課程名稱︰物理化學二─量子力學 課程性質︰化學系大二必修 課程教師︰鄭原忠 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰5/17/2011 考試時限（分鐘）：120min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Physical Chemistry II: Quantum Chemistry Mid-term Exam #3 Date: 5/17/2011 ** Keep your answers brief. Extra wrongs will result in deduction of points. (h'=h bar 下標以_表示) 1. (10%) State the vaiational theorem and give a simple proof. (hint: superposition principle). 2. (20%) The allyl radical, C3H3+, is a linear molecule with three electrons in three π orbitals. We will apply the Hückcl theory to describe the π system in this problem. (a) Give the definitions of the Hückel α and βintegrals. You need to give the physical meanings and mathematical formulas, and clearly describe the functions in the definitions. (b) Write down the secular equation for the π molecular orbital energy. (c) Find the energy level of the molecular orbitals in terms of the Hückel α and β integrals and draw the energy level diagram. What is the HückeI ground-state energy of the π system? (d) On the energy level diagram, denote the HOMO →LUMO transition. What is the wavelength of the lowest-energy electronic transition? 3. (15%) One day, Obi-Wan gave Anakin the following Hamiltonian to study a molecular system with N_a atoms and N_e electrons: ^ Na h'^2 Na Na e^2 Ne h'^2 Na Ne Z_a *e^2 H = Σ ___ ▽a^2+ Σ Σ ___________ + Σ ___ ▽i^2 - Σ Σ ___________ a=1 2Ma a=1 b=1 4πε0R_ab i=1 2me a=1 i=1 4πε0 r_ia Na Ne e^2 + Σ Σ ____________ i=1 j=1 4πε0 r_ij (a) It is obvious that Obi-Wan, though strong with the Force, was not very good in Physics, and had made several mistakes. Can you help Anakin out by correcting the mistakes and give the physical meaning to each term in H !? (b) Obi-Wan taught Anakin how to use the Born-Oppenheimer approximation to simplify the time-independent Schrodinger equation. What is the Born-Oppenheimer approximation? Give the mathematical expression as well as the physical justifications for thc approximation. (c) Anakin applied the Born-Oppenheimer approximation to write down the electronic Schrodinger equation and nuclear Schrodinger equation. Can you show the derivations? Clearly define the variables and operators in your formulas. 4. Consider oxygen atom in the configuration [He]2s^2 2p^4. (a) (3%) What is the number of states that belong to this electron configuration? (b) (%) Omitting spin-orbital couplings, this electronic configuration splits into three terms: 1s, 1D, 3P. Give the number of states in each of the terms. (c) (3%) According to Hund's rules, which term is the ground state? (d) (5%) When spin-orbital couplings are considered, the states further split according to their total angular moment quantum numbers J=L+S. This yields five term symbols for the [He]2s^2 2p^4 configuration. Give the five term symbols. 5. In this problem we consider low-energy excited states of two elecfrons inside a one-dimensional box with length a, which can he approximated as products of the n=1 and n =2 particle-in-a-box eigenfunctions. In other words, the electronic configuration is that having one electron occupying the n = 1 level and the other electron occupying the n = 2 level. Note that for a single particle in a box, the energy level E_n =n^2h^2/8ma^2 , and the eigenfunction Φ_n(x) = Nsin(nπx/a ) (a) (2%) The spatial part ot the two-electron wavefunction can be written as 1/√2[φ_1(1)φ_2(2)±φ_1(2)φ_2(1)] Explain why simple product such as φ_1(1)φ_2(2) is not a valid two-electron wavefunction. (b) (4%) Consider the spin part of the wavefunction. Using the Pauli exclusion principle, show that there are four possible spin-adapted wavefunctions for this configuration. Give the four full (spatial+spin) wavefunctions. (c) (2%) Separate the four states into a triplet manifold and a singlet manifold. (d) (5%) Define the integrals J=∫∫[φ_1(x_1)]^2*[e^2/(4πε0│x_1-x_2│)]*[φ_2(x_2)]^2dx_1dx_2 and K=∫∫[φ_1(x_1)φ_2(x_2)]*[e^2/(4πε0│x_1-x_2│)]* [φ_1(x_1)φ_2(x_2)]^2dx_1dx_2 Evaluate the energies of the singlet and triplet states to showthat 1E=E_0+J+K and 3E=E_0+J-K. What is E_0 ? (e) (2%) According to the previous results, what is the multiplicity of the first excited state of the two-electron system? (J,K>0): 6. (20%) Consider the electronic structure of the Hydrogen molecule within the minimal basis set model using Ψ= c_1 1s_A+c_2 1s_B as the LCAO molecular orbitals for H2. (a) Use symmetry to determine the coefficients for the two possible LCAO MOs and give the normalized wavefunctions for both MOs in terms of the overlap integral S = ∫1S_A1S_B dτ. Show that the two MOs are orthogonal to each other, and that the lower and higher energy MOs can be denoted as the 1σ_g and1σ*_u orbitals, respectively. You should clearly explain your reasoning. (b) Give the Slater determinant representing the ground-state wavefunction of the hydrogen molecule in terms of the 1σ_g and1σ*_u MOs and spin α/β functions. (c) Expand the Slater determinant to give the ground-state wavefunction in terms of AOs. Denote the covalent contributions and the ionic contributions. Explain why this Slater determinant fails miserably in describing the dissociation of the hydrogen molecule. (d) Describe what the Hartree-Fock limit is. For H2 model, state the means to go beyond the Hartree-Fock limit by giving the expression of an improved many-electron wavefunction. 7. Bonus questions (10 points): Give the LS terms that arise from the d2 atomic electronic configuration. --

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