http://web.uconn.edu/~ch351vc (MAIN MENU)
Quantum Chemistry (including some Statistical Mechanics)


This is the main site for Chemistry 351, Advanced Physical Chemistry.

This is a site devoted to theoretical chemistry subjects. The contributions have been added as we came upon these topics in class. The order is irrelevant.

All of the texts are written in greater detail than normal, on the theory that most chemistry students are not enamoured of mathematics and/or physics, so providing detailed steps in mathematical derivations would appear to be helpful.

Some of the readings are in Computer Guided Reading format. CGR is a scheme in which students/readers answer questions while they read which, when answered correctly, open up more text for reading. The earliest CGR texts (below) were composed using LaTeX2HTML, resulting in equations which appeared in browsers as gif files. This is not the best scheme, and a new (~2002-3) scheme has been introduced in which readings are presented to the reader in pdf form, which is clearer and easier to read (and print). Depending on work load, I may convert the older readings to pdf form, since students appear to prefer this form.

If ever anyone reads any of these materials and learns something, I will have done something of value.

I would like to thank all the students who have labored mightily to help expunge the inevitable typos which infested these materials.

I would also like to thank my wife who has suffered my insane desire to improve this site.

These materials are proof certain that "time on task" is not a necessary nor a sufficient condition for learning.
Note added May 2009. With my upcoming retirement, I wish to make sure that these documents are not lost when the University upgrades this computer, and I fail to respond to the inevitable query about "keeping" these materials.
Therefore, I've converted most of this material to pdf form, and inserted them into the digital commons,which is a new form of publication. The editing of these materials may have resulted in errors, as I unfortunately know, so if you find one, please, contact me and I will attempt correction.
Further, the links below which concern CGR are being changed, since CGR failed completely, to route the reader to the digital commons site appropriately.
Thank you for your understanding.




These are a dsp2 orbital (hybrid) shown in CountourPlot3D format and Countour form.







There is an error in this drawing which I've just discovered. Can you find it? If so, can you e-mail me using this link. See the following: Infra-Red Microwave Spectra, Overtones, Degeneracy and Thermal Populations all in one diagram
After long (and sorry) experience, the following warning is issued. It can not be said that the site is error free. Therefore, all messages concerning what's wrong with this site are gratefully appreciated.

Thanks.
This is a link to undergraduate pchem problems

Here is an UpToDate List of Physical Constants From England
Guided (CGR) and normal Readings:
  1. (new CGR) Why Study Thermodynamics, Statistical Mechanics and Quantum Mechanics ?

  2. Review of Selected (Relevant) Elementary Mechanics
    1. Classical Mechanics Related to the H-atom Problem. here
    2. The Classical Kepler problem at digitalcommons.uconn.edu/chem_educ/13 .
    3. Ellipsoidal Coordinates at digitalcommons.uconn.edu/chem_educ/5/ are useful in the H2+ problem. This is the one electron simplest molecule possible.


  3. Introduction to Fourier Series (and assorted related topics)
    A good overview can be found at Wikipedia as well as in standard texts.
    1. Introduction to Fourier Series, etc.
      1. Minimum Error in an Expansion(CGR) Derivation
        See also at digitalcommons.uconn.edu/chem_educ/27/
      2. (CGR) Dirac Notation and Vector Spaces Part One
      3. (CGR) Introduction to the Classical Wave Equation Part Three
        See also at digitalcommons.uconn.edu/chem_educ/29/
    2. Fourier Integrals
      1. (CGR) Introduction to Fourier Integrals Part One
      2. (CGR) Introduction to Fourier Integrals Part Two
      3. (CGR) Introduction to Fourier Integrals Part Three
      See also at digitalcommons.uconn.edu/chem_educ/28/
    3. Ehrenfest's Theorem
      See also at digitalcommons.uconn.edu/chem_educ/53/


  4. The Particle in a Box discussed in relation to square, rectangular, and circular boxes can be found at at digitalcommons.uconn.edu/chem_educ/12/
    See also at digitalcommons.uconn.edu/chem_educ/25/ for a discussion of ladder operators applied to this problem.

