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

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

Dunitz et al. (1981)

Dunitz et al. (1981)

Surprising only for its beauty

Lone "Pair"of N atom

Dunitz et al. (1981)

Bond Cross SectionsMissing Bond?

H

H

H

H

HH

Pathological Bonding

Dunitz et al. (1981)

MissingBond !

BentBonds !

In three weeks we’ll understand these pathologies.

Pathological Bonding

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).

Is there a Better Bond Theory, maybe even a Quantitative one?

YES!Chemical Quantum

Mechanics

Erwin Schrödinger (Zurich,1925)Schrödinger Wave Equation

(1926)

http://www.zbp.univie.ac.at/schrodinger/lebensbilder/bilder9.htm

Age 38

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?

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

- Stockholm

PaulDirac

WernerHeisenberg

ErwinSchrödinger

AIP Emilio Segre Visual Archives, Peierls Collection

December 1933

Schrödinger Equation

H = E ???

Leipzig (1931)

AIP Emilio Segre Visual Archives, Peierls Collection

WernerHeisenberg

FelixBloch

1952 (NMR)

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)

“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

First we’ll learn howto findand use it.

Later we learn what it means.

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

Schrödinger Equation

H = E

(for “stationary” states)time-independent

( E times )(NOT H times )

=

H = E

Kinetic Energy + Potential Energy = Total Energy

Given - Nothing to do with (Couloumb is just fine)

Hold your breath!

H = E

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

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.

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 :

What's Coming?1 Particle, 1 Dimension

Molecules & Bonds

Functional Groups & Reactivity

1-Electron Atoms (3 Dimensions)

Many Electrons & OrbitalsSept 25 Exam

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

+

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

From “Jeopardy” Approachto Recipe for Solution of Schrödinger Equation

Using Guessed Total Energies

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.

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

End of Lecture 7Sept 16, 2009

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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