The 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)]... 

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. 

The Bom-Oppenheimer approximation is usually very good. For the hydrogen molecule the error is of the order of 10 ", and for systems with heavier nuclei, the approximation becomes better. As we shall see later, it is only possible in a few cases to solve the electronic part of the Schrodinger equation to an accuracy of 10 ", i.e. neglect of the nuclear-electron coupling is usually only a minor approximation compared with other errors. 

Let us first review the Bom-Oppenheimer approximation in a bit more detail. The total Hamilton operator can be written as the kinetic and potential energies of the nuclei and electrons. 

For the majority of systems the Bom-Oppenheimer approximation introduces only very small errors. Once the Bom-Oppenheimer approximation is made, the... 

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an... 

Corrections involving nuclei (with the nuclear spin I replacing the electron spin s) are analogous to the above one- and two-particle terms in eqs. (8.29-8.30), with the exception of those involving the nuclear mass, which disappears in the Bom-Oppenheimer approximation (which may be be considered as the Mnucieus oo limit). 

The spin-spin term is analogous to that in (8.30), while the whole term describing the orbit-orbit interaction disappears owing to the Bom-Oppenheimer approximation. The... 

The only term surviving the Bom-Oppenheimer approximation is the direct spin-spin coupling, as all the others involve nuclear masses. Furthermore, there is no Fermi-contact term since nuclei cannot occupy the same position. Note that the direct spin-spin coupling is independent of the electronic wave function, it depends only on the molecular geometry. 

V X is the curl operator). Only the kinetic energy of the electrons are considered within the Bom-Oppenheimer approximation, and the generalized momentum becomes... 

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

It is first transfonned to mass-dependent coordinates by a G matrix eontaining the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, this is in line with the Bom-Oppenheimer approximation that the electrons follow the nucleus). 

Assumption of a rigorous separation of nuclear and electronic motions (Bom-Oppenheimer approximation). In most cases this is a quite good approximation, and there is a good understanding of when it will fail. There are, however, very few general techniques for going beyond the Bom-Oppenheimer approximation. 

Within the Bom-Oppenheimer approximation, the last term is a constant. It is seen that the Hamilton operator is uniquely determined by the number of electrons and the potential created by the nuclei, V e, i.e. the nuclear charges and positions. This means that the ground-state wave function (and thereby the electron density) and ground state energy are also given uniquely by these quantities. 

In this volume dedicated to Yngve Ohm we feel it is particularly appropriate to extend his ideas and merge them with the powerful practical and conceptual tools of Density Functional Theory (6). We extend the formalism used in the TDVP to mixed states and consider the states to be labeled by the densities of electronic space and spin coordinates. (In the treatment presented here we do not explicitly consider the nuclei but consider them to be fixed. Elsewhere we shall show that it is indeed straightforward to extend our treatment in the same way as Ohm et al. and obtain equations that avoid the Bom-Oppenheimer Approximation.) In this article we obtain a formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electtons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, we also obtain exact equations for Aese one-particle functions. 

Within the Bom-Oppenheimer approximation to molecular structure, the electronic Schrodinger equation... 

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