A NAME: ________________________________________________________________ Chemistry 3502/5502 Exam IV November 25, 2009 1) Fil in the blank on each question with the corect 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 corect answer to every fil-in-the-blank problem. There is no partial credit. No answer wil be used more than once. There are answers that are not used, however. 3) On the short-answer problem, show your work in ful. 4) You should try to go through all the problems once quickly, saving harder ones for later. 5) There are 24 fil-in-the-blank problems. Each is worth 3 points. The short-answer problem is worth 28 points. 6) There is no penalty for guesing. 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 corection fluid wil not be eligible for regrade under any circumstances. Score on Next Page after Grading 2 NAME: ________________________________________________________________ Fil in the numbered boxes on the HF calculation flowchart (from lecture 28) with the appropriate steps from the answer list (use the leters—don’t write the phrases). Chose amolecular geometry q (0) Does the curent geometry saify opiizain criteria? Outpt dat for ptimized gemety Outpt dat for unotimize gemety yes no yes no Chose nw geometry acrdig to ptiization alrith no yes 1 2 3 4 5 6 7 8 B X L Z J U E R 3 NAME: ________________________________________________________________ 9. An integral equal to –1: ___N____ 10. An operator H = h 1 + h 2 + h 3 where h 1 ψ 1 = 4ψ 1 , h 2 ψ 2 = 2ψ 2 , and h 3 ψ 3 = 1ψ 3 . If ψ 1 , ψ 2 , and ψ 3 are normalized, what is < ψ 1 ψ 2 ψ 3 | H | ψ 1 ψ 2 ψ 3 >? ___W____ 11. A generic density matrix element P µν : ___H____ 12. The exchange integral K ab involving orbitals a and b: ___Y____ 13. A Hartre-product many-electron wave function: ____DD___ 14. A generic Fock matrix element F µν (atomic units): ___A____ 15. The Coulomb integral J ab betwen an electron in orbital a and another electron in orbital b: ___T____ 16. A generic overlap matrix element S µν : ___HH____ 17. An integral equal to zero: ___O____ 18. A generic 4-index integral ( µν | λσ ): ___K____ 19. An antisymmetric, many-electron wave function with normalization implicit: ___BB____ 4 NAME: ________________________________________________________________ The following 5 questions refer to a HF/STO-6G calculation on neutral hydroxylamine, H 2 NOH. The atomic numbers of H, N, and O are 1, 7, and 8 respectively. 20. By what factor wil the number of one-electron integrals over primitive basis functions exced the number of one-electron integrals over contracted functions? ___S____ 21. As a linear combination of how many contracted basis functions wil each molecular orbital be expresed? ___AA____ 22. What is a reasonable value for the final HF energy in a.u.? ___V____ 23. Ignoring symmetry and the turnover rule, how many two-electron integrals over contracted basis functions would need to be evaluated in the calculation? ___I____ 24. How many occupied orbitals wil be used to construct the Slater determinantal many-electron wave function that would result from a restricted Hartre-Fock calculation? ___GG____ 5 NAME: ________________________________________________________________ Hückel Theory Consider the simplest possible Hückel system, ethylene, H 2 C=CH 2 , which has 2 π electrons. How many basis functions are needed to cary out a Hückel theory calculation of the molecular orbitals of ethylene? What are the basis functions, specificaly? There are 2 basis functions. They are 2p z 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 al matrix elements that wil appear in the secular determinant for ethylene? To what experimental quantities do α and β refer, specificaly? S 11 = S 22 = 1, S 12 = 0, H 11 = H 22 = α, H 12 = β α is the negative of the ionization potential of the methyl radical (the energy of an electron in a fre 2p z 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 a 2 – b 2 = ( a + b ) (a – b ) to be helpful. ! "#E$ $"#E =0 The solution to this secular equation is ! 0="#E () 2 #$ 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. 6 NAME: ________________________________________________________________ What does Hückel theory predict for the singlet-triplet spliting 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 wil 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 diference is 2β and that is the singlet-triplet spliting. 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. 7 NAME: ________________________________________________________________ A: ! µ– 1 2 " 2 #–Z k k nuclei $µ 1 r k # +P %& %& $µ#%& ( ) – 1 2 µ%#& ( ) ’ ) * + , R: Optimize molecular geometry? B: Choose a basis set S: 36 C: 21 T: ! a1 ()"" b2 () 1 r 12 a1 () b2 () dr1 () dr2 () D: ! 1s H a 1s H b where H a and H b are the two H atoms in water U: Is new density matrix P (n) sufficiently similar to old density matrix P (n–1) ? E: Koopmans’ theorem V: –130.505 204 660 F: 16 W: 7 G: ! " 1 2 # i 2 " Z k r ikk=1 M $ X: Compute and store al overlap, one- electron, and two-electron integrals H: ! 2a µi i occupied MOs " #i Y: ! a1 ()"" b1 () r 12 a2 () b2 () dr1 () dr2 () I: 13 4 Z: Construct and solve Hartre-Fock secular equation J: Construct density matrix from occupied MOs AA: 13 K: ! " µ ##1()" $ 1 () r 12 " % 2 () " & 2 () dr1 () dr2 () BB: ! "=# 12 # 3 !# N where the various χ i are one-electron spin orbitals L: Gues initial density matrix P (0) CC: π M: 41.818 911 429 DD: ! "=# 12 !# N where the various ψ i are one-electron orbitals N: ! "2p x,N 2p x,N EE: Replace P (n–1) with P (n) O: ! 2p x,N 2p z,O where N and O are both on the x axis FF: 21 4 P: The Born-Oppenheimer approximation GG: 9 Q: 16 4 HH: ! " µ r () " # r ()$ dr Laura Gagliardi 3502_S06_Exam_4a_key