1. For copyright notice see final page of this file
Chemistry 125: Lecture 11 Sept. 27, Orbital Correction and Plum-Pudding Molecules
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. The same is true for more complicated molecules, because Schrödinger was right.
For copyright notice see final page of this file

2. What's Coming for Next Exam?
Atoms
Orbitals for Many-Electron Atoms (Wrong!)
Recovering from the Orbital Approximation
Molecules
Plum-Pudding Molecules (the "United Atom" Limit)
Understanding Bonds (Pairwise LCAO)
"Energy-Match & Overlap"
Reality: Structure (and Dynamics) of XH3 Molecules
Reactivity
HOMOs and LUMOs
Recognizing Functional Groups
Payoff for
Organic
Chemistry!
How Organic Chemistry Really Developed (Intro)

3. Multiply 1-e Wave Functions2
a(r1,q1,f1)  b(r2,q2,f2) =
Multiply 1-e Wave Functions
Y(r1,q1,f1,r2,q2,f2)
attr to R. Feynman: “Thereﾕs a reason physicists are so successful with what they do, and that is they study the hydrogen atom and the helium ion and then they stop.”
Orbitals can’t work!

4. Tricks for Salvaging Orbitals

5. r 1s = K e-r/2 2Z r  r nao Z - effective ! ! Pretty Crude
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
Zeff 3s Na
Zeff 2s Zeff 2p C
1s = K e-r/2
2s slightly
“less screened”
than 2p
r  r 2Z
nao r Z ! 2p 2s 1s !
but vice versa for Na
(subtle)

6. Self-Consistent Field (SCF)
1. Find approximate orbitals for all electrons
(e.g. using Zeff)
2. Calculate potential energy due to 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 improve
the orbital for another electron.
Improve all orbitals one by one.
Cycle back to improve 1st orbital further, etc. etc.
Quit ?
when orbital shapes (and energy) stop changing

7. True Energy < SCF Energy
Still Wrong! because real electrons are not fixed clouds. They keep out of each other’s way by correlating their motions.
True Energy < SCF Energy
What do people do about this error?

8. Conceal the residual error after full SCF calculation to the “Hartree-Fock” limit by giving it a fancy name:
"Correlation Energy"
Where to get correct energy (& total electron density)?
by Experiment
or by a Much More Complex Calculation:
e.g. “Configuration Interaction” (CI) or
“Density Functional Theory” (DFT)

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

10. Energy Magnitudes Should Chemists care about the error in Orbital Theory?8 6 2 4 ~
12C Nucleus (2  109)
Fortunately nuclear energy
is totally unchanged
during chemistry!
Loses 0.1 amu (E = mc2)
0.001% change in nuclear energy would overwhelm all of Coulomb. - C • C
C+6 + + - + - - - + + + -
Coulombville
C Core (2  104)
correlation error ≈ bond
C Atom (3  103)
Changes in
"correlation energy"
can be ~10-15%
of Bond Energy.
log (Energy Change kcal / mole)
1/2  4 Single Bonds (2  102)
C "Correlation Energy" (102)
"Non-bonded" Contacts (1-20) 
Correlation
Orbital Theory
is fine for
Qualitative
Understanding
of Bonding. -2
52Å! (2  10-6)

11. Orbitals can't be “true” for >1 electron, because of e-e repulsion
but we'll use them
to understand bonding, structure, energy, and reactivity *
* Resort to experiments or fancy calculation for precise numbers.

12. 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)

13. How Lumpy is the N Atom? Spherical ! (2px)2 = K x2 e-
(2py)2 = K y2 e-
(2pz)2 = K z2 e-
Total = K(x2 + y2 + z2) e-
Total = K(r2) e-
Spherical !
[from an Organic Text]

14. 2px2 + 2py2 depends on (x2+y2) It is thus symmetricalF N
TFDCB ?
C N Triple Bond
cross section
is round
not clover-leaf
nor diamond!
2px2 + 2py2 depends on (x2+y2) It is thus symmetrical
about the z axis

15. Atoms Molecules 3-Dimensional Reality (H-like Atoms) Hybridization
Orbitals for Many-Electron Atoms (Wrong!)
Recovering from the Orbital Approximation
Molecules
Understanding Bonds (Pairwise LCAO-MOs)
“Overlap & Energy-Match"
First an aside on computer-generated MOs:
Plum-Pudding MOs (the "United Atom" Limit)

16. What Spreads and Shapes Atomic Orbitals?
Potential Energy
scales r
(via )
double the nuclear charge
Kinetic Energy
creates nodes
(Schrödinger) 2s 4d

17. Ways of Looking at an Elephant
"Blind monks examining an elephant" by Itcho Hanabusa. 1888
Ukiyo-e print illustration from Buddhist parable

