Abstract
This is a quite compact and important chapter. The attentive reader should be able to work through it rather quickly after having refreshed her or his basic knowledge in the previous two chapters. We shall make here the first steps to generalize the methods successfully applied for the H atom, and allow for an interaction potential which is no longer strictly proportional to 1/r. This forms the basis for understanding the physics behind the periodic system of the elements which is summarized in Sect. 3.1. The most simple multi-electron systems are the alkali atoms. Their energy levels are discussed in Sect. 3.2 phenomenologically and analyzed qualitatively and quantitatively, briefly explaining quantum defect theory. In Sect. 3.3 we introduce time independent perturbation theory – a tool that will be used later on quite frequently – and illustrate it by way of example for the energies of alkali atoms.
The Schrödinger equation for the hydrogen atom has been solved in a fully analytic manner. This was possible due to the special mathematical properties of the 1/r Coulomb potential. We introduce now step by step deviations from this particular simple case, and aim at describing more and more subtle effects – and later on also more complex problems as they are observed in spectroscopic and dynamic studies of atoms, molecules and clusters.
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Notes
- 1.
Strictly speaking, one should even introduce an additional quantum number j for the total angular momentum, which becomes increasingly relevant as Z gets larger. We have already familiarized ourselves with it in the context of the Stern-Gerlach experiment in Sect. 1.9.5 – and shall come back to it in detail in Sect. 6.2.5.
- 2.
This holds at least for the smaller alkali atoms where relativistic effects play a minor role and spin-orbit interaction can be treated as a small perturbation.
- 3.
The \(r^{n^{\ast}}\) factor used here in contrast to (2.119) can improve the convergence.
- 4.
This is a rather crude choice. It leads, however, to qualitatively correct wave functions. The radius of the ionic core for Na+ is typically given in the literature as \(0.095\operatorname{nm}=1.8a_{0}\). At this distance this screened potential V S (r) is about −1.1/r.
References
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Authors and Affiliations
Acronyms and Terminology
- a.u.:
-
‘atomic units’, see Sect. 2.6.2.
- DFT:
-
‘Density functional theory’, today one of the standard methods for computing atomic and molecular electron densities and energies (see Sect. 10.3).
- IP:
-
‘Ionization potential’, of free atoms or molecules (in solid state physics the equivalent is called “workfunction”).
- MQDT:
-
‘Multichannel quantum defect theory’, advanced form of QDT for the interpretation of complex atomic and molecular spectra, especially of highly excited Rydberg states (see Sect. 3.2.6).
- NIST:
-
‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html.
- ODE:
-
‘Ordinary differential equation’.
- QDT:
-
‘Quantum defect theory’, interprets experimental spectra by phase shifts in the radial wave functions and makes predictions for scattering processes (see Sect. 3.2.6).
- UV:
-
‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between \(100\operatorname{nm}\) and \(400\operatorname{nm}\) according to ISO 21348 (2007).
- VIS:
-
‘Visible’, spectral range of electromagnetic radiation. Wavelengths between \(380\operatorname{nm}\) and \(760\operatorname{nm}\) according to ISO 21348 (2007).
- VUV:
-
‘Vacuum ultraviolet’, spectral range of electromagentic radiation. Part of the UV spectral range. Wavelengths between \(10\operatorname{nm}\) and \(200\operatorname{nm}\) according to ISO 21348 (2007).
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Hertel, I.V., Schulz, CP. (2015). Periodic System and Removal of ℓ Degeneracy. In: Atoms, Molecules and Optical Physics 1. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54322-7_3
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