Molecular structure adiabatic approximation

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

Before getting into a deeper analysis of the concept of resonance, we must define precisely what we understand by chemical structure . One of the most basic concepts in molecular quantum mechanics is the one of potential energy surface (PES). It allows us to define a molecular structure as an arrangement of nuclear positions in space. The definition of molecular structure depends on the validity of the Bom-Oppenheimer approximation for a given state. Actually, its validity is limited to selected portions of the entire Bom-Oppenheimer PES. When a state is described by one PES, we call it an adiabatic state. It is clear that the concept of chemical structure , depends on the existence of a previously defined molecular structure . Only adiabatic states have a molecular structure . From now on, we will always be dealing with adiabatic states. 

By the Born-Oppenheimer adiabatic approximation we obtain a molecular model in which the potential energy depends on structural variables of the nuclear framework only, whereas it is independent of the position of the molecule in space. Correspondingly it is convenient to use generalized coordinates which are divided into two classes, the internal coordinates determining the relative positions of the N atoms,... 

Ceo fullerides exhibit several interesting phenomena related with the presence of strong correlations such as high temperature superconductivity or antiferromagnetism. The existence of non-conventional behaviors can be anticipated from the fact that both the electron-phonon and electron-electron interactions are, respectively, comparable and much larger than the narrow bandwidth predicted by standard electronic structure calculations (a few hundreds of meV). These systems are therefore close to a metal-insula-tor transition of the Mott-Hubbard type and the validity of the adiabatic approximation, assumed in most electronic structure calculations, can be questioned. From the experimental point of view the band derived from the molecular LUMO is usually seen as a quite broad feature, much wider that the theoretical estimates, in photoemission experiments. However, the observation of the band dispersion has proven elusive in ARPES studies until very recently. In a recent joint experimental and theoretical paper, Yang et al. [158] have reported the first photoemission measurement of the band dispersion for a K-doped Qo monolayer deposited on a Ag(lll) substrate. The results have been compared with ab initio calculations performed with SIESTA. In those calculations the Ag substrate was modeled by a slab con-... 

Without the adiabatic approximation, questions about the molecular 3-D structure of the benzene molecule could only be answered in a very enigmatic way. For example ... 

Fig. 14.11. The reaction H2 + OH -> H2O + H (within the vihrational adiabatic approximation). Three sets of the vibrational numbers (roH. t HH) = (0,0), (1, 0), (0, 1) were chosen. Note that the height and position of the barrier depend on the vibrational quantum numbers assumed. An excitation of H2 decreases considerably the barrier height Source T. Dunning, Jr. and E. Krala, from Advances in Molecular Electronic Structure Theory, ed. T. Dunning, Jr., JAI Press, Greenwich, C T (1989), courtesy of the authors.

This matrix was introduced by F. T. Smith [25] for the treatment of non-adiabatic (diabatic) couplings in atomic collisions. It is now familiar also in molecular structure problems, to indicate local breakdowns of the Born-Oppenheimer approximation. Within the hyperspherical formalism, it has been introduced in the three-body Coulomb problem [20] and in chemical reactions [21-24], see also Section 3. Also, from equation (A4)... 

Nuclei are thousands times heavier than the electrons. As an example let us take the hydrogen atom. From the conservation of momentum law, it follows that the proton moves 1840 times slower than the electron. In a polyatomic system, while a nucleus moves a little, an electron travels many times through the molecule. It seems that a lot can be simplified when assuming electronic motion in a field created by immobile nuclei. This concept is behind what is called adiabatic approximation, in which the motions of the electrons and the nuclei are separated. Only after this approximation is introduced, can we obtain the fundamental concept of chemistry the molecular structure in 3D space. 

Mohallem, J. R., Tostes, J. G. (2002). The adiabatic approximation to exotic leptonic molecules Further analysis and a nonlinear equation for conditional amplitudes. Journal of Molecular Structure Theochem, 580, 27. 

Whereas the Born-Handy formula holds only in the framework of adiabatic approximation, the Eq. 28.58 is quite general and holds in the whole scale (o/Ae including the non-adiabatic cases where the B-O approximation is broken. We denote it the generalized or extended Bom-Handy formula. It is highly significant because, as we will see further, it defines the molecular and crystallic structure. 

Which of these surface rearrangements takes place upon cleavage is determined by the surface electronic structure. This follows within the adiabatic approximation which is applicable for a wide range of molecular systems and is based on the fact that the ratio me I Mis small. Here, me and M are the electron and the mean nuclei masses, respectively. In the zeroth-order approximation, which corresponds to infinite masses of nuclei, one can neglect their kinetic energies and consider the states of the electronic subsystem at fixed nuclei positions specified by a multidimensional vector R. Then the electronic energy E(R) plays the role of a potential in which the nuclei move. The equilibrium positions of the nuclei, Rq, are found at the minimum... 

The study of molecular systems using quantum mechanics is based on the Born-Oppenheimer approximation. This approximation relies on the fact that the electrons, because of their smaller mass, move much faster than the heavier nuclei, so they follow the motion of the nuclei adiabatically, whereas the latter move on the average potential of the former. The Born-Oppenheimer approximation is sufficient to describe most chemical processes. In fact, our notion of molecular structure is based on the Born-Oppenheimer approximation, because the molecular structure is formed by nuclei being placed in fixed positions. There are, however, essential nonadiabatic processes in nature that cannot be described within this approximation. Nonadiabatic processes are ubiquitous in photophysics and photochemistry, and they govern such important phenomena as photosynthesis, vision, and charge-transfer reactions. 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

The adiabatic picture developed above, based on the BO approximation, is basic to our understanding of much of chemistry and molecular physics. For example, in spectroscopy the adiabatic picture is one of well-defined spectral bands, one for each electronic state. The structure of each band is then due to the shape of the molecule and the nuclear motions allowed by the potential surface. This is in general what is seen in absorption and photoelectron spectroscopy. There are, however, occasions when the picture breaks down, and non-adiabatic effects must be included to give a faithful description of a molecular system [160-163],... 

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