Chemistry 125: Lecture 3 Double Minima, Earnshaw’s Theorem, and Plum Puddings The double-well potential of the ozone molecule and its structural equilibrium continues the discussion of Lewis structures and chemical forces. The inability of inverse-square force laws to account for stable arrangements of charged particles is established by Earnshaw's Theorm, which may be visualized by means of lines of force. J.J. Thomson circumvented Earnshaw’s prohibition on Coulombic structure by postulating a “plum-pudding” atom, with negative corpuscles embedded in a sphere of diffuse positive charge. When Rutherford showed that the positive nucleus was not diffuse, Thomson and G.N. Lewis concluded incorrectly that Coulomb’s law was invalid at small distances. Synchronize when the speaker finishes saying “…from the slides at the end last time…” Synchrony can be adjusted by using the pause(||) and run(>) controls. For copyright notice see final page of this file

Equilibrium vs. Resonance H C O O H HC O O H H C O O HC O O Two Species Two Species? H C O O HC O O One Nuclear Geometry! One Species! (Evidence: Infrared Spectroscopy) LORE (Evidence: Electron Paramagnetic Resonance) LORE: That which is learned; learning, scholarship, erudition. Also, in recent use, applied to the body of traditional facts, anecdotes, or beliefs relating to some particular subject (Oxford English Dictionary)

2. Structures in which all first-row atoms have filled octets are generally important; however, resulting formal charges and electronegativity differences can make appropriate nonoctet structures comparably important. From a good Text “empirical rules for assessing the relative importance of the resonance structures of molecules and ions. 1. Resonance structures involve no change in the positions of nuclei; only electron distribution is involved. 3. The more important structures are those involving a minimum of charge separation, particularly among atoms of comparable electronegativity. Structures with negative charges assigned to electronegative atoms may also be important.” (our depiction of) ^ LORE

From Number of Valence Electrons we would like to predict: Constitution (valence numbers for different atoms) Reactivity Charge Distribution  

O 2 O 3 O O O Equilateral Triangle O O O O O O O O OO O O O + Double Bond Open Trivalent O is positive.

What is Ozone’s Structure? O O O + _ OO O Ring O O O + _ Open A Problem in 4 Dimensions! (3 distances + energy) symmetrical single minimum?

Graph Help Be sure you can do the problems, but you don't have to hand them in. (Click for an answer key)Click USGS

Ivanic, Atchity, Ruedenberg 1997  Ring Open  4-Dimensional Structure- Energy Plot 3 / Constrained by assuming symmetry R R 12 = R 23 Requires e.g. R 12, R 23,  Energy Energy Contours “Steepest-Descent” Path Pass Between Valleys

O3O3 More Constrained 4-Dimensional Structure- Energy Plot Distance along Steepest-Descent Curve Energy (kcal/mol) Ring Open / R 12 ≠ R 23 gives higher E  symmetrical "resonant” structure Pass

Ozone What of charge distribution is “predicted” by Lewis bookkeeping? + in middle - on ends? O O O + _ O O O + _ Open symmetrical single minimum?

*) Energy of a proton on the “molecular surface” Suface Potential* of Open Ozone (from Quantum-Mechanics) HIGH (+ 25 kcal/mole) (-16 kcal/mole) LOW + in middle - on ends? YES!

From Number of Valence Electrons we would like to predict: Constitution (valence numbers for different atoms) Structure (distances & angles ) (we’ll test this later) Energy Content (we’ll test this later) Reactivity (at least for H 3 N: BH 3 ) Charge Distribution (at least qualitatively for O 3, H 3 N-BH 3 )    ~ ~

Lewis Dot Structure Attempts to provide a “physical” basis for valence rules. New: Reactivity from unshared pairs (both “hooks” from the same atom) Convenient for electron bookkeeping (molecular charge; “formal” atomic charges; qualitatively realistic, at least in the case of O 3 ) Stability and “Resonance”?

What’s so great about octets? How bad are sestets? How bad are structures with formal charge separation? How bad is “bad” charge separation? from Wiki : “I have a question when drawing these structures. Is it more ‘important’ to try to fill the octet or to have lowest formal charge on as many atoms, especially C, as possible? and WHY?”

Is it at all True? Force Laws? Are there e-pairs between nuclei and unshared on some atoms?

In systems governed by inverse-square force laws there can be no local minimum (or maximum) of potential energy. Earnshaw's Theorem (1839) by permission Sheffield University Samuel Earnshaw ( )

Visualizing Earnshaw - Coulomb's Electrostatics “Lines of Force”MagneticElectrostatic

Faraday/Davy/Phillips young Michael Faraday by permission Alfred Bader Collection

Can show magnitude (as well as direction) of Force 2-D (Flatland) force magnitude  line density Circumference  r 2 Force  line density  1/r

Can show magnitude (as well as direction) of Force 3-Dimensions Surface  r 2 Force  line density  1/r 2 force magnitude  line density In 3D such Diagrams Work only for Inverse Square Forces!

A positive particle has a local maximum or minimum of energy only at the location of another charged particle, never in free space. A positive particle has a local maximum or minimum of energy (peak or valley) only at the location of another charged particle, never in free space.

In systems governed by inverse-square force laws there can be no local minimum (or maximum) of potential energy in free space. Earnshaw's Theorem (The only “stationary” points are saddle points.)

Levitator by Martin Simon (UCLA) Eppur sta fermo “and yet it stands still”

J.J. Thomson ( ) Electron (1897) Plum-Pudding Atom © Cavendish Laboratpry, Cambridge University

"[We can] solve the special case where the corpuscles are confined to a plane." Thomson's Model of Electron Configuration "consider the problem as to how 1…2…3…n corpuscles would arrange themselves if placed in a sphere filled with positive electricity of uniform density…" “distributed in the way most amenable to mathematical calculation” in Thomson, Corpuscular Theory of Matter (1907)

    Vortex Lattice Models (Greg Blonder  

"[We can] solve the special case where the corpuscles are confined to a plane." Thomson's Model of Electron Configuration "consider the problem as to how 1…2…3…n corpuscles would arrange themselves if placed in a sphere filled with positive electricity of uniform density…" "the equilibrium of eight corpuscles at the corners of a cube is unstable." in

“I have ever since regarded [the cubic octet] as representing essentially the arrangement of electrons in the atom” G. N. Lewis (1923) Was Lewis ignorant of Earnshaw's Theorem? “Electric forces between particles which are very close together do not obey the simple law of inverse squares which holds at greater distances.” G. N. Lewis (1916)

The Electron in Chemistry J. J. Thomson (1923) “… if [electron-nuclear attraction] were to vary strictly as the inverse square of the distance we know by Earnshaw's theorem than no stable configuration in which the electrons are at rest or oscillating about positions of equilibrium is possible... Couloumb r c r c … then a number of electrons can be in equilibrium about a positive charge without necessarily describing orbits around it.” I shall assume that the law of force between a positive charge and an electron is expressed by the equation F =F = Ee r2r2 1 atomic length scale (for distances r smaller than c, the force changes sign.)

End of Lecture 3 Sept 8, 2008 Copyright © J. M. McBride 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).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