Energies of Atomic Orbitals in Many-Electron Systems

Energies of Atomic Orbitals in Many-Electron Systems 

CHAPTER 6 Quantum Theory and the Electronic Structure of Atoms 

Student Annotation Two electrons in the same orbital with opposite spins are said to have paired spins. 

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In Chapter 2, the development of atomic orbitals for many-electron atoms was built upon an understanding of the orbitals obtained from the exact solution for the one-electron hydrogen atom. Similarly, it is useful to examine the electronic structure for the simplest molecular species—the hydrogen molecule ion (Hj)—to begin our discussion of molecular orbitals. Like the H atom, the ion contains only one electron. The difference is the presence of two nuclei instead of one. The potential energy of the system is a sum of the Coulomb attraction of the electron to each of the two nuclei and Coulomb repulsion between the positively charged nuclei. 

There is no unequivocal answer to the question as to which is the better method. Calculations by the VB method are likely to be more reliable than those by the MO method, but in practice are much more difficult to carry out. For many-electron molecules the MO procedure is simpler to visualize because we combine atomic orbitals into molecular orbitals and then populate the lower-energy orbitals with electrons. In the VB method, atomic orbitals are occupied, but the electrons of different atoms are paired to form bonds, a process that requires explicit consideration of many-electron wave functions. To put it another way, it is easier to visualize a system of molecular orbitals containing N electrons than it is to visualize a hybrid wave function of N electrons. 

Whereas, for non-Coulombic potentials, one can define nr and , n is then no longer related simply to the binding energy. Indeed, for a complex, many-electron atom, it is not at all obvious how one should set about quantising the system, since there is no guarantee that the orbits of individual electrons will close.6 In fact, conservation of the angular momentum for individual electrons is, at best, only an approximation. It would hold exactly for central fields. Even then, the same simple, precise relationship between n and as for H is not to be expected for many-electron atoms. As we shall see, the very meaning of n (the principal or most important quantum number) becomes less clear-cut for many-electron systems. In a nutshell the n and quantum numbers of... 

In 1927, Burrau calculated the energy of Hj and Heitler and London treated the hydrogen molecule. In 1928, the Heitler-London or valence bond method was applied to many electron systems, and simultaneously Hund and Mulliken started the development of the molecular orbital theory. In 1931, Slater expressed the v/avefunctions of complex molecules in terms of Slater determinants made up of linear combinations of atomic orbitals. Thus, the Golden Age was born. 

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