Born-Oppenheimer conditions

The 2+/3+ transition in the Marcus model or its successors involves only small Inner Sphere changes in solvation molecule coordinates in the radial direction to and from the ion. Electron transfer under FC or Born-Oppenheimer conditions demands that an activation energy in the outer Continuum should resist one-electron transfer via a continuum inertial term X = (e2/2r )(l/n2 - l/e0) where r is an effective intermediate reactant-product radius. To avoid error, X can be... 

We now consider a derivation of a SERS charge transfer enhancement in terms of the polarizability components. If we view the molecule-metal distance to be a normal mode of the combined system and assume that the zero-order wave functions of the molecule or metal have been chosen so as to satisfy the Born-Oppenheimer conditions, then we may follow the Herzberg-Teller theory, adopting Eq. (40) and Eqs. (46)-(51). 

When the Drude particles are treated adiabatically, a SCF method must be used to solve for the displacements of the Drude particle, d, similarly to the dipoles Jtj in the induced dipole model. The implementation of the SCF condition corresponding to the Born-Oppenheimer approximation is straightforward and the real forces acting on the nuclei must be determined after the Drude particles have attained the energy minimum for a particular nuclear configuration. In the case of N polarizable atoms with positions r, the relaxed Drude particle positions r + d5CF are found by solving... 

Experimental probes of Born-Oppenheimer breakdown under conditions where large amplitude vibrational motion can occur are now becoming available. One approach to this problem is to compare theoretical predictions and experimental observations for reactive properties that are sensitive to the Born-Oppenheimer potential energy surface. Particularly useful for this endeavor are recombinative desorption and Eley-Rideal reactions. In both cases, gas-phase reaction products may be probed by modern state-specific detection methods, providing detailed characterization of the product reaction dynamics. Theoretical predictions based on Born-Oppenheimer potential energy surfaces should be capable of reproducing experiment. Observed deviations between experiment and theory may be attributed to Born-Oppenheimer breakdown. 

Yang-Mills field is conditioned by the finiteness of the basic Born-Oppenheimer set. Detailed topics are noted in the introductory Section I. 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

Finally, Freed and Jortner discuss, in general terms, the influence of external perturbations on radiationless processes. They show under what conditions the external perturbation has either no effect, a small, or a large effect on the radiationless transitions in the statistical, intermediate, and resonance coupling limits, respectively. An interesting aspect of their analysis is the demonstration that the widely used Born-Oppenheimer and molecular eigenstate basis sets provide complimentary pictures, and hence are completely equivalent. 

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

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-... 

As it was shown above, the condition of validity of usual Born-Oppenheimer s approach (see wave function (59)) is the inequality (58) which practically coincides with inequality... 

So, the description of the theory of electron tunneling transfer is logically accomplished in this chapter - it describes the methods of calculation of the electron matrix element, whereas the methods of calculation and the form of the vibration part of the transition probability was represented in Chapter 2. Besides, in Chapter 3 the procedure of the calculation of the rate constant of tunneling transfer in the conditions of the violation of Born-Oppenheimer s approach is examined. The basic results of this chapter may be formulated as follows. 

In this approach, the external potential displacements that are responsible for a transition from stage (i) to stage (ii) create conditions for the subsequent CT effects, in the spirit of the Born-Oppenheimer approximation. Clearly, the consistent second-order Taylor expansion at M°(co) does not involve the coupling hardness t A B and the off-diagonal response quantities of Eqs. (168) and (170), which vanish identically for infinitely separated reactants. However, since the interaction at Q modifies both the chemical potential difference and the... 

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. 

While studies of specific acid catalysis of redox cofactors shed light on the intricacies of the electron transfer process [54], the conditions required for preprotonation of the cofactor are highly acidic (pH < 0), and would not generally be found in biological systems. There are, however, systems such as Qb reduction in the Rhodobacter sphaeroides reaction center [55], where kinetic data indicate proton transfer prior to or simultaneous with electron transfer. This would seem to indicate that a general acid process is operative. At first glance, this sort of mechanism would seem to be contrary to the Born-Oppenheimer approximation. This apparent paradox can be avoided, however, if quantum chemical (nonadiabatic) processes are considered. 

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