Single Born-Oppenheimer approximation

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

The Born-Oppenheimer approximation applies here just as it does for a single molecule. Calculate the electronic energy as a function of nuclear configuration. The problem is shown schematically in Fig. 4.5. Corresponding to certain configurations... 

The well-known Born-Oppenheimer approximation (BOA) assumes all couplings Kpa between the PES are identically zero. In this case, the dynamics is described simply as nuclear motion on a single adiabatic PES and is the fundamental basis for most traditional descriptions of chemistry, e.g., transition state theory (TST). Because the nuclear system remains on a single adiabatic PES, this is also often referred to as the adiabatic approximation. 

Adiabatic photoreaction Within the Born-Oppenheimer approximation , a reaction of an excited state species that occurs on a single potential-energy surface . Compare diabatic photoreaction. 

In order to keep the presentation as simple as possible, we will limit the discussion to triatomic molecules and motion on a single PES (Born-Oppenheimer approximation). Moreover, only one chemical arrangement channel is considered in the reaction ABC A -F BC. Formal extensions... 

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

Beam studies have until recently been largely confined to systems in which the dynamics are governed by a single potential surface. The use of classical trajectory studies and adiabatic correlation diagrams in predicting the reaction path are both implicitly founded on the Born-Oppenheimer approximation which allows us to deal with only one electronic state during... 

Another assumption made in the straightforward application of classical mechanics to atomic motions is the Born-Oppenheimer approximation. That is, for each atomic geometry there is a single electronic potential energy surface under whose influence the atoms move. In reality, there are geometries at which more than one surface can play a role. The simplest such case is where two potentials (in the so-called diabatic picture) cross each other. 

This method differs fundamentally from the other calculational methods previously mentioned. The Born-Oppenheimer approximation says that one can separate the nuclear from the electronic motions in a molecule, and the previously discussed quantum mechanical methods have to do with the electronic system, after the nuclear positions have been established (or assumed). To determine structures by such methods, one must repeat the calculation for a number of different nuclear positions, and locate the energy minimum in some way. Unless the structure is known at the outset, one therefore requires not just a single calculation, but many calculations, in order to determine the actual structure. 

In this thesis work. Dr. Ren also studied the non-adiabatic effect in the F -I- D2 reaction, where the F ( Pi/2) is expected to be non-reactive according to the Bom-Oppenheimer approximation. He measured accurately the population ratio of F( P3/2) and F ( Pi/2) in the beam using synchrotron radiation single photon autoionization, then determined the relative reactivity of F and F with D2. For the first time, he found that F ( Pi/2) is more reactive than F( P3/2) at low collision energy, providing a clear case of the breakdown of Born-Oppenheimer approximation. This is the first accurate experimental measurement of the non-adiabatic effects of this important system. 

The Born-Oppenheimer approximation is to assume that a many body state, /), may be factorized as a single, direct product of an electronic state, i R ), and a nuclear state vi) associated with the electronic state, i R ) ... 

To calculate the amplitude of these transitions we adopt the Born-Oppenheimer approximation and factorize J) as a single, direct product of the electronic and nuclear degrees of freedom... 

---