Polyelectronic

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

It is useful to represent the polyelectronic wave function of a compound by a valence bond (VB) structure that represents the bonding between the atoms. Frequently, a single VB structure suffices, sometimes it is necessary to use several. We assume for simplicity that a single VB stiucture provides a faithful representation. A common way to write down a VB structure is by the spin-paired determinant, that ensures the compliance with Pauli s principle (It is assumed that there are 2n paired electrons in the system)... 

A symmetry that holds for any system is the permutational symmetry of the polyelectronic wave function. Electrons are fermions and indistinguishable, and therefore the exchange of any two pairs must invert the phase of the wave function. This symmetry holds, of course, not only to pericyclic reactions. 

Tie hydrogen molecule is such a small problem that all of the integrals can be written out in uU. This is rarely the case in molecular orbital calculations. Nevertheless, the same irinciples are used to determine the energy of a polyelectronic molecular system. For an ([-electron system, the Hamiltonian takes the following general form ... 

Figure 7.1 Orbital energies, E, typical of a polyelectronic atom...

Since s = j only, j is not a very useful quantum number for one-electron atoms, unless we are concerned with the fine detail of their spectra, but the analogous quantum number J, in polyelectronic atoms, is very important. 

The hydrogen atom and one-electron ions are the simplest systems in the sense that, having only one electron, there are no inter-electron repulsions. However, this unique property leads to degeneracies, or near-degeneracies, which are absent in all other atoms and ions. The result is that the spectrum of the hydrogen atom, although very simple in its coarse structure (Figure 1.1) is more unusual in its fine structure than those of polyelectronic atoms. For this reason we shall defer a discussion of its spectrum to the next section. 

Had we been dealing with a polyelectron system, there would have been extra terms in the total Hamiltonian to take account of the electron-electron repulsion. These would have also been collected into Hg. 

The first step is to work out e in terms of the one- and two-electron operators and the orbitals. .., For a polyatomic, polyelectron molecule, the electronic Hamiltonian is a sum of terms representing... 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

A many-electron atom is also called a polyelectron atom. 

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

The equation determining the optimum orbitals of polyelectronic systems in the case of the SCF and MCSCF theories can be written in the form ... 

We present here numerical results illustrating that the solutions of the radial equations (eq.(5) for the hydrogen-like case and eq.(14) for polyelectronic atoms) are weakly dependent of e in a finite volume. 

In the case of polyelectronic atoms we have calculated the Jum and Gum parameters as described in the preceding section (see above, the 1.4) i.e. using the normalised orbitals resulting from a RHF calculation of the atom in a gaussian basis (11). 

The method of assuring the antisymmetry of a system of electrons, asjfpr example in a polyelectronic atom, is to construct what is often called the Slater determinant.1 If the N elections are numbered 1,2,3,... and each can occupy a state a, b, c,.... the determinant... 

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