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7: Approximation Methods

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    The Schrödinger equation for realistic systems quickly becomes unwieldy, and analytical solutions are only available for very simple systems - the ones we have described as fundamental systems in this module. Numerical approaches can cope with more complex problems, but are still (and will remain for a good while) limited by the available computer power. Approximations are necessary to cope with real systems. Within limits, we can use a pick and mix approach, i.e. use linear combinations of solutions of the fundamental systems to build up something akin to the real system. There are two mathematical techniques, perturbation and variation theory, which can provide a good approximation along with an estimate of its accuracy. These two approximation techniques are described in this chapter.

    • 7.1: Slater's Rules for Shielding
      Slater's rules allow you to estimate the effective nuclear charge from the real number of protons in the nucleus and the effective shielding of electrons in each orbital "shell" (e.g., to compare the effective nuclear charge and shielding 3d and 4s in transition metals). Slater's rules are fairly simple and produce fairly accurate predictions of things like the electron configurations and ionization energies.
    • 7.2: The Variational Method
      In this section we introduce the powerful and versatile variational method and use it to improve the approximate solutions we found for the helium atom using the independent electron approximation.
    • 7.3: Linear Variational Method and the Secular Determinant
      A special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known) .
    • 7.4: Trial Functions Can Be Linear Combinations of Functions with Variational Parameters
      An alternative approach to the general problem of introducing variational parameters into wavefunctions is the construction of a  wavefunction as a linear combination of other functions each with one or multiple parameters that can be varied
    • 7.5: Perturbation Theory Expresses the Solutions in Terms of Solved Problems
      Perturbation theory is the single most important method of solving problems in quantum mechanics, and is widely used in atomic physics, condensed matter and particle physics. Perturbation theory is another approach to finding approximate solutions to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbation" parts.
    • 7.E: Approximation Methods (Exercises)
      These are homework exercises to accompany the chapter.

    7: Approximation Methods is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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