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Unit 4: Quantum Chemistry: Atomic Structure and Spectra  

In this unit, you will start to apply quantum mechanics to solve real chemistry problems, thus finally approaching quantum chemistry.  Your studies will begin with the simplest system, the hydrogen-like atoms (or ions), which is comprised by a nuclei and a single electron orbiting around it.  As additional electrons enter the system under study, the wavefunction describing the system will become more complex, and you will use approximation techniques to simplify your calculations.  You will discover how a particular atomic structure results in a well-defined ground state electronic configuration.  This electronic configuration is responsible for the atom photon absorption and emission spectra, as well as its ionization energy, electron affinity, and degenerate energy levels.

Unit 4 Time Advisory
Time Advisory: This unit will take you 15 hours to complete.

☐    Subunit 4.1: 5 hours

☐    Subunit 4.2: 6 hours

☐    Subunit 4.3: 4 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

  • Describe the structure of the hydrogen-like systems.
  • Derive the eigenvalues of hydrogen-like systems.
  • Explain the difference in terms of kinetic and potential energy between single-electron and multi-electrons atoms/ions.
  • Describe the shielding phenomena, and explain how it affects energy calculations.
  • Explain and apply the Pauli Principle, Aufbau Principle, and Hund’s rule to derive electronic configurations.
  • Derive term symbols for atoms and ions.
  • Apply techniques of approximation to energy calculations.

4.1 The Structure of Single-electron Systems (Hydrogen-like Systems)  

  • Reading: Boston University: Professor Dan Dill’s “One-Electron Atom” and “One-Electron Atom Radial Functions” Link: Boston University: Professor Dan Dill’s “One-Electron Atom” and “One-Electron Atom Radial Functions” (PDF)
     
    Instructions: For Professor Dill’s notes, click on the links above, scroll down the webpage to the italic headings: “One-Electron Atom” and “One-Electron Atom Radial Functions.”  To open these PDF files, click on the hyperlink next to these headings.  Read the entire documents (10 pages and 8 pages, respectively).  While reading the material, please follow the mathematical derivation of the eigenvalues for a one-electron system, shell amplitudes, and radial functions.  Studying this resource should take approximately 5 hours to complete.  Note that these resources also cover the material you need to know for subunits 4.1.1–4.1.4.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.1.1 One-Electron Atoms/Ions (Hydrogen-Like Atoms/Ions)   Note: This subunit is covered by the readings assigned beneath subunit 4.1.  In particular, please focus on pages 1–5 of Professor Dan Dill’s “One-Electron Atom” to learn about the quantum mechanics of systems consisting of a single electron and an atomic nucleus.

4.1.2 Effective Potential Energy   Note: This subunit is covered by the readings assigned beneath subunit 4.1.  In particular, please focus on pages 5–6 of Professor Dan Dill’s “One-Electron Atom” to learn about the effective potential energy in atomic systems.

4.1.3 Atomic Orbitals and Their Energies   Note: This subunit is covered by the readings assigned beneath subunit 4.1.  In particular, please focus on pages 7–10 of Professor Dan Dill’s “One-Electron Atom” to learn about the energy of atomic orbitals.

4.1.4 Shell Amplitudes and Shell Energies   Note: This subunit is covered by the readings assigned beneath subunit 4.1.  In particular, please focus on Professor Dan Dill’s “One-Electron Atom Radial Functions” to learn about shell amplitudes and their energies.

4.2 The Structure of Multiple-Electron Atoms  

  • Reading: Boston University: Professor Dan Dill’s “Many-Electron Atoms: Fermi Holes and Fermi Heaps” Link: Boston University: Professor Dan Dill’s “Many-Electron Atoms: Fermi Holes and Fermi Heaps” (PDF)
     
    Instructions: Please click on the link and scroll down the webpage to the italic heading, “Many-Electron Atoms: Fermi Holes and Fermi Heaps.”  To open the PDF file, click on the link next to the heading.  Read the entire PDF (17 pages).  While reading the material, pay close attention on how the introduction of extra electrons changes the terms of the Schrodinger equation.  Studying this resource should take approximately 2 hours to complete.  Note that this reading also covers the material you need to know for subunits 4.2.1–4.2.6.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Reading: Southern Methodist University: Professor Werner Horsthemke’s “Physical Chemistry II Lecture Notes” Link: Southern Methodist University: Professor Werner Horsthemke’s “Physical Chemistry II Lecture Notes” (PDF)
     
    Instructions: Please click on the “Physical Chemistry II Lecture Notes” link, scroll down the webpage to the section, “Lecture Notes,” and click on the “PC2Set5.pdf” link.  This set of notes will open as a PDF file, and you can read the entire document (29 pages).  Make sure you understand how the Pauli Principle, the Aufbau Principle, and Hund’s Rule and their application derive the electronic configuration of atoms.  Studying this resource should take approximately 2 hours to complete.  Note that this reading also covers the material you need to know for subunits 4.2.1–4.2.6.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.1 The Orbital Approximation   Note: This subunit is covered by the readings assigned beneath subunit 4.2.  In particular, please focus on pages 1–6 of Professor Dan Dill’s “Many-Electron Atoms: Fermi Holes and Fermi Heaps.”

