For copyright notice see final page of this file Chemistry 125: Lecture 7 Sept. 16, 2009 Quantum Mechanical Kinetic Energy 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. For copyright notice see final page of this file

Exam 1 - Friday, Sept. 25 ! Covers Lectures through Wednesday, Sept. 23 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. 21 Get-aquainted with Erwin at Thursday Discussion Exam Review 8-10 pm Wednesday, Sept. 23, Room TBA

Dunitz et al. (1981)

Surprising only for its beauty Dunitz et al. (1981)

Pathological Bonding Missing Bond? Dunitz et al. (1981) Lone "Pair" of N atom H Missing Bond? Bond Cross Sections Dunitz et al. (1981)

In three weeks we’ll understand these pathologies. Pathological Bonding Bent Bonds ! Missing Bond ! In three weeks we’ll understand these pathologies. Dunitz et al. (1981)

but they are hopelessly crude when it comes to describing 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

Schrödinger Wave Equation (1926) Erwin Schrödinger (Zurich,1925) http://www.zbp.univie.ac.at/schrodinger/lebensbilder/bilder9.htm Age 38 Schrödinger as “97 pound weakling” the year before Debye, in his ETH seminar, suggested that he work on a wave equation rather than the “unimportant” stuff he was doing at Univ. of Zurich.

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." H y = E y

December 1933 - Stockholm Erwin Schrödinger Paul Werner Dirac L-R: Heisenberg's mother, Schrodinger's wife Anny, Dirac's mother, Dirac, Werner Heisenberg, Erwin Schrodinger; Stockholm train station Schroedinger 1887-1961 Dirac 1902-1984 (5th youngest) Heisenberg 1901-1976 age 31 at prize (2nd youngest) Paul Dirac Werner Heisenberg AIP Emilio Segre Visual Archives, Peierls Collection

Schrödinger Equation H y = E y Wave Function ???

Leipzig (1931) Felix Bloch 1952 (NMR) Werner Heisenberg Bloch b. 1905, Zurich. 21 in 1926 undergraduate at ETH, Ph.D. with Heisenberg at Leipzig George Placzek, b. 1905 Brno Guido Gentile Italian Rudolph Peierls b. 1907 Berlin G. Wick Victor Weisskopf b. 1908 Vienna Fritz Sauter b. 1906 Innsbruck? Werner Heisenberg AIP Emilio Segre Visual Archives, Peierls Collection

Felix Bloch & Erich Hückel on  (1926) 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 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. 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?” 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. Schrödinger’s Grave Alpbach, Austria Schrödinger Lecture “What is Matter” (1952) Wikipedia by permission from Supposé CD Erwin Schrödinger Was ist Materie?

First we’ll learn how to findand use it. Later we learn what it means.

? Y a Function of What? Named by "quantum numbers" (e.g. n,l,m ; 1s ; 3dxy ; s p p*) Function of 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

H y = E y Schrödinger Equation  time-independent (for “stationary” states) Schrödinger Equation H y = E y  (NOT H times y ) ( E times y )

y H y = E y H = E = Kinetic Energy + Potential Energy = Total Energy Given - Nothing to do with y (Couloumb is just fine) Hold your breath!

 Kinetic Energy? mi vi2 i Const  1 2 Fine for our great grandparents (adjusts for desired units) Sum of classical kinetic energy over all particles of interest.

 y y y y Kinetic Energy! C C 2 xi2 yi2 zi2 + d2 dx2 i h2 8p2 1 mi Note: Involves the shape of y, not just its value. One particle, One dimension: d2 dx2 y 1 m C C y Curvature of m

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 ] Given : To Find : TY a Function of the position(s) of the particle(s) Such that HY/Y is the same (E) everywhere AND Y remains finite!!! (single-valued, continuous, Y 2 integrable)

What's Coming? 1 Particle, 1 Dimension 1-Electron Atoms (3 Dimensions) Sept 25 Exam Many Electrons & Orbitals Molecules & Bonds Functional Groups & Reactivity

The Jeopardy Approach -C/m -C/m ’’ ” Answer Problem  = sin (x) C/m Kinetic Energy Problem mass and Potential Energy(x) y C Curvature of m Independent of x  Const PE (particle in free space)  = sin (x) C/m sin (x) C m - sin (x)  = sin (ax) a2 C/m ’’ higher kinetic energy ( a > 1  shortened wave) sin (ax) C m - a2 sin (ax)  1 / 2 -C/m  = ex Const PE > TE a = 3 Negative kinetic energy! -C/m  = e-x ex C m ex ” Not just a mathematical curiosity. Actually happens for all electrons bound to nuclei! V NOT your great grandparent’s 1/2 mv2. E (at large distance, where 1/r ceases changing much)

Potential Energy from Arbitrary  Shape via Kinetic Energy + _ (x) x Curving toward = 0  Positive Positive Zero Negative ? Curvature Amplitude Potential Energy Curving away from = 0  Negative The potential energy function for this  must be a double minimum. • • Total Energy

From “Jeopardy” Approach to Recipe for Solution of Schrödinger Equation Using Guessed Total Energies

Rearranging Schrödinger to give a formula for curve tracing. y Curvature of m + V = E C y Curvature 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.

Danger Negative Kinetic Energy (Curve Away from Baseline) 100 kcal/mole 2.5Å Too Cold Nodes and Quantization in One Dimension from Erwin Meets Goldilocks (for Wiki see Monday Problem Set) Erwin Meets Goldilocks Just Right! Guess 21 kcal/mole 20.74 kcal/mole Guess 20 kcal/mole Too Hot

End of Lecture 7 Sept 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

Could there be a lower-energy Psi? 100 kcal/mole 2.5Å 4.15 kcal/mole 12.45 kcal/mole Erwin Meets Goldilocks NODES 0 because of sign change Could there be an energy between? Could there be a lower-energy Psi? More Energy  More Curvature More Nodes 20.74 kcal/mole