CHEM 408 – Sp06 3/31/2006 Exam II – Prep Questions The following is a collection of questions of the sort that I used for a 408 exam last year. The coverage here isn’t complete, but most of the questions are applicable to this year’s. Exceptions are #7, 9, 13, & 17, which are on details that we didn’t really discuss and I wouldn’t ask this time around. Elementary Quantum Mechanics & 1e Atoms: 1. Is 23)(axexf−= an eigenfunction of the operator xO ∂∂= /ˆ? If so, what is its eigenvalue? ____________________________________________________________________________________ 2. What is the expectation value of x (<x>) for an electron in the 1s state of a H atom? (Explain.) ____________________________________________________________________________________ 3. ),,( zyxψ is the wavefunction of a single particle in 3 dimensions. (a) what are the units of ψ expressed in the SI system (m, kg, s, …)? (b) What does it mean to say that the wavefunction is normalized? (Be explicit.) ____________________________________________________________________________________ 4. What is qualitatively incorrect about the following sketch as a representation of the radial distribution function of the 3s orbital of H? r00P(r) ____________________________________________________________________________________ 5. A mathematical expression of the variational theorem is: 1ˆˆEddHHapp≥ΨΨΨΨ=><∫∫∗∗ττ Describe the meaning of this expression in words and comment on the importance of this theorem in electronic structure calculations. ____________________________________________________________________________________ 6. Consider the radial distribution function of the 1s orbital of a H atom. (a) Sketch this function as realistically as possible. [Hint: The function has a maximum at r = a]. (b) Describe carefully the physical meaning of the maximum of this function.CHEM 408 – Sp05 3/31/2006 2(b) What would be the radius of the corresponding maximum in a plot of the radial distribution of the 1s orbital of Be3+ (Z=4)? (c) What is the energy of the 1s orbital in Be3+? (d) What can you say about the radial distribution and the energy of an electron in the 1s orbital of neutral Be? Multi-Electron Atoms: 7. Consider the electronic wavefunction of a He atom: )1()2(1)1()1(1)2,1(βαss=Ψ (a) Does this wavefunction possess the proper exchange symmetry? (b) Ignoring spin effects, discuss why the function )2(1)1(1)2,1( ss=Ψ where “1s” represents the 1s orbital on He+ is not a very accurate description of the ground state of a He atom. (c) Again ignoring spin effects, explain why the function )2()1()2,1(11ψψ=Ψ where ψ1 is any 1-electron orbital cannot be an exact representation of the electronic ground state of He. ____________________________________________________________________________________ 8. Discuss how the Pauli principle is related to the symmetry requirements on electronic wavefunctions. ____________________________________________________________________________________ 9. What is the meaning of the expression: baψψ=Ψ )2,1( ? ____________________________________________________________________________________ 10. Consider the electronic ground state of the element S (Z=16). (a) What is its electronic configuration? (b) What is its spin multiplicity? ____________________________________________________________________________________ 11. We often write the ground state configurations of the elements C and N as: C: 1s2 2s2 2p2 N: 1s2 2s2 2p3 but by using the same symbols for both atoms, for example “2s”, we do not mean that the orbitals are identical in the two cases. Discuss what is different about the 2s orbitals in C and N, both with respect to their spatial extent (orbital exponent) and their energy? ____________________________________________________________________________________CHEM 408 – Sp05 3/31/2006 3Molecular Electronic Structure: 12. Discuss the Born-Oppenheimer approximation and its importance in chemistry. ____________________________________________________________________________________ 13. The Born-Oppenheimer approximation for the total wavefunction of a molecule can be summarized by the equations: );()(),(NeelNnucNeRrRRrrrrrrψψ≅Ψ );()();(ˆNeelNelNeelelRrRERrHrrrrrψψ= NelTHHˆˆˆ−= {})()()(ˆNnucNnucNelNRERRETrrrψψ=+ Describe the physical meaning of );(NeelRrrrψ and )(NelREr in these equations. ____________________________________________________________________________________ 14. Using the variables defined in the figure at the right, express the electronic Hamiltonian for the diatomic ion LiH2+ in atomic units and identify the meaning of the various terms involved. ____________________________________________________________________________________ 15. Answer the following questions about Hückel molecular orbital theory. (a) What are the main assumptions made by the theory. (b) How are the overlap τϕϕdji)1()1(∫ and “hamiltonian” τϕϕπdhji)1(ˆ)1(∫ integrals evaluated? (c) What is the HMO secular determinant that would be needed to solve for the orbital energies of cyclobutadiene, c-C4H4? ____________________________________________________________________________________ 16. Define the following terms: (a) basis set (a) Slater-type orbital (b) Gaussian type orbital (c) contracted Gaussian function ____________________________________________________________________________________ 17. Write the expressions for the following integrals between two molecular orbitals ψi and ψj: (a) The overlap integral Sij. 3+1+1rr2rrARrBRree3+3+1+1+1rr2rrARrBRreeCHEM 408 – Sp05 3/31/2006 4(b) The Coulomb integral Jij. (c) The exchange integral Kij. ____________________________________________________________________________________ 18. Gaussian-type functions are almost universally employed as basis functions for electronic structure calculations. What are the good and bad features of these functions compared to Slater-type