1. For copyright notice see final page of this file
Chemistry 125: Lecture 7 Sept. 16, Quantum Mechanical Kinetic Energy
After pointing out several discrepancies between electron difference density results and Lewis bonding theory, the course introduces quantum mechanics. The wave function Ψ, which beginning students find confusing, was equally confusing to the physicists who created quantum mechanics. The Schroedinger equation reckons kinetic energy through the shape of Ψ. When Ψ curves toward zero, kinetic energy is positive; but when it curves away, kinetic energy is negative! A simple tool allows finding Ψ for one-dimensional problems.
For copyright notice see final page of this file

2. Exam 1 - Friday, Sept. 25 !
Covers Lectures through Wednesday, Sept. 23
Including:
Functional Groups
X-Ray Diffraction
1-Dimensional Quantum Mechanics
& 1-Electron Atoms
(Sections I-V of quantum webpage
& Erwin Meets Goldilocks )
IMPORTANT PROBLEMS
therein due Monday, Sept. 21
Get-aquainted with Erwin at Thursday Discussion
Exam Review 8-10 pm Wednesday, Sept. 23, Room TBA

3. Dunitz et al. (1981)

4. Surprising only for its beauty
Dunitz et al. (1981)

5. Pathological Bonding Missing Bond? Dunitz et al. (1981)
Lone "Pair"
of N atom H
Missing Bond?
Bond Cross Sections
Dunitz et al. (1981)

6. In three weeks we’ll understand these pathologies.
Pathological Bonding
Bent
Bonds !
Missing
Bond !
In three weeks we’ll understand these pathologies.
Dunitz et al. (1981)

7. but they are hopelessly crude when it comes to describing
Lewis Pairs/Octets provide a pretty good bookkeeping device for keeping track of valence
but they are hopelessly crude
when it comes to describing
actual electron distribution.
There is electron sharing (~5% of Lewis's prediction).
There are unshared "pairs" (<5% of Lewis's prediction).

8. Is there a Better Bond Theory, maybe even a Quantitative one?
YES!
Chemical Quantum
Mechanics

9. Schrödinger Wave Equation (1926)
Erwin Schrödinger (Zurich,1925)
Age 38
Schrödinger as “97 pound weakling” the year before Debye, in his ETH seminar, suggested that he work on a wave equation rather than the “unimportant” stuff he was doing at Univ. of Zurich.

10. Felix Bloch, Physics Today (1976)
"Once at the end of a colloquium I heard Debye saying something like: Schrödinger, you are not working right
now on very important problems anyway. Why don't you
tell us sometime about that thesis of de Broglie?
"So in one of the next colloquia, Schrödinger gave a beautifully clear account of how de Broglie associated
a wave with a particle…When he had finished, Debye
casually remarked that he thought this way of talking was
rather childish… he had learned that, to deal properly
with waves, one had to have a wave equation.
It sounded rather trivial and did not seem to make a
great impression, but Schrödinger evidently thought
a bit more about the idea afterwards."
~gasse/gesch1.html

11. "Just a few weeks later he gave another talk in the colloquium, which he started by saying: My colleague Debye suggested that
one should have a wave equation:
Well, I have found one."
H y = E y

12. December 1933 - Stockholm Erwin Schrödinger Paul Werner Dirac
L-R: Heisenberg's mother, Schrodinger's wife Anny, Dirac's mother, Dirac, Werner Heisenberg, Erwin Schrodinger; Stockholm train station
Schroedinger Dirac (5th youngest)
Heisenberg age 31 at prize (2nd youngest)
Paul
Dirac
Werner
Heisenberg
AIP Emilio Segre Visual Archives, Peierls Collection

13. Schrödinger Equation
H y = E y
Wave Function
???

14. Leipzig (1931) Felix Bloch 1952 (NMR) Werner Heisenberg
Bloch b. 1905, Zurich. 21 in 1926 undergraduate at ETH, Ph.D. with Heisenberg at Leipzig
George Placzek, b Brno
Guido Gentile Italian
Rudolph Peierls b Berlin
G. Wick
Victor Weisskopf b Vienna
Fritz Sauter b Innsbruck?
Werner
Heisenberg
AIP Emilio Segre Visual Archives, Peierls Collection

15. Felix Bloch & Erich Hückel on 
(1926)
Gar Manches rechnet Erwin schon
Mit seiner Wellenfunktion.
Nur wissen möcht man gerne wohl,
Was man sich dabei vorstell'n soll.
Erwin with his Psi can do
calculations, quite a few.
We only wish that we could glean
an inkling of what Psi could mean.

