1

Section 1. H2

+

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

H2+ exact

electronic energies

Ground state:Re = 2.00 Bohr

De = (-0.500 - -0.603) EhDe = 0.103 Eh = 2.79 eV

source: Ira Levine, Quantum Chemistry, 5th ed., Prentice Hall, 20002

H2+ electronic wave functions

source: unrestricted Hartree-Fock 6-31G(d,p) plus {s,p,d}with ζ=0.55.

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

Orbital angular momentun:λ is the quantum number for z-axis angular momentum(like ml).

λ = 0, 1, 2, … and <Lz > = ±λħ.

Wave functions of diatomic molecules are labeled according to λ.

λ: 0 1 2ψ: σ π δ

Subscript "g" or "u" (gerade or ungerade)tells inversion symmetry.

3

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

Section 2: minimum-basis LCAO theory of H2+

ϕ1 sa =1√π (

ζ

a0 )3/2

e−ζ|r−rA|/a0

MO ψ = ca ϕ1 sa+cbϕ1 sb

ψg and ψu are graphed along z with x=0 and y=0.

ψg = (2+2 Sab )−1/2(ϕ1 sa + ϕ1 sb)

ψu = (2−2 S ab)−1 /2(ϕ1 sa − ϕ1 sb)

4

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

Section 2: LCAO H2+ energy

E g =H aa+H ab1 + S ab

E u =H aa−H ab1 − S ab

Results graphed are for ζ=1. Optimization gives

ζ = 1.24Re=2.00 BohrDe=0.087 Hartree

5

E g =H aa+H ab

1 + S abE 1 s =−

12

Hartree (ζ=1)

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

6

E u =H 11−H 121 − S12

ψu = (2−2 S12)−1/2

(ϕ1 − ϕ2)

E g =H 11+H 121 + S12

ψg = (2+2 S1 2)−1 /2 (ϕ1 + ϕ2)

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

Summary of homonuclear solutions:

ψu

ψg

7

Section 5. Homonuclear Diatomic Molecular OrbitalsSigma Orbitals

source: http://www.mpcfaculty.net/mark_bishop/molecular_orbital_theory.htm

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

8

Section 5. Homonuclear Diatomic Molecular OrbitalsPi Orbitals

source: http://www.mpcfaculty.net/mark_bishop/molecular_orbital_theory.htm

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

9

Homonuclear Diatomic Molecular Orbital Energiespairwise LCAO energy levels secondary sigma(s) - sigma(p

z) mixing

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

secondary σ-σ interactiongreater for smaller Z

E(3σg) > E(1πu) before O2

All Es ↓ as Z ↑

10

Section 6. Homonuclear Diatomic Molecular Orbital EnergiesEngel and Reid's Figure 12.19 (23.19)

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

Notable predictions: N2 triple bond O2 triplet ground state

11

E (Eh): -1.685 -1.327 -0.641 -0.641 -0.596 -0.454 -0.454 +0.446

MO: 2σg

2σu* 1π

u(x) 1π

u(y) 3σ

g1π

g*(x) 1π

g*(y) 3σ

u*

F1 2s 0.648 0.768 0.000 0.000 0.225 0.000 0.000 -0.283

F1 2pz -0.107 0.086 0.000 0.000 0.639 0.000 0.000 0.825

F1 2px 0.000 0.000 0.683 0.000 0.000 0.734 0.000 0.000

F1 2py 0.000 0.000 0.000 0.683 0.000 0.000 0.734 0.000

F2 2s 0.648 -0.768 0.000 0.000 0.225 0.000 0.000 0.283

F2 2pz 0.107 0.086 0.000 0.000 -0.639 0.000 0.000 0.825

F2 2px 0.000 0.000 0.683 0.000 0.000 -0.734 0.000 0.000

F2 2py 0.000 0.000 0.000 0.683 0.000 0.000 -0.734 0.000

F2RHF using STO-3G minimum basisSCF eigenvalues (Hartree) and eigenvectorsF-F is on the z axis. R = 1.315 Angstroms

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

12

F2RHF using STO-3G minimum basis

LCAO-MO valence orbitals

13

Section 8. Heteronuclear bondingHF LCAO-MO with H1s and F2pz AOs

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

14

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

Section 8. Heteronuclear bondingHF LCAO-MO with H1s and F2pz and F 2s AOs, 4 electrons

ψ = c1ϕH 1 s + c2ϕF 2 pz+ c3ϕF 2 s

15

Section 9. electrostatic potential of hydrogen fluoride

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

16

Section 9. electrostatic potential of hydrogen fluoride

Chapter 12 (23): Chemical Bonding in Diatomic Molecules

17

electrostatic potential of ethyltrifluoracetate

electrostatic potentialon electron density surface

DFT B3LYP 6-31G*

Chapter 12 (23): Chemical Bonding in Diatomic Molecules