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3502_HWkey7 - Chem 3502/5502 Physical Chemistry II(Quantum...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Answers to Homework Set 7 From lecture 25: Using the third root of the secular equation for the allyl system, verify the orbital coefficients given in eq. 25-16 The second root was E = α . Plugging that value into the linear equations a i H ki " ES ki ( ) i = 1 N # = 0 $ k recalling that for the allyl system H 11 = H 22 = H 33 = α , H 12 = H 21 = H 23 = H 32 = β , H 13 = H 31 = 0, S 11 = S 22 = S 33 = 1, and all other S values are 0, we have a 1 " " # 2 $ ( ) •1 [ ] + a 2 $ " # 2 $ ( ) • 0 [ ] + a 3 0 – " # 2 $ ( ) • 0 [ ] = 0 a 1 $ " # 2 $ ( ) • 0 [ ] + a 2 " " # 2 $ ( ) •1 [ ] + a 3 $ " # 2 $ ( ) • 0 [ ] = 0 a 1 0 – " # 2 $ ( ) • 0 [ ] + a 2 $ " # 2 $ ( ) • 0 [ ] + a 3 " " # 2 $ ( ) •1 [ ] = 0 These equations simplify to a 1 2 " + a 2 " = 0 a 1 " + a 2 2 " + a 3 " = 0 a 2 " + a 3 2 " = 0 If we subtract the third equation from the first, we obtain a 1 2 " # a 3 2 " = 0 which gives a 1 = a 3 If we use this relationship in the second equation above we have
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HW7-2 a 1 " + a 2 2 " + a 1 " = 0 which gives a 2 = " 2 a 1 Normalization requires that a i 2 i = 1 3 ! = 1 which leads to the final result a 13 = 1 2 , a 23 = 2 2 , a 33 = 1 2 QED. From lecture 26: What is the Hartree-product wave function for 2 non-interacting quantum mechanical harmonic oscillators (QMHOs) of reduced mass 1 a.u. in a potential having a force constant of 1 a.u., where the first QMHO is in the ground state and the second is in the first excited state? Determine the energy of the two QMHO system as an expectation value of the Hartree-product wave function. Is the correct Hamiltonian for this system separable into one-QMHO terms? If the QMHOs were interacting, explain how you could use perturbation theory to determine the energy of the system correct to first order (you don’t have to actually do it, just explain how to do it).
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