Chem 125 Lecture 12 10/1/08 This material is for the exclusive use of Chem 125 students at Yale and may not be copied or distributed further. It is not readily understood without reference to notes from the lecture.

Ethane & Methanol (Spartan 6-31G*)

7 Pairs of Valence Electrons C C HH HH H H O C H HH H Compare MOs to AOs of Ar (7 electron pairs)

2s CH 3 Orbital Energy Occupied Vacant HOMO-6 CH 3 OH Orbital Energy Occupied Vacant Rotated 90° Pedantic Note: with two “heavy” atoms there are two boring “core” orbitals. For the purpose of making atomic analogies to study valence-level molecular orbitals, we’ll use the atomic 1s orbital to stand for the set of molecular core orbitals. Thus we start with 2s rather than 1s for valence-level MOs, which will in truth include tiny nodes around the heavy nuclei.

HOMO-5 2p z CH 3 Orbital Energy CH 3 OH Orbital Energy

HOMO-4 2p x CH 3 Orbital Energy CH 3 OH Orbital Energy

HOMO-3 2p y CH 3 Orbital Energy CH 3 OH Orbital Energy

HOMO-2 CH 3 Orbital Energy CH 3 OH Orbital Energy 3s

HOMO-1 3d xz CH 3 Orbital Energy CH 3 OH Orbital Energy

HOMO 3d yz CH 3 Orbital Energy CH 3 OH Orbital Energy

LUMO 3d z 2 CH 3 Orbital Energy CH 3 OH Orbital Energy

LUMO+1 3p z CH 3 Orbital Energy CH 3 OH Orbital Energy

LUMO+3 LUMO+2 3p y CH 3 Orbital Energy CH 3 OH Orbital Energy

LUMO+2 LUMO+3 3p x CH 3 Orbital Energy CH 3 OH Orbital Energy

LUMO+4 3d xy CH 3 Orbital Energy

LUMO+5 3d x 2 -y 2 CH 3 Orbital Energy

LUMO+6 LUMO+4 4f CH 3 Orbital Energy CH 3 OH Orbital Energy

1-Fluoroethanol

Wire

1s (F) Core 1

1s(O) Core 2

1s(C 1 ) Core 3

1s(C 2 ) Core 4

1s(valence)

2p x

2p y rotate

2p y rotate

2sp z (up)

2sp z (down)

3d xy

e-density contours of H 2 Single “United Atom” distorted by a fragmented nucleus Which contour should we use? The Plum-Pudding View of Molecular Orbitals Shows Generality of Kinetic-Energy-Based Clouds Atoms with weak bonding But One Must Probe Harder to Gain a Qualitative Understanding of Chemical Bonds

Pairwise LCAO MOs 1 √2 ( AO a + AO b )  (x 1,y 1,z 1 ) = SUM (L inear C ombination ) of AOs (like hybridization, but with two atoms) Why is this form sensible? “True” molecular orbitals extend over entire molecules, but we want to understand local bonds as

H 2 at Great Distance 1 2 ( AO A 2 + AO B 2 )    (x 1,y 1,z 1 ) = H 2 at Bonding Distance? 1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = + AO A  AO B error?negligible!

Overlap (A  B) Creates Bonding If we approximate a molecular orbital as a sum of atomic orbitals: and square to find electron density: then subtract the average of the atom electron densities: we find bonding, the difference electron density due to overlap: Looks very good near nuclei (A near A, B near B) “By-product” of squaring a sum. A completely different instance of multiplying! (NOT two electrons) “By-product” of squaring a sum.  < (normalization) < Shifts e-density from atoms _ to overlap region.

in Ain B Wells far apart Wells far apart Total Energy of Particle "Mixing" localized   s for double minimum Wells close together in AB Antibonding Holds A & B together Black line is energy Blue line is  Bonding! Stabilzation of Particle e-Density Grows e-Density Shrinks

AA 2 BB 2 Where is  A  B significant? no yesa littleno!  b small yes! Where is  A 2 significant? At the center 2  A  B is as large as  A 2 +  B 2 Electron Density nearly Doubled! “Overlap Integral” (   A  B ) measures net change from atoms. Region of Significant Overlap 

92.9% of Total Electronic Energy (almost all of which was already present in the atoms) High accuracy is required to calculate correct value of the Bond Energy, the difference between atoms and molecule. (Cf. X-ray difference density) Total e-Density Difference Density 1s (atomic) 52% Bond Energy 0.02 e/a o 3 Coutoured at e/a o 3 Coutoured at e/a o 3 State-of-the-art 40 years ago Laws & Lipscomb, Isr. J. Chem. 10, 77 (1970)

Total e-Density Difference Density 1s (atomic) 52% s (optimized exponent) 73% 0.04 Bond Energy Very crudest model shows most of bond. General spread increases bonding density/stabilization. shift from atom to bond larger shift from atom to bond Adjust molecular orbital to lower the energy. This makes it more realistic, because the true energy is the lowest possible according to the “variational principle”.)

1s (optimized exponent) 73% 0.04 General spread increases bonding density/stabilization. Directed spread improves bonding density. larger shift from atom to bond Total e-Density Difference Density Hybridized + SCF (96.7% 1s; 0.6% 2s; 2.7% 2p) 76% Bond Energy % 1sHybrid: 96.7% 1s 0.6% 2s 2.7%2p Helps overlap but at the cost of 3% n=2 character larger shift from beyond nucleus to bond

Total e-Density Difference Density Hybridized + SCF (96.7% 1s; 0.6% 2s; 2.7% 2p) 76% Bond Energy some correlation 90% 0.11 Density ~unchanged much better energy Directed spread improves bonding density. (How so?)

Pairwise LCAO-MO Looks like atoms (especially near nuclei) (the Main Event for electrons; ~ 6x larger than bond) <1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = Virtues: Builds up e-density between nuclei (through Overlap - the source of Bonding) Hybridizing AOs provides flexibility (unlimited if you use all H-like AOs in hybrid) Easy to formulate and understand (but keep it simple - valence shell is fairly good) Smooths  to lower kinetic energy [though ultimate contraction toward nuclei raises it again]

Pairwise LCAO-MO <1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = <1 2 (AO A 2 + AO B AO A AO B ) == Atoms Bond (overlap / product) >1>1 >1>1 Anti

Overlap & Energy-Match

End of Lecture 12 Oct. 1, 2008