18. Ways of Looking at a Molecule (or a Molecular Orbital)
from set
of atoms
Molecule
as one atom
Molecule
as atoms
Set of
~normal atoms
Single “United Atom”
Atoms with small
bonding distortion
(~0.05 Lewis)
distorted by
fragmenting
the nucleus
J. J. Thomson's
Plum Pudding!
(backwards)
Nuclei embedded in
a cloud of electrons
dispersed and “noded”
by kinetic energy
Which
contour
should
we use?
e-density
contours
of H2
(worth a quick look)

19. How the Plums Distort Hydrogen-Like Kinetic-Energy Puddings

20. Methane & Ammonia Spartan 6-31G. calculates good SCF MOs (on my laptop
We want to understand them visually.

21. 4 Pairs of Valence Electrons
Compare MOs
to AOs of Ne
(4 electron pairs with n=2) H C N H
4 Pairs of Valence Electrons

22. Contour Level 0.001 e/Å3 CH4 NH3 Boring! 1s 8 valence e-  4 MOs.. ..
energy
Three “degenerate”
Molecular Orbitals
Two “degenerate”
Molecular Orbitals
CH4
NH3 1s
"Core" Orbitals
Like 1s of C/N
Tightly Held
Little Distortion
Boring!
We'll focus on
Valence
Orbitals
Contour Level e/Å3

23. .. 2s
Contour Level 0.03 e/Å3

24. .. 2s
“Spherical”
node

25. ..
CH4
NH3
2px

26. ..
CH4
NH3
CH4
NH3
2py

27. ..
CH4
NH3
CH4
NH3
2py

28. CH4 NH3 CH4 NH3 2pz HOMO .. Lewis's "unshared pair"
+Unoccupied Orbitals
HOMO
Highest Occupied Molecular Orbital ..
CH4
NH3
CH4
NH3
2pz
Lewis's "unshared pair"
No proton stabilizes this lobe.

29. Lowest Unocupied Molecular Orbital
"HUMO?" ..
LUMO
Lowest Unocupied Molecular Orbital ..
CH4
NH3 3s

30. .. 2s
CH4
NH3 3s
LUMO

31. .. .. .. .. ..
CH4
NH3
3dx2-y2

32. ..
CH4
NH3
3dx2-y2

33. .. .. .. .. ..
CH4
NH3
3dxy

34. ..
CH4
NH3
3dxy

35. .. .. .. ..
3dz2
3dz2
CH4

36. Ethane & Methanol (Spartan 6-31G*)

37. 7 Pairs of Valence ElectronsH
7 Pairs of Valence Electrons
Compare MOs
to AOs of Ar
(7 electron pairs) O C H

38. Rotated 90°
CH3CH3
Orbital
Energy
Occupied
Vacant 2s
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-6
CH3OH
Orbital
Energy
Occupied
Vacant

39. CH3CH3
Orbital
Energy
CH3OH
2pz
HOMO-5

40. CH3CH3
Orbital
Energy
CH3OH
2px
HOMO-4

41. CH3CH3
Orbital
Energy
CH3OH
2py
HOMO-3

42. CH3CH3
Orbital
Energy
CH3OH 3s
HOMO-2

43. CH3CH3
Orbital
Energy
CH3OH
3dxz
HOMO-1

44. CH3CH3
Orbital
Energy
CH3OH
3dyz
HOMO

45. CH3CH3
Orbital
Energy
CH3OH
LUMO
3dz2
LUMO

46. CH3CH3
Orbital
Energy
CH3OH
LUMO+1
3pz
LUMO+1

47. CH3CH3
Orbital
Energy
CH3OH
LUMO+2
3py
LUMO+3

48. CH3CH3
Orbital
Energy
CH3OH
LUMO+3
3px
LUMO+2

49. CH3CH3
Orbital
Energy
LUMO+4
3dxy

50. CH3CH3
Orbital
Energy
LUMO+5
3dx2-y2

51. CH3CH3
Orbital
Energy
CH3OH
LUMO+6 4f
LUMO+4

52. 1-Fluoroethanol

53. Wire

54. Core 1
1s (F)

55. Core 2
1s(O)

56. Core 3
1s(C1)

57. Core 4
1s(C2)

58. 1s(valence)

59. 2px

60. 2py
rotate

61. 2py
rotate

62. 2spz (up)

63. 2spz(down)

64. 3dxy

65. Single “United Atom” distorted by a fragmented nucleus Atoms with
The Plum-Pudding View of Molecular Orbitals Shows Generality of Kinetic-Energy-Based Clouds
e-density
contours
of H2
But One Must Probe Harder to Gain a
Useful Understanding of Chemical Bonds
Single “United Atom”
distorted by a
fragmented nucleus
Atoms with
weak bonding
Which
contour
should
we use?

66. End of Lecture 11 Sept. 27, 2010
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