16. Even Schrödinger was never comfortable with what  really means:
“etwa so wie Cervantes einmal den Sancho Panza, sein liebes Eselchen auf dem er zu reiten pflegte, verlieren läßt. Aber ein paar Kapitel später hat der Autor das vergessen und das gute Tier ist wieder da.
“Once Cervantes had Sancho Panza lose
the well-loved little donkey he rode on.
But a couple chapters later the author
had forgotten and the good beast
reappeared.
Nun werden sie mich vielleicht zuletzt fragen, ja was sind denn nun aber
wirklich diese Korpuskeln, diese
Atome - Moleküle.
Now you will perhaps in conclusion
ask me, “So what are they then, I mean
really, these corpuscles – these atoms –
molecules?”
Ehrlich müßte ich darauf bekennen,
ich weiß es sowenig, als ich weiß,
wo Sancho Panzas zweites Eselchen hergekommen ist.
I must admit honestly, on this subject
I know just as little, as I know
where Sancho Panza’s second little donkey came from.
Schrödinger’s Grave
Alpbach, Austria
Schrödinger Lecture
“What is Matter”
(1952)
Wikipedia
by permission from Supposé CD
Erwin Schrödinger Was ist Materie?

17. First we’ll learn how to findand use it. Later we learn what it means.

18. ? Y a Function of What? Named by "quantum numbers"
(e.g. n,l,m ; 1s ; 3dxy ; s p p*)
Function of
Function of Particle Position(s)
[and sometimes of time and "spin"]
N particles  3N arguments!
[sometimes as many as 4N+1]
We focus first on one particle, one dimension,
then three dimensions (one atomic electron),
then atoms with several electrons,
then molecules and bonding,
finally functional groups & reactivity

19. H y = E y Schrödinger Equation  time-independent
(for “stationary” states)
Schrödinger Equation
H y = E y 
(NOT H times y )
( E times y )

20. y H y = E y H = E = Kinetic Energy + Potential Energy = Total Energy
Given - Nothing to do with y
(Couloumb is just fine)
Hold your breath!

21.  Kinetic Energy? mi vi2 i Const  1 2 Fine for our great grandparents
(adjusts for desired units)
Sum of classical
kinetic energy over all particles of interest.

22.  y y y y Kinetic Energy! C C 2 xi2 yi2 zi2 + d2 dx2 i h2 8p2 1 mi
Note:
Involves the shape of y,
not just its value.
One particle,
One dimension: d2
dx2 y 1 m C C y
Curvature of m

23. Solving a Quantum Problem
Given : a set of particles
their masses & their potential energy law
[ e.g. 1 Particle/1 Dimension : 1 amu & Hooke's Law ]
Given :
To Find : TY
a Function of the position(s) of the particle(s)
Such that HY/Y is the same (E) everywhere
AND Y remains finite!!!
(single-valued, continuous, Y 2 integrable)

24. What's Coming? 1 Particle, 1 Dimension 1-Electron Atoms (3 Dimensions)
Sept 25 Exam
Many Electrons & Orbitals
Molecules & Bonds
Functional Groups & Reactivity

25. The Jeopardy Approach -C/m -C/m ’’ ” Answer Problem  = sin (x) C/m
Kinetic Energy
Problem
mass and
Potential Energy(x) y C
Curvature of m
Independent of x
 Const PE
(particle in free space)
 = sin (x)
C/m
sin (x) C m
- sin (x)
 = sin (ax)
a2 C/m ’’
higher kinetic energy
( a > 1  shortened wave)
sin (ax) C m
- a2 sin (ax)
 1 / 2
-C/m
 = ex
Const PE > TE
a = 3
Negative kinetic energy!
-C/m
 = e-x ex C m ex ”
Not just a mathematical curiosity.
Actually happens for all electrons bound to nuclei! V
NOT your great grandparent’s 1/2 mv2. E
(at large distance, where 1/r ceases changing much)

26. Potential Energy from Arbitrary  Shape
via Kinetic Energy + _
(x) x
Curving
toward = 0
 Positive
Positive
Zero
Negative ?
Curvature
Amplitude
Potential Energy
Curving away
from = 0
 Negative
The potential energy function for this  must be a
double minimum. • •
Total Energy

27. From “Jeopardy” Approach to Recipe for Solution of Schrödinger Equation Using Guessed Total Energies

28. Rearranging Schrödinger to give a formula for curve tracing.y
Curvature of m
+ V = E C y
Curvature of m
(V- E) =
Curves away from 0 for V>E; toward 0 for V<E.
Since m, C, V(x) are given, this recipe allows tracing (x) in steps, from initial (0) [= 1], with initial slope [0], and a guessed E.

29. Danger
Negative
Kinetic
Energy
(Curve Away from Baseline)
100 kcal/mole
2.5Å
Too Cold
Nodes and Quantization in One Dimension from Erwin Meets Goldilocks (for Wiki see Monday Problem Set)
Erwin Meets Goldilocks
Just Right!
Guess 21 kcal/mole
20.74 kcal/mole
Guess 20 kcal/mole
Too Hot

30. End of Lecture 7 Sept 16, 2009
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J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0

31. Could there be a lower-energy Psi?
100 kcal/mole
2.5Å
4.15 kcal/mole
12.45 kcal/mole
Erwin Meets Goldilocks
NODES
0 because of sign change
Could there be an energy between?
Could there be a lower-energy Psi?
More Energy 
More Curvature
More Nodes
20.74 kcal/mole