Double-minimum potentials generate one-dimensional bonding, A different technique is needed to

address multi-dimensional problems. Solving Schroedinger’s three-dimensional differential equation might

have been daunting, but it was not, because the necessary formulas had been worked out more than a

century earlier in connection with acoustics. Acoustical “Chladni” figures show how nodal patterns relate to

frequencies. The analogy is pursued by studying the form of wave functions for “hydrogen-like” one-

electron atoms. Removing normalizing constants from the formulas for familiar orbitals reveals the

underlying simplicity of their shapes.

Chemistry 125: Lecture 10Sept 20, 2010

More Dimensions: Chladni Figures and One-Electron Atoms

For copyright notice see final page of this file

Preliminary

Reward for Finding Y

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 energyand raised second energy

Both are~ same as

single-minimumsolution

“Splitting”

@ 0

.6Å

@ 1

.3Å

in A in B

Wells farapart

Wells farapartTo

tal E

nerg

y of

Par

ticl

e"Mixing" localized Ys for double minimum

Wells closetogether

in AB

Antibonding

HoldsA & B

together

Black line is energy

Blue line is

Bonding!lower Kinetic Energy!

Sta

bili

zati

onof

Par

ticl

e

reduced curvature

increased curvature

For bonding between 3D coulombic atoms, see Lecture 12.

Dynamics:Tunneling

Dynamics:Tunneling

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.

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.

NegativeKineticEnergy

1.4 kcal/molesplitting

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

Well-to-Well time 510-14 sec

Splitting (kcal/mole)Assertion : based on

time-dependent quantum mech.

Dynamics:Tunneling

Reward for Finding Y

Knowledge of Everythinge.g.

Allowed EnergiesStructureDynamicsBonding

Reactivity (coming later)

Morse Quantization

"Erwin" can find Ys 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 from = 0 in “forbidden” continuum.

Why is this not

satisfactory?

“Erwin” even

handles MultipleMinima

Erwin’s 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 / dxi2) / contributions.

Unfortunately:

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

Acoustics (1803)

e.g. Chladni Figures in 2 Dimensions

SandCollectsin Nodes

Touch inDifferent

Places

Bow inDifferent

Places

dryice

Click for Short Chladni Movie (3MB)

Click for Longer Chladni Movie (9.5MB)

CO2

brass plate

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

e.g. Daniel

Bernoulli(1700-1782)

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

2/x2 + 2/x2 + 2/x2

(abbreviated 2)the Laplacian Operator

s for one-electron atomsinvolve

Laplace’s “Spherical Harmonics”

(3D-Analogues of Chladni Figures)

3-Dimensional H-Atom Wavefunctions

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

Adrien-Marie Legendre(1752 -1833)

() is the normalized“Associated Legendre Function”

Edmond Laguerre(1834-1886)

R(r) are the normalized“Associated Laguerre Polynomials”

with contributions from other old-time mathematicians

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

1

simplifies V(r,q,f) =r

c

Name Y byquantum numbers (n > l ≥ m)

or by nickname (1s, etc.)

Y = Rnl(r) Qlm(q) Fm(f)

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

f

q

Y Table for H-like AtomsY = R(r) Q(q) F(f)

1s

r r2Znao

Allows writing the same e-/2

for any nuclear charge (Z)and any n.

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

x

y

z

n

er

f

q

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

Squaring gives a number, Z

3 per unit volume (units of probability density)

Why r instead of r?

e-density at H nucleus (r = 0)

>1e/Å3!

Q: How much less dense is 2s at nucleus?

r 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 total probability

constant) 1/6

216(0.044 Å)

Å (1sC) 0.1 0.2

(0.044 Å)

Different Å scalesCommon Å scale

(1sH ~2 e/Å3)

+5

Increasing nuclear chargesucks standard 1s function

toward the nucleus(renormalization keeps

total probabilityconstant)

Summary r r2Z

nao

What would the exponential part

of……. look like?

+5

C2s

1sC(0) is 216 times 1sH(0)! E

lect

ron

Den

sity

(e/

Å3 )

H1s

C1s

0.5 1.00.1 0.2

Common Å scale

100

200

300

400

2

(but smoothed by vibration)

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 table vacant?

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

4) Do the 6 atomic orbital problemsClick Here

2

2 2

Y Table for H-like Atoms

1s = K e-r/2

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

Shape of H-like Y

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

Simpler (!) than Erwin 1-D Coulombic

x

y

z

n

er

f

qr cos() = z

Spherical node at r = 2

Planar node at z = 2

The angular part of a p orbital

Polar Plot of cos(q) [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(q)

re-r/2 cos()

Find Max:

= 0

dre-r/2)/d r

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

r

2p Contour Plot

•r

•

?

(max for C+6 = 14 e/Å3)

Polar Coordinates

Exam 1 - Friday, Sept. 24 !Session 1

10:15-11:15 Room 111 SCL

Session 210:30-11:30 Room 160 SCL

Extra Review/Help SessionWednesday 8:00-10:00 pm

Room 119 WLH (McBride)

Atom-in-a-Box

Shape of H-like Y

Specialthanks to

Dean Dauger(physicist/juggler) http://dauger.com

Dean at AppleWorld Wide

Developers Conference2003

permission D. Dauger

End of Lecture 9Sept 20, 2010

(see Lecture 10 for description of Atom-in-a-Box)

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