Born-Oppenheimer assumption

An alternative strategy is to synthesize a molecular wave function, on chemical intuition, and progressively modify this function until it solves the molecular wave equation. However, chemical intuition fails to generate molecular wave functions of the required spherical symmetry, as molecules are assumed to have non-spherical three-dimensional structures. The impasse is broken by invoking the Born-Oppenheimer assumption that separates the motion of electrons and nuclei. At this point the strategy ceases to be ab initio and reduces to semi-empirical quantum-mechanical simulation. The assumed three-dimensional nuclear framework is no longer quantum-mechanically defined. The advantage of this model over molecular mechanics is that the electron distribution is defined quantum-mechanically. It has been used to simulate the H2 molecule. 

Within the framework of the Born-Oppenheimer assumption, studies of isotope effects provide augmental information to a knowledge of force fields. Such information is especially valuable for the condensed states (5), where spectroscopic methods fail to provide accurate information. 

The assumption that E(Q ) can be used as an effective potential which regulates the vibratory internuclear motion in a molecule is known as the Born-Oppenheimer assumption. According to this assumption E of Eq. (2.15) is thus an eigenvalue of the total Schrodinger equation after the translation has been split off. This means that the total energy of a molecule is given by... 

Potential Energy Hypersurfsoes.—So far in this chapter the emphasis has been on how one describes the results of reactive encounters between species in selected quantum states. Now it is time to examine what the fundamental factors are that control the collision dynamics, and therefore lead to different detailed results. This discussion will be based on the foundation of the Born-Oppenheimer assumption that is, that the motions of the nuclei during a collision are determined at all (or... 

For a single atom the nucleus is fixed at the origin of coordinates. For molecules, a big simplification results from the fact that electrons rearrange so much faster than nuclei that the positional coordinates of the nuclei can be kept fixed in the calculation of electronic energies, and the molecular wavefiinction depends only on the coordinates of electrons. This is called the Born-Oppenheimer assumption (there are no potential energy terms for nuclei in the hamiltonian 3.39). The total electronic energy is the expectation value of the hamiltonian operator, equation 3.8 ... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

It should also be mentioned that a theoretical model using an empirical LEPS potential energy surface has successfully been used to reproduce the vibrational population distribution of the products of this surface reaction.40 This approach confines itself to the assumptions of the Born-Oppenheimer approximation and underscores one of the major questions remaining in this field do we just need better Born Oppenheimer potential surfaces or do we need a different theoretical approach ... 

One branch of chemistry where the use of quantum mechanics is an absolute necessity is molecular spectroscopy. The topic is interaction between electromagnetic waves and molecular matter. The major assumption is that nuclear and electronic motion can effectively be separated according to the Born-Oppenheimer approximation, to be studied in more detail later on. The type of interaction depends on the wavelength, or frequency of the radiation which is commonly used to identify characteristic regions in the total spectrum, ranging from radio waves to 7-rays. 

Each of the approaches is based on the premise that it makes sense to focus on the Born Oppenheimer potential for the OH stretch for fixed bath variables. Such a potential has vibrational eigenvalues, and for example h times the transition frequency of the fundamental is simply the difference between the first excited and ground state eigenvalues. Thus in essence this is an adiabatic approximation the assumption is that the vibrational chromophore is sufficiently fast compared to the bath coordinates. To the extent that the h times frequency of the chromophore is large compared to kT, and those of the bath are small compared to kT, this separation of time scales exists and so this should be a reasonable approximation. For water, as discussed earlier, some of the bath variables (librations) have frequencies somewhat larger than kT/h, and... 

The assumption of weak electronic coupling may not be valid for vibrational levels near the region where the reactant and product surfaces intersect. If the extent of electronic coupling is sufficient (tens of cm ), the timescale for electron transfer for vibrational levels near the intersectional region will approach the vibrational timescale, electronic and nuclear motions are coupled, and the Born-Oppenheimer approximation is no longer valid. 

