Born-Oppenheimer electronic theory description

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

So, the description of the theory of electron tunneling transfer is logically accomplished in this chapter - it describes the methods of calculation of the electron matrix element, whereas the methods of calculation and the form of the vibration part of the transition probability was represented in Chapter 2. Besides, in Chapter 3 the procedure of the calculation of the rate constant of tunneling transfer in the conditions of the violation of Born-Oppenheimer s approach is examined. The basic results of this chapter may be formulated as follows. 

The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born-Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as ... 

Orbital models are almost invariably the first approximation for the electronic structure of molecules within the Born-Oppenheimer approximation. By considering the motion of each electron in the averaged field of the remaining electrons in the systems in the self-consistent field theory of Hartree and Fock, a model is obtained in which each electron is described by an spin-orbital, a description which is familiar to all chemists. 

In this chapter some aspects of the present state of the concept of ion association in the theory of electrolyte solutions will be reviewed. For simplification our consideration will be restricted to a symmetrical electrolyte. It will be demonstrated that the concept of ion association is useful not only to describe such properties as osmotic and activity coefficients, electroconductivity and dielectric constant of nonaqueous electrolyte solutions, which traditionally are explained using the ion association ideas, but also for the treatment of electrolyte contributions to the intramolecular electron transfer in weakly polar solvents [21, 22] and for the interpretation of specific anomalous properties of electrical double layer in low temperature region [23, 24], The majority of these properties can be described within the McMillan-Mayer or ion approach when the solvent is considered as a dielectric continuum and only ions are treated explicitly. However, the description of dielectric properties also requires the solvent molecules being explicitly taken into account which can be done at the Born-Oppenheimer or ion-molecular approach. This approach also leads to the correct description of different solvation effects. We should also note that effects of ion association require a different treatment of the thermodynamic and electrical properties. For the thermodynamic properties such as the osmotic and activity coefficients or the adsorption coefficient of electrical double layer, the ion pairs give a direct contribution and these properties are described correctly in the framework of AMSA theory. Since the ion pairs have no free electric charges, they give polarization effects only for such electrical properties as electroconductivity, dielectric constant or capacitance of electrical double layer. Hence, to describe the electrical properties, it is more convenient to modify MSA-MAL approach by including the ion pairs as new polar entities. 

In order to explore the proton conductivity, quantum molecular dynamics methods are extensively used. There are two different types of ab initio molecular dynamics based on two theories Born-Oppenheimer and Car-Parrinello. Both of them use an explicit treatment of the electron interactions. The bonding state of the system can change along the simulation. Therefore, these methods are of great interest to obtain information concerning the transfer of protons. They play a central role in the description of dynamic phenomena, using ab initio or empirical level of theory. 

In principle, it is known that the theory of quantum mechanics gives a complete description of a system at an atomic level [1]. In practice, the equations that result are impossible to solve, either analytically or numerically, except in a very few cases. It is usual, therefore, to invoke a number of simplifications. The first is the Born-Oppenheimer approximation which states that the dynamics of electrons and nuclei can be treated separately because of the large disparity in their masses. This leads to a two-step procedure in which the electronic problem is solved first and the nuclear problem is dealt with afterwards [2]. 

At first sight it might be surprising that the Hessian matrix, which after all in oh initio molecular orbital theory is inherently quantum mechanical, is amenable to a purely classical treatment. This is because the Born-Oppenheimer approximation (Born and Oppenheimer 1927 Wikipedia 2010) allows for a pretty good separation of the electronic and nuclear motion, allowing the latter to be treated classically. A quantum mechanical description of the simple harmonic oscillator leads to quantized energy levels given by... 

At the same time that Heisenberg was formulating his approach to the helium system, Born and Oppenheimer indicated how to formulate a quantum mechanical description of molecules that justified approximations already in use in treatment of band spectra. The theory was worked out while Oppenheimer was resident in Gottingen and constituted his doctoral dissertation. Born and Oppenheimer justified why molecules could be regarded as essentially fixed particles insofar as the electronic motion was concerned, and they derived the "potential" energy function for the nuclear motion. This approximation was to become the "clamped-nucleus" approximation among quantum chemists in decades to come.36... 

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