Born-Oppenheimer expansion

If the system can sample regions of nuclear configuration space for which the e d(q) and ead(q) PESs are close to one another, more terms in (58) must be included. If, however, the system s energy is low enough for the nuclear motion on Ejad(q) not to sample regions for which this surface comes close to the next higher one, it suffices to include the two terms n = i, j in the Born-Oppenheimer expansion ... 

This expansion is known as the Born-Oppenheimer expansion. Formally, Eq. (5) is exact, since the set j(r, R) is complete. It is only when the expansion is truncated that approximations are introduced. The Born-Oppenheimer expansion certainly provides a perfectly valid ansatz if f(r, R) describes a bound state solution of the full Schrodinger equation... 

It has been argued, see, for instance, that the ansatz (5) may not be justified for describing continuum states. There are sufficiently many arguments which allow one to take the pragmatic stand point that the Born-Oppenheimer expansion is valid also for continuum states, see, for instance, Ref. 3. 

Klein, M., Martinez, A., Seiler, R., Wang, X. P. (1992). On the Born-Oppenheimer expansion for polyatomic molecules. Communications in Mathematical Physics, 143, 607. 

In this approach, the external potential displacements that are responsible for a transition from stage (i) to stage (ii) create conditions for the subsequent CT effects, in the spirit of the Born-Oppenheimer approximation. Clearly, the consistent second-order Taylor expansion at M°(co) does not involve the coupling hardness t A B and the off-diagonal response quantities of Eqs. (168) and (170), which vanish identically for infinitely separated reactants. However, since the interaction at Q modifies both the chemical potential difference and the... 

If the eigenfunctions of the full Hamiltonian (11.102) are expressed as an expansion of the Born-Oppenheimer solutions,... 

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

Configuration interaction is conceptually the simplest method for solving the time-independent electronic Schrodinger equation H ) = i ) under the Born-Oppenheimer approximation. The electronic wavefunction j f) is approximated by a linear expansion of iV-electron basis functions (where N is the number of electrons in the system), i.e.,... 

CSFs into the wavefunction expansion. Although unattainable in molecular calculations, the second limiting case, corresponding to full Cl for a complete orbital set, is called the complete Cl expansion s. The eigenvalues of the complete Cl expansion are the exact energies within the clamped-atomic-nucleus Born-Oppenheimer approximation. A correspondence may then be established with the bracketing theorem between the lowest eigenvalues of a limited CSF expansion and those of the exact complete Cl expansion. This is illustrated schematically in Fig. 2. 

We begin with some general considerations of perhaps lesser-known, but important, features of exact electronic wavefunctions. Our motive is to establish a theoretical framework together with a reasonably consistent notation in order to carry through the spin-coupled VB and other expansions of the total wavefunction. We consider an atomic or molecular system consisting of N electrons and A nuclei. We assume the Born-Oppenheimer separation and write the Hamiltonian operator for the motion of the electrons in the form ... 

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. 

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