Note that the formula for the rate constant in VTST is exactly the same as in TST (compare Equations 6.1, 6.4 and 6.5). In TST the dividing surface is defined by the saddle point in the Born-Oppenheimer electronic energy surface (the maximum along the MEP from reactants to products), while in VTST it is defined as that surface which leads to the minimum value of the rate constant. In both approaches the dividing surface separates product space from reactant space. The assumption in VTST is that a given transition state in equilibrium with reactants will pass through... 

Both assumptions seem reasonable in the case of Born-Oppenheimer wavefunctions. Accordingly, the transition probability can be written ... 

Breakdown of the Born-Oppenheimer Approximation. The B-O approximation is based on the independence of the motions of nuclei and electrons. This is generally a reasonable assumption, except at the crossing point of two electronic states where a minor nuclear displacement is linked to the transition between two electronic states (Figure 3.31). 

By assuming electronic vibrational motions to be separable, as in the Born-Oppenheimer approximation (this assumption for tight-bond electrons such as covalent bonds is inaccurate and will be corrected later) we can find the electronic interaction between the two harmonic oscillators as a function of vibrational coordinates. First, we shall consider the vibration of the center atom only and take the force constant to be... 

The theory of multi-oscillator electron transitions developed in the works [1, 2, 5-7] is based on the Born-Oppenheimer s adiabatic approach where the electron and nuclear variables are divided. Therefore, the matrix element describing the transition is a product of the electron and oscillator matrix elements. The oscillator matrix element depends only on overlapping of the initial and final vibration wave functions and does not depend on the electron transition type. The basic assumptions of the adiabatic approach and the approximate oscillator terms of the nuclear subsystem are considered in the following section. Then, in the subsequent sections, it will be shown that many vibrations take part in the transition due to relative change of the vibration system in the initial and final states. This change is defined by the following factors the displacement of the equilibrium positions in the... 

For a direct absorption, x and g label the excited and ground states, respectively. These states are, in turn, coupled by a dipole moment operator p, and are assumed to be of Born-Oppenheimer type, i.e. the electronic contributions are separated from those of the nucleus. With these assumptions, the interactions between these states result from vibrational overlaps between the ground and excited state with the transition probability Wg x and the density of the final states pf. In general terms, we now can evaluate the transition probability, which mainly depends upon two parameters ... 

An unbiased simulation may use a truncated basis set that represents the lowest complex surface harmonics of the atomic valence shell on a Born-Oppenheimer framework with the correct relative atomic masses. For small molecules, of less than about fifteen atoms, the nuclear framework could perhaps even be generated computationally without assumption. The required criterion is the optimal quenching of angular momentum vectors. The derivation of molecular structure by the angular-momentum criterion will be demonstrated qualitatively for some small molecules. 

The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

Most ab initio calculations on molecules with more than three electrons have involved the assumption of the Born-Oppenheimer (B-O) approximation.1 How-... 

In this chapter we describe the various stages of the factorisation process. Following the separation of translational motion by reference of the particles coordinates to the molecular centre of mass, we separate off the rotational motion by referring coordinates to an axis system which rotates with the molecule (the so-called molecule-fixed axis system). Finally, we separate off the electronic motion to the best of our ability by invoking the Born-Oppenheimer approximation when the electronic wave function is obtained on the assumption that the nuclei are at a fixed separation R. Some empirical discussion of the involvement of electron spin, in either Hund s case (a) or (b), is also included. In conclusion we consider how the effects of external electric or magnetic fields are modified by the various transformations. 

Separation of Electronic and Nuclear Motion. Because, in general, electrons move with much greater velocities than nuclei, to a first approximation electron and nuclear motions can be separated (Born-Oppenheimer theorem [3]). The validity of this separation of electronic and nuclear motions provides the only real justification for the idea of a potential-energy curve of a molecule. The eigenfunction Y for the entire system of nuclei and electrons can be expressed as a product of two functions F< and T , where is an eigenfunction of the electronic coordinates found by solving Schrodinger s equation with the assumption that the nuclei are held fixed in space and Yn involves only the coordinates of the nuclei [4]. 

---