Born-Oppenheimer electronic states

Another approach widely used for nonadiabatic reactions is the diabatic one. The channel Hamiltonians Hex and H determining the zeroth-order Born-Oppenheimer electron states of the donor A and acceptor B and the perturbations Vt and Vf leading to the forward and reverse electron transitions, respectively, are separated... 

Since the interaction of the electron with the medium polarization is strong, in the reference model it was usually included in the zeroth-order Hamiltonians determining the Born-Oppenheimer electron states ... 

A convenient analogy between the multilevel and two-level cases is generated by separating the electronic and nuclear wavefunctions of the multilevel system.21 We define ipa(r,R) and b(r,R) as the electronic wavefunctions associated with the ground and excited Born-Oppenheimer electronic states of the system. The Hamiltonian with the light on is then given by... 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

Note that the formula for the rate constant in VTST is exactly the same as in TST (compare Equations 6.1, 6.4 and 6.5). In TST the dividing surface is defined by the saddle point in the Born-Oppenheimer electronic energy surface (the maximum along the MEP from reactants to products), while in VTST it is defined as that surface which leads to the minimum value of the rate constant. In both approaches the dividing surface separates product space from reactant space. The assumption in VTST is that a given transition state in equilibrium with reactants will pass through... 

As mentioned in Section II.A, the Pgl process is ideal for the application of the optical model. This is clear in the classical and semiclassical Pgl theory,24,25 for which opacity and cross-section formulas are completely equivalent to those given earlier in this chapter. The quantal optical model is also rigorously related to the elastic component of the quantal Pgl theory. Miller49 has shown that T(r), identified in Pgl as the autoionization width of the excited electronic state, may be accurately obtained by a standard Born-Oppenheimer electronic structure calculation as... 

In the absence of perturbation the stationary electronic states of a molecule with n electrons and JV nuclei are described in terms of the eigenfunctions 17 > to the unperturbed Born-Oppenheimer electronic Hamiltonian... 

The Born-Oppenheimer approximation states that the electrons are able to adjust themselves instantaneously to tlie motions of the nuclei. The motions of the nuclei are in this approximation therefore not able to induce electronic transitions, an assumption that is also known as tire adiabatic approximation. The electrons thus create an effective electronic potential in which the nuclei move, and for a given electronic state tire valuation in the electronic energy with respect to the nuclear configuration defines a potential energy surface for the electronic state. The electronic Schrodinger equation can be written as... 

H. We understood H to be complete and including electronic as well as nuclear degrees of freedom, and in which case the states are the true nonadiabatlc vlbronic eigenstates of the system and hence the properties are the exact ones. Nothing prevents us, however, to introduce the adiabatic approximation and to assume the wave functions to be products of electronic and nuclear (vibrational) parts. In this case, the Born-Oppenheimer electronic plus vibrational properties will appear. We can even reduce the accuracy to the extent that we adopt the electronic Hamiltonian, work with the spectrum of electronic states, and thus extract the electronic part of the properties. In all these cases, the SOS property expressions remain unchanged. 

The Born-Oppenheimer approximation states that a diatomic s electronic energy depends only on the internuclear separation. Use this information to sketch and explain the relative location of the first few vibrational levels for H2 and D2. 

The molecular mechanics method is used to calculate molecular structures, conformational energies, and other molecular properties using concepts from classical mechanics. Electrons are not explicitly included in the molecular mechanics method, which is justified on the basis of the Born-Oppenheimer approximation stating that the movements of electrons and the nuclei can be separated. Thus, the nuclei may be viewed as moving in an average electronic potential field, and the molecular mechanics method attempts to describe this field by its force field. ... 

Because the nuclei are several orders of magnitude more massive than the electrons and will therefore move more slowly than the electrons, the Born-Oppenheimer approximation states that the nuclear and electronic motions of the molecule can be treated separately. Essentially, we can assume that the nuclei in the molecule are stationary and solve the equation solely for the electronic motion. This causes the nuclear or first term in Equation (10.13) to drop out. 

The general problem of isolated-molecule nonradiative relaxation may be stated as follows. A Born-Oppenheimer molecular state in electronic state manifold A is prepared in a molecule by photon absorption from some lower... 

The proper quantumdynamical treatment of fast electronic transfer reactions and reactions involving electronically excited states is very complex, not only because the Born-Oppenheimer approximation brakes down but... 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

Chemical reactions of molecules at metal surfaces represent a fascinating test of the validity of the Born-Oppenheimer approximation in chemical reactivity. Metals are characterized by a continuum of electronic states with many possible low energy excitations. If metallic electrons are transferred between electronic states as a result of the interactions they make with molecular adsorbates undergoing reaction at the surface, the Born-Oppenheimer approximation is breaking down. 

From the point of view of associative desorption, this reaction is an early barrier reaction. That is, the transition state resembles the reactants.46 Early barrier reactions are well known to channel large amounts of the reaction exoergicity into product vibration. For example, the famous chemical-laser reaction, F + H2 — HF(u) + H, is such a reaction producing a highly inverted HF vibrational distribution.47-50 Luntz and co-workers carried out classical trajectory calculation on the Born-Oppenheimer potential energy surface of Fig. 3(c) and found indeed that the properties of this early barrier reaction do include an inverted N2 vibrational distribution that peaks near v = 6 and extends to v = 11 (see Fig. 3(a)). In marked contrast to these theoretical predictions, the experimentally observed N2 vibrational distribution shown in Fig. 3(d) is skewed towards low values of v. The authors of Ref. 44 also employed the electronic friction theory of Tully and Head-Gordon35 in an attempt to model electronically nonadiabatic influences to the reaction. The results of these calculations are shown in... 

Fig. 4. Accumulating evidence is starting to show that molecules which undergo large amplitude vibration can interact strongly with metallic electrons in collisions and reactions at metal surfaces. This suggests that the Born-Oppenheimer approximation may be suspect near transition states of reactions at metal surfaces.

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