Diatomic Molecules. Atom -> Molecule. Diatomic Molecules - Lukas Schott

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Diatomic Molecules

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Outline

1.

Introduction and motivation

1.

Overview of natural molecules (and sneak preview)

2.

Reasons of interest

2.

A Brief History of Molecules

3.

Born Oppenheim approximation

4.

Examples for electron orbits

1.

Linear Combination of Atomic Orbits (LCAO) with H2+

2.

LCAO of Molecular Orbits with H2

3.

Short introduction to other Approximations

5.

Nuclei vibration and rotation levels

6.

Current research

7.

Summary

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1.1 Overview of natural molecules

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1.2 Reasons of interest

•

LEGO bricks of all molecules e.g. amino acid,

sugar, gasoline (chemistry and biology)

–

Mathematical basics for large molecules

–

Experimental tests of theoretical approximations

–

Further possible bindings

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2 A Brief History of Molecules

•

Etymology: molecule from moles (=mass barrier)

•

~400 B.C.

Platon and Empedocles: Elements: Water, Fire…

•

~350 B.C.

Democritus Leucippus : atoms = small particles with

properties

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•

Etymology: molecule from moles (=mass barrier)

•

~400 B.C.

•

~350 B.C.

properties

•

1625

Descartes: Resurgence of atomic theory

•

1808

Daltons law of Definite and Multiple Proportions

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•

~400 B.C.

•

~350 B.C.

properties

•

1625

•

1808

Daltons law of Definite and Multiple Proportions

•

1811

Avogadro: atoms->molecules-> “particles”

•

1903

Gilbert N. Lewis cube modulation

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•

~400 B.C.

•

~350 B.C.

properties

•

1625

•

1808

Daltons law of Definite and Multiple Proportions

•

1811

Avogadro: atoms->molecules-> “particles”

•

1903

•

1916

Lewis structure -> electron pair bond

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•

~400 B.C.

•

~350 B.C.

properties

•

1625

•

1808

Daltons law of Definite and Multiple Proportions

•

1811

Avogadro: atoms->molecules-> “particles”

•

1903

•

1916

Lewis structure -> electron pair bond

•

1927

Born, Oppenheimer, London, Heitler, Pauling… QM

description

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Pentacene

IBM picture on

molecular

level of nano tube

using AFM (atomic

force microscope

AFM

Atomic

Force

Microscope

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•

Estimations & principle:

–

No spin

–

Not relativistic: Bohr:

–

=>

terms negligible for the electrons

–

Nuclei velocity relative to electron movement very slow:

•

Plan:

Hamiltonian Neglect nuclei velocity “Solve” electron wave equa. Separation & rewrite Born Oppen-heimer Negelct Matrix elements

3 Born Oppenheimer

Approximation

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•

Configuration:

•

Hamiltonian

•

SGE:

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•

Neglect

for now!

(Electronic SGE)

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•

Neglect

for now!

•

Examples later

☺

•

Now pretend to know a solution

:

•

orthonormal & complete set

(Electronic SGE)

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•

Separation:

•

New rewritten SGL

•

Multiply by and integrate over r

Matrix elements: Non-adiabatic

coupling terms (NACT)

Born Oppenheimer Equation

Sum

convention

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Born Oppenheimer Equation

Vibration levels

Rotation levels

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Non adiabatic coupling terms

Both:

Collapse of the BO

approximation

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“The underlying physical laws necessary for the

mathematical theory of a large part of physics

and the whole of chemistry are thus completely

known, and the difficulty is only… finding the

right approximation.”

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4.1 LCAO-Linear Combination of

Atomic Orbits with H2+

To solve:

Step overview:

A. Neglect nuclei movement

B. Estimate a wave function

C. Set the distance of nuclei to parameter R

D. Normalisation -> interaction integral

E. Energy expectation value <H>

F. Minimize Energy

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4.1 LCAO precision

Exact values

LCAO values

E(R) for

the stable

symmetric

and

unstable

asymmetri

c

Overview

of effects

on the

LCAO

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“Nothing is solved”

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4.2 Other approximations for H2+

•

Adjusted wave function

+

λ

z: charge distribution drawn out in z- direction -> polarisation

+radial probability density

η

(R).

