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.

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