Born-Oppenheimer energy surface

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

The success of any molecular simulation method relies on the potential energy function for the system of interest, also known as force fields [27]. In case of proteins, several (semi)empirical atomistic force fields have been developed over the years, of which ENCAD [28,29], AMBER [30], CHARMM [31], GRO-MOS [32], and OPLSAA [33] are the most well known. In principle, the force field should include the electronic structure, but for most except the smallest systems the calculation of the electronic structure is prohibitively expensive, even when using approximations such as density functional theory. Instead, most potential energy functions are (semi)empirical classical approximations of the Born-Oppenheimer energy surface. 

S. Lifson and P. S. Stern,/. Chem. Phys., 77, 4542 (1982). Born-Oppenheimer Energy Surfaces of Similar Molecules Interrelations between Bond Lengths, Bond Angles, and Frequencies of Normal Vibrations in Alkanes. 

In Quantum Mechanical calculations, the energy is computed from the exact hamiltonian. It is then possible to build a Born-Oppenheimer energy surface which can be used later to perform lattice dynamics or to study the reaction path of a displacive phase transition. These methods give access to the electron density, the spin density and the density of states which are useful to predict electric and optical properties as well as to analyze the bonding. Recently, methods combining a quantum mechanical calculation of the potential and the Molecular Dynamics scheme have been developed after the seminal work of R. Car and M. Parrinello. 

Lifson, S.,and Stern, P.S., 1982, Born-Oppenheimer energy surface of similar molecules interrelations between bond-lengths, bond angles and frequencies of normal vibrations in alkanes,... 

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. 

When considering reaction paths on the PE surfaces of excited states, as required for the rationalization of photochemistry [4], two major additional complications arise. First, reliable ab initio energy calculations for excited states are typically much more involved than ground-state calculations. Secondly, multi-dimensional surface crossings are the rule rather than the exception for excited electronic states. The concept of an isolated Born-Oppenheimer(BO) surface, which is usually assumed from the outset in reaction-path theory, is thus not appropriate for excited-state dynamics. At surface crossings (so-called conical intersections [5-7]) the adiabatic PE surfaces exhibit non-differentiable cusps, which preclude the application of the established methods of mathematical reaction-path theory [T3]. As an alternative to non-differentiable adiabatic PE surfaces, so-called diabatic surfaces [8] may be introduced, which are smooth functions of the nuclear coordinates. However, the definition of these diabatic surfaces and associated wave functions is not unique and involves some subtleties [9-11]. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

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... 

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. 

Central to the modern approach to chemical reactivity as dynamics on a potential energy surface, is the Born-Oppenheimer approximation.9... 

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. 

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