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solns21b-ps2-Jan11

# solns21b-ps2-Jan11 - Chemistry 21b Problem set 2 Out 12...

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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 / radicalbig 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 find the optimal value of ζ . From this, calculate the values of E, < ˆ T >, and < ˆ V > at ζ opt . Show, from the numerical results in example 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 satisfied at this value of ζ , without having to work through equations (10.31-10.32).

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solns21b-ps2-Jan11 - Chemistry 21b Problem set 2 Out 12...

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