Born-Oppenheimer approximation definition

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

The mathematical definition of the Born-Oppenheimer approximation implies following adiabatic surfaces. However, software algorithms using this approximation do not necessarily do so. The approximation does not reflect physical reality when the molecule undergoes nonradiative transitions or two... 

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

In the crude Born-Oppenheimer approximations, the oscillator strength of the 0-n vibronic transition is proportional to (FJ)2. Furthermore, the Franck-Condon factor is analytically calculated in the harmonic approximation. From the hamiltonian (2.15), it is clear that the exciton coupling to the field of vibrations finds its origin in the fact that we use the same vibration operators in the ground and the excited electronic states. By a new definition of the operators, it becomes possible to eliminate the terms B B(b + b ), BfB(b + hf)2. For that, we apply to the operators the following canonical transformation ... 

The structure of approximate reasoning is not simple. Consider the Born-Oppenheim approximation (separability of electronic and nuclear motions due to extreme mass difference), which in application produces "fixed nuclei" Hamiltonians for individual molecules. In assuming a nuclear skeleton, the idealization neatly corresponds to classical conceptions of a molecule containing localized bonds and definite structure. All early quantum calculations, and the vast majority to date, invoke the approximation. In 1978, following decades of quiet assumption, Cambridge chemist R. G. Woolley asserted ... 

In the very same way as the Born-Oppenheimer approximation allows the definition of a potential energy surface for a Van der Waals molecule, it enables, too, the concept of an interaction tensor field. This is a field dependent on the relative coordinates of the monomers and transforming as a tensor under rotation of the complex as a whole. (The potential energy surface is an example of a rank zero interaction tensor field). In the case of tensor fields it is also convenient to base the theory on irreducible tensors and to use an expansion in terms of a complete set of functions of the five angular coordinates describing a Van der Waals dimer. 

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... 

The interpretation of the MCD terms by means of electronic and magnetic properties of the electronic states of a molecule was formulated in terms of the Rigid Shift Model which assumes in a Born-Oppenheimer approximation that the band shape in the presence of the field is unchanged compared to the shape in the absence of the field, just shifted in energy as a result of the Zeeman interaction. The definitions of the Ai, Bq, and Cq MCD parameters and Dq are... 

A more rigorous definition of a biradicaloid geometry employs the concept of natural orbitals and their occupancies, which are well defined at all levels of quantum mechanical description of molecules in the Born-Oppenheimer approximation at a biradicaloid geometry, two of the ground state natural orbital occupation munbers are approximately equal to unity (the others, of course, are close to two or close to zero). 

The middle of the twentieth century marked the end of a long period of determining the building blocks of chemistry chemical elements, chemical bonds, and bond angles. The lists of these are not definitely closed, but future changes will be more cosmetic than fundamental. This made it possible to go one step further and begin to rationalize the structure of molecular systems, as well as to foresee the structural features of the compounds to be synthesized. The crucial concept is based on the Born-Oppenheimer approximation and on the theory of chemical bonds and resulted in the spatial structure of molecules. The great power of such an approach was first proved by the construction of the DNA double helix model by Watson and Crick. The first DNA model was built from iron spheres, wires, and tubes. 

Let us look first at the transition of the original definitions as integrals over the charge density, Eqs. (4.5), (4.6) and (4.8), to quantum mechanics that we will illustrate for the example of the electric dipole moment. In the Born-Oppenheimer approximation, Section 2.2, the electrons in a molecule form a continuous charge distribution whereas the discrete nuclear charges are located at fixed points Rk- The expression, Eq. (4.5) for the a-component of the electric dipole moment can therefore be rewritten as... 

It is very important to remember that this definition of a PES is based on the assumption that the atomic positions can be exactly specified, which is the ultimate condition for the structure or shape of a molecule. This means adoption of the Born-Oppenheimer (B.O.) approximation, in which the nuclei are viewed as stationary point charges, whereas the electrons are described quantum mechanically [5]. This approximation is justified by the fact that the electrons are much lighter than the nuclei and hence are moving faster. The classical nature of the atomic nuclei is usually a valid approximation, but the zero-point vibrational energy of molecules or the tunneling effect, for example, make it evident that it does not always hold. 

The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable. A formal definition of the Born-Oppenheimer principle can be made by considering the time-independent Schrodinger equation of a molecule, which is of the form... 

Note that the potential matrix V( (R) is a diagonal matrix by definition, in contrast to W(s)(R). Again, in analogy to the common treatment in Sec. 3.1, we call (23) the group-Born Oppenheimer adiabatic approximation or briefly the group-adiabatic approximation. This approximation assumes that the states within the manifold g are much stronger coupled to each other — e.g. via the presence of a conical intersection of the potential surfaces — than to the rest of the electronic space. 

Here E and E are the exact energies of the two individual molecules A and B when they are isolated, while E" is the exact energy of the supersystem (molecular complex, for example). Theoretically, these quantities can be obtained from the exact solution of the Schrodinger equation for the corresponding systems. (We remain within the nonrelativistic Born-Oppenheimer model.) This requires the definition of the Hamiltonians H", H and H" , and one feels challenged to handle these Hamiltonians in a common (e.g., perturbational) scheme. This point is not at all trivial especially if approximate model Hamiltonians are used. In what follows we shall consider this issue emphasizing the points where the second quantized approach can help to clarify the situation. 

The main scientific disciplines of quantum chemistry and solid-state physics were developed by way of a mathematical simplification or approximation of the Schrodinger equation, known as the Born-Oppenheimer (B-O) approximation [1]. It does not only give the basis of almost all quantum chemical calculations, but it also provides the very concept of molecular structure [2]. There are two main contemporary trends in quantum chemistry that put a question mark over the B-O approximation and its role in the definition of (molecular) structure theories based on the incorporation of the centre-of-mass (COM) problem and applications in connection with the Jahn-Teller (J-T) effect. 

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