We covered part of this on the 1st exam. I put it in the review here again because the
later material builds on the early material. Also you should be able to list the ideal
models, energy expressions, energy eigenfunctions, quantum numbers, angular
momentum (if applicable for the translational, vibrational, rotational, and electronic
ideal energy states.
Quantum Theory can be written down in terms of the postulates. You should be familiar
with the postulates and understand all that they define. Postulate one tells about the
wavefunction and the idea that all the information about the system is in the
wavefunction. The Born interpretation, tells us about d What are the restrictions
on the wavefunctions that are solutions to an eigenvalue eqn. of a physical observable.
Postulate 2 defines the operators in the position representation where x = x, and p = ihd/dx in x dimension. Be able to determine if a particular function is normalized. Be
able to show if two eigenfunctions are orthogonal.
Most of what we are studying are solutions to Schrodinger’s Equation to get the energies
and the wavefunctions (energy eigenfunctions) for the ideal models which model
translational movement (particle in a box)
vibrational movement (harmonic oscillator)
rotational motion (particle on a sphere)
electronic energies (hydrogen atom)
What are the allowed values for the quantum numbers for each of these? We are also
looking at the energy differences between the energy levels in each of these, since this
corresponds to the energy of the photon that would be absorbed or emitted if the system
changed its energy by emitting or absorbing a photon.
The particle in a box case models translational motion, translation of electrons in dye
molecules for instance. How can you determine the wavelength of light absorbed for a
particle in box. What are the allowed values of the quantum #. How do we get
degeneracy (define) for a two or higher dimensional box? Be able to recognize and the
energy and wavefunctions (energy eigenfucntions) for this model.
The ideal model for vibrational motion is the harmonic oscillator. Be able to set up the
Schrodinger equation. What polynomial makes up part of the energy eigenfunctions. Be
able to write out the complete energy eigenfunction and be able to set up the integral that
finds expectation values like the mean displacement or the mean squared displacement.
What is the spacing between the energy eigenvalues. What are the allowed values of the
quantum #. Remember that the solutions for the energy eigenstates can also be applied to
the case of vibration of molecules by using the reduced mass. Be able to show if two
eigenfunctions are orthogonal. What is the difference between the energy eigenstates in
a harmonic oscillator as compared to a particle in a box? What is the energy, wavelength
or frequency of light needed to make a transition between two energy states?
Rotational Motion of a diatomic molecule (Rigid Rotor) is described by the Schrodinger
Eqn. Be able to come up with the Schrodinger eqn in either Cartesian or Spherical Polar
Coordinates. What is din each of these? What are the moment of inertia, center of
mass, and reduced mass? What is the angular momentum. What is the magnitude of the
angular momentum, and the magnitude of the zth component of the angular momentum?
What is the Schrodinger eqn for the particle on a sphere. What are the solution to the 
and parts or the names of the solutions. What are the limits on r,  and ? The overall
wavefunction is the Spherical Harmonic which is governed by 2 quantum #s. What are
the allowed values for those two quantum #s. What are the forms of the eigenvalues for
the energy of a rigid rotator? Be able to calculate the energy of any level or the
difference in the energy between two levels. What is the degeneracy of the lth level?
You should know Schrodinger’s equation for the hydrogen atom, and to recognize the
solutions, the energy eigenfunctions and energies that are the solutions? You should
know what well known polynomial is a solution to the radial part of the wave eqn. for the
hydrogen atom, and what well known polynomial (see particle on a sphere) is a solution
to the angular part. Which quantum number(s) are associated with the radial and angular
part of the solution? What is a hydrogenic atom? The total eigenfunctions for the
hydrogenic atoms are called orbitals. Each orbital is defined by three quantum numbers
n, l, and ml. What are the limitations on these quantum numbers? What are these three
quantities associated with? What are their names? Be able to write the wavefunctions.
