Complex Atoms
The multi-electron, central force problem is one that does not have an exact solution. Approximations must be applied, and some aspects of these approximations are common to all multi-particle problems, particularly those of nuclear physics. Atoms can rig
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Complex Atoms
The multi-electron, central force problem is one that does not have an exact solution. Approximations must be applied, and some aspects of these approximations are common to all multi-particle problems, particularly those of nuclear physics. Atoms can rightly be thought of as the building blocks of our material world. Understanding how quantum mechanics describes atoms, which justifies so much that you have been taught in chemistry and modern physics courses, is the goal for the rest of this text. In this chapter, the hardest part of that broad effort will be attempted which is trying to understand the energy-level structure of an isolated, multi-electron atom. That will be what is meant by the “solution” to the problem at hand. The problem, as presented, will rapidly grow in complexity and seem to be impossible to handle. But then the complexity will shrink as so many terms of interest are shown to be equal to others or zero. The problem being considered is to find the total energy of all of the electrons of a multi-electron atom under the influence of the Coulomb attraction of the electrons to the nucleus and the mutual repulsion of each of the other electrons. Solving the Schr¨odinger equation is not the approach to take. Rather one starts by evaluating the energy assuming that you know the wave functions. The variational procedure tells you that if you then modify the assumed wave functions in any way at all that lowers the energy, both the wave functions and energy are closer to the “correct” ones. Interestingly, how one performs this procedure is not really very important. The optimum method is the Hartree–Fock procedure which will be briefly described in the last section of this chapter. What is important is understanding how one evaluates the energy, what approximations are used in making that evaluation, and which quantum numbers can be used to describe the states available to the atom. After all, when doing atomic physics research, the atom is often in an excited state, so these are every bit as important to understand as the ground state of the atom.
R.L. Brooks, The Fundamentals of Atomic and Molecular Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-1-4614-6678-9 2, © Springer Science+Business Media New York 2013
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2 Complex Atoms
2.1 Shell Model of the Atom Consider forming a system of several electrons electrostatically bound to an infinitely heavy nucleus of charge Ze. The electrons also electrostatically repel each other, so a Hamiltonian for the system can be written that, in atomic units, looks like
or
H=
N N Z 1 ∇2 − i − + 2 | r | | r − rj | i i=1 i>j=1 i
H=
N N Z 1 ∇2 − i − + 2 r r i i=1 i>j=1 ij
(2.1)
where ri is the position of electron i with respect to the nucleus. ri and rij are defined by the corresponding expression in the equation above. While the Hamiltonian is not complete, all additional terms can quite successfully be treated as perturbations. The Schr¨odinger equation for this Hamiltonian is hopelessly complicated
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