Chemistry 433 Computational Chemistry Fall Semester 2002 Dr. Rainer Glaser

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1 Chemistry 433 Computational Chemistry Fall Semester 2002 Dr. Rainer Glaser Second 1-Hour Examination Electron Correlation Methods Wednesday, November 6, 2002, 11:00-11:50 Name: Answer Key Question 1. Electron Correlation. 20 Question 2. Approaches to EC. 20 Question 3. Perturbation Theory. 20 Question 4. Multilevel Theories. 20 Question 5. Situations. 20 Total 100 1

2 Question 1. Electron Correlation. (20 points) (a) What is electron correlation? Consider the electron correlation in helium atom to illustrate the fundamental problem of Hartree-Fock theory. Be brief and to the point. Drawings might help. (4 p.) The probabilities of finding each electron at a specific place are independent of each other. And that is wrong. If one electron is at a certain place, then the other electron will try to avoid being at that same place at all cost. The square of the molecular orbital therefore does not give the correct electron density. The true electron density will be somewhat different from the HF computed electron density. Also, of course, the electron-electron repulsion will be reduced and the total energy will be lowered. Often one focuses on this correlation effect, the lowering of the energy. (b) What is the main effects of electron correlation on the electron density of helium atom? In particular, will helium be smaller or larger once electron correlation is turned on? (2 points) Smaller. Correlated electron motion reduces ee-repulsion and allows the electrons to get closer to the nucleus. [This effect cannot be achieved with correlated methods that do not optimize MOs.] (c) What are the main effects of electron correlation on the electron density distribution of the dinitrogen molecule? (4 points) Electron density of moved out of the triple bond region. Electron density shift behind the atoms. Increase of the distance of the electron density in the triple bond region from the internuclear axis. 2

3 (d) What do the terms Hartree-Fock Limit and CI Limit mean? What do you need to do to reach (or at least get close to) the these limits? Draw a diagram in which the vertical axis is total energy. Draw horizontal lines for the energy computed at RHF/STO-3G, RHF/6-311G**, the HF Limit and the CI Limit. (10 points) Hartree-Fock Limit: Lowest energy achievable in a Hartree-Fock calculation of a given atom or molecule. To approach this limit one increases the basis set in the number and types of basis functions. Configuration Interaction Limit: Lowest energy achievable in a CI calculation of a given atom or molecule. To approach this limit one increases the basis set in the number and types of basis functions. To approach this limit one increases the number of excitations in the number (choice of active space) and types (single, double, ) excitations. Total Energy Diagram: RHF/STO-3G RHF/6-311G** HF-Limit CI-Limit 3

4 Question 2. Approaches to Electron Correlation. (20 points) The basic idea of all electron correlation theories is the same and involves the mixing in of excited configurations into the ground state configuration. Let s take a look at the molecule dilithium, Li Li, to illustrate some of these ideas. (a) This molecule has a total of _ 6 _ electrons; _ 4 _ electrons are in the core and _ 2 _ electrons are in the valence. Suppose we use a minimal basis set, how many molecular orbitals are occupied and how many orbitals are in the virtual space? (6 points) AOs in Li minimum basis set: 1s, 2s, 2p-set = 5 AOs Total number of basis functions: 10 Number of occupied MOs: 3 Number of MOs in virtual space: 7 (b) In a full CI calculation for Li Li, you need to consider single excitations, double excitations, up to what kind of excitation? sextet (2 points) (c) Use the usual MO level diagram (horizontal lines with up & down arrows) and illustrate the ground state configuration, one configuration generated by a single excitation, one configuration generated by a double excitation, one configuration generated by a triple excitation, and one configuration generated by a quadruple excitation. Do show the correct number of MO lines in each picture assuming that a minimal basis set is used. (5 points) Ground Single Double Triple Quadruple State Excitation Excitation Excitation Excitation 4

5 (d) Now let s use a simpler molecule, H H, and consider only single and double excitations in the MOs provided by a minimal basis set. So, the ground state is σ 2 1, the single excited configuration is σ 1 1 σ 1 2 and the double excited configuration is σ 2 2. The correlated wavefunction will be a linear combination Ψ = c GS Ψ(σ 2 1 ) + c SE Ψ(σ 1 1 σ 1 2 ) + c DE Ψ(σ 2 2 ) and the target is the determination of the correlation coefficients c GS, c SE, and c DE. Respond to the following questions. (7 points) Describe in principle how the correlation coefficients are determined in CISD theory. In particular, is an iterative procedure necessary? Use an iterative procedure to find the correlation coefficients that yield the lowest energy. Describe in principle how the correlation coefficients are determined in MP2 theory. In particular, is an iterative procedure necessary? Non-iterative!! Correlation coefficients are approximated in a fixed manner. In CISD theory, are the molecular orbitals used in the ground state and in the excited states the same or are they different? How are the molecular orbitals determined? The same! The Hartree-Fock MOs of the ground state are used. CIS and CID theory are CI theories that consider only single or double excitations, respectively. Write the correlated wavefunctions for these two cases. CIS: Ψ = c GS Ψ(σ 1 2 ) + c SE Ψ(σ 1 1 σ 2 1 ) CID: Ψ = c GS Ψ(σ 1 2 ) + c DE Ψ(σ 2 2 ) 5

