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Hypothetical Potential Energy Surface

Chapter 15 (26): Computational Chemistry

stationary pointsall ν s >0 :

minimum

one ν im:transition statesaddle point

multiple ν im:hilltop

2

Ethane conformationsHartree-Fock theory, 6-31G(d) basis set

Source: Hyperchem calculationHF 6-31G(d)geometry relaxed at each angle


staggered

eclipsed

3

Potential Energy Surface for F( 2P3/2,

2P1/2 ) + CH4 → FH + CH3


MP2 with 6-311+G(2df,2pd) basis setSource: Cipriano Rángel, et al., J. Phys. Chem. A, 109 (7), 1441 -1448, 2005. 4

Thermodynamic versus kinetic control

Engel's Figures 15.4,5,6

∆E relates to thermodynamic control. Ea relates to kinetic control.


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Section 4: Hartree Fock, Configuration Interaction (CI), and Exact Dissociation of H2


HF is inaccurate for homolytic bond cleavageE.g.,

H2 (1Σg) →2H (2S1/2)

RHF: badUHF: poor

CI corrects for correlation.

Full-CI E > exact E because of small 6-31G(d,p) basis set.

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atom ormolecule

EHF (Hartree)

Ecorrelation(Hartree)

Percent

He -2.862 -0.043 1.5C -37.689 -0.153 0.4F -99.409 -0.313 0.3Ar -526.817 -0.725 0.1H2 -1.132 -0.043 3.8

CH4 -40.219 -0.291 0.7

H2O -76.067 -0.364 0.5

H2O2 -150.861 -0.520 0.5

CO -112.796 -0.520 0.5HF -100.074 -0.373 0.4

source: Lowe's Quantum Chemistry, Tables 11-2 and 11-4.

Electron correlation is everything except relativity that is not in HF.

Correlation energy increases in magnitude with increasing number of electrons.


E correlation ≡ E nonrelativistic − E HF

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Hartree-Fock is incorrect for homolytic bond cleavage.

ΔE (kJ/mol) HF Experiment HF errorCH3 – CH3 276 406 -130

CH3 – F 289 477 -188

F – F -163 184 -347

Hartree-Fock is better for reactions that maintain electron pairs. A+H+ → AH+

source: Engel's Tables 15.2 and 15.4. 6-311+G(d,p) basis

H+ affinity (kJ/mol) HF Experiment HF errorammonia -50 -38 -12pyridine 29 29 0trimethylamine 50 46 4


Section 4: Limiting Hartree-Fock results

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Section 6: Correlation energy, continued


source: Modern Quantum Chemistry, by Szabo and Ostlund, Dover, 1996, section 4.3.

basis CISD Full CI exact correlationset with the stated energy with

basis set infinite basis

H2 at R=1.4a0 6-31G(d,p) -0.034 same as CISD -0.041BeH2 double zeta -0.074 -0.076 -0.14H2O 39-STO -0.28 -0.30 -0.37

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CI-Singles and Doubles is not size consistent.Example He monomer and dimer with {1s,2s} basis.

CISD for He monomer

CISD for HeaHeb dimer

CISD for separated Hea Heb monomers includes four additional double excitations.

Figures are from Engel's chapter 15.

CIS and full CI are size consistent. 10

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

▪ Walter Kohn and Lu Sham, 1964expanded ρ(x,y,z) in electron orbitals (like MO theory)Kohn-Sham equations for orbitals and their energies (SCF)

Chapter 15 (26): Computational ChemistryDensity Functional Theory (DFT)

Walter KohnNobel 1998www.nobelprize.org

[− 12

∇2− ∑

nuclei A

Z A

|⃗r− R⃗A|+∫

ρ( r⃗ ' )|⃗r − r⃗ '| d r⃗ ' + v xc ]ψi

KS( r⃗ ) = ϵi

KSψi

KS( r⃗ )

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Kohn-Sham and Hartree-Fock orbitals for ethene

HOMO-LUMO0.24 Eh

HOMO-LUMO0.55 Eh

source: GAMESS, images from wxMacMolPlt.12


Density Functional Theory, results

mean absolute errors

Bond Lengths1 Atomization2 Dipole3

Method (Angstrom) (kJ/mol) (Debye)HF 0.022 310 0.21B3LYP 0.004 9.2 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 6-31G(d). 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.

