## (solution) Chem 120A, Fall 2015 Problem Set 10

This is physical Chemistry Homework Problems. Please see attachment.

1. Consider a collection of distinguishable andindependent (i.e. non-interacting) harmonicoscillators, all with the same characteristic frequency .  If there are 4oscillators in the collection,how many Hamiltonian eigenstatesare there with a total system energy less thanor equal to 8?w?What about 8 oscillators and 16?w?  What about 12 oscillators and 24?w?

2.Using your results from question 1, explain why it would be extremely difficult to find the exact ground state in a large collection of oscillators that interacted with each other.

3.  Consider the lowest energy triplet state (in which both electrons are spin up) of the hydrogenmolecule for a bond distance of 0.75bohr.Inspired by mean field theory, you decide to construct as your approximate wave function a Slater determinant in which the two molecular orbitals are the +and ?linear combinations of the hydrogen atom 1s orbitals centered on each of the two nuclei. Assuming both electrons lie on the internuclear axis and that one of them is held fixed 1/6of the wayin between the nuclei, evaluate the probabilities of finding the other electron 0/9, 1/9, 2/9, ?, up to8/9 of the way in between the nuclei relative to finding it at the far nucleus(i.e. at 9/9). Explain whatyou see in your results.

Hint:  When computing relative probabilities (i.e. ratios of probabilities), the normalization of the wave function does not matter.

Chem 120A, Fall 2015 Problem Set 10 Due in G08 Gilman on Dec 9 at 3pm 1. Consider a collection of distinguishable and independent (i.e. non-interacting) harmonic oscillators, all with the same characteristic frequency . If there are 4 oscillators in the collection, how many Hamiltonian eigenstates are there with a total system energy less than or equal to 8?? What about 8 oscillators and 16?? What about 12 oscillators and 24?? 2. Using your results from question 1, explain why it would be extremely difficult to find the exact ground state in a large collection of oscillators that interacted with each other. 3. Consider the lowest energy triplet state (in which both electrons are spin up) of the hydrogen molecule for a bond distance of 0.75 bohr. Inspired by mean field theory, you decide to construct as your approximate wave function a Slater determinant in which the two molecular orbitals are the + and ? linear combinations of the hydrogen atom 1s orbitals centered on each of the two nuclei. Assuming both electrons lie on the internuclear axis and that one of them is held fixed 1/6 of the way in between the nuclei, evaluate the probabilities of finding the other electron 0/9, 1/9, 2/9, ?, up to 8/9 of the way in between the nuclei relative to finding it at the far nucleus (i.e. at 9/9). Explain what you see in your results. Hint: When computing relative probabilities (i.e. ratios of probabilities), the normalization of the wave function does not matter.

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