Hypothetical Potential Energy Surface Ethane conformations Hartree-Fock theory, basis set stationary points all ν s >0: minimum eclipsed one ν im: transition state saddle point multiple ν im: hilltop 1 Source: Hyperchem calculation geometry relaxed at each angle 2 Thermodynamic versus kinetic control Potential Energy Surface for F( 2P3/2, 2P1/2 ) + CH4 FH + CH3 E relates to thermodynamic control. MP2 with 6-311+G(2df,2pd) basis set Source: Cipriano Rángel, et al., J. Phys. Chem. A, 109 (7), 1441-1448, 2005. staggered Ea relates to kinetic control. Engel's Figures 15.4,5,6 3 4
E correlation E nonrelativistic E Section 4: Hartree Fock, Configuration Interaction (CI), and Exact Dissociation of H2 is inaccurate for homolytic bond cleavage E.g., H2 (1Σg) 2H (2S1/2) Electron correlation is everything except relativity that is not in. Correlation energy increases in magnitude with increasing number of electrons. R: bad U: poor CI corrects for correlation. Full-CI E > exact E because of small 6-31G(d,p) basis set. 5 406 477 184 Ecorrelation (Hartree) Percent -0.043-0.153-0.313-0.725-0.043 1.5 0.4 0.3 0.1 3.8 CH4 H2O -40.219-0.291 0.7-76.067-0.364 0.5 H2O2-150.861-0.520 0.5 CO -112.796-100.074-0.520-0.373 0.5 0.4 6 source: Lowe's Quantum Chemistry, Tables 11-2 and 11-4. basis set Hartree-Fock is incorrect for homolytic bond cleavage. 276 289-163 E (Hartree) -2.862-37.689-99.409-526.817-1.132 Section 6: Correlation energy, continued Section 4: Limiting Hartree-Fock results ΔE (kj/mol) CH3 CH3 CH3 F F F atom or molecule He C F Ar H2 error -130-188 -347 H2 at R=1.4a0 6-31G(d,p) BeH2 double zeta H2O 39-STO CISD Full CI with the stated basis set -0.034 same as CISD -0.074-0.076-0.28-0.30 exact correlation energy with infinite basis -0.041-0.14-0.37 source: Modern Quantum Chemistry, by Szabo and Ostlund, Dover, 1996, section 4.3. Hartree-Fock is better for reactions that maintain electron pairs. A+H+ AH+ H+ affinity (kj/mol) ammonia -50 pyridine 29 trimethylamine 50-38 29 46 source: Engel's Tables 15.2 and 15.4. 6-311+G(d,p) basis error -12 0 4 7 8
CI-Singles and Doubles is not size consistent. Density Functional Theory (DFT) Example He monomer and dimer with {1s,2s} basis. CISD for He monomer Figures are from Engel's chapter 15. DFT focuses on ρ(x,y,z) rather than on Ψ(all electrons' x,y,z) Walter Kohn (1923-2016) and Pierre Hohenberg theoretical basis, 1963 CISD for HeaHeb dimer Walter Kohn Nobel 1998 www.nobelprize.org Walter Kohn and Lu Sham, 1964 expanded ρ(x,y,z) in electron orbitals (like MO theory) Kohn-Sham equations for orbitals and their energies (SCF) CISD for separated Hea Heb monomers includes four additional double excitations. CIS and full CI are size consistent. [ ZA 1 2 + 2 r RA nuclei A ρ( r ' ) r r ' d r ' + v xc ] KS KS ψks i ( r ) = ϵi ψ i ( r ) 9 10 Kohn-Sham and Hartree-Fock orbitals for ethene Density Functional Theory, results mean absolute errors Bond Lengths1 Method (Angstrom) 0.022 B3LYP 0.004 HOMO-LUMO 0.24 Eh HOMO-LUMO 0.55 Eh Dipole3 (Debye) 0.21 0.03 1. for 32 molecules containing only first-row atoms. Basis set 6-311G(d,p). Cramer, Essentials of Computational Chemistry, Table 8.5. 2. Atomization energies were calculated for the G2 set: 55 molecules. Basis. Cramer, Essentials of Computational Chemistry, Table 8.1. 3. Average error in Debye for CO, H2O, H2S, NH3, PH3 and SO Cohen, Chemical Physics Letters, 299, 465-472, 1999. 11 source: GAMESS, images from wxmacmolplt. Atomization2 (kj/mol) 310 9.2 12
Section 8: model selection Warren Hehre presents four methods Uncorrelated 1 Hartree Fock with the 3-21G basis set basis set is defined for H - Cs (55) Section 8: model selection, continued Most bond lengths are good from all three methods. Tables 15.9 and 15.10, C-X bonds method /3-21G / DFT B3LYP/ 2 Hartree-Fock with the basis set larger and polarized basis set, better than 3-21G basis set is defined for H - Zn (30) Correlated (post-hartree Fock) 3 MP2/ I am skipping MP2. mean absolute error (Å) 0.013 0.014 0.006 4 DFT B3LYP/ about twice as slow as / computation time proportional to N 3 13 14 ΔE of bond cleavage is much better with correlated methods. dipole moments (D) reflect geometry and charge distribution reaction energies (kj/mol) 1 debye = 3.33564 10-30 C m molecule CO H2O 3-21G -0.40 2.39 DFT B3LYP -0.26 2.20 0.06 2.09 (CRC) 0.11 1.86 3-21G DFT B3LYP experiment bond cleavage CH3-CH3 2 CH3 285 293 406 406 F-F 2 F -121-138 176 159 H2S NH3 1.41 1.41 1.43 0.98 isomerization 1.76 1.92 1.91 1.47 acetic acid methyl formate 54 54 50 75 PH3 SO2 0.87 0.88 0.96 0.57 ethanol dimethyl ether 25 29 21 4 2.29 2.19 1.77 1.63 mean abs err 0.45 0.41 0.28 3.3 7.9 4.2 9.6 2.9 8.8 2.8 7.3 conformational n-butane gauche methylcyclohexane equatorial axial 15 source: Engel, Quantum Chemistry and Spectroscopy, Tables 15.11, 15.12, 15.14. 16
comparison of and correlated activation energies Section 9: Graphical Models HOMO and LUMO are important in chemical reactivity errors in Ea (kj/mol) for 12 organic elementary gas-phase reactions. average error maximum error 3-21G 6-31(d) 40 60 160 130 DFT B3LYP 25 90 frontier orbitals for concerted additions HOMO source: Levine, Quantum Chemistry, 5th ed., page 703. activation energies (kj/mol) calculated with the basis set reaction DFT B3LYP 205 121 130 167 84 84 CH3NC CH3CN 192 HCO2CH2CH3 HCO2H + C2H4 293 172 222 LUMO 159 167,184 ethene + butadiene source: Hehre's Table 15.17 (Engel, Quantum Chemistry and Spectroscopy). 17 but not ethene + ethene Engel's figures 15.30 and 15.31. 18 singlet oxygen adds to diphenylisobenzofuran electron density surfaces can show bonds and lone pairs DPBF is used to trap singlet oxygen. O 2 adds across the furan ring. diborane B2H6 molecule e density (0.12 surface) e density (0.08 "bond" surface) LUMO ammonia NH3 pyramidal planar Pictures from Spartan DFT B3LYP 6-31G* HOMO Spartan image 19 Engel's Figure 15.39 20
electrostatic potential near H correlates with acidity Spartan. PM3 geometry. electrostatic potential. 21