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Section 8: model selectionWarren Hehre presents four methodsUncorrelated

1 Hartree Fock with the 3-21G basis setbasis set is defined for H - Cs (55)

2 Hartree-Fock with the 6-31G(d) basis setlarger and polarized basis set, better than 3-21Gbasis set is defined for H - Zn (30)

Correlated (post-Hartree Fock)3 MP2/6-31G(d) I am skipping MP2.

4 DFT B3LYP/6-31G(d)about twice as slow as HF/6-31G(d)computation time proportional to N3

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Section 8: model selection, continued

Most bond lengths are good from all three methods.

Tables 15.9 and 15.10, C-X bonds

method mean absolute error (Å)

HF/3-21G 0.013

HF/6-31G(d) 0.014

DFT B3LYP/6-31G(d) 0.006

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dipole moments (D) reflect geometry and charge distribution 1 debye = 3.33564×10-30 C·m

moleculeHF

3-21GHF

6-31G(d)DFT B3LYP

6-31G(d)Experiment

(CRC)

CO -0.40 -0.26 0.06 0.11

H2O 2.39 2.20 2.09 1.86

H2S 1.41 1.41 1.43 0.98

NH3 1.76 1.92 1.91 1.47

PH3 0.87 0.88 0.96 0.57

SO2 2.29 2.19 1.77 1.63

mean abs err 0.45 0.41 0.28


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ΔE of bond cleavage is much better with correlated methods.

reaction energies (kJ/mol)

HF 3-21G

HF 6-31G(d)

DFT B3LYP6-31G(d)

experiment

bond cleavage

CH3-CH3→2 CH3 285 293 406 406

F-F → 2 F -121 -138 176 159

isomerization

acetic acid → methyl formate 54 54 50 75

ethanol → dimethyl ether 25 29 21 4

conformationaln-butane →gauche 3.3 4.2 2.9 2.8methylcyclohexane equatorial → axial

7.9 9.6 8.8 7.3


source: Engel, Quantum Chemistry and Spectroscopy, Tables 15.11, 15.12, 15.14.

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reaction HF DFT B3LYP ExperimentCH3NC → CH3CN 192 172 159

HCO2CH2CH3 → HCO2H + C2H4 293 222 167,184

205 121 130

167 84 84

comparison of HF and correlated activation energies

source: Hehre's Table 15.17 (Engel, Quantum Chemistry and Spectroscopy).

activation energies (kJ/mol) calculated with the 6-31G(d) basis set

errors in Ea (kJ/mol) for 12 organic elementary gas-phase reactions.

source: Levine, Quantum Chemistry, 5th ed., page 703.

HF HF DFT B3LYP3-21G 6-31(d) 6-31G(d)

average error 40 60 25maximum error 160 130 90


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Section 9: Graphical ModelsHOMO and LUMO are important in chemical reactivity

ethene + butadiene but not ethene + ethene

frontier orbitals for concerted additions

Engel's figures 15.30 and 15.31.


HOMO

LUMO

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singlet oxygen adds to diphenylisobenzofuranDPBF is used to trap singlet oxygen. O

2 adds across the furan ring.

Spartan image

HOMO

LUMO


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electron density surfaces can show bonds and lone pairsdiborane B2H6molecule e density (0.12 surface) e density (0.08 "bond" surface)

ammonia NH3pyramidal planar

Engel's Figure 15.39


Pictures from SpartanDFT B3LYP 6-31G*

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electrostatic potential near H correlates with acidity

Spartan. PM3 geometry. HF 6-31G(d) electrostatic potential.


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