Presentation on theme: "Chem 125 Lecture 7 9/17/08 Preliminary This material is for the exclusive use of Chem 125 students at Yale and may not be copied or distributed further."— Presentation transcript:
Chem 125 Lecture 7 9/17/08 Preliminary This material is for the exclusive use of Chem 125 students at Yale and may not be copied or distributed further. It is not readily understood without reference to notes from the lecture.
Exam 1 - Friday, Sept. 26 ! Covers Lectures through Wednesday, Sept. 24 Including: Functional Groups X-Ray Diffraction 1-Dimensional Quantum Mechanics & 1-Electron Atoms (Sections I-V of quantum webpage & Erwin Meets Goldilocks)Erwin Meets Goldilocks IMPORTANT PROBLEMS therein due Monday, Sept. 22 Get-aquainted with Erwin at Thursday Discussion Exam Review 8-10 pm Wednesday, Sept. 24, Room TBA Im working on checking off Wiki contributions and hope to make personalized scorecards (including homeworks) available via Postem at ClassesV2 sometime tomorrow.
Dunitz et al. (1981) Pathological Bonding Å ! for average positions Typically vibrating by ±0.050 Å in the crystal
Dunitz et al. (1981) Surprising only for its beauty
Lone "Pair" of N atom Dunitz et al. (1981) Bond Cross Sections Missing Bond? H H H H H H Pathological Bonding
Dunitz et al. (1981) Missing Bond ! Bent Bonds ! In three weeks well understand these pathologies. Pathological Bonding
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
~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 Paul Dirac Werner Heisenberg Erwin Schrödinger AIP Emilio Segre Visual Archives, Peierls Collection December 1933
Schrödinger Equation H = E ???
Leipzig (1931) AIP Emilio Segre Visual Archives, Peierls Collection Werner Heisenberg Felix Bloch 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 Panzas 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? Schr ö dinger Lecture What is Matter (1952) Even Schr ö dinger was never comfortable with what really means:
First well learn how to find and use it. Later we learn what it means.
of What? Named by "quantum numbers" (e.g. n,l,m ; 1s ; 3d xy ; 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 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 classical kinetic energy over all particles of interest. (adjusts for desired units) m i v i 2 i Const 1 2 Fine for our great grandparents
Kinetic Energy! 2 x i 2 2 y i 2 2 z i mimi i h2h2 8 2 d2d2 dx2dx2 1 m C 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 particles their masses & their potential energy law [ e.g. 1 Particle/1 Dimension : 1 amu & Hooke's Law ] To Find : a 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)
The Jeopardy Approach Answer ( ) Problem mass and Potential Energy(x) = sin (x) = sin (ax) = e x Kinetic Energy = e -x C/m Indep. of x Const PE (particle in free space) a 2 C/m higher kinetic energy - 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) C m - a 2 sin (ax) sin (ax) C m ( a > 1 shortened wave) exex exex C m NOT your great grandparents 1/2 mv 2. (at large distance, where 1/r ceases changing much) 1 / 2
+ 0 _ (x) x Potential Energy Total Energy Curving toward = 0 Positive Curving away from = 0 Negative Potential Energy from Shape via Kinetic Energy The potential energy function for this must be a double minimum. Positive Zero Negative ? Curvature Amplitude