The Polyelectronic Atom

Within the hydrogen atom, the lower the value of n, the more stable will be the orbital. For the hydrogen atom, the energy depends only upon n for atoms with more than one electron the quantum number / is important as well. 

There are 21 + I orbitals of each type, that is, one s, three p, five d, and seven / orbitals, etc, per set This is also equal to the number of values that m, may assume for a given value of f, since m, determines the orientation of orbitals, and obviously the number of orbitals must be equal to the number of ways in which they are oriented. 

There are n types of orbitals in the nth energy level, for example, the third energy level has s, p, and d orbitals. 

There are n — / — I nodes in the radial distribution functions of all orbitals, for example, the 3s orbital has two nodes, the 4d orbitals each have one. 

There are / nodal surfaces in the angular distributional functions of all orbitals, for example, s orbitals have none, d orbitals have two. 

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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)... 

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... 

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. 

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. 

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. ... 

Equation (2.1) cannot be solved exactly for a polyelectronic atom A because of complications resulting from interelectronic repulsions. We therefore use approximate solutions which are obtained by replacing A with a fictitious atom having the same nucleus but only one electron. For this reason, atomic orbitals are also called hydrogen-like orbitals and the orbital theory the monoelectronic approximation. 

The energy scale is approximate. We only need remember that for a polyelectronic atom, the orbital energy within a given shell increases in the order s, p, d and that the first three shells are well separated from each other. However, the 4s and 3d orbitals have very similar energies. As a consequence, the 3d, 4s and 4p levels in the first-row transition metals all function as valence orbitals. The p orbitals are degenerate (i.e. the three p AOs of the same shell all have the same energy), as are the five d orbitals. 

The above rule can readily be extended to other polyelectronic systems, like the tt system of benzene (6), or to molecules bearing lone pairs as in formamide (7). In this latter case, calling n, c, and o, respectively, the tt atomic orbitals of nitrogen, carbon, and oxygen, the VB wave function describing the neutral covalent structure is given by Equation 3.10 ... 

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