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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.