Chem 125 Lecture 109/24/08

This material is for the exclusive use of Chem 125 students at Yale and may not

be copied or distributed further.

It is not readily understood without reference to notes from the lecture.

Exam 1 - Friday, Sept. 26 !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 116 WLH

Tomorrow Night (Thursday) 7-10 pmRooms 113 and 116 WLH

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)

Schrödinger Equation

Energy (ev)

Formula

Which shell (1 or 2) has higher density?

12

Which shell contains more stuff (probability)?

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 ≈ 300 kcal/mole

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 24, 2008

Good luck on the exam.