer approximation, and assume that the wavefunctions for the individual modes may be separated. Quite apart from the symmetry-related selection rules, there is one further very important factor that determines the intensity of individual vibrational bands in electronic transitions, and that is the geometries of the two electronic states concerned. Relative intensities of different vibrational components of an electronic transition are of importance in connection with both absorption and emission processes. The populations of the vibrational levels obviously affect the relative intensities. In addition, electronic transitions between given vibrational levels in upper and lower states have a specific probability, determined in part... 

From the general considerations presented in the previous section, one can expect that the many-body non-adiabatic wave function should fulfill the following conditions (1) All particles involved in the system should be treated equivalently (2) Correlation of the motions of all the particles in the system resulting from Coulombic interactions, as well as from the required conservation of the total linear and angular momenta, should be explicitly incorporated in the wave function (3) Particles can only be distinguishable via the permutational symmetry (4) The total wave function should possess the internal and translational symmetry properties of the system (5) For fixed positions of nuclei, the wave functions should become equivalent to what one obtains within the Born-Oppenheimer approximation and (6) the wave function should be an eigenfunction of the appropriate total spin and angular momentum operators. 

Gain a deeper understanding of the Born-Oppenheimer approximation and... 

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). 

In the framework of the Born-Oppenheimer approximation, radiationless transitions from one surface to another are impossible. (See, e.g., Michl and BonaCit -Koutecky, 1990.) It is therefore necessary to go beyond the Born-Oppenheimer approximation and to include the interaction between different electronic molecular states through the nuclear motion in order to be able to describe such transitions. Using the time-dependent perturbation theory for the rate constant of a transition between a pair of states one arrives at... 

The Born-Oppenheimer Approximation and Separation of Temporal Scales. 

What are the advantages of the Born Oppenheimer approximation and when is it necessary to go beyond [Separation of electronic and vibrational wavefunctions, existence of PES calculation of IC and ISC rate constants, near-degenerate states]... 

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