Born-Oppenheimer approximation nuclear Schrodinger equation

The purpose of most quantum chemical methods is to solve the time-independent Schrodinger equation. Given that the nuclei are much more heavier than the electrons, the nuclear and electronic motions can generally be treated separately (Born-Oppenheimer approximation). Within this approximation, one has to solve the electronic Schrodinger equation. Because of the presence of electron repulsion terms, this equation cannot be solved exactly for molecules with more than one electron. 

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... 

This approximation of treating electronic and nuclear motions separately is the Born-Oppenheimer approximation. In the rest of this chapter, we consider the electronic Schrodinger equation. 

We now consider the nuclear motions of polyatomic molecules. We are using the Born-Oppenheimer approximation, writing the Hamiltonian HN for nuclear motion as the sum of the nuclear kinetic-energy TN and a potential-energy term V derived from solving the electronic Schrodinger equation. We then solve the nuclear Schrodinger equation... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrodinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrodinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

Computational methods typically employ the Born-Oppenheimer approximation in most electronic structure programs to separate the nuclear and electronic parts of the Schrodinger equation that is still hard enough to solve approximately. There would be no potential energy (hyper)surface (PES) without the Born-Oppenheimer approximation -how difficult mechanistic organic chemistry would be without it ... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

Some theoretical purists tend to view molecular mechanics calculations as merely a collection of empirical equations or as an interpolative recipe that has very little theoretical Justification. It should be understood, however, that molecular mechanics is not an ad hoc approach. As previously described, the Born-Oppenheimer approximation allows the division of the Schrodinger equation into electronic and nuclear parts, which allows one to study the motions of electrons and nuclei independently. From the molecular mechanics perspective, the positions of the nuclei are solved explicitly via Eq. (2). Whereas in quantum mechanics one solves, which describes the electronic behavior, in molecular mechanics one explicitly focuses on the various atomic interactions. The electronic system is implicitly taken into account through judicious parametrization of the carefully selected potential energy functions. 

In most electronic-structure calculations, the Born-Oppenheimer approximation is invoked. This leads to flow charts like that of Figure 1, where the electronic properties are determined for a given set of nuclear positions. This is done by solving some kind of time-independent electronic Schrodinger equation,... 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

The Born-Oppenheimer approximation states that the electrons are able to adjust themselves instantaneously to tlie motions of the nuclei. The motions of the nuclei are in this approximation therefore not able to induce electronic transitions, an assumption that is also known as tire adiabatic approximation. The electrons thus create an effective electronic potential in which the nuclei move, and for a given electronic state tire valuation in the electronic energy with respect to the nuclear configuration defines a potential energy surface for the electronic state. The electronic Schrodinger equation can be written as... 

Born and Oppenheimer" showed in 1927 [9] that to a very good approximation the nuclei in a molecule are stationary with respect to the electrons. This is a qualitative expression of the principle mathematically, the approximation states that the Schrodinger equation (chapter 4) for a molecule may be separated into an electronic and a nuclear equation. One consequence of this is that all ( ) we have to do to calculate the energy of a molecule is to solve the electronic Schrodinger equation and then add the electronic energy to the internuclear repulsion (this latter quantity is trivial to calculate) to get the total internal energy (see section 4.4.1). A deeper consequence of the Born-Oppenheimer approximation is that a molecule has a shape. 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schrodinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximalion is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

When the Born-Oppenheimer approximation is used we concentrate on the electronic motions the nuclei are considered to be fixed. For each arrangement of the nuclei the Schrodinger equation is solved for the electrons alone in the field of the nuclei. If it is desired to change the nuclear positions then it is necessary to add the nuclear repulsion to the electronic energy in order to calculate the total energy of the configuration. 

It turns out to be useful to separate the energy functional into three different parts, which can be recognized from the Schrodinger equation a kinetic energy part 1 p], the attraction between nuclei and electrons <, [Pl and the electron-electron repulsion [/ ] (the nuclear-nuclear repulsion adds a constant energy term to the total energy in the Born-Oppenheimer approximation). The electron-electron repulsion energy is further divided into two parts the classical electron-electron repulsion term J p, i. e. 

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