  5. The Harmonic Oscillator
    1. (CGR) The Harmonic Oscillator, Classically
      See also at digitalcommons.uconn.edu/chem_educ/19/
    2. (CGR) The Harmonic Oscillator, Quantum Mechanically, Introduction (I)
    3. (CGR) The Harmonic Oscillator, Quantum Mechanically, Introduction (II)
      See also at digitalcommons.uconn.edu/chem_educ/17/ and at digitalcommons.uconn.edu/chem_educ/15/
    4. Vibrations in a diatomic molecule (transformation to center of mass) Full text here
    5. The Schrodinger Equation for the Harmonic Oscillator
    6. Solving Hermite's Differential Equation
      1. Hermite Polynomials via the Method of Frobenius at digitalcommons.uconn.edu/chem_educ/17/
      2. (CGR) Hermite Polynomials (Part 1)
      3. (CGR) Hermite Polynomials (Part 2)
      4. (CGR) Hermite Polynomials (Part 3)
      5. (CGR) Hermite Polynomials (Part 4)
      See also at digitalcommons.uconn.edu/chem_educ/17/
    7. (CGR) Continued Fraction Method for Hermite's Differential Equation
      I've combined the Hermite, Laplace and Legendre equation solutions using continued fractions into one manuscript. /5 See at digitalcommons.uconn.edu/chem_educ/77/
    8. The Harmonic Oscillator Using Ladder Operators
      1. The Harmonic Oscillator Using Ladder Operators at digitalcommons.uconn.edu/chem_educ/19/
      2. (CGR) The Harmonic Oscillator Using Ladder Operators (I)
      3. (CGR) The Harmonic Oscillator Using Ladder Operators (II)
      4. (CGR) The Harmonic Oscillator Using Ladder Operators (III)
      5. (CGR) The Harmonic Oscillator Using Ladder Operators (IV)
      See also at digitalcommons.uconn.edu/chem_educ/19/

    9. Rotations
      1. (CGR) Planar Rotations(I)
      2. CGR) 3 Dimensional Compounded Rotations(II)
      3. (CGR) Equivalent Axis for 3 Dimensional Compounded Rotations(II)
    10. The Rigid Rotor and Angular Momentum
      1. (CGR) Center of Mass and Moment of Inertia (diatomic)
      2. A particle moving about a central point in space.
      3. Rotations in the center of mass system for 2 particle systems (CGR) Classical Diatomic Rotational Energy
      4. The Schrodinger Equation for the Rigid Rotor Text herePDF, October 2003
      5. Variable Separation in Quantum Chemistry in 2 and 3 dimensions May, 2004
      6. Legendre Polynomials, etc.
        1. (CGR) Heuristic Derivation of Legendre Polynomials (I)
        2. (CGR) dz2 orbitals
        3. (CGR) Laplace's Equation in Spherical Polar Coordinates
        4. (CGR) Imaginary Representation of Spherical Harmonics
        5. (CGR) Legendre's Equation (I)
        6. (CGR) Legendre's Equation (II)
        7. (CGR) Legendre Polynomials and Schmidt Orthogonalization
        8. (CGR) Legendre's Equation and the Method of Frobenius
        9. (CGR) Legendre's Equation and the Rodrigues' Formula
      7. (CGR) Moment of Inertia (Polyatomics)
      8. (CGR) Continued Fraction Method for Legendre's Differential Equation
      9. (CGR) Continued Fraction Method for Legendre's Differential Equation (Part 2)
      10. (CGR) Euler Angles
        1. (CGR) Euler Angles(Part 1)
        2. Euler Angles(Part 2)
      11. Angular Momementum
        1. (CGR) Angular Representation of Angular Momentum (I)
        2. (CGR) Angular Representation of Angular Momentum (II)
        3. (CGR) Angular Representation of Angular Momentum (III)
        4. (CGR) Angular Representation of Angular Momentum (Ladder Operators) (I)
        5. (CGR) Angular Representation of Angular Momentum (Ladder Operators)(II)