6 Question 3. Perturbation Theory. (20 points) (a) Explain briefly what the following terms mean. MP2(fc) (2 points) MP2 = second-order Mφller Plesset perturbation theory, singles and doubles frozen core = only valence electrons are included in correlation treatment MP3(full) (2 points) MP3 = third-order Mφller Plesset perturbation theory, singles, doubles and triples full = all electrons are included in correlation treatment MP4(full,SDQ) (4 points) MP4 = forth-order Mφller Plesset perturbation theory SDQ = single, double, and quadruple excitations are included full = all electrons are included in correlation treatment How does MP2 scale with N, the number of basis functions? (1 point) N 5 How does MP4(SDTQ) scale with N, the number of basis functions? (1 point) N 7 MP4(fc)/6-311G*//MP2(full)/6-311G* (4 points) MP4(fc)/6-311G* energy calculation using MP2(full)/6-311G* optimized structure 6

7 (b) An MP4(full)/6-311G*//MP2(full)/6-311G* calculation of N 2 produced the following output. (The complete output is posted on the course web site.) Circle the RHF, MP2, MP3, and MP4 energies in the output. Look at the numbers and comment on anything you notice and that you think is interesting. (6 points) SCF Done: E(RHF) = A.U. after 1 cycles Convg = D-10 -V/T = S**2 = Range of M.O.s used for correlation: 1 36 NBasis= 36 NAE= 7 NBE= 7 NFC= 0 NFV= 0 NROrb= 36 NOA= 7 NOB= 7 NVA= 29 NVB= 29 Spin components of T(2) and E(2): alpha-alpha T2 = D-01 E2= D-01 alpha-beta T2 = D-01 E2= D+00 beta-beta T2 = D-01 E2= D-01 ANorm= D+01 E2= D+00 EUMP2= D+03 R2 and R3 integrals will be kept in memory, NReq= DD1Dir will call FoFMem 1 times, MxPair= 56 NAB= 28 NAA= 0 NBB= 0. MP4(D)= D-01 MP4(S)= D-02 MP4(R+Q)= D-02 T4(AAA)= D-03 T4(AAB)= D-02 Time for triples= seconds. MP4(T)= D-01 E3= D-01 EUMP3= D+03 E4(DQ)= D-02 UMP4(DQ)= D+03 E4(SDQ)= D-01 UMP4(SDQ)= D+03 E4(SDTQ)= D-01 UMP4(SDTQ)= D+03 Observations? Comments? Insights? The MP3 energy is higher than the MP2 energy! This is an example of an oscillation in the energy as the order of the perturbation treatment increases. Oscillation are a first indication that perturbation theory might not be good enough for the problem at hand. 7

8 Question 4. G1 and G2 Multilevel Theories. (20 points) (a) Describe the basic frustration in the approaches to electron correlation and outline the concept used by the multilevel approaches to resolve this frustration. Use a diagram if it helps. (8 points) Want: The best method in conjunction with the best basis set. Frustration: Scaling is exponential with the number of basis functions. Use best method with modest basis set. Estimate basis set effects at lower levels and assume transferability. Some empirical terms. (b) Describe in detail how G1 and G2 differ. State exactly how this/these additional term(s) are computed (method, basis set, based on what structure). (6 points) G2 includes a term that estimates the effects of diffuse functions and multiple polarization at the same time; requires computation at MP2/6-311+G(3df,2p)//MP2(full)/6-31G* G2 includes an additional empirical correction term n β (c) You are an associate editor of J. Phys. Chem. handling a manuscript in which G1 and G2 data are reported for a homolysis. One referee recommends publication as is. The other referee recommends rejection unless the authors also report G3 results. How will you decide? (6 points.) G2 is a better method than G3. Publish as is. 8

9 Question 5. Situations. (20 points) (a) You want to compute the activation barrier for the Diels-Alder reaction between butadiene and ethene to within 5 kcal/mol. Do you need small, large or very large basis sets? Do you have to include electron correlation in the computation of structures? Do you have to include electron correlation in the computation of energies? Which method would you suggest to use? Briefly argue why. Many bonds change at the same time; correlation effects on structure are important. The request for 5 kcal/mol accuracy is not very strict and there is no need to go to multilevel methods. Optimize at MP2(full)/6-31G* Follow with single point calculations at higer levels Multiple polarization might be useful as many angles can change (b) You want to compute the homolytic dissociation energy of hydrogenperoxide to within 1 kcal/mol. Do you need small, large or very large basis sets? Do you have to include electron correlation in the computation of structures? Do you have to include electron correlation in the computation of energies? Which method would you suggest to use? Briefly argue why. This ia a very difficult problem! Your basic nightmare! Homolysis! Lots of lone pairs on atoms. Lone pairs on adjacent atoms. Hellas, the molecule is small. Hit it with everything you ve got! 9