Born-Oppenheimer potential surfaces,

Evaluating the energy e for different values of R gives the effective potential for the nuclei in the presence of the electron. This function is called the Born-Oppenheimer potential surface or just the potential surface. In order to evaluate e(R) we have to determine HAA, HAB, and SAB. These quantities, which can be evaluated using elliptical coordinates, are given by... 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

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 ... 

As pointed out in Chapter 2, nuclear motion takes place on the Born-Oppenheimer potential surface. The motion of the center of mass (corresponding to translation) rigorously separates from the other motions of the atoms. Translational motion may be subject to a potential corresponding to the fact that the molecule... 

The eigenvalue E(R) in equation (2) yields the Born-Oppenheimer potential surface if the nuclear positions, R, are all varied. In particular, because the energy obtained is that of the lowest energy state, that is, the ground electronic state, the surface is the ground-state potential-energy surface. If we know E(R) accurately, 1 hen we could predict the detailed atomic forces and the chemical behavior of the entire system. 

The ability to predict accurate potential surfaces means that we are in a position to investigate the nature of the kinetic barriers, including the activated complexes, of key surface processes. In fact, Born-Oppenheimer potential surfaces can be used not only with the transition-state approach to kinetics but also with the much more general and exact collision theory (e.g., scattering S-matrix ilieory). While methods based on collision and scattering theory have pointed out deficiencies in the traditional transition-state theory (TST), they have also served to uphold many of TST s simple claims. In turn, new generalized transition state theories have been born. For complex systems, the transition-dale approach, while admittedly approximate, has been well established. 

While the activated complex is not a true minimum, its structure is uniquely and precisely determined by the Born-Oppenheimer potential surface. To ascertain the structure of the activated complex, we must appraise carefully the topography of the potential surface describing the reaction sketched in Eq. 13. It is one of the aims of this Chapter to show what can be done to unravel the details of the activated complex in reactions such as reaction 13. 

The reader may easily verify that Eqs. (A4) are formal solutions of Eq. (A2) by differentiating. We consider the strong-field limit, where we further assume that the duration of the pulses is short compared with wavepacket motion on either the ground or excited Born-Oppenheimer potential surface. This situation may be described in terms of n pulses, familiar from the NMR and... 

There are certain interactions due to nuclear motion which are not readily apparent from the appearance of the potential curves. A typical example is the Renner-Teller effect in triatomic molecules, in which the two components of an orbitally degenerate state split into two states of different symmetry upon bending motion, as seen in Fig. 27. A strong coupling between the two components is observed, however, and spectral patterns can be understood only by coupling the electronic and nuclear motions, as has been pointed out already in numerous textbooks. From an ab initio point of view, one can treat this problem in analogy to the previous examples by calculating the Born-Oppenheimer potential surfaces and the interaction matrix element, in this case < Pa 5/34 Pb> and (and the same for Pg) as... 

A disadvantage of this technique is that isotopic labeling can cause unwanted perturbations to the competition between pathways through kinetic isotope effects. Whereas the Born-Oppenheimer potential energy surfaces are not affected by isotopic substitution, rotational and vibrational levels become more closely spaced with substitution of heavier isotopes. Consequently, the rate of reaction in competing pathways will be modified somewhat compared to the unlabeled reaction. This effect scales approximately as the square root of the ratio of the isotopic masses, and will be most pronounced for deuterium or... 

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. 

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. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

Because the electronic distribution of a system is determined by I2 for a specific solution of Schrodinger s equation, definition (D1) allows us to determine molecular character directly from the form of the system s wavefunction f, corresponding to some definite point on the Born-Oppenheimer potential-energy surface.3... 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

In any given region of the Born-Oppenheimer potential-energy surface, we can judge which structure of (5.6a)-(5.6d) is best by determining which perturbative decomposition in (5.7) is numerically most rapidly convergent. 

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

For a quantum center where the perturbing electric field is almost constant, at least neglecting local atomic interactions typically described by short-range potentials such as the Lennard-Jones one, we can write [26] the perturbed Hamiltonian matrix H of the quantum center on the Born-Oppenheimer (BO) surface as... 

Figure 4.1. Schematic representation of Born-Oppenheimer potential energy surfaces. Using the photochemical nomenclature, the ground-state surface of a closed-shell system, which is the lowest singlet surface, is labeled So, followed by S Sj, etc. in order of increasing energies. The triplet surfaces are similarly labeled T, Tj,...

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