    11. The H Atom
      1. Atomic Units
        1. Atomic Units at digitalcommons.uconn.edu/chem_educ/6/ under my name.
        2. Atomic Units at digitalcommons.uconn.edu/chem_educ/7/ under my name.
      2. The Schrodinger Equation for the H-atom
        The Bohr Atom as precursor to the H-atom, can be found at digitalcommons.uconn.edu/chem_educ/3/
      3. Understanding Orbitals (Graphical)
        1. (CGR) Understanding Atomic Orbitals(Part 1)
        2. (CGR) Understanding Atomic Orbitals(Part 2)
        3. (CGR) Understanding Atomic Orbitals(Part 3)d-orbitals
        4. (CGR) Understanding Atomic Orbitals(Part 4)d-orbitals
      4. Trial and Error solutions for the H-atom
      5. Factoring the Schrodinger Equation for the H-atom (radial/angular decomposition)
      6. Radial Ladder Operators and the Radial Equation
      7. Ladder Operators and the Runge Lenz Vector
        The following materials have been superceded by materials at digitalcommons.uconn.edu/chem_educ/14/ where the precursor matters for using the Runge Lenz vector are addressed. Another manuscript is under preparation for this material (6/2018)
      8. Factoring the Schrodinger Equation for the H-atom
      9. The Radial Equation and its Solution
        1. Laguerre Polynomial Introduction at digitalcommons.uconn.edu/chem_educ/24/
        2. (CGR) Laguerre Polynomials(Part 1)
        3. (CGR) Laguerre Polynomials(Part 2)
        4. (CGR) Laguerre Polynomials(Part 3)
        5. (CGR) Laguerre Polynomials(Part 4)
        6. (CGR) Laguerre Polynomials(Part 5)
      10. (CGR) Continued Fraction Solution for Laguerre Polynomials


    12. The H2+ Problem
      1. Coordinate Systems and the diatomic molecule Elliptical Coordinate Transformation
      2. The Schrodinger Equation for the H2+ cation free radical
      3. (CGR) The Laplacian in Elliptical Coordinates
      4. Separating the Schrodinger Equation into Three (Linked) equations
      5. The constants of the motion for the H2+ system
        1. (CGR) Three Body Constant of the Motion (I)
        2. (CGR) Three Body Constant of the Motion (II)
        3. (CGR) Three Body Constant of the Motion (III)
        4. (CGR) Three Body Constant of the Motion (IV)
      6. Solving the exact problem
      7. Solving the approximate problem. (Please note that the following two entries were recently found to be in error, and a new version has been published at digital commons.)
        1. Molecular Orbital Theory for H2+ Using Maple (I) (s-orbitals to sigma)
        2. Molecular Orbital Theory for H2+Using Maple (II)
        Commentary
        1. (CGR) LCAO's are not Eigenfunctions of the H2 + Hamiltonian (Part 1)
        2. (CGR) LCAO's are not ... (Part 2)
        3. (CGR) LCAO's are not ... (Part 3)


    13. Polyelectronic Atoms
      1. Helium as a correlation problem
      2. Perturbation schemes
      3. Variational schemes
      4. Hartree-Fock type schemes
        1. (Postscript)Hartree Fock Methods Illustrated Using Helium (not CGR)May 2004
        2. (PDF)Hartree Fock Methods Illustrated Using Helium (not CGR) May 2004
      5. "Exact" Solutions
        1. The new pdf version of the Hylleraas Derivations at digitalcommons.uconn.edu/chem_educ/8/
        2. Gronwall Derivation
          1. Gronwall Derivation

    14. Polyelectronic Molecules
      1. Huckel Theory
        1. (CGR) Huckel MO Theory (Part 1)
        2. (CGR) Huckel MO Theory (Part 2)
        3. (CGR) Huckel MO Theory (Part 3)
        4. (CGR) Matrix Huckel MO Theory (Part 1)
        5. (CGR) Matrix Huckel MO Theory (Part 2)
      2. What Constitutes a Good MO?
        1. (CGR) MO Theory (Part 1)
        2. (CGR) MO Theory (Part 2)
        3. (CGR) MO Theory (Part 3)
      3. Woodward Hoffmann Electrocylic Reaction Mechanisms Using Huckel Theory
        1. (CGR) Introduction to Electrocyclic Ring Closures
        2. (CGR) Exchange Integrals
        3. (CGR) The Hamiltonian
        4. (CGR) Conrotatory Hamiltonian
        5. (CGR) Disrotatory Hamiltonian
        6. (CGR) Integration and Summary
      4. Butadiene
        1. (CGR) Butadiene using Huckel Theory and MAPLE
        2. (CGR) Butadiene using Huckel Theory and MAPLE
      5. Hartree Fock Schemes
      6. A Valence Bond versus Molecular Orbitals discussion can be found here (rejected J.Chem.Ed.)


