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Unformatted text preview: B Chemistry 3502/5502 Exam IV November 25, 2009 1) Fill in the blank on each question with the correct answer, by letter, from the list provided on the last page of the exam (you may tear the list off if you like). 2) There is one correct answer to every fill-in-the-blank problem. There is no partial credit. No answer will be used more than once. There are answers that are not used, however. 3) On the short-answer problem, show your work in full. 4) You should try to go through all the problems once quickly, saving harder ones for later. 5) There are 24 fill-in-the-blank problems. Each is worth 3 points. The short-answer problem is worth 28 points. 6) There is no penalty for guessing. 7) Please write your name at the bottom of each page. 8) Please mark your exam with a pen, not a pencil. If you want to change an answer, cross your old answer out and circle the correct answer. Exams marked with pencil or correction fluid will not be eligible for regrade under any circumstances. Score on Next Page after Grading
NAME: ________________________________________________________________ 2 Fill in the numbered boxes on the HF calculation flowchart (from lecture 28) with the appropriate steps from the answer list (use the letters—don’t write the phrases).
1 B Choose a molecular geometry q(0)
2 X 3 L 4 Z 5 EE 6 J no 7 U Choose new geometry according to optimization algorithm
8 yes R
yes no no Does the current geometry satisfy the optimization criteria?
yes Output data for unoptimized geometry Output data for optimized geometry NAME: ________________________________________________________________ 3 9. A Hartree-product many-electron wave function: ____DD___ 10. The Coulomb integral Jab between an electron in orbital a and another electron in orbital b: ___T____ 11. The exchange integral Kab involving orbitals a and b: ___Y____ 12. An operator H = h1 + h2 + h3 where h1ψ1 = 4ψ1, h2ψ2 = 2ψ2, and h3ψ3 = 1ψ3. If ψ1, ψ2, and ψ3 are normalized, what is < ψ1ψ2ψ3 | H | ψ1ψ2ψ3 >? ___W____ 13. A generic Fock matrix element Fµν (atomic units): ___A____ 14. A generic overlap matrix element Sµν: ___HH____ 15. A generic density matrix element Pµν: ___H____ 16. A generic 4-index integral ( µν | λσ ): ___K____ 17. An integral equal to zero: ___O____ 18. An integral equal to –1: ___N____ 19. An antisymmetric, many-electron wave function with normalization implicit: ___BB____ NAME: ________________________________________________________________ 4 The following 5 questions refer to a HF/STO-6G calculation on neutral hydroxylamine, H2NOH. The atomic numbers of H, N, and O are 1, 7, and 8 respectively. 20. How many occupied orbitals will be used to construct the Slater determinantal many-electron wave function that would result from a restricted Hartree-Fock calculation? ___GG____ 21. As a linear combination of how many contracted basis functions will each molecular orbital be expressed? ___AA____ 22. Ignoring symmetry and the turnover rule, how many two-electron integrals over contracted basis functions would need to be evaluated in the calculation? ___I____ 23. By what factor will the number of one-electron integrals over primitive basis functions exceed the number of one-electron integrals over contracted functions? ___S____ 24. What is a reasonable value for the final HF energy in a.u.? ___V____ NAME: ________________________________________________________________ 5 Hückel Theory Consider the simplest possible Hückel system, ethylene, H2C=CH2, which has 2 π electrons. How many basis functions are needed to carry out a Hückel theory calculation of the molecular orbitals of ethylene? What are the basis functions, specifically? There are 2 basis functions. They are 2pz orbitals, one on each carbon, where the z axis is the axis orthogonal to the plane of the atoms (i.e., the p orbitals forming the π system). In terms of 0, 1, α, and β, what are the specific values of all matrix elements that will appear in the secular determinant for ethylene? To what experimental quantities do α and β refer, specifically? S11 = S22 = 1, S12 = 0, H11 = H22 = α , H12 = β α is the negative of the ionization potential of the methyl radical (the energy of an electron in a free 2pz orbital) and β is one half the rotational barrier in ethylene. Write the Hückel theory secular equation for ethylene. What values of E permit solution of the secular equation? You may find the equation a2 – b2 = ( a + b ) (a – b ) to be helpful. "#E $ $ =0 "#E The solution to this secular equation is ! 0 = (" # E ) # $ 2 = (" # E + $)(" # E # $)
which is satisfied by E = α + β and E = α – β . The first root is lower in energy since α and β are negative quantities. 2 ! NAME: ________________________________________________________________ 6 What does Hückel theory predict for the singlet-triplet splitting in ethylene? Explain your answer. The energy of the singlet is computed from placing the two ethylene π electrons in the lowest energy orbital. Given the energy determined above, that makes the total energy 2α + 2β . Making the triplet will require removing one electron from the lowest energy orbital and moving it to the higher energy orbital (since we can’t have two electrons of the same spin in the same orbital). So, now we have the energy of each orbital taken once and added together, which gives α + β + α – β = 2α . The difference is 2β and that is the singlet-triplet splitting. For those interested in the chemistry, notice that this is, by definition of 2β , equal to the rotational barrier in ethylene. This is exactly what one expects, since the triplet has one electron in the bonding orbital and one in the antibonding, there is no net π bond, which is the same thing that happens at the rotational transition state: the π bond is destroyed. NAME: ________________________________________________________________ 7 A: nuclei 1 1 µ – " 2 # – $ Zk µ # 2 rk k R: Optimize molecular geometry? ' * 1 + $ P%& ) (µ# %&) – (µ% #&) , ( + 2 %& B:
! Choose a basis set 21
1sH a 1sH b where H and H are a b S: T: U:
"" a(1)b(2) 1 a(1)b(2) dr (1) dr (2) r12 C: D: the two H atoms in water
! Is new density matrix P(n) sufficiently similar to old density matrix P(n–1) ? –130.505 204 660 7 Compute and store all overlap, oneelectron, and two-electron integrals
"" a(1)b(1) 1 a(2)b(2) dr (1) dr (2) r12 E: F: G: H: Koopmans’ theorem 16
MZ 1 " # i2 " $ k 2 k =1 rik
occupied MOs V: W: X: 2 " aµi a#i
i Y: Z:
! I: J: 134 Construct density matrix from occupied MOs
1 " % (2) "& (2) dr (1) dr (2) r12 Construct and solve Hartree-Fock secular equation 13 AA: BB: K: ## "µ (1) " $ (1) " = # 1# 2 # 3 ! # N where the various χi are one-electron spin
" = #1#2 ! #N where the various ψi are one-electron orbitals L:
! Guess initial density matrix P(0) 41.818 911 429
"2p x ,N 2p x ,N 2p x ,N 2p z ,O where N and O are ! CC:
! M: N: O:
! ! Replace P(n–1) with P(n) 214 9 FF: GG: HH: both on the x axis P: Q: The Born-Oppenheimer approximation 164 $ "µ (r ) " # (r )dr ! NAME: ________________________________________________________________ ...
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- Fall '08