several tricks (z-effective and self consistent field) allow one to correct approximately for the...

Several tricks (“Z-effective” and “Self Consistent Field”) allow one to correct approximately

for the error in using orbitals when there is electron-electron repulsion. Residual error is

hidden by naming it “Correlation energy.” J.J. Thomson’s Plum-Pudding model of the atom

can be modified to visualize the form of molecular orbitals. There is a close analogy in form

between the molecular orbitals of CH4 and NH3 and the atomic orbitals of neon, which has the

same number of protons and electrons. The underlying form, dictated by kinetic energy, is

distorted by pulling protons out of the Ne nucleus to play the role of H atoms.

Chemistry 125: Lecture 11Sept. 28, 2009

Orbital Correction and Plum-Pudding Molecules

For copyright notice see final page of this file

What's Coming for Next Exam?

MoleculesPlum-Pudding Molecules (the "United Atom" Limit)

Understanding Bonds (Pairwise LCAO)"Energy-Match & Overlap"

Reality: Structure (and Dynamics) of XH3 Molecules

AtomsOrbitals for Many-Electron Atoms (Wrong!)Recovering from the Orbital Approximation

Payoff forOrganic

Chemistry!

ReactivityHOMOs and LUMOs

Recognizing Functional Groups

How Organic Chemistry Really Developed (Intro)

2-e Wave Function

(r1,1,1,r2,2,2)

a(r1,1,1) b(r2,2,2)

=?

Multiply 1-e Wave Functions

2

2 2

No way can electrons be independent!

They repel one another.

Tricks for SalvagingOrbitals

Pretend that the other electron(s) just reduce the nuclear charge for the orbital of interest.

"Clementi-Raimondi" values for Zeff(best fit to better calculations as of 1963)

Atom Z Zeff 1s

He 2 1.69

2s

2p

Z - effective

Zeff 2s Zeff 2p

C 6 5.67 3.22 3.14

Zeff 3s

Na 11 10.63 6.57 6.80 2.51

!

!

2s slightlyless screened

than 2p

vice versa for Na

Pretty

Crude

r2Znao

1s = K e-/2

(subtle)

1s

Self-Consistent Field (SCF)1. Find approximate orbitals for all electrons

(e.g. using Zeff)

2. Calculate potential from fixed, point protons and fixed clouds for all electrons but one.

3. Use this new potential to calculate an. an..improved orbital for that one electron.

4. Repeat steps 2 and 3 to improvethe orbital for another electron.

. . . Improve all orbitals one by one.

Quit When orbital shapes stop changing

Cycle back to improve 1st orbital further, etc. etc.

Still Wrong!because real electronsare not fixed clouds.

They keep out of each other’s way by correlating their motions.

True Energy < SCF EnergyWhat do people do about this error?

"Correlation Energy"

Conceal the residual error after full SCF calculation to the “Hartree-Fock” limit

by giving it a fancy name:

Where to get correct energy (& total electron density)?

by Experiment

or by a Whopping Calculation:

e.g. “Configuration Interaction” (CI)

or

“Density Functional Theory” (DFT)

If we’re really lucky, "Correlation Energy"might be Negligible.

"Non-bonded" Contacts (1-20)

+

++ ++ +C+6

-- ---

Energy Magnitudes

Should Chemists care about the error in Orbital Theory?

-2log

(Ene

rgy

Cha

nge

kca

l / m

ole

C Core (2 104)

1/2 4 Single Bonds (2 102)

He•He @ 52Å! (2 10-6)

Changes in "correlation energy"

can be ~10-15% of Bond Energy.

Orbital Theory is fine for Qualitative

Understanding of Bonding.

C "Correlation Energy" (102)

-C••

••C

12C Nucleus (2 109)Loses 0.1 amu (E = mc2)Fortunately nuclear energy

is totally unchangedduring chemistry!

0.001% change in nuclear energy would overwhelm all of Coulomb.

correlation error ≈ bond

8

0

6

2

4

~

C Atom (3 103)

Orbitals can't be “true”for >1 electron, because of e-e repulsion

butwe'll use themto understand

bonding, structure,energy, and reactivity

* Resort to experiments or fancy calculation for precise numbers.

*

If we use orbitals, how should we reckon total electron density?

