Bom-Oppenheimer surfaces

In the adiabatic limit, the coupling F is strong, so that one may consider the transition between the two quantum states a continuous motion of the system on a single Bom-Oppenheimer surface (called the adiabatic state) that is the lowest eigenvalue of the 2 x 2 matrix in Eq. (18). 

In fact these calculations did not treat the time scales correctly because they generally fixed most features of the atomic structure of solvent and then calculated the resulting electronic structure, for fixed potential drop across the interface. (A recent calculation [34] that takes more detailed account of the electronic structure of the electrode than these early calculations also suffers from this defect.) In fact, of course, in the Bom-Oppenheimer approximation, the electronic structure should be recalculated for each atomic configuration in an ensemble of atomic configurations that follow the Bom-Oppenheimer surface. This became possible with Car-Parrinello... 

A calculated PES, which we might call a Bom-Oppenheimer surface, is normally the set of points representing the geometries, and the corresponding energies, of a collection of atomic nuclei the electrons are taken into account in the calculations as needed to assign charge and multiplicity (multiplicity is connected with the number of unpaired electrons). Each point corresponds to a set of stationary nuclei, and in this sense the surface is somewhat unrealistic (see Section 2.5). 

Symbol potential energy on a Bom-Oppenheimer surface (i.e. in a PES diagram) is denoted in Chapter 2 by E. Other common designations are V (origin obscure) and PE, and sometimes U, but this latter is best reserved for internal energy. Equation potential energy is the integral over the relevant distance of the force, itself usually a function of distance. 

Separation of the movement of the nuclei and electrons. This is possible because the electrons move much more rapidly (smaller mass) than the nuclei. The position of the nuclei is fixed for the calculation of the electronic Schrodinger equation (in MO calculations the nuclear positions are then parameters, not quantum chemical variables). Bom-Oppenheimer surfaces are energy vs. nuclear structure plots which are (n+l)-dimensional, where n is 3N-6 with N atoms (see potential energy surface). 

Millam, J. M. Bakken, V. Chen, W. Hase, W. L. Schlegel, H. B. Ab initio classical trajectories on the Bom-Oppenheimer surface Hessian-based integrators using fifth-order polynomial and rational function fits, J. Chem. Phys. 1999, 111, 3800-3805. 

F( Py2) state but concludes that the adiabatic picture is largely correct. The issue of whether a reaction can be described by a single Bom-Oppenheimer surface is of considerable interest in chemical dynamics [10], and it appears that the effect of multiple surfaces must be considered to gain a complete picture of a reaction even for as simple a model system as the F + H2 reaction. 

For two Bom-Oppenheimer surfaces (the ground state and a single electronic excited state), the total photodissociation cross section for the system to absorb a photon of energy co, given that it is initially at a state x) with energy Eq can be shown, by simple application of second-order perturbation theory, to be [89]... 

Starting from the energy functional E p], see Eq. (5), the first task is to compute the electronic ground-state energy t/, which depends parametrically on the ionic coordinates (Ri), and to define one point on the Bom-Oppenheimer surface of the system... 

HOMO and LUMO occupation, and thus leaves the Bom-Oppenheimer surface. Needless to say, this problem is especially relevant for metal clusters, because these systems have a comparatively smaller HOMO-LUMO gap than other clusters or molecules. Nevertheless, for finite systems a vanishing HOMO-LUMO gap is the exception and not the rule, even for metallic elements, and experience based on many computations has shown that extensive adiabatic simulations can be performed for metal clusters. 

Ab initio molecular dynamics methods can roughly be divided into two classifications Born-Oppenheimer Molecular Dynamics and Car-Parrinello Molecular Dynamics . In both simulations, the wavefunction is propagated with the changes in the nuclear coordinates. In the Born-Oppenheimer MD approach, the forces on each of ions are explicitly calculated at each MD time step. As such, the system directly follows the Bom-Oppenheimer surface. The primary drawback of the Born-Oppenheimer MD approach relates to the fact that time-intensive electronic structure calculations must be converged... 

Electrons are kept on the Bom-Oppenheimer surface by means of explicit electronic stmcture optimization after each MD step. Evaluation of force and energy from first principles is always the most computationally expensive part of BOMD. As a result, the efficiency of the MD step itself has no impact on the speed of the calculation. 

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