Two-electron orbit

We first note that an isolated atom with an odd number of electrons will necessarily have a magnetic moment. In this book we discuss mainly moments on impurity centres (donors) in semiconductors, which carry one electron, and also the d-shells of transitional-metal ions in compounds, which often carry several In the latter case coupling by Hund s rule will line up all the spins parallel to one another, unless prevented from doing so by crystal-field splitting. Hund s-rule coupling arises because, if a pair of electrons in different orbital states have an antisymmetrical orbital wave function, this wave function vanishes where r12=0 and so the positive contribution to the energy from the term e2/r12 is less than for the symmetrical state. The antisymmetrical orbital state implies a symmetrical spin state, and thus parallel spins and a spin triplet. The two-electron orbital functions of electrons in states with one-electron wave functions a(x) and b(x) are, to first order,... 

The matrix elements r o are quite straightforward to evaluate. Before leaving them, however, it is worthwhile to make some qualitative observations about them. First, the Condon-Slater rules dictate that for the one-electron operator r, the only matrix elements that survive are those between determinants differing by at most two electronic orbitals. Thus, only absorptions generating singly or doubly excited states are allowed. 

Absorption in the ultraviolet (k 100 - 400 nm) and the visible (A. 400 - 800 nm) is primarily the result of transitions in the electronic state of the molecule. In such a process, the transition dipole moment would be proportional the overlap in the densities of the charge distributions between the two electron orbitals involved in a transition. The periodic displacement of electrons from one state to another will cause the charge distribution to be anisotropic, with net negative and positive contributions in certain locations within the molecule. The result is the formation of a dipole moment. Very often, dye molecules that absorb in the visible are dispersed within a sample or attached to the molecules of a sample and are used to monitor its degree of alignment. However, since the relative orientation of such a dye molecule to the molecular axes of the constituent sample molecules is often unknown, the interpretation of these measurements can be difficult. 

Currently available numerical results indicate that the one-dimensional heUum atom is completely chaotic. The best-known semiclassical quantization procedure for completely chaotic systems is Gutzwiller s trace formula (see Section 4.1.3), which is based on classical periodic orbits. Therefore we search for simple periodic orbits of the one-dimensional he-hum atom. Since a two-electron orbit is periodic if the orbits ni t), 0i t)) and (ri2(t), 2( )) of the first and second electron have a common period, the periodic orbits of the one-dimensional model can be labelled with two integers, m and n, which count the 27r-multiplicity of the angle variables 0i and 02 after completion of the orbit. Therefore, if for some periodic orbit... 

Clearly, the two-electron orbitals require a double-expansion the p < v restriction is introduced to avoid double-counting of the configurations. 

Fig. 3. A schematic representation of two-electron orbitals crossing as a function of nuclear coordinate. The curve is redrawn from Dunlap and Mei (1983) where it was the crossing of the doubly degenerate 7r and 2a in the region of internuclear separation 1.0-4.5 bohr, and for occupations corresponding to the Z and A, states which are degenerate in spatial symmetry-restricted Xa-like methods. The one-electron energy scale is from —4 to — 14eV.

It is somewhat artificial, however, to describe the effects of ionization so simply. The configuration of the ion is certainly not to be described accurately in terms of the same two-electron orbitals which were appropriate for the molecule. The presence of the positive charge results in altered a values—the effective electronegativities of the constituent atoms—so that slightly altered atomic orbitals have to be used in the linear combination method. Further, the effects of electrostatic polarization in systems of mobile electrons can assume significant proportions, which amounts to saying that the positive hole left by the ith electron does not correspond exactly to It is hardly surprising, therefore, that... 

Electrons partly because of their very small size are impossible to locate at any particular time. It is however possible to locate a region or volume where the electron is most likely to be found. This region is called Orbital. Each orbital can hold a maximum of two electrons. Orbitals can be divided into s, p", d, f types. Each type of orbital has its own characteristic shape. 

Chapters 2, 3, and 5 form the core of this book. Perturbations are defined and simple procedures for evaluating matrix elements of angular momentum operators are presented in Chapter 2. Chapter 3 deals with the troublesome terms in the molecular Hamiltonian that are responsible for perturbations. Particular attention is devoted to the reduction of matrix elements to separately evaluable rotational, vibrational, and electronic factors. Whenever possible the electronic factor is reduced to one- and two-electron orbital matrix elements. The magnitudes and physical interpretations of matrix elements are discussed in Chapter 5. In Chapter 4 the process of reducing spectra to molecular constants and the difficulty of relating empirical-parameters to terms in the exact molecular Hamiltonian are described. Transition intensities, especially quantum mechanical interference effects, are discussed in Chapter 6. Also included in Chapter 6 are examples of experiments that illustrate, sample, or utilize perturbation effects. The phenomena of predissociation and autoionization are forms of perturbation and are discussed in Chapters 7 and 8. 

The following is an example of the SCF method in practice in the treatment of the two electrons present in a helium atom. Because helium has two electrons orbiting a -1-2 nucleus, it presents itself as the three-body problem shown in Figure 4.9, where the nucleus is presumed to be at rest and therefore sits at the origin of the coordinate system. The Hamiltonian for the helium atom includes three potential energy terms an attractive force between electron I and the nucleus (r ), an attractive force between electron 2 and the nucleus (r2), and the electron-electron repulsion between the two electrons (r 2), as shown in Equation (4.8). 

Figure 10 Crystal structure of BagPrsIis (a) and the interaction scheme of three linearly oriented dz2 orbitals as well as the a bonding three-center-two-electron orbital (b)...

We are interested in studying the correlation energy of the two-electron orbitals of particles 1 and 2. Thus, turning to finite separations 21 = 2 — i we consider the Schrodinger equation... 

In the following we assume the separation r2i to be large enough to exclude overlap of electron orbitak localized at different particles. In this case it is permissible to omit all exchange terms in the two-electron orbitals v>(i-j, r ) and to write... 

The two-electron orbital v(i (,i y) is a linear combination of products of one-electron orbitals localized at different particles. Applying second order perturbation theory to the Schrodinger equation (7.4), the energy of the orbital specified by (7.6) is found to be... 

The simplest way to make a two-electron orbital wave function satisfy the antisym-metrization requirement is to add a second term that is the negative of the first term with the coordinate labels interchanged, giving... 

There is an important fact about electrons that we can see in Eq. (18.2-6). If the spin orbitals fi and f2 are the same function, the antisymmetric wave function is the difference of two identical terms and vanishes. Therefore, a given spin orbital cannot occur more than once in any term of an antisymmetrized two-electron orbital wave function. This is an example of the Pauli exclusion principle, which applies to orbital wave functions for any number of electrons In an antisymmetrized orbital wave function, the same spin orbital cannot occur more than once in each term. Another statement of the Pauli exclusion principle is In an antisymmetrized orbital wave function, no two electrons can occupy the same spin orbital. 

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