Density of electron 1 = 1 2(x1,y1,z1)

Density of electron 2 = 2 2(x2,y2,z2)

Total density (x,y,z) = 1 2(x,y,z) + 2

2(x,y,z)

(Sum, not Product. Not a question of joint probability)

How Lumpy is the N Atom?

Total = K(r2) e-

(2px)2 = K x2 e-

(2py)2 = K y2 e-

(2pz)2 = K z2 e-

Total = K(x2 + y2 + z2) e-

Spherical ![from an Organic Text]

TFDCBC

CC

C

F

N

is roundnot clover-leafnor diamond!

C N Triple Bond

2px2 + 2py

2 depends on (x2+y2) It is thus symmetrical

about the z axis

cross section

?

MoleculesUnderstanding Bonds (Pairwise LCAO-MOs)

“Overlap & Energy-Match"

Atoms3-Dimensional Reality (H-like Atoms)

HybridizationOrbitals for Many-Electron Atoms (Wrong!)Recovering from the Orbital Approximation

First an aside on computer-generated MOs:

Plum-Pudding MOs (the "United Atom" Limit)

What gives Atomic Orbitals their Shape?

Potential Energyscales r

(via )

Kinetic Energy

creates nodes(Schrödinger)

4d

2s

double the nuclear charge

Ways of Looking at an Elephant

Set of~normal atoms

Atoms with smallbonding distortion

(~0.05 Lewis)

Single “United Atom”

Ways of Looking at a Molecule(or a Molecular Orbital)

e-densitycontours

of H2

Whichcontourshould

we use?

Moleculefrom setof atoms

Moleculeas one atom

distorted byfragmentingthe nucleus

Nuclei embedded ina cloud of electrons

dispersed and “noded”by kinetic energy

J. J. Thomson'sPlum Pudding!

(backwards)

Moleculeas atoms

(worth a quick look)

How the PlumsDistort Hydrogen-Like

Kinetic-Energy Puddings

Methane&

Ammonia

Spartan 6-31G* calculates good SCF MOs(on my laptop!)

We want to understand them visually.

4 Pairs of Valence Electrons

H

C HHH

NH H

H

Compare MOsto AOs of Ne

(4 electron pairs with n=2)

1sCH4 NH3

"Core" OrbitalsLike 1s of C/NTightly Held

Little Distortion

Contour Level 0.001 e/Å3

We'll focus onValenceOrbitals

Boring!

..

.. ......

.... ..

8 valence e-

4 MOs8 valence e-

4 MOs

ene

rgy

Three “degenerate”Molecular Orbitals

2s.... ....

...... ..

2s.... ....

...... ..

“Spherical”node

2px

.. ......

..

CH4 NH3

.. ....

CH4 NH32py

.. ..

......

CH4 NH3

.. ....

CH4 NH32py

.. ..

......

CH4 NH3

.. ....

CH4 NH32pz

HOMOLewis's "unshared pair"

......

CH4 NH3

.... .... ..

+Unoccupied Orbitals +Unoccupied Orbitals

CH4 NH3

3s LUMO"HUMO?"

..

.. ......

.... ..

2sCH4 NH3

3s LUMO"HUMO"

...... ..

..

.. ....

CH4 NH33dx2-y2

..

.. ......

.... ..

CH4 NH33dx2-y2

...... ..

..

.. ....

CH4 NH33dxy

..

.. ......

.... ..

CH4 NH33dxy

...... ..

..

.. ....

CH4

3dz2 3dz2.... ....

Ethane&

Methanol

(Spartan 6-31G*)

7 Pairs ofValence

Electrons

C

C

HH

H H

H

H

OC

HH H

H

Compare MOsto AOs of Ar

(7 electron pairs)

2s

CH3CH3

OrbitalEnergy

Occupied

Vacant

HOMO-6CH3OHOrbitalEnergy

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

2pz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO-4

2px

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO-3

2py

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO-2

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

3s

HOMO-1

3dxz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO

3dyz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO

LUMO

3dz2

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+1

LUMO+1

3pz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+3

LUMO+2

3py

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+2

LUMO+3

3px

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+4

3dxy

CH3CH3

OrbitalEnergy

LUMO+5

3dx2-y2

CH3CH3

OrbitalEnergy

LUMO+6

LUMO+4

4f

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

End of Lecture 11Sept. 28, 2009

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