Basicity Born-Oppenheimer approximation

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 Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

The basic problem is to solve the time-independent electronic Schrodinger equation. Since the mass of the electrons is so small compared to that of the nuclei, the dynamics of nuclei and electrons can normally be decoupled, and so in the Born-Oppenheimer approximation the many-electron wavefunction P and corresponding energy may be obtained by solving the time-independent Schrodinger equation in which the nuclear positions are fixed. We thus solve... 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

The Born-Oppenheimer approximation was developed in 1927 by the physicists Max Born (German) and J. Robert Oppenheimer (American), just one year after Schrodinger presented his quantum treatment of the hydrogen atom. This approximation method is the foundation for all of molecular quantum mechanics, so you should become familiar with it. The basic idea of the Born-Oppenheimer approximation is simple because the nuclei are so much more massive than the electrons, they can be considered fixed for many periods of electronic motion. Let s see if this is a reasonable approximation. Using H2 as a specific example, we estimate the velocity of the electrons to be roughly the same as that of an electron in the ground state (w = 1) of the hydrogen atom. From the Bohr formula, V = we calculate an electron velocity of 2.2 X 10 m sec . We... 

For example, the Born-Oppenheimer approximation is ubiquitous. The separation of the electronic and nuclear motion is most often an excellent approximation. However, it is also fundamental to the concept of molecular structure. The model of fixed nuclei surrounded by electrons which accommodate almost instantly any change in the nuclear positions is basic to qualitative and quantitative discussions of molecular structure. 

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]. 

Reactions of the type (37) are not suitable for study in electrochemical cells ( ) because they inTOlve the isotopically-mixed species HD. Howeirer, a reaction like (36) can be investigated directly in an electrochemical cell. The basic idea is to compare an experimental value of K for reaction (36) with two calculated (by Wolfsberg, et al.) values of K, one obtained invoking the Born-Oppenheimer approximation, and the other obtained without invoking the Born-Oppenheimer approximation. In this way the reliability of the Born-Oppenheimer approximation for hydrogen-isotope-exchange reaction can be tested experimentally. 

The purpose of this review article is to present a comprehensive account of what is generally known as the Born-Oppenheimer approximation, its meaning, its implications, its properties, and very importantly also its limitations and how to cure them. This approximation and the underlying idea have been a milestone in the theory of molecules and actually also of electronic matter in general. Still today this approximation is basic to all molecular quantum mechanics and even in those cases where it fails, it remains the reference to which we compare and in terms of which we discuss this failure. 

The approximation of separating electronic and nuclear motions is called the Born-Oppenheimer approximation and is basic to quantum chemistry. [The American physicist J. Robert Oppenheimer (1904-1967) was a graduate student of Born in 1927. During World War II, Oppenheimer directed the Los Alamos laboratory that developed the atomic bomb.] Born and Oppenheimer s mathematical treatment indicated that the true molecular wave function is adequately approximated as... 

In Chap. 2, we formulate our basic framework of the chemical theory based on the Born-Oppenheimer approximation. We briefly discuss how valid or how accurate the Born-Oppenheimer approximation for bound states is. Also a theory of electron scattering by polyatomic molecules within the Born-Oppenheimer framework (or the so-called fixed nuclei approximation) is presented. This is one of the typical theories of electron dynamics, along with the theory of molecular photoionization. 

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. 

Density Functional Theory allows the replacement of the complicated N electron wave function and the associated Schrodinger equation by the much simpler electron density and its associated scheme to determine it. Indeed, for an A -electron system and within the Born-Oppenheimer approximation, the wave function is dependent on 3N spatial variables and N spin variables, a total of 4iV variables. The electron density on the other hand is a function of only three spatial coordinates. There is a long status of models using the electron density as the basic variable, and this use was legitimized in 1964 with the Hohenberg-Kohn theorems. The first theorem states that the external potential v(r) is determined, within a trivial additive constant, by the electron density p(r). For an isolated system, the external potential v(r) is the potential due the nuclei of charge Z and position R acting on the electrons is... 

In the present section all basic information required for understanding rotational spectroscopy is provided. We give as already assumed the Born-Oppenheimer approximation [21], which allows the separation of nuclear and electronic motion, as well as the separation of the various nuclear motions themselves (vibrational, rotational, translational). We therefore focus only on the quantum mechanics elements related to the rotational motion. 

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