Schrodinger equation Born-Oppenheimer approximation

Both molecular and quantum mechanics methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrodinger equation (1) gives the wave functions and energies of a molecule. 

Within the Born-Oppenheimer approximation discussed earlier, you can solve an electronic Schrodinger equation... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent 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... 

Potential energy surface for a chemical reaction can be obtained using electronic structure techniques or by solving Schrodinger equation within Born-Oppenheimer approximation. For each geometry, there is a PE value of the system. 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

One way to simplify the Schrodinger equation for molecular systems is to assume that the nuclei do not move. Of course, nuclei do move, but their motion is slow compared to the speed at which electrons move (the speed of light). This is called the Born-Oppenheimer approximation, and leads to an electronic Schrodinger equation. 

Almost all studies of quantum mechanical problems involve some attention to many-body effects. The simplest such cases are solving the Schrodinger equation for helium or hydrogen molecular ions, or the Born— Oppenheimer approximation. There is a wealth of experience tackling such problems and experimental observations of the relevant energy levels provides a convenient and accurate method of checking the correctness of these many-body calculations. 

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). 

Based on first principles. Used for rigorous quantum chemistry, i. e., for MO calculations based on Slater determinants. Generally, the Schrodinger equation (Hy/ = Ey/) is solved in the BO approximation (see Born-Oppenheimer approximation) with a large but finite basis set of atomic orbitals (for example, STO-3G, Hartree-Fock with configuration interaction). 

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 ... 

Within the Born-Oppenheimer approximation, the Schrodinger equation for a whole molecular system can be divided into two equations. The electronic Schrodinger equation needs to be solved separately for each different (fixed) set of positions for the nuclei making up the system and gives the electronic wavefunction and the electronic... 

Schrodinger equations for atoms and molecules use the the sum of the potential and kinetic energies of the electrons and nuclei in a structure as the basis of a description of the three dimensional arangements of electrons about the nucleus. Equations are normally obtained using the Born-Oppenheimer approximation, which considers the nucleus to be stationary with respect to the electrons. This approximation means that one need not consider the kinetic energy of the nuclei in a molecule, which considerably simplifies the calculations. Furthermore, the... 

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 both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

An important modification of the general Schrodinger equation (Eq. 2.10) is that based on the Born-Oppenheimer approximation[l ], which assumes stationary nuclei. Further approximations include the neglect of relativistic effects, where they are less important, and the reduction of the many-electron problem to an effective one-electron problem, i. e., the determination of the energy and movement... 

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. 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of 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. 

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