Chem 125 Lecture 99/22/08

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.

Reward for Finding

Knowledge of Everythinge.g.

Allowed EnergiesStructureDynamicsBonding

Reactivity

Single- vs. Double MinimumFor Hooke's Lawthe Blue Energy

is too Lowand

the Red Energyis too High.

The Correct LowestEnergy must lie

between these values.

Single-Mimimum

Actuallythis is aDouble-

Minimum.

The Blue and Reds are correct!

What if the wells were further apart?

Closer wells givelowered minimum energy

and higher next energy

~ same as single-minimum

solution

“Splitting”

in A in B

Wells farapart

Wells farapartT

otal

Ene

rgy

of P

arti

cle

"Mixing" localized s for double minimum

Wells closetogether

in AB

Antibonding

HoldsA & B

together

Black line is energy

Blue line is

Bonding!Sta

bilz

atio

nof

Par

ticl

e

Dynamics:Tunneling

Dynamics:Tunneling

The word "Tunneling" is one of my pet peeves:

It is misleading and mischievous because it suggests that there is something weird about the potential energy in a double minimum.

.

In fact it simply involves the same negative kinetic energy that one sees in the tails of EVERY bounded wavefunction.

The word reveals naiveté about quantum mechanics.

1.4 kcal/molesplitting

~410-14 sec to get from well to well.

Well-to-Well time 510-14 sec

Energy (kcal/mole)Assertion

from time-dependent q. mech.

Dynamics:Tunneling

Reward for Finding

Knowledge of Everythinge.g.

Allowed EnergiesStructureDynamicsBonding

ReactivityComing

soonAfter Exam,

Atoms, Molecules

Morse Quantization

"Erwin" can find s for any complicated V(x)

7 Å

and rank them by energy / "curvature" / # of nodes

Don’t cross 0 in “forbidden” continuum.

Don’t slope out and away in “forbidden” continuum.

What’s wrong with this picture?

EvenMultipleMinima

This curve-tracing recipe won't work in more dimensions (e.g. 3N).

But Schrödinger had no trouble finding solutions for the 3-dimensional H atom, because they were familiar from a long tradition of physicists studying waves.

When there are many curvatures, it is not clear how to partition the kinetic energy among the different (d2 / dxi

2) / contributions to Etotal.

E. F. F. Chladni(1756-1827)

Acoustics (1803)

e.g. Chladni Figures in 2 Dimensions

SandCollectsin Nodes

Touch inDifferent

Places

Bow inDifferent

Places

Click for Short Chladni Movie (3MB)

Click for Longer Chladni Movie (9.5MB)

Crude Chladni Figures

3 Diameters / 1 Circle3 Circles

1 Diameter / 2 Circles

4 Diameters / 1 Circle from in-class demo

Chladni’sNodal

Figures for a

Thin Disk

Portion inside outer circular nodeCf. http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html

(1,2)

Chladni’sNodal

Figures for a

Thin Disk

Number of Diametrical Nodes

Num

ber of Circular N

odes

PITCH

47 Patterns!

"These pitch relationships agree approximately with the squares of the following numbers:"

Frequency ≈ (Diametrical Nodes + 2 Circular Nodes) 2

Note: Increasing number of ways to get a higher frequencyby mixing different numbers of circles and lines

8 Lines

4 Circles

2 Circles4 Lines3 Circles

2 Lines

1 Circle6 Lines

Num

ber of Circles

Number of Diameters

1 Circle2 Lines

Great Mathematicians Worked on Chladni’s 2-D Problems:

e.g. Daniel Bernoulli

s for one-electron atomsinvolve

“Spherical Harmonics”

(3D-Analogues of Chladni Figures)

3-Dimensional H-Atom Wavefunctions

(,,) = R(r) () ()

Adrien-Marie Legendre(1752 -1833)

() is the normalized“Associated Legendre Polynomial”

Edmond Laguerre(1834-1886)

R(r) is the normalized“Associated Laguerre Function”

Available from other old-time mathematicians

Table for H-like Atoms V(x,y,z) =sqrt(x2 + y2 + z2)

1

simplifies V(r,,) =r

c

Name byquantum numbers (n > l ≥ m)

or by nickname (1s, etc.)

= Rnl(r) lm() m()

product of simple functionsof only one variable each

and

(x,y,z) is very complicated

change coordinate

system:x,y,z r

x

y

z

n

er

Table for H-like Atoms = R(r) () ()

1s

r2Znao

Why instead of r?

Allows using the same e2

for any nuclear charge (Z)and any n.

= K e-/2N.B. No surprise forCoulombic Potential

x

y

z

n

er

Note: all contain (Z / ao)3/2

Squaring gives a number, Z

3 per unit volume (units of probability density)

r2Znao

exp-

r = 2Z

nao

r1H = 2

0.53Å

r1C = 12

0.53Å

All-Purpose Curveshrunk by Z; expanded by n

Å (1sH)

(0.26 Å)

0.5 1.0

Increasing nuclear chargesucks standard 1s function

toward the nucleus

0.1Å (1sC) 0.2

(renormalization keeps probability density

constant) 1/6

6(0.044 Å)

Å (1sC) 0.1 0.2

(0.044 Å)

Different Å scalesCommon Å scale

H1s

C1s

Rel

ativ

e E

lect

ron

Den

sity

+5

0.5 1.0

Increasing nuclear chargesucks standard 1s function

toward the nucleus

0.1 0.2

(renormalization keeps total probability

constant)

Common Å scale

Summary r2Znao

What would the exponential part

of……. look like?

C2s

+5

For Wednesday:

1) Why are there no Chladni Figureswith an odd number of radial

nodes? (e.g. 3 or 5 radii)

2) Why are the first two cells [(0,0) and (1,0)] in Chladni's tables vacant?

3) Compare 1sH with 2sC+5 in Energy

4) Do the 6 atomic orbital problemsClick Here

2

2 2

Table for H-like Atoms

1s = K e-/2

2s = K'(2-) e-/2

Shape of H-like

= K'''( cos()) e-/22pz zGuess what 2px and 2py look like.

Simpler (!) than Erwin 1-D Coulombic

x

y

z

n

er

The angular part of a p orbital

Polar Plot of cos() vs.

0.5- 0.5- 1 0 1• •

•

••

•

•

•

= 0°

0.86

0.86

0.71

0.710.5

0.5

= ±30° = ±60° = ±45° = ±90°

+

cos2() vs.

e-/2 cos()

Find Max:

=0

de-/2)/d

-e-/2/2 + e-/2 (-/2 + 1) e-/2

Polar 2p Contour Plot

•

•

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

Link to Video of Dauger Juggling at Apple Developer's Conference

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)

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.

End of Lecture 9Sept 23, 2008