Quiz07 - Physical Chemistry (II) Quiz 03 May 21, 2008 1....

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Physical Chemistry (II) Quiz 03 May 21, 2008 1. (20%) Write down the Hamiltonian for H 2 molecule and briefly describe how the Schrödinger equation can be solved based on Born-Oppenheimer approximation. 2. (25%) For H 2 molecule, if the VB wavefuntion is applied to solve the ground state electronic energy for H 2 , show that 1 2 2 1 VB s J K E E S + = + + , where * * A B B A S s s dr s s dr = = , * * 1 2 2 1 12 * * 1 2 1 2 12 1 1 1 1 1 (1)1 (2) 1 (1)1 (2) 1 1 1 1 1 (1)1 (2) 1 (1)1 (2) A B A B A B B A B A A B J drdr s s s s r r r R drdr s s s s r r r R = - - + + = - - + + ∫∫ ∫∫ , and * * 1 2 1 2 12 * * 1 2 2 1 12 1 1 1 1 1 (1)1 (2) 1 (1)1 (2) 1 1 1 1 1 (1)1 (2) 1 (1)1 (2) A B B A A B B A A B A B K drdr s s s s r r r R drdr s s s s r r r R = - - + + = - - + + ∫∫ ∫∫ . 3. (20%) For 2 H + molecule, construct the wavefunctions using the LCAO-MO method, i.e., let ψ g = c g (1S A +1S B ) and u = c u (1S A - 1S B ), and determine the energies of the two states in terms of j , k and S (in atomic units) according to * * * * ˆ ˆ and g g u u g u g g u u H d H d E E d d = = τ ψ ψ , where * * 2 1 1 1 1 1 [1 (1 ) ] R A A B B B A s s s s j dr dr R e r r R - = = = - + , * * 1 1 1 1 (1 ) R A B B A A B s s s s k dr dr e R r r - = = = + , and 2 * * 1 1 1 1 (1 ) 3 R A B B A R S s s dr s s dr e R - = = = + + ; R is the separation between two nuclei (in unit of Bohr radius a 0 ). 4. (10%) Derive one of the above expressions for the key parameters ( j , k or S ) in terms of R using elliptic coordinate system. Hint : 1 1 1 ,1 A B r r A B s e s e π - - = = ,
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3 2 2 , , ( ) 8 A B A B r r r r R dr d d d R R λ μ μ φ + - = = = - 1 , 1 1 0 2 with and φ π ≤ ∞ - ≤ 5. (15%) Work out the ground-state molecular terms for 2 H + , 2 He + , and B 2. 6.
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Quiz07 - Physical Chemistry (II) Quiz 03 May 21, 2008 1....

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