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Advanced Physical Chemistry
Review for the Final Exam
You should know the selection rules for atomic transitions, and be able to calculate the energy re
quired for a particular transition. Why do the selection rules arise. What is the transition dipole
moment for a transition between states the initial and final state. Remember that group theory
can help you here. Do magnetic dipole transitions and electric quadrupole transitions have the
same selection rules?
In the case of the hydrogen atom, why is there fine structure on the spectrum, that is how does
spin-orbit coupling come about and change the spectrum . Be able to use the
expression for the interaction energy, and how does it relate to the total energy. Be able to write
a term symbol. What is Russel Saunders coupling, jj coupling? What is the multiplicity?
Be able to write out the hamiltonian for a many electron atom. What is the Coulomb integral?
What is the exchange integral? What does it mean for a wavefunction to be symmetric,
antisymmetric? Why is the average of the electron-electron repulsion energy lower for the
asymmetric spatial wavefunction than the symmetric spatial wavefunction. What are the
selection rules for symmetric and asymmetric states? What about with regard to the
multiplicities. Be able to state the Pauli Exclusion Principle. Note that one statement of the
Pauli Principle is derived from the He atom spectrum, the allowed states are all antisymmetrical
under particle interchange. What is the purpose of the Slater determinant. Be able to use it.
What is a spinorbital? What are STO’s. What is the difference between them and the electronic
configuration? What is the Zeeman Effect and the Stark Effect?
Be able to explain the general idea between an SCF-Hartree Fock calculation. What is the core
Hamiltonian. Remember that the Coulomb operator represents the Coulombic interaction of
electron 1 with electron 2 in the orbital. The exchange operator takes into account the effects of
spin correlation. Remember that the Hartree-Fock SCF atomic orbitals are not the most refined
orbitals that can be obtained. They are rooted in the orbital approximation and therefore to an
approximate central field form of the potential. The true wavefunctions should depend explicitly
on the instantaneous separations of the electrons, not merely their distances from the nucleus.
What is the Born-Oppenheimer Approximation? How does this approximation allow for the
Schrodinger equation for H2+ to be solved? Be able to write the Hamiltonian for diatomic
molecules. How can the molecular potential energy curve for H2+ be calculated? What
coordinate system allow the electronic Schrodinger eqn for H2+ to be solved exactly? Remember
that the basis for molecular orbital calculations, MO calculations, is the LCAO. The AO used in
these expansion constitute the basis set for the calculation. In principle an infinite basis set
should be used to give the best representation, but this is not possible.
In chapter 9 then we largely talked about the methods used to construct the basis sets and use
them to solve the electronic Schrodinger eqn. under the Born-Oppenheimer Approximation. Two
main approaches to solve the electronic Schrodinger eqn., ab initio, and semiempirical methods.
What is the basis of each calculation, and what are the similarities and differences of the two
techniques. The Hartree Fock-SCF method is the starting point of many of the ab initio methods.
What is the crucial complication in all electronic structure calculations, remember in HF this is
treated in an average way. Essentially the HF method used the variation theory to calculate the
correction to the core hamiltonian (no electron repulsion included). What does it mean to be self
consistent in this application? Here the spinorbitals are used. The n lowest energy spinorbitals
are called the occupied orbitals, for a finite number m of spinorbitals what is the number of
virtual orbitals. Remember that there should be an infinite number of spin orbitals which are
eigenfunctions of the Fock operator, but have to cut it off somewhere.
