Interelectronic repulsion Hamiltonian

We note that three spin-allowed electronic transitions should be observed in the d-d spectrum in each case. We have, thus, arrived at the same point established in Section 3.5. This time, however, we have used the so-called weak-field approach. Recall that the adjectives strong-field and weak-field refer to the magnitude of the crystal-field effect compared with the interelectron repulsion energies represented by the Coulomb term in the crystal-field Hamiltonian,... 

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian That is, we wish to find orbitals that minimize (4 hp H I hp). By applying variational calculus, one can show that each such orbital i/f, is an eigenfunction of its own operator hi defined by... 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

[Pg.271]

The examination takes place in two stages, one corresponding to the formal interelectronic repulsion component of the Hamiltonian HER and the second to the spin-orbit coupling term Hes. As will be pointed out, in principle, and in certain cases in practice, it is not proper to separate the two components. However, the conventional procedure is to develop HLS as a perturbation following the application of Her. That suffices for most purposes, and simplifies the procedures. Any interaction between the d- or/-electron set and any other set is ignored. It is assumed that it is negligible or can be taken up within the concept of an effective d-orbital set. 

If in atoms it is natural to split the Hamiltonian into an upper -turbed part with the desired characteristics and a non-negative definite perturbation J, in molecules, if one considers an unperturbed part for which the lower part of the spectrum is known, the perturbation is not positive. If one chooses the perturbation in the same way as in atoms i. e. the interelectronic repulsion, the unperturbed spectrum has no known spectrum. This dilemma was faced by Bazley and Fox, who suggested a method of truncations combined with a splitting of the unperturbed Hamiltonian into parts corresponding to the different nuclei. 

Almost all approaches to many-electron wave functions, for both atoms and molecules, involve their formulation as products of one-electron orbitals. If the interelectronic repulsion term in (6.23) is small compared with the other terms, the Hamiltonian is approximately separable into independent operators for each electron and the two-electron wave function, f (1, 2), can be written as a simple product of one-electron functions,... 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

We shall make use of the effective Hamiltonian formalism [14] that enables us to isolate effects of interest from irrelevant complications. We divide the electronic Hamiltonian into a strong part H° and a weak part H, and we shall suppose that H° is simple enough to be solved exactly. The Hamiltonian including the cubic field and interelectronic repulsion only is the usual choice for H in the case of the 3d group ions. Then H should include all other interactions (spin-orbit coupling, lower symmetry fields, electron-phonon interaction, external fields, strain etc). The most important assumption is that the perturbations, described by the H Hamiltonian (in particular the JT interaction) must be smaller relative to the initial splitting due to H°. In the case of the 3d metal ions the assumption is usually well justified. 

In the Pariser-Parr-Pople scheme, the so-called zero differential overlap approximation is used, and the u-electron system is treated as a nonpolarizable core. The interelectronic repulsions are explicitly taken into account in the total Hamiltonian. Resonance integrals, core integrals, and electronic repulsion integrals are given empirically, and Coulomb penetration integrals are neglected. ... 

It should be emphasized that OEMO theory does not treat explicitly interelectronic repulsions, which are reproduced by the two electron part of a complete hamiltonian operator, as well as internuclear repulsions. These effects are partially accounted for by virtue of the empirical evaluation of matrix elements in the OEMO method and will be grouped undo-the heading steric effects . It is obvious that steric effects will tend to favor uncongested structures. It is then apparent that the OEMO theory will lead to incorrect predictions when steric effects become a dominant influence. 

The Schrddinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules is not separable in any coordinate system and cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

Suppose we take the interelectronic repulsions in the li atom as a perturbation on the remaining terms in the Hamiltonian. By the same steps used in the treatment of helium, the unperturbed wave functions are products of three hydrogenlike functions. For the ground state,... 

The simplest semiempirical w-electron theory is the free-electron molecular-orbital (FE MO) method, developed about 1950. Here the interelectronic repulsions l/r,y are ignored, and the effect of the cr electrons is represented by a particle-in-a-box potential-energy function V" = 0 in a certain region, while V = oo outside this region. With the interelectronic repulsions omitted, in (16.1) becomes the sum of Hamiltonians for each electron hence (Section 6.2)... 

LFT is a parametric approach in which the symmetry of the complex is treated explicitly but the bonding is handled implicitly through the ligand field parameters. These parameters describe the three contributions to the overall Hamiltonian, FI the ligand field, Hup, interelectronic repulsion, Z/ er and spin orbit coupling, Hps- The relative importance of each of these terms depends on the element s position in the periodic table. 

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