4.2.2 Pauli Principle   Note: This subunit is covered by the readings assigned beneath subunit 4.2.  In particular, please focus on pages 7–10 of Professor Dan Dill’s “Many-Electron Atoms: Fermi Holes and Fermi Heaps.”

4.2.3 The Pauli Exclusion Principle   Note: This subunit is covered by the readings assigned beneath subunit 4.2.  In particular, please focus on pages 11–17 of Professor Dan Dill’s “Many-Electron Atoms: Fermi Holes and Fermi Heaps.”

4.2.4 Aufbau Principle   Note: This subunit is covered by the readings assigned beneath subunit 4.2.  In particular, please focus on page 5.17 of Professor Werner Horsthemke’s “Physical Chemistry II Lecture Notes” to learn about the Aufbau Principle.

4.2.5 Hund’s Rule   Note: This subunit is covered by the readings assigned beneath subunit 4.2.  In particular, please focus on pages 5.13, 5.18, and 5.23 of Professor Werner Horsthemke’s “Physical Chemistry II Lecture Notes” to learn about Hund’s Rule.

4.2.6 Spin-Orbit Coupling   Note: This subunit is covered by the readings assigned beneath subunit 4.2.  In particular, please focus on pages 5.24–5.29 of Professor Werner Horsthemke’s “Physical Chemistry II Lecture Notes” to learn about Spin-Orbit Coupling.

4.2.7 The Helium Atom  

  • Reading: MIT’s OpenCourseWare: Professor Griffin and Professor Voorhis’s “Helium Atom” Links: MIT’s OpenCourseWare: Professor Griffin and Professor Voorhis’s “Helium Atom” (PDF)
     
    Instructions: Please click on the MIT link above, scroll down the webpage to “Lecture 25” (note: the title of this lecture is different, but the Helium Atom file will open up), and click on the PDF link to download the lecture.  Read the entire document (8 pages).  As you read through this resource, repeat the approximation technique used to solve this two-electron system.  Studying this resource should take approximately 1 hour to complete.
     
    Terms of Use: Robert Guy Griffin and Troy Van Voorhis, Physical Chemistry 5.61, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed August 16, 2012). License: Creative Commons BY-NC-SA 3.0. The original version can be found here.

4.2.8 Term Symbols  

  • Reading: University of California Davis: UC Davis ChemWiki’s “Atomic Term Symbols” Link: University of California Davis: UC Davis ChemWiki’s “Atomic Term Symbols” (HTML)
     
    Instructions: Please click on the link to and read “Atomic Term Symbols.”  This resource will teach you how to derive the term symbols.  Studying this resource should take approximately 1 hour to complete.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.3 Introduction to Techniques of Approximation  

  • Reading: University of Southampton: Professor Chris-Kriton Skylaris’s “Perturbation Theory” Link: University of Southampton: Professor Chris-Kriton Skylaris’s “Perturbation Theory” (PDF)
     
    Instructions: Please click on the link above, and under the heading “Perturbation Theory,” select the “Lecture notes” link to access the PDF file.  Please read this entire chapter of lecture notes (42 pages).  Studying this resource should take approximately 4 hours to complete.  Note that this resource also covers the material you need to know for subunits 4.3.1–4.3.3.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: The Saylor Foundation’s “Assessment 8” Link: The Saylor Foundation’s “Assessment 8” (DOC)
     
    Instructions: Complete the attached assessment questions to check your understanding of the material covered thus far. Once you have completed the assessment, you may check your answers against the “Answer Key” (DOC).
     
    Completing this assessment should take approximately 1 hour.

4.3.1 Time-Independent Perturbation Theory   Note: This subunit is covered by the readings assigned beneath subunit 4.3.  In particular, please focus on pages 2–12 to learn about time-independent perturbation theory.

4.3.2 Time-Dependent Perturbation Theory   Note: This subunit is covered by the readings assigned beneath subunit 4.3.  In particular, please focus on pages 12–27 to learn about time-dependent perturbation theory.

4.3.3 Applications of Perturbation Theory   Note: This subunit is covered by the readings assigned beneath subunit 4.3.  In particular, please focus on pages 28–41 to learn about applications of perturbation theory.