Bom—Oppenheimer Approximation

Quasiclassical Trajectory Calculations on a H -I- D2 Reaction at 2.20 eV in. The Extended Bom-Oppenheimer Approximation... 

The approximation involved in Eq. (B.17) is known as the Bom-Oppenheimer approximation and this equation is called the Bom-Oppenheimer equation. 

Within the Bom-Oppenheimer approximation, the electronic wave function R)ei, is well defined, throughout the reaction and may be written analogously [cf. Eq. (6)]... 

The fact that now is not zero will affect the ordinaiy Bom-Oppenheimer approximation. To show that, we consider Eq. (15) for M = 1, once for a real... 

The extended Bom-Oppenheimer approximation based on the nonadiabatic coupling terms was discussed on several occasions [23,25,26,55,56,133,134] and is also presented here by Adhikari and Billing (see Chapter 3). 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

The Bom-Oppenheimer approximation is not peculiar to the Huckel molecular orbital method. It is used in virtually all molecular orbital calculations and most atomic energy calculations. It is an excellent approximation in the sense that the approximated energies are very close to the energies we get in test cases on simple systems where the approximation is not made. 

In summary, we have made three assumptions 1) the Bom-Oppenheimer approximation, 2) the independent particle assumption governing molecular orbitals, and 3) the assumption of n-molecular orbital theory, but the third is unique to the Huckel molecular orbital method. 

In the general case of an electronic Hamiltonian for atoms or molecules under the Bom-Oppenheimer approximation,... 

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

The Bom-Oppenheimer approximation is valid because the electrons adjust instantaneously to any nuclear motion they are said to follow the nuclei. For this reason Eg can be treated as part of the potential field in which the nuclei move, so that... 

It follows from the Bom-Oppenheimer approximation that the total wave function ij/ can be factorized ... 

Accurate intensity measurements have been made in many cases and calculations of r — r" made, including the effects of anharmonicity and even allowing for breakdown of the Bom-Oppenheimer approximation. 

A fully theoretical calculation of a potential energy surface must be a quantum mechanical calculation, and the mathematical difflculties associated with the method require that approximations be made. The first of these is the Bom-Oppenheimer approximation, which states that it is acceptable to uncouple the electronic and nuclear motions. This is a consequence of the great disparity in the masses of the electron and nuclei. Therefore, the calculation can proceed by fixing the location... 

There are phenomena such as the Renner and the Jahn-Teller effects where the Bom-Oppenheimer approximation breaks down, hut for the vast majority of chemical applications the Born-Oppenheimer approximation is a vital one. It has a great conceptual importance in chemistry without it we could not speak of a molecular geometry. 

The Bom-Oppenheimer approximation shows us the way ahead for a polyelec-tronic molecule comprising n electrons and N nuclei for most chemical applications we want to solve the electronic time-independent Schrodinger equation... 

The first step is to make use of the Bom-Oppenheimer approximation, so I separate the nuclear and the electronic terms ... 

But we can carry forward the knowledge of the Bom-Oppenheimer approximation gained from Chapter 2 and focus attention on the electronic problem. Thus... 

In Chapter 4,1 discussed the concept of an idealized dihydrogen molecule where the electrons did not repel each other. After making the Bom-Oppenheimer approximation, we found that the electronic Schrddinger equation separated into two independent equations, one for either electron. These equations are the ones appropriate to the hydrogen molecule ion. 

Breakdown of the Bom-Oppenheimer approximation is responsible for the small but non-zero permanent electric dipole moment of HD (2 x 10 Cm, Trefler and Gush, 1968) but otherwise the effect is negligible to chemical accuracy. 

The derivations given above related to a single particle in a constant magnetic induction. For a molecule within the Bom-Oppenheimer approximation, the derivation is similar except that we take the nuclei to be fixed in space. There is a nuclear and an electronic contribution to each property. 

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