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