After discussion of how increased nuclear charge affects the energies of one-electron atoms and...
31
After discussion of how increased nuclear charge affects the energies of one-electron atoms and discussion of hybridization, this lecture finally addresses the simple, inconvenient truth that multi-electron systems cannot be properly described in terms of one-electron orbitals. Chemistry 125: Lecture 10 Sept. 23, 2009 Reality and the Orbital Approximation For copyright notice see final page of this file
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After discussion of how increased nuclear charge affects the energies of one-electron
atoms and discussion of hybridization, this lecture finally addresses the simple, inconvenient
truth that multi-electron systems cannot be properly described in terms of one-electron
orbitals.
Chemistry 125: Lecture 10Sept. 23, 2009
Reality and the Orbital Approximation
For copyright notice see final page of this file
Exam 1 - Friday, Sept. 25 !Session 1
10:15-11:15 Room 111 SCL
Session 210:30-11:30 Room 160 SCL
Review/Help SessionsTonight 8:00-10:00 pm Room 207 WLH (McBride)
and Tomorrow Night (Thursday) 7-10 pm
Room 211 WLH (Eugene then Peer Tutors)
The angular part of a p orbital
Polar Plot of cos() [radius] vs. [angle]
0.5- 0.5- 1 0 1• •
•
••
•
•
•
= 0°
0.86
0.86
0.71
0.710.5
0.5
= ±30° = ±60° = ±45° = ±90°
+
2()
e-/2 cos()
Find Max:
=0
de-/2)/d
-e-/2/2 + e-/2 (-/2 + 1) e-/2
Polar 2p Contour Plot
•
•
?
(max for C+6 = 14 e/Å3)
Atom-in-a-Box
Shape of H-like
Specialthanks to
Dean Dauger(physicist/juggler) http://dauger.com
Dean at AppleWorld Wide
Developers Conference2003
permission D. Dauger
QuickTime™ and aMPEG-4 Video decompressor
are needed to see this picture.
Information from Atom-in-a-Box
r2 R(r)2
ProbabilityDensity
SurfaceWeighting
Where is the density highest?What is the most likely distance?
n,l,m (nickname)
SchrödingerEquation
Energy (eV)1 eV ~ 23 kcal/mole
Formula for
Radius = 122 mm
Shell thickness ~1.6 mm
Rubber Basketballdensity ~0.9 g/ml
Lead Sinkerdensity 13.4 g/ml
SurfaceWeighting
Which is heavier?
r2
Which shell (1 or 2) has higher density?
12
Which shell contains more stuff (probability)?
Shell 2 has ~ 3 the radius
~9 the volume of 1.
Information from Atom-in-a-Box
Single Slice
3D 2D
at different levels
near
far
Information from Atom-in-a-Box
Nodes (Shape & Energy)
?3d4d
Cf.
Scaling H-like forChanging Nuclear Charge (Z)
Size
e-Density
Energy
Scaling Size with Z
r2Z
nao
Increasing Z shrinks wave function(makes r smaller for same)
H+ : C+6 : K+19 = 1 : 1/6 : 1/19
Scaling Size with Z : 1s
H+ : C+6 : K+19 = 1 : 1/6 : 1/19
Scalinge-Density
with ZNormalization:
d= 1(reason for most constants)
Table for H-like Atoms
Note: Z3
H+ : C+6 : K+19 = 1 : 216 : 6859
(Helps X-ray find heavieratoms more easily;H very difficult)
Scaling Kinetic Energy with Z
F(Zr)
Z F'(Zr)
'
Z2 F"(Zr)
"
" Z2
Scaling Potential Energy with Z
Distance Shrinkage 1/Z (thus 1/r Z )
V at fixed distance Z
Coulomb's Law V Zer
V Z2
Scaling Total Energy with Z (and n)
E = -RZ2
n2
Independent of l , m (e.g. 3s = 3p = 3d)for 1-electron atoms
R ~ 314 kcal/mole(ionization energy of H)
As we saw for 1-D Coulomb
1
2
35
E=0
4
n =
Scaling H-like forChanging Nuclear Charge (Z)
Size
e-Density
Energy
1/Z
Z3
Z2
(n/Z)
/n2
Physicist’s 2p (m=1)with “orbital
angular momentum”
Information from Atom-in-a-Box
Superposition (a kind of hybridization)
Chemist’s 2py
complex
numbers
Multiplying and AddingWave Functions
Multiply “pieces” to create 1-electron wave function for atom:
(,,) = R(r) () ()
“ORBITAL”
Add orbitals of an atom to create a “hybrid” atomic orbital:
2py + 2pz = hybrid orbitalFunction of what?
Position of one electron!
Change Orientation by Hybridization
a 2py + b 2pz (a weighted sum)
2pz 2py25%50%75%
50:50 mixture of pz and py?Other mixtures of pz and py?
Orientation
0.000.020.040.060.090.110.180.250.330.501.00
Change Shape by Hybridization
spn = a 2 + b 2px
(spn)2 = a2 22 + b2 2px2 + 2ab 2 2px
b2
a2n
(a weighted sum)
Maximumextension
for sp1
hybrid
(see Web& A-i-B)
E
ShapeWhat would happen to 2s in an electric field?
1.00 4 2 3 924 (Pure 2p)
Change Shape by Hybridization
spn = a 2 + b 2px
(spn)2 = a2 22 + b2 2px2 + 2ab 2 2px
b2
a2n
(a weighted sum)
E
Multiplying and AddingWave Functions
Multiply “pieces” to create 1-electron wave function for atom:
(,,) = R(r) () ()
“ORBITAL”
Add orbitals of one atom to create a “hybrid” atomic orbital:
2s + 2pz = hybrid orbitalAllows adjusting to new situations (e.g. electric field) while
preserving the virtues of real solutions for the nuclear potential.
(..function of what? )r1,1,1,r2,2,2
2-e Wave Function
“An Orbital is... a One-ElectronWave Function”
a(r1,1,1) b(r2,2,2)
=?
Multiply 1-e Wave Functions
2
2 2
If so - Orbital Paradise
Total e-density (x,y, z) = 1 2(x, y, z)
+ 2 2(x, y, z)
+ …
Total e-Energy = 1 + E2 + …
e.g. Ne (1s)2 (2s)2 (2px)2 (2py)2 (2pz)2
(3Ne variables) = 1 (x1, y1, z1) 2 (x2, y2, z2)
…
Whole = Sum of Parts
Two (or more) Electrons: a Problem in Joint Probability
Prob (A and B) = Prob (A) Prob (B)
like tossing two coins for two heads
IF the events are independent
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.
End of Lecture 10Sept 23, 2009
Good luck on the exam.
Copyright © J. M. McBride 2009. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).
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