probset2_11

probset2_11 - 10-3, that this procedure for H + 2 directly...

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Chemistry 21b Problem set # 2 Out: 12 January 2011 Due: 19 January 2011 Problems are worth: 1=25; 2=30; 3=20; 4=25. 1. Problem 10-17 McQuarrie, page 551, which shows that the simplest wavefunction for H + 2 , Ψ + = 1 / r 2(1 + S )(1 s A + 1 s B ), does not even qualitatively satisfy the virial theorem. 2. A somewhat extended version of problem 10-20 McQuarrie, pp. 551-552, about what is called the Scaling Theorem within the Born-Oppenheimer approximation. Start, as the text suggests, by deriving formulae for < ˆ T > and < ˆ V > when all of the coordinates are multiplied, or scaled, by a constant parameter ζ . Here’s the point. Now take the the results you get, namely < ˆ T > and < ˆ V > for arbitrary ζ in terms of < ˆ T > | ζ =1 and < ˆ V > | ζ =1 , and write the expression for the total energy E . Treat ζ as a variational parameter and Fnd the optimal value of ζ . ±rom this, calculate the values of E, < ˆ T >, and < ˆ V > at ζ opt . Show, from the numerical results in example
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Unformatted text preview: 10-3, that this procedure for H + 2 directly yields an optimal ζ opt = 1 . 238 and the energy listed in Table 10.2. Show also that the virial theorem is satisFed at this value of ζ , without having to work through equations (10.31-10.32). 3. Problem 10-26 McQuarrie, page 553 (this helps explain the LCAO-MO virtual orbital diagram for H 2 , ±ig. 10.28 McQuarrie, page 536). 4. In general, a diatomic molecule does not possess a zero-point rotational energy such as we will Fnd is present in the vibrational degrees of freedom. However, in the case of H 2 (or any other molecule with equivalent H atoms, such as H 2 O), there is an e²ective zero-point rotational energy. Explain why. What about the isotopologues HD and D 2 ? Responsible TA for Problem Set #2: Scarlett Dong ([email protected])...
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This note was uploaded on 01/03/2012 for the course CH 21b taught by Professor List during the Fall '10 term at Caltech.

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