What is the ionization energy (what is the value of n what the electron in a hydrogenic
atom is ionized)?. What are the following: shell, subshell, and orbital. In general, the
number of orbitals in a shell of principal quantum number n is n2, so in a hydrogenic
atom, each shell is n2 fold degenerate. Remember the total wavefunction is a product of
the spatial parts defined by n,l,ml and the spin part defined by ms. Also, the symmetry of
the entire wavefunction needs to be antisymmetric for a fermion like an electron.
Remember that the  wavefunction is associated with spin up and  with spin down.
You should know what letters correspond to the different number values of n and l. What
is the value of ml for the pz orbital? Remember the px and the py orbitals are linear
combinations the complex orbitals corresponding to ml = +/-1. What is a nodal plane of
an orbital. How many total nodes, radial nodes, and angular nodes are there in a
particular orbital? What is the general equation for the energy eigenstates of hydrogenic
atoms? When an electron undergoes a transition from an orbital to a different orbital this
also means that it may undergo a change in energy state as well. It undergoes a change of
energy E and discards the excess energy (or absorbs the necessary energy) as a photon
of electromagnetic energy with a frequency  such that h. Are all transitions
between atomic states possible by absorption or emission of a photon? How about
through collision? Think about conservation of momentum. What is the momentum of a
photon? What is a selection rule. What are the selection rules for hydrogenic atoms? Z
is?
You should be able to write down the Schrodinger equation for a many-electron atom.
Can this be solved analytically? In the orbital approx., the solution to Schrodinger’s
equation for the many electron atom is assumed to be a product of the hydrogenic
eigenfunctions, but with nuclear charges that are changed by the presence of the other
electrons. Each electron is thought of as occupying its “own” orbital. In the orbital
approximation, the electronic structure of the atom is expressed by reporting its
configuration which is built up using the Pauli exclusion principle. Aufbau’s principle,
and Hund’s rule. You should know the definitions of the Pauli Exclusion Principle,
effective atomic number, valence electrons, Aufbau principle, and Hund’s Rule. Know
how to “build-up” the order of occupation of hydrogenic orbitals, that is the atomic
electronic configuration for the ground state of a many-electron or ion. Remember the
special exceptions and that the s electrons are lost before the d electrons for a metal
cation. Why do the s, p, and d orbitals with the same principal quantum number, of a
many-electron atom have different energies? The actual wavefunction of a manyelectron atom is a very complicated function of the coordinates of all of the electrons. In
the orbital approximation, we suppose that a reasonable first approx. to the exact
wavefunction is obtained by taking the product of hydrogenic orbitals with nuclear
charges that are modified by the presence of all of the other electrons, hence the
configuration. What is an atomic term symbol? What information does it provide
beyond the atomic electronic configuration?
You should be able to write down Schrodinger’s equation for the hydrogen molecule, H2
This can be solved analytically, and the solution is a one electron molecular orbital. For
molecules with more than one electron, the solutions are difficult, and approximations
must be made. Two approximations were looked at: Valence Bond Theory, and
Molecular Orbital Theory, MO Theory. In MO theory, orbitals which describe electrons
being able to move throughout the entire molecule (MOLECULAR ORBITALS) are
approximated by a linear combination of the atomic orbitals of the atoms which make up
the molecule. In valence bond theory, we consider the valence electrons to be shared.
Here the bonding is described in terms of the more localized atomic orbitals which give
rise to the shapes of the molecules as they overlap and bond. The approximate
wavefunctions are the sum and difference of the product of the atomic wavefunctions.
Hybrid orbitals explain more fully the shapes and the fact that all the bonds in molecules
like CH4 are equal. You should be able to use hybrid orbitals to explain bonding for
molecules similar to the way we did in class for ethylene or ethyne.
