Wave function Born-Oppenheimer approximation

Both inoleciilar and qiiantnin mechanics in ethods rely on the Born-Oppenheimer approximation. In qnantiinn mechanics, the Schrddmger equation (1) gives the wave function s and energies of a inolecii le. 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

The quantity, V(R), the sum of the electronic energy Egjg. computed in a wave function calculation and the nuclear-nuclear coulomb interaction V(R,R), constitutes a potential energy surface having 3N independent variables (the coordinates R). The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

Equation (28) is the set of exact coupled differential equations that must be solved for the nuclear wave functions in the presence of the time-varying electric field. In the spirit of the Born-Oppenheimer approximation, the ENBO approximation assumes that the electronic wave functions can respond immediately to changes in the nuclear geometry and to changes in the electric field and that we can consequently ignore the coupling terms containing... 

Separation of the wave functions describing the motion of nuclei and the wave function describing the electrons (Born-Oppenheimer approximation). This approximation is based on the fact that the nuclear particles are much heavier than the ele ctrons, and therefore much slower than the latter. In such a situation, the electronic wave function can be found at fixed positions of the nuclei. 

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

Taking the Born-Oppenheimer approximation into consideration, the molecular wave function may be written as... 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

Born-Oppenheimer approximation (physchem) The approximation, used in the Born-Oppenheimer method, that the electronic wave functions and energy levels at any instant depend only on the positions of the nuclei at that instant and not on the motions of the nuclei. Also known as adiabatic approximation. born ap an.hT-mar 3,prak s3,ma shan J... 

Background Philosophy. Within the framework of the Born-Oppenheimer approximation (JJ ), the solutions of the Schroedin-ger equation, Hf = Ef, introduce the concept of molecular structure and, thereby, the total energy hyperspace provided that the electronic wave function varies only slowly with the nuclear coordinates, electronic energies can be calculated for sets of fixed nuclear positions. The total energies i.e. the sums of electronic energy and the energy due to the electrostatic re-... 

Using the Born-Oppenheimer approximation, electronic structure calculations are performed at a fixed set of nuclear coordinates, from which the electronic wave functions and energies at that geometry can be obtained. The first and second derivatives of the electronic energies at a series of molecular geometries can be computed and used to find energy minima and to locate TSs on a PES. 

Before we can discuss the recent developments further, we must discuss terminology. This will also provide a guide to earlier literature as well as to the classification we use in our review of recent papers (Section lOd). As mentioned, the total wave function in the Born-Oppenheimer method. is expressed as a product of electronic and vibrational wave functions. What has resulted is that different types of electronic wave functions have been used (which is not necessarily confusing), and that in many cases a particular selection has been called the Born-Oppenheimer approximation (which has led to confusion). We discuss here only the two predominant choices of... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Up to now, we have been discussing many-particle molecular systems entirely in the abstract. In fact, accurate wave functions for such systems are extremely difficult to express because of the correlated motions of particles. That is, the Hamiltonian in Eq. (4.3) contains pairwise attraction and repulsion tenns, implying that no particle is moving independently of all of the others (the term correlation is used to describe this interdependency). In order to simplify the problem somewhat, we may invoke the so-called Born-Oppenheimer approximation. This approximation is described with more rigor in Section 15.5, but at this point we present the conceptual aspects without delving deeply into the mathematical details. 

To deduce whether a transition is allowed between two stationary states, we investigate the matrix element of the electric dipole-moment operator between those states (Section 3.2). We will use the Born-Oppenheimer approximation of writing the stationary-state molecular wave functions as products of electronic and nuclear wave functions ... 

The Born-Oppenheimer approximation separates the molecular wave function into a product of electronic and nuclear wave functions ... 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

As indicated in Figure 5-1, the observed bands are sometimes rather broad. This may be surprising, since it seems to indicate that the electronic energies are poorly defined. However, the explanation is that the wave function of the molecule is a function not only of the electronic motions but the rotational and vibrational motions as well. Assuming the Born-Oppenheimer approximation, we have for the total wave function,... 

We have used the Born Oppenheimer approximation to factor 4 0/3, I,ma into electronic and nuclear parts and have further assumed that the former are orthogonal to enable us to reduce V. Both wave functions may be approximated by products of electronic, nuclear rotation and vibrational wave functions. The last of these may be factored out at once, and... 

In addition, we assume, for the systems of interest here, that the electronic motion is fast relative to the kinetic motion of the nuclei and that the total wave functions can be separated into a product form, with one term depending on the electronic motion and parametric in the nuclear coordinates and a second term describing the nuclear motion in terms of adiabatic potential hypersurfaces. This separation, based on the relative mass and velocity of an electron as compared with the nucleus mass and velocity, is known as the Born-Oppenheimer approximation. 

We have considered the case of vibrational motion of the photofragments accompanied by slow relative motion. We have developed the adiabatic approach to evaluate the nuclear wave-function (Jp and obtained eqs. 74 and 96. Note, that instead of a system of electrons and nuclei (Born-Oppenheimer approximation), we considered here only nuclear motion of a polyatomic system with several degrees of freedom, one of which is "fast" relative to the others. 

The relevant question regarding secondary IEs on acidity is the extent to which IEs affect the electronic distribution. How can an inductive effect be reconciled with the Born-Oppenheimer approximation Although the potential-energy function and the electronic wave function are independent of nuclear mass, an anharmonic potential leads to different vibrational wave functions for different masses. Averaging over the ground-state wave function leads to different positions for the nuclei and thus averaged electron densities that vary with isotope. This certainly leads to NMR isotope shifts (IEs on chemical shifts), because nuclear shielding is sensitive to electron density.16... 

If the wave functions of individual species are separable (Born-Oppenheimer approximation), and if there is a weak energy coupling between the system of long-lived, chemically reacting particles with other degrees of freedom, the total probability distribution function p(t, r, v, e) of the system can be separated ... 

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