The Born Oppenheimer picture

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and 

The sets of normal modes obtained in this way are in principle different for different potential surfaces and can be related to each other by a unitary rotation in the nuclear coordinate space (see further discussion below). An important simplification is often made at this point We assume that the normal modes associated with the two electronic states are the same, xa b = Ua except for a shift in their equilibrium positions. Equation (12.19) is then replaced by 

A schematic view of the two potential surfaces projected onto a single normal mode is seen in Fig. 12.3. The normal mode shifts express the deviation of the equilibrium configuration of electronic state from some specified reference configuration (e.g. the ground state equilibrium), projected onto the normal mode directions. Other useful parameters are the single mode reorganization energy defined by the inset to Fig. 12.3, 

To summarize, the Bom-Oppenheimer states are of the form /) (r, R)x ,v(R) where the vibrational wavefunction is an eigenstate of the nuclear 

Hamiltonian associated with the electronic state n. In the harmonic approximation these Hamiltonians are separable, h a, so that x ,v(R) = 

v(R) are nuclear wavefunctions associated, for each electronic state n, with a nuclear potential surface given by Ef(R). These nuclear potential surfaces are therefore different for different electronic states, and correspond within the harmonic approximation to different sets of normal modes. Mathematically, for any given potential surface, we first find the corresponding equilibrium position, that is, the minimum energy configuration and make the harmonic approximation by 

To summarize, the Bom-Oppenheimer states are of the form ( (r,R)x ,v(R) where the vibrational wavefimction x ,v(R) is an eigenstate of the nuclear Hamiltonian associated with the electronic state n. In the harmonic approximation these Hamiltonians are separable, Hg = Yla na, so that x ,v(R) = 

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We deal first with electronic-nuclear coupling in systems with few atoms, and therefore few degrees of freedom for nuclear motions, so that we can concentrate on the first mentioned challenge. The structure and properties of a molecule in stationary states are well described within the Born-Oppenheimer picture in which the disparity in masses of nuclei and the electron, with mn me and the similarity of Coulomb forces on nuclei and electrons, Fn Fe, mean that within a short time interval At, changes in velocities satisfy Avn — Fn/(mnAt) Ave = Fnj(mnAt) so that an interaction involving small velocities to begin with, and lasting a short time, would be described by slow nuclei. The well known Born-Oppenheimer prescription is then to construct the electronic Hamiltonian Hq for fixed nuclear positions Q = (Ri,. ..,Rn), to calculate electronic states Q) for electron... 

The CICD rate as a function of the Kr-Kr distance was used in the nuclear wave packet simulation of the process within the Born-Oppenheimer picture... 

Hamiltonian (12.15) is the same as that inferred from the Born-Oppenheimer picture in the Condon approximation, under the assumption that different potential surfaces are mutually related by only rigid vertical and horizontal shifts. In spite... 

The trimer states, which in most cases can be called Efimov trimers, are interesting objects. Their existence can be seen from the Born-Oppenheimer picture for two heavy atoms and one light atom in the gerade state. Within the Born-Oppenheimer approach the three-body problem reduces to the calculation of the relative motion of the heavy atoms in the effective potential created by the light atom. For the light atom in the gerade state, this potential is + (/ ), found in the previous subsection. The Schrodinger equation for the wavefunction of the relative motion of the heavy atoms, Xv(R), reads... 

In the Born-Oppenheimer picture, electronic transitions occur between rovi-brational levels of two electronic states n and m. The excitation energy is the energy difference between the two levels yielding the spectral position of... 

How important the breakdown of the Born-Oppenheimer approximation is in limiting our ability to carry out ab initio simulations of chemical reactivity at metal surfaces is the central topic of this review. Stated more provocatively, do we have the correct theoretical picture of heterogeneous catalysis. This review will restrict itself to a consideration of experiments that have begun to shed light on this important question. The reader is directed to other recent review articles, where aspects of this field of research not mentioned in this article are more fully addressed.10-16... 

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... 

Born-Oppenheimer approximation The Born Oppenheimer approximation consists of separating the motion of nuclei from the electronic motion. An often used physical picture is that the nuclei being so much heavier than electrons may be treated as stationary as the electrons move around them. The Schrodinger equation can then be solved for the electrons alone at a definite internuclear separation. The Born-Oppenheimer approximation is quite good... 

Another assumption made in the straightforward application of classical mechanics to atomic motions is the Born-Oppenheimer approximation. That is, for each atomic geometry there is a single electronic potential energy surface under whose influence the atoms move. In reality, there are geometries at which more than one surface can play a role. The simplest such case is where two potentials (in the so-called diabatic picture) cross each other. 

The notion of molecular structure based on frozen nuclear conformations is limited. From quantum mechanics, we know that this view is not correct because nuclear vibrations about a conformational energy minimum cannot be eliminated. Moreover, despite the appealing picture produced by the Born—Oppenheimer approximation, nuclei are indeed quantum mechanical particles, and the Heisenberg principle applies to them as well as to electrons. 

Abstract. The Born-Oppenheimer approximation, introduced in the 1927 paper On the quantum theory of molecules , provides the foundation for virtually all subsequent theoretical and computational studies of chemical binding and reactivity, as well as the justification for the universal ball and stick picture of molecules as atomic centers attached at fixecl distances by electronic glue. 

The concepts of energy surfaces for molecular motion, equilibrium geometries, transition structures and reaction paths depend on the Bora-Oppenheimer approximation to treat the motion of the nuclei separately from the motion of the electrons. Minima on the potential energy surface for the nuclei can then be identified with the classical picture of equilibrium structures of molecules saddle points can be related to transition states and reaction rates. If the Born-Oppenheimer approximation is not valid, for example in the vicinity of surface crossings, non-adiabatic effects are important and the meaning of classical chemical structures becomes less clear. Non-adiabatic effects are beyond the scope of this chapter and the discussion of energy surfaces and optimization will be restricted to situations where the Bom-Oppenheimer approximation is valid. 

Electronic spectra arise from transitions between electronic states of different quantum numbers induced by electromagnetic radiation with ultraviolet or visible (UV/vis) light. The term electronic spectra implies the Born-Oppenheimer (BO) picture of molecules where the electronic and nuclear degrees of freedom are separated. Similarly, the description of the spectra in terms of particular electronic states is valid solely in a small region of the... 

One obvious question is whether the nuclear and electronic motion can be separated in the fashion which is done in most models for molecule surface scattering and also in the above-mentioned treatment of electron-hole pair excitation. The traditional approach is to invoke a Born-Oppenheimer approximation, i.e., one defines adiabatic potential energy surfaces on which the nuclear dynamics is solved — either quantally or classically. In the Bom-Oppenheimer picture the electrons have had enough time to readjust to the nuclear positions. Thus the nuclei are assumed to move infinitely slowly. For finite speed, nonadiabatic corrections therefore have to be introduced. Thus, before comparison with experimental data is carried out we have to consider whether nonadiabatic processes are important. Two types of nonadiabatic processes are possible—one is nonadiabatic transitions in the gas phase from the lower adiabatic to the upper surface (as discussed in Chapter 4). The other is the nonadiabatic excitation of electrons in the metal through electron-hole pair excitation. 

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. 

In both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

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