What is the difference between a RHF wavefunction and a restricted open-shell formalism and a
UHF formalism. The HF procedure works well for atoms due to spherical symmetry, but for
molecules, this symmetry is not there. Must expand the spatial parts of the spinorbitals in terms
of another basis set. From a set of M basis functions , obtain M linearly independent spatial
wavefunctions. End up solving the Roothaan equations, which are solved form the secular
determinant. One and two electron integrals are part of the matrix. For M basis functions, there
are M4 two-electron orbitals to evaluate. A complete set of basis functions must be used to
represent the spinorbitals exactly, which again means an infinite # would give the energy in the
variational expression, the Hartree Fock limit. But a finite basis is always used leading to
basis-set truncation errors. Choose bases wisely. Why use the STO’s as basis functions, why use
the GTO’s. Why are GTO’s special. What is a primitive Gaussian and a contracted Gaussian
function? What is meant by an s-type, p-type etc Gaussian? What is the minimal basis set? A
DZ, TZ and SV basis set or a DZP? What is meant by STO-3G? What is meant by the 3-21G
basis set or the 6-31G** basis set? What is configuration interaction, how can one correct for
this, CI? Talked about the perturbation theory methods as well MPPT, MP3. In the
semiempirical methods, you should be able to write a Huckel molecular orbital secular
determinant. What is the differential overlap? You should know the acronyms ZDO, CNDO,
INDO, MINDO, MINDO, NDDO, MNDO, AM1 and PM3. What are the main differences
between the different methods?
In the last part of the class, we had a quick review of thermodynamics. You should be able to
give the 1st and 2nd laws of thermodynamics, know the definitions of dU, dH, dA, dG. From
chapter 1 in the Statistical Thermodynamics book you should be able to calculate the number of
ways of getting a particular configuration assuming identical but distinguishable particles. What
happens to the number of ways with regard to each of the allowed configurations. Get a
predominant configuration. How can one find the predominant configuration? What is Stirlings
approximation? From a consideration of dW/d 0, one can derive the Boltzmann distribution.
The Boltzmann distribution gives a description of the equilibrium configuration in an isolated ass
embly of distinguishable units. From it we can calculate the relative populations for the various
levels ni/nj. Be able to do his. We can also get the fraction of units in a particular state from the
Boltzmann distribution law in the form ni = N exp(-ei/kT) / z, where z is the partition function.
What is the definition of the partition function? Know this both in terms of states and energy
levels. Remember that the degeneracy must be included for one. We also realize that here for
the first time for polyatomic gases there are many contributions to the energy. This must be
reflected in the partition functions.
ztot = ztrans * zrot * zvib * z elec
Remember that the reason there is a lower case z here is that the actual magnitude of z is entirely
independent of the number of units present since the sum is over the quantum states or the
quantum levels characteristic of single units.
We also determined that Boltzmann Statistics could be applied to the case of ordinary gas
molecules which are indistinguishable since there is free translation. So no way to distinguish
based on placement. Remember that Boltzmann statistics still apply in the dilute gas
approximation. What is that? The expression for W is different, but the predominant
configuration is still a Boltzmann-like expression:
ni = N exp(-ei/ kT) / z
Expressing the partition function for an N unit assembly became: Z = zN (distinguishable case) or
Z = zN/N! (indistinguishable). Putting his in terms of all of the degrees of freedom for the energy
gives: Ztot = ztransN/N! * zrot N * zvib N * z elec N\
What is this expression useful for. Why does only ztrans have N! ?
Interested in the form of the partition function for N units in order to calculate macroscopic value
s of Thermodynamic variables from statistical considerations AND it turns out that theses thermo
dynamic parameters are expressible in terms of the partition function. There is an expression for
U (internal energy kT2(dlnZ/dT), for S (entropy) klnZ + E/T, for A (Helmholtz free energy) -kTln
Z, and other thermodynamic functions. Remember the form of Z will depend on the type of
assembly (ensemble).
Lastly putting it all together, the expression for the energy states or levels derived from Quantum
Mechanics are put into the partition function expressions. What Quantum systems correspond to
the translational, rotational, and vibrational energy states? What is used for the electronic energy?
Note that the energy differences between these states can be determined form Spectroscopic
Measurements. Thus one can conceivably derive Macroscopic Thermodynamic Variables from
Spectroscopic information. You should be familiar with the fundamental and exact expressions
used to calculate the values of the various partition functions for the various degrees of freedom.
Overall
The final examination will be at 10:00 in this room on Tuesday, May. The problem part of the
exam will be from the new material. From the Quantum book expect a problem related to the
energies associated with spin orbit coupling and the selection rule and/or you may need to
perform a semi-empirical calculation using Huckel MO. From the new material, be prepared to u
se combinatorics like we did in class and/or calculate and use the Boltzmann distribution and
partition functions corresponding to the various degrees of freedom.