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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.
Chemistry 125: Lecture 7Sept. 16, 2009
Quantum MechanicalKinetic Energy
For copyright notice see final page of this file
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Exam 1 - Friday, Sept. 25 !
Exam Review 8-10 pm Wednesday, Sept. 23, Room TBA
Covers Lectures through Wednesday, Sept. 23
Including:
Functional GroupsX-Ray Diffraction
1-Dimensional Quantum Mechanics & 1-Electron Atoms
(Sections I-V of quantum webpage& Erwin Meets Goldilocks )
Get-aquainted with Erwin at Thursday Discussion
IMPORTANT PROBLEMStherein due Monday, Sept. 21
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Dunitz et al. (1981)
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Dunitz et al. (1981)
Surprising only for its beauty
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Lone "Pair"of N atom
Dunitz et al. (1981)
Bond Cross SectionsMissing Bond?
H
H
H
H
HH
Pathological Bonding
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Dunitz et al. (1981)
MissingBond !
BentBonds !
In three weeks we’ll understand these pathologies.
Pathological Bonding
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Lewis Pairs/Octets provide a pretty good bookkeeping device
for keeping track of valencebut 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).
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Is there a Better Bond Theory, maybe even a Quantitative one?
YES!Chemical Quantum
Mechanics
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Erwin Schrödinger (Zurich,1925)Schrödinger Wave Equation
(1926)
http://www.zbp.univie.ac.at/schrodinger/lebensbilder/bilder9.htm
Age 38
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www.uni-leipzig.de/ ~gasse/gesch1.html
"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."
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?
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Well, I have found one."
"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:
H = E
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- Stockholm
PaulDirac
WernerHeisenberg
ErwinSchrödinger
AIP Emilio Segre Visual Archives, Peierls Collection
December 1933
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Schrödinger Equation
H = E ???
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Leipzig (1931)
AIP Emilio Segre Visual Archives, Peierls Collection
WernerHeisenberg
FelixBloch
1952 (NMR)
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Felix Bloch & Erich Hückel on 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.
(1926)
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“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.
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.
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?”
Schrödinger Lecture“What is Matter”
(1952)
by permission from Supposé CDErwin Schrödinger Was ist Materie?
Even Schrödinger was never comfortable with what really means:
Schrödinger’s GraveAlpbach, Austria
Wik
iped
ia
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First we’ll learn howto findand use it.
Later we learn what it means.
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of What?
Named by "quantum numbers"(e.g. n,l,m ; 1s ; 3dxy ;
Function of Particle Position(s)[and sometimes of time and "spin"]
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
N particles 3N arguments![sometimes as many as 4N+1]
?
Function of
a Function
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Schrödinger Equation
H = E
(for “stationary” states)time-independent
( E times )(NOT H times )
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=
H = E
Kinetic Energy + Potential Energy = Total Energy
Given - Nothing to do with (Couloumb is just fine)
Hold your breath!
H = E
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Kinetic Energy?
Sum of classicalkinetic energy
over all particles of interest.
(adjusts for desired units)
mi vi2
i
Const 12 Fine for our great grandparents
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Kinetic Energy!2
xi2
2
yi2
2
zi2
+ +1mi
i
h2
82
d2
dx21
mC
C
Curvature of
m
One particle, One dimension:
Note:
Involves the shape of , not just its value.
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Solving a Quantum Problem
Given : a set of particlestheir masses & their potential energy law
[ e.g. 1 Particle/1 Dimension : 1 amu & Hooke's Law ]
Ta 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)
Given :
To Find :
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What's Coming?1 Particle, 1 Dimension
Molecules & Bonds
Functional Groups & Reactivity
1-Electron Atoms (3 Dimensions)
Many Electrons & OrbitalsSept 25 Exam
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The Jeopardy ApproachAnswer
()
Problemmass and
Potential Energy(x)
= sin (x)
= sin (ax)
= ex
Kinetic Energy
= e-x
C/mIndependent of x
Const PE(particle in free space)
a2 C/mhigher kinetic energy’’
-C/m
-C/m
Const PE > TE
”Not just a mathematical curiosity.
Actually happens for all electrons bound to nuclei!
Negative kinetic energy!
C Curvature of
m
- sin (x)
sin (x)Cm
- a2 sin (ax)
sin (ax)Cm
( a > 1 shortened wave)
ex
ex
Cm
NOT your great grandparent’s 1/2 mv2.
(at large distance, where 1/r ceases changing much)
1 / 2
a = 3
E
0
V
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+
0
_
(x
)
x
Potential E
nergy
Total Energy• •
Curvingtoward = 0 Positive
Curving awayfrom = 0 Negative
Potential Energy from Arbitrary Shape
The potential energy function for this
must be a double minimum.
PositiveZero
Negative?
CurvatureAmplitude
via Kinetic Energy
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From “Jeopardy” Approachto Recipe for Solution of Schrödinger Equation
Using Guessed Total Energies
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Rearranging Schrödinger to give a formula for curve tracing.
C
Curvature of
m
+ V = E
CCurvature 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.
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100 kcal/mole
2.5Å0
Nodes and Quantizationin One Dimension
from Erwin Meets Goldilocks(for Wiki see Monday Problem Set)
Too Cold
Too Hot
Just Right!
20.74 kcal/moleGuess 21 kcal/mole
Guess 20 kcal/mole
DangerNegativeKineticEnergy
(Curve Away from Baseline)
DangerNegativeKineticEnergy
(Curve Away from Baseline)
Erwin Meets Goldilocks
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End of Lecture 7Sept 16, 2009
Copyright © J. M. McBride 2009. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).
Use of this content constitutes your acceptance of the noted license and the terms and conditions of use.
Materials from Wikimedia Commons are denoted by the symbol .
Third party materials may be subject to additional intellectual property notices, information, or restrictions.
The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0
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100 kcal/mole
2.5Å0
20.74 kcal/mole
Erwin Meets Goldilocks
Could there be a lower-energy Psi?
4.15 kcal/mole
12.45 kcal/mole
Could there be an energy between?
NODES0 because of sign change
More Energy
More Curvature
More Nodes