Atoms/atomic polyelectronic

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)... 

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. 

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. 

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... 

The theory of chemical shifts of the nuclei of polyelectronic atoms is complicated and certainly does not yet produce results in quantitative agreement with theory. It is conceivable that a more qualitative use of these parameters might be more appropriate to the problem in hand and an example of this sort is illustrated in Fig. 8, due to Lauterbur (75), where the chemical shifts of the Si and nuclei in analogous com-... 

The treatment of atoms with more than one electron (polyelectronic atoms) requires consideration of the effects of interelectronic repulsion, orbital penetration towards the nucleus, nuclear shielding, and an extra quantum number (the spin quantum number) which specifies the intrinsic energy of the electron in any orbital. The restriction on numbers of atomic orbitals and the number of electrons that they can contain leads to a discussion of the Pauli exclusion principle, Hund s rules and the aufbau principle. All these considerations are necessary to allow the construction of the modern form of the periodic classification of the elements. 

Polyelectronic atom, 20-43 Polyhalide ions, 839-843 Polyhalogen cations, 848 Polymers, and phosphazene polymers, 773-775 Polynuclear carbonyl complexes, 633-639... 

The aim of molecular orbital theory is to provide a complete description of the energies of electrons and nuclei in molecules. The principles of the method are simple a partial differential equation is set up, the solutions to which are the allowed energy levels of the system. However, the practice is rather different, and, just as it is impossible (at present) to obtain exact solutions to the wave equations for polyelectronic atoms, so it is not possible to obtain exact solutions for molecular species. Accordingly, the application of molecular orbital theory to molecules is in a regime of successive approximations. Numerous rigorous mathematical methods have been utilised in the effort to obtain ever more accurate solutions to the wave equations. This book is not concerned with the details of the methods which have been used, but only with their results. 

T. Mercouris, Y. Komninos, C.A. Nicolaides, The state-specific expansion approach to the solution of the polyelectronic time-dependent Schrodinger equation for atoms and molecules in unstable states, in C.A. Nicolaides, E. Brandas (Eds.), Unstable States in the Continuous Spectra, Part I Analysis, Concepts, Methods, and Results, Vol. 60 of Advances in Quantum Chemistry, Academic Press, 2010, pp. 333 405. 

Hartree s method was to write a plausible approximate polyelectronic wavefunc-tion (a guess ) for an atom as the product of one-electron wavefunctions ... 

Because the Schrodinger equation cannot be solved exactly for polyelectron atoms, it has become the practice to approximate the electron configuration by assigning electrons to hydrogen-like orbitals. These orbitals are designated by the same labels as for hydrogen s orbitals and have the same spatial characteristics described in the previous section, Orbitals. ... 

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