η

<1 contraction

1.

LCAO simple

2.

LCAO contraction

η

3.

LCAO contraction & polarisation

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4.3 Approximation for H2 LCAO

•

Hamiltonian

•

Solving Idea?

–

Linear combination of atomical

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4.3 Approximation for H2 LCAO

i)

- for

two 1-s H-atom functions

- for

includes two H- atoms

ii)

- Contains non binding factor

-higher energy

Symmetric position, asymmetric spin

Antisymmetric position, symmetric spin

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4.4 Heitler London Approximation

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5.Vibration and rotation levels

•

Rotational symmetry

•

We know the solution

•

Born Oppenheimer for nuclei & two body

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•

This reduces the equation to

•

New: For small l,

has a minima

depending on l at r

l

.

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•

Solving this leads us to

:

–

Effective electrical energy

Visible light

1

–

Vibration energy

infrared

2

–

Rotation energy

far infrared &microwaves

3

1 2 3

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6. Current Research - conceptual

•

MPIK – nuclear physics early Universe H

2

formation

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•

2

formation

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•

2

formation

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•

2

formation

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•

2

formation

Star origin, molecular clouds, fast ion neutral reactions -> molecules

•

Atom-molecule dark state from Bose-Einstein-Condensate

–

BEC (|a>) + Laser beam -> frees Rb atoms to Rb2 molecules

(photoassociation)

–

Second laser beam: certain frequency ->suppression of photoassociation

–

Dark state

~

c

1

|g> - c

2

|a>

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7. Summary

•

Natural diatomic molecules Br I N Cl H O F

•

Born Oppenheimer

•

LCAO (H2+)

•

Heitler London

•

Vibration & rotation levels

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8. References

Literature

•  Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No. 792 (6 April 1929)

•  Schwabl, Franz (2002). Quantenmechanik. 6. Auflage, Berlin: Springer Verlag

•  Born, Max & Oppenheimer, Robert (1927), Analen der Physik – Zur Quantenheorie der Molekeln. Vierte Folge Band 84, Leipzig: Verlag von Joh. Ambrosius Barth

•  Lewis, Gilbert. N., (April 1916), The Atom and the Molecule. Boston. Verlag unklar

•  Demtröder, Wolfgang, (Februar 2005), “Experimentalphysik 3- Atome Moleküle und Festkörper”. Auflage 3, New York: Springer Verlag

•  Lector of Jack Simpson, PhD in Toronto, Canada Area, Lecture to Born and Oppenheimer Approximation

History

•  http://en.wikipedia.org/wiki/History_of_molecular_theory

•  Gilbert N. Lewis Original Work:

http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr216.3-lewispub-19160400-07.html

•  BBC News Penatacine AFM Image http://news.bbc.co.uk/2/hi/8225491.stm

Born Oppenheimer & solutions

•  Technische Universität Braunschweig: http://www.pci.tu-bs.de/aggericke/Lehre/Quantenmechanik/gdanitz/node4.html

•  University of Oxford Department of Physics:

http://www2.physics.ox.ac.uk/sites/default/files/2011-09-16/born_oppenheimer_pdf_31916.pdf

•  New York University: http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_10/node1.html

•  Universität Jena: http://www.theochem.uni-jena.de/teaching/WS1112/vf/VL06.pdf

•  Paul Dirac Quote: http://en.wikiquote.org/wiki/Paul_Dirac

Current research

•  MPIK HEIDELBERG & http://www.hkreckel.de/molecular_hydrogen.html

•  Ultra Cold Gases and Molekular Superposition University of Innsbruck:

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•

Justification of neglecting the first NACT term

insignifican

t

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•

Justification for the second NACT term

•

Estimation with oscillation wave function:

•

Let

be a typical nucleus amplitude

terms of size

insignifica

nt

Hamiltonian Neglect nuclei

velocity “Solve” electron wave equa. Separation & rewrite Born Oppen-heimer Negelct Matrix elements