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Chem 125 Lecture 7
9/17/08
Preliminary
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 !
Covers Lectures through Wednesday, Sept. 24
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. 22
I’m
working on with
checking
offatWiki
contributions
and
Get-aquainted
Erwin
Thursday
Discussion
hope to make personalized scorecards (including homeworks)
Exam Review 8-10 pm Wednesday, Sept. 24, Room TBA
available via Post’em at ClassesV2 sometime tomorrow.
Pathological Bonding
0.002 Å !
for average
positions
Typically vibrating
by ±0.050 Å
in the crystal
Dunitz et al. (1981)
Surprising only for its beauty
Dunitz et al. (1981)
Pathological Bonding
Lone "Pair"
of N atom
H
H
H
H
H
H
Bond Cross Sections
Dunitz et al. (1981)
Missing Bond?
Pathological Bonding
Missing
Bond !
Bent
Bonds !
In three weeks we’ll understand these pathologies.
Dunitz et al. (1981)
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).
Is there a Better Bond Theory,
maybe even a Quantitative one?
YES!
Chemical
Quantum
Mechanics
Erwin Schrödinger (Zurich,1925)
Wave Equation
(1926)
http://www.zbp.univie.ac.at/schrodinger/lebensbilder/bilder9.htm
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."
www.uni-leipzig.de/ ~gasse/gesch1.html
"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."
Hy = E y
December 1933 - Stockholm
Paul
Dirac
Werner
Heisenberg
Erwin
Schrödinger
AIP Emilio Segre Visual Archives, Peierls Collection
Schrödinger Equation
Hy =Ey
???
Leipzig (1931)
1952 (NMR)
Felix
Bloch
Werner
Heisenberg
AIP Emilio Segre Visual Archives, Peierls Collection
Felix Bloch & Erich Hückel on
(1926)
y
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.
Even Schrödinger was never
comfortable with what  really means:
“etwa
somüßte
wie Cervantes
einmal
denzuletzt
Sancho
Nun
werden
sie
vielleicht
Ehrlich
ichmich
darauf
bekennen,
Panza,
sein
liebes
Eselchen
aufweiß,
dem
fragen,
jaes
was
sind
denn
nun
aberer zu
ich weiß
sowenig,
als ich
reiten
pflegte,
verlieren
läßt. Aber
ein paar
wirklich
diese
Korpuskeln,
diese
wo Sancho
Panzas
zweites
Eselchen
Schrödinger
Lecture
Kapitel später
hat der Autor
das vergessen und
Atome
- Moleküle.
hergekommen
ist.
das gute Tier
ist wieder
da.
“What
is Matter”
NowCervantes
you will
perhaps
conclusion
I must
admit
honestly,
onin
this
subject
“Once
had
Sancho
Panza
lose
(1952)
the
well-loved
little
donkey
hethen,
rode Ion.
ask
me,just
“Soas
what
are
mean
I know
little,
asthey
I know
But
a couple
latersecond
author
really,
thesechapters
corpuscles
–the
these
atoms –
where
Sancho
Panza’s
little
had
forgotten
and
the
good
beast
molecules?”
donkey came from.
reappeared.
First we’ll learn how
to findand use it.
Later we learn what it
means.
 )
?
Function of What?
Named by "quantum numbers"
(e.g. n,l,m ; 1s ; 3dxy ; s p p*)
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
time-independent
(for “stationary” states)
Schrödinger Equation
Hy=Ey
(NOT H times y )
( E times y )
Hy=Ey
Hy
=
y
=E
Kinetic Energy + Potential Energy = Total Energy
Hold your breath!
Given - Nothing to do with y
(Couloumb is just fine)
Kinetic Energy?
Const 

i
(adjusts for
desired units)
1m v2
2 i i
Sum of classical
kinetic energy
over all particles
of interest.
Fine for our great grandparents
Kinetic Energy!
h2
8p2

i
1 2y 2y 2y
+
+
mi xi2
yi2 zi2
y
One particle, One dimension:
C
1 d2y
m dx2
y
Note: Involves. …
the shape of y,
not just its value.
C
m
Curvature of
y
y
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 ]
To Find : 
a Function of the position(s) of the particle(s)
Such that H/ is the same (E) everywhere
AND  remains finite!!!
(single-valued, continuous,  2 integrable)
What's Coming?
1 Particle, 1 Dimension
1-Electron Atoms (3 Dimensions)
Sept 26 Exam
Many Electrons & Orbitals
Molecules & Bonds
Functional Groups & Reactivity
The Jeopardy Approach
(y )
C
Curvature of
y
m
 = sin (x)
 = sin (ax)
mass and
Potential Energy(x)
y
C/m
Indep. of x  Const PE
C - sin (x)
m 2 sin (x)
( a > 1  shortened wave)
a C/m
C - a2 sin (ax)
m
 = ex
Problem
Kinetic Energy
Answer
sin (ax)
-C/m
(particle in free space)
’’
higher kinetic energy
 1 / 2
Const PE > TE
Negative kinetic energy!
=
e-x
-C/m
m e
C ex
x
”
Not just a mathematical curiosity.
your great
grandparent’s
mv2. to nuclei!
Actually NOT
happens
for all
electrons 1/2
bound
(at large distance, where 1/r ceases changing much)
Potential Energy from  Shape via Kinetic Energy
(x)
+
Curving
toward = 0
 Positive
Curvature
Amplitude
Curving away
from = 0
 Negative
Positive
Zero
Negative
?
x
0
•
_
•
The potential energy
function for this 
must be a
double minimum.
Total Energy
End of Lecture 7
Sept 17, 2008