Adiabatic dynamics Born-Oppenheimer approximation

Formally, the Hamiltonian describing a collection of atoms with electrons of coordinates r and nuclei of coordinates R is 

The well-known Born-Oppenheimer approximation (BOA) assumes all couplings Kpa between the PES are identically zero. In this case, the dynamics is described simply as nuclear motion on a single adiabatic PES and is the fundamental basis for most traditional descriptions of chemistry, e.g., transition state theory (TST). Because the nuclear system remains on a single adiabatic PES, this is also often referred to as the adiabatic approximation. 

For much of the discussion in this chapter, the BOA is assumed valid so that the bond making/breaking is simply described by motion of nuclei on a multidimensional ground state PES. For example, dissociation of a molecule from the gas phase is described as motion on the PES from a region of phase space where the molecule is far from the surface to one with the adsorbed atoms on the surface. Conversely, the time-reversed process of associative desorption is described as motion on the PES from a region of phase space with the adsorbed atoms on the surface to one where the intact molecule is far from the surface. For diatomic dissociation/associative desorption, this PES is given as V(Z, R, X, Y, ft, cp, ), where Z is the distance of the diatomic to the surface, R is the distance between atoms in the molecule, X and Y are the location of the center of mass of the molecule within the surface unit cell, ft and cp are the orientation of the diatomic relative to the surface normal and represent the thermal distortions of the hh metal lattice atom 

There are now very many DFT calculations for chemical systems in which only regions of the PES around the transition states are explored, especially the barriers V for activated processes, since only this region of the PES is necessary to describe the overall chemical rate within transition state theory (TST) (see the chapter by Bligaard and Nprskov in this book). TST requires zero point corrected adiabatic barriers V (0), but the zero point corrections are generally 0.1 eV for most chemistry at surfaces. Therefore, the distinction between V and V (0) will usually be ignored in this chapter. 

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In the usual Born-Oppenheimer picture, the sum of A (Ai,A2,9) and V(q) is the adiabatic potential for the molecular coordinate q. For the computation of the adiabatic potential, q is treated as a fixed parameter. In the dynamic Born-Oppenheimer approximation discussed above, we interpret g as a classical dynamical variable, with the result that the molecular vibrations are described by the Hamiltonian function... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

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.)...

Consider now the influence of the high-frequency fluctuations in the environment only (is 3> B). Since the frequencies of the fluctuations are much higher than the typical spin-dynamics frequencies, one may eliminate these high-frequency fluctuations using the adiabatic (Born-Oppenheimer) approximation, as described, e.g., by Leggett et al. [8]. 

Solving for the dynamics of such system is a highly complex problem. The first simplification amounts to assume an adiabatic separations of nuclei and electrons motions. In this approximation a partial factorization of the total system wavefunction is performed and we consider the ions fixed from the point of view of the electrons. This will lead us to the Born-Oppenheimer approximation. 

Beam studies have until recently been largely confined to systems in which the dynamics are governed by a single potential surface. The use of classical trajectory studies and adiabatic correlation diagrams in predicting the reaction path are both implicitly founded on the Born-Oppenheimer approximation which allows us to deal with only one electronic state during... 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

The Born-Oppenheimer adiabatic approximation represents one of the cornerstones of molecular physics and chemistry. The concept of adiabatic potential-energy surfaces, defined by the Born-Oppenheimer approximation, is fundamental to our thinking about molecular spectroscopy and chemical reaction djmamics. Many chemical processes can be rationalized in terms of the dynamics of the atomic nuclei on a single Born Oppenheimer potential-energy smface. Nonadiabatic processes, that is, chemical processes which involve nuclear djmamics on at least two coupled potential-energy surfaces and thus cannot be rationalized within the Born-Oppenheimer approximation, are nevertheless ubiquitous in chemistry, most notably in photochemistry and photobiology. Typical phenomena associated with a violation of the Born-Oppenheimer approximation are the radiationless relaxation of excited electronic states, photoinduced uni-molecular decay and isomerization processes of polyatomic molecules. 

Thus this book describes the recent theories of chemical dynamics beyond the Born-Oppenheimer framework from a fundamental perspective of quantum wavepacket dynamics. To formulate these issues on a clear theoretical basis and to develop the novel theories beyond the Born-Oppenheimer approximation, however, we should first learn a basic classical and quantum nuclear dynamics on an adiabatic (the Born-Oppenheimer) potential energy surface. So we learn much from the classic theories of nonadiabatic transition such as the Landau-Zener theory and its variants. 

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