    15. Time Dependence and Transitions

    16. Magnetic Resonance
      1. Invitation to Magnetism, Magnetic Resonance, and More at digitalcommons.uconn.edu/chem_educ/9/
      2. More on Magnetic Resonance at digitalcommons.uconn.edu/chem_educ/10/
    17. Pauli Spin Matrices
      1. Pauli Spin Matrices
      2. Pauli Spin Matrices For NMR
    18. Here are advanced readings, using a new CGR scheme. Comments about this new scheme would be greatly appreciated.
      1. Introduction to Perturbation Theory (non-degenerate)March 2003
        1. (CGR) Introduction to Time Dependent Perturbation Theory (Part One)March 2003
        2. (CGR) Introduction to Time Dependent Perturbation Theory (Part Two)March 2003





    Elementary (non-guided) Readings:
    1. DeMoivre's (Euler's) Theorem Doing Trigonometry the Easy Way (DeMoivre's Theorem)
    2. Stirling's Approximation for the Factorial (N!) Sum -> Integral Derivation
    3. The Gauss Integral A Classic Derivation of a Wonderful Integral
    4. The Gauss Integral(II) Related Integrals (for Stat. Mech., Kinetic Theory, etc..
    5. Legendre Transformation
    6. Sponaneity
    7. Derivation of Boltzmann Distribution without Lagrange Multipliers

    Non-elementary (non-guided) Readings:
    1. Ladder Operator for the Particle in a Box
    2. Generating Functions for Legendre Polynomials
    3. Orbital Allignment as a pdf

    If you encounter problems with the any aspect of this server, please, please pop me a note. You can send me a message using this address and your own mailer. And, PLEASE give your electronic address in any correspondence so that I can properly respond.

    Chemistry 352 readings follow:

    Note, these are not Computer Guided Readings, since they are just simple pdf's. As we finish them, I will convert them to pdf's. cwd
    1. Laplacian in Spherical Polar Coordinates (detailed) Please check this for typos before I convert it to Computer Guided Reading format. Thanks. cwd
    2. Commutators and Time Dependence
    3. Ladder Operator for the Particle in a Box
    4. Matrix Representations
      The next three are from last semester (I think). Please check that I am right.
    5. Planar Rotations(I)
    6. 3 Dimensional Compounded Rotations(II)
    7. Equivalent Axis for 3 Dimensional Compounded Rotations(II)
    8. Moment of Inertia in 2 & 3 Dimensions (corrected by G. Churchill and reposted December 2004) http://digitalcommons.uconn.edu/chem_educ/22/
    9. Euler Angles and their representationas infinitesimal rotations http://digitalcommons.uconn.edu/chem_educ/23/
    10. Intro to Perturbation Theory
    11. The Morse Potential for Vibration
    12. Intro to Normal Modes (not ready for prime time.)
    13. Degenerate Perturbation Theory
    Thank you for reading any of the above.
    Carl W. David

    The following materials are also available:
    Beginning Discussion of Mathematics Prerequisites for Physical Chemistry (a work in progress, comments cheerfully accepted)

    If you are interested in how these questions were created, i.e., Perl, CGI, HTML, Latex2HTML, Java, etc., etc., etc., then you may wish to look at a "book" which explains at various levels how these questions can be implemented. Since this book is constantly changing, please forgive its rather informal style.
    If you are interested in computer assisted testing, and/or physical chemistry problems using the WWW, then please look at this site which contains a large (and increasing) supply of physical chemistry problems in a computer assisted testing mode.



    Note added, June 2006. The CGR format has failed completely, i.e., students do not learn more using this complicated form of reading, at least in the small statistical sample I used, and the idea is being abandoned. As a result, the CGR texts are being re-formated to PDF files. Since I'm close to retirement age, and this site will disappear upon that happy occasion, I am relocating the pdf files to an archival server which the University has just provided. How long that lasts, well, I won't be here to see, will I?

    Ein kluger Mensch lernt von seinen Fehlern, ein klügerer Mensch von den Fehlern von Anderen!

    Frank W. David, June 4, 1904 to June 10, 1960
    If you wish to comment to C. W. David about this material, assumptions you (or I) are making, inconsistencies in the phraseology of the material, objections to the material, etc., etc., etc., you may use this space for that purpose.Please include your e-mail address explicitly. Thank you.
    e-mail addresss = at (18.119.161.216)

    Last updated May 2, 2003