Born-Oppenheimer potential-energy surface

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... 

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 ... 

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. 

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...

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,...

So far, our discussion has focussed on stationary quantum chemical methods, which yield results for fixed atomic nuclei, i.e. for frozen molecular structures like minimum structures on the Born-Oppenheimer potential energy surface. Processes in supramolecular assemblies usually feature prominent dynamical effects, which can only be captured through explicit molecular dynamics or Monte Carlo simulations [95-98]. Molecular dynamics simulations proved to be a useful tool for studying the detailed microscopic dynamic behavior of many-particle systems as present in physics, chemistry and biology. The aim of molecular dynamics is to study a system by recreating it on the computer as close to nature as possible, i.e. by simulating the dynamics of a system in all microscopic detail over a physical length of time relevant to properties of interest. 

First-principles simulations are techniques that generally employ electronic structure calculations on the fly . Since this is a very expensive task in terms of computer time, the electronic structure method is mostly chosen to be density functional theory. Apart from the possibility of propagating classical atomic nuclei on the Born-Oppenheimer potential energy surface represented by the electronic energy V (R ) = ji(R ), another technique, the Car-Parrinello method, emerged that uses a special trick, namely the extended Lagrangian technique. The basic idea... 

Figure 1. Photoabsorption between two Born-Oppenheimer potential energy surfaces. (A) The Franck-Condon wavepacket, arising out of = nx L X >s shown on the lower surface, and /j>(t) on the upper], grazes (0) several times on the way to dissociation. The result is an absorption band with some limited vibrational structure. (B) Direct dissociation leading to a broad, featureless absorption band. (Reproduced, with permission, from Ref. 1.)...

Figure 14. Model ground-state Born-Oppenheimer potential energy surface. 

There have been previous model studies of these systems [61]. These studies, while including the effects of environment, did not address the question of the effect of a promoting vibration. These reactions are inherently electronically nonadia-batic, while the formulation we have thus far presented included evolution only on a single Born-Oppenheimer potential energy surface. We have developed a model system to allow the extension of the Quantum Kramers methodology to such systems, and we now describe that model. 

Car and Parrinello [97,98] proposed a scheme to combine density functional theory [99] with molecular dynamics in a paper that has stimulated a field of research and provided a means to explore a wide range of physical applications. In this scheme, the energy functional [ (/, , / , ] of the Kohn-Sham orbitals, (/(, nuclear positions, Ri, and external parameters such as volume or strain, is minimized, subject to the ortho-normalization constraint on the orbitals, to determine the Born-Oppenheimer potential energy surface. The Lagrangian,... 

Figure 1 Schematic picture of the relationship between Born-Oppenheimer potential energy surfaces, the rotation-vibration Hamiltonian, and the observed spectroscopy and dynamics.

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