In MO theory, the approximate overall wavefunction is simply a sum (bonding) and
difference of the atomic orbitals, thus the term LCAO-MO is used. You should be able to
normalize an LCAO-MO. One should note, and this was not explicitly stated, that we
have been considering only the shape part of the orbitals explicitly, and have left out the
spin. The spin enters since overall the wavefunction must be antisymmetric. There are
many different types of MO that can be formed. The type (bonding or antibonding) and
shape of the orbitals is determined by the way that the atomic orbitals overlap. For
example, if the overlap results in electron density between the atoms with a lowering of
the energy as compared with the separated atoms then this is a bonding MO. If there are
nodes between the atoms, it is most likely an antibonding MO. If the electron density,
has cylindrical symmetry about the internuclear axis and lies along the internuclear axis
then it is a bond.. If it is off axis and has angular momentum then it is a bond.
You should be able to give a molecular orbital energy level diagram for homonuclear and
heteronuclear diatomics, and give ground and excited molecular electronic configurations
based upon the molecular Aufbau principle as applied to the molecular orbitals the
overall molecular wavefunction can then be approximated as a product of these oneelectron molecular orbitals, hence the configuration. Be familiar with the terms core
orbitals, valence orbitals, and virtual orbitals. Why does the ordering of the MO levels
change when going from N2 to O2. You should know how to calculate the bond order and
relative bond strength for a given configuration. For heteronuclear diatomics, you should
be able to figure out which wavefunction makes more of a contribution to the bonding
and antibonding orbitals based on the ionization energies of the atoms bonding to form
the diatomic.
The Variation Principle allows one to find the coefficients in the linear combinations
used to build the molecular orbitals. The basis for this is that the true grnd state energy is
always less than or equal to the value of the expectation energy derived from an
acceptable trial wavefunction. What makes a trial wavefunction acceptable? It must
satisfy the requirements for any wavefunction (continuous, 1st and 2nd derivative, satisfy
boundary conditions). The Variation Principle is more generally applicable as an
approximation method to solve for the energy levels of a nonstandard Hamiltonian. The
following terms should be familiar: trial wavefunction, coulomb integral, resonance
integral, overlap integral, secular determinant, and secular equations. The basis of the
approximation is to minimize the energy with respect to some parameter. You should
know the approximations assumed in the Huckel Approximation with the variation
principle to explain the extra bonding energy due to delocalization and aromaticity, and
to set up the secular determinant for a conjugated molecule. Know the terms HOMO,
highest occupied molecular orbital, LUMO, lowest unoccupied molecular orbital. What
is the  electron binding energy?
Review for Chp.14
This chapter is essentially the application of the ideal theoretical models for atomic and
molecular energy learned earlier and largely involves the interaction of electromagnetic
radiation with an atom or molecule. This is called spectroscopy, and it is a very powerful
tool that can be used to both identify and quantify molecular and atomic species. We
must remember that the use of electromagnetic radiation, in general causes a
transition in the molecule and so the spectrum (spectra is the plural form) is the
absorbance or intensity vs the wavelength, frequency, or energy of light involved in the
transition the E! So we must be very familiar with For instance, in IR absorbance
spectroscopy, the peaks that occur in the spectrum appear at the wavenumber which
corresponds to the energy change associated with moving from the ground vibrational
state to the 1st excited vibrational state (v=0 to v=1). Based on our ideal theoretical
model, Ev = (v+ ½) ho or (v + ½) ħ, so:
Ephoton =  = E1 - E0 =(1 + ½) h0 - (0 + ½) h0 = h0 = hc/
so the wavelength at which there would be a peak is  or in 1/in cm-1 if is in cm.
We discussed how the different parts of the electromagnetic spectrum probed (through
absorption or emission of a photon) the various types of molecular energies that a
molecule possesses. You must be familiar with this.
c ,  = h, wavenumbers = 1/, and velocity = c/n
Remember that you will have a problem associated with filling in the ideal models, the
energy eigenstates (eigenfunctions), the energy eigenvalues, the quantum numbers for
each of the types of molecular energy we spoke about.
You will also likely have a problem associated with the variation principle and Huckel
MO Theory.
Other problems will be similar to concepts covered in the homework or on Quizzes.
The format of the exam will be similar to the last exam.