Chem 125 Lecture 8 9/22/06 Projected material 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. 29 ! Covers Lectures through next Wednesday Including: Functional Groups X-Ray Diffraction 1-Dimensional Quantum Mechanics (Sections I-IV of webpage & Erwin Meets Goldilocks)Erwin Meets Goldilocks IMPORTANT PROBLEMS therein due Wednesday Get-aquainted session this afternoon 4-5:30 in Lecture Room Exam Review 7-9 pm Tuesday, Room WLH 208 Other Help Available Wednesday 8-10 PM, RTBA Thursday 7-10:30 PM, RTBA

HELIX w S S vw S Curious Intensity Sequence B-DNA R. Franklin (1952)

Offset Double Helix repeated pair pattern

BASE STACKING B-DNA R. Franklin (1952) w S S vw S MAJOR & MINOR GROOVES HELIX DIAMETER

X-Ray Diffraction Old-Style Electron Density Map Contours connect points of equivalent density.

K Penicillin 3-D Map (1949) K

1 e/Å 3 contours Rubofusarin No H? High e-Density No : on O! Stout & Jensen "X-Ray Structure Determination (1968) 5 e/Å 3 7 e/Å 3 long short intermediate No : Bonds! Spherical Atoms

“Seeing” Bonds with Difference Density Maps Observed e-Density - Atomic e-Density (experimental) (calculated) sometimes called Deformation Density Maps

Spherical Carbon Atoms Subtracted from Experimental Electron Density (H not subtracted) Triene ~0.2 e ~0.1 e H ~1 e

Triene plane of page cross section partial double bond

Leiserowitz ~0.1 e ~0.3 e ~0.2 e C C C C Why not? Bent bonds from tetrahedral C ?

Lewis Bookkeeping Integrated Difference Density (e) How many electrons are there in a bond? Bond Distance (Å) Berkovitch-Yellin & Leiserowitz (1977)

Bonding Density is about 1/20 th of a “Lewis”

Tetrafluorodicyanobenzene CC C C F N CC C C F N F F Dunitz, Schweitzer, & Seiler (1983) unique

TFDCB C CC C F N is round not clover-leaf nor diamond! C N Triple Bond

TFDCB Where is the C-F Bond? C CC C F N Unshared Pair!

The Second Great Question

Compared to what? What d'you think of him? Exactly! Compared with what, sir? 1) RESONANCE STABILIZATION 2) DIFFERENCE DENSITY

TFDCB Where is the C-F Bond? C CC C F N Unshared Pair! Need to subtract F instead of “unbiased ” spherical F

Dunitz et al. (1981) Pathological Bonding  Å ! for average positions Typically vibrating by ±0.050 Å in the crystal

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

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

Dunitz et al. (1981) Pathological Bonding Missing Bond ! Bent Bonds !

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

Erwin Schrödinger (Zurich,1925)

~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 the he thought this way of talking was rather childish… he had learned that, to deal pro- perly 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 (1933) Paul Dirac Werner Heisenberg Erwin Schrödinger

Schrödinger Equation H  = E 

Leipzig (1931) American Institute of Physics Werner Heisenberg Felix Bloch Victor Weisskopf

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)

 Function of What? Named by "quantum numbers" (e.g. n,l,m ; 1s ; 3d xy ;  Function of Particle Position(s) [and time and "spin"] We focus first on one dimension, then three dimensions (one electron), then many-electron atoms, then many atoms, & finally functional groups. N particles  3N arguments! [sometimes 4N+1]

Schrödinger Equation H  = E  (for “stationary” states) time-independent

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

Kinetic Energy? m i v i 2  i Const 

Kinetic Energy! 22 xi2xi2  22 yi2yi2  22 zi2zi2  ++ 1 mimi  i h2h2 8282  d2d2 dx2dx2  1 m C  One particle;One dimension:

Kinetic Energy! 22 xi2xi2  22 yi2yi2  22 zi2zi2  ++ 1 mimi  i h2h2 8282  d2d2 dx2dx2  1 m C  C  Curvature of  m One particle;One dimension:

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)

Much Harder for Many Particles Is it worth our effort?

Reward for Finding  Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity

The Jeopardy Approach Answer Problem  = sin (x)  = sin (ax)  = e x Kinetic Energy  = e -x C/m particle in free space a 2 C/m shorter wave  higher energy ’’ - C/m Const PE > TE ’’ (actually appears for electrons bound to nuclei at large distance, where 1/r ceases changing much!)

Rearranging Schr ö dinger to give a formula for curve tracing. C  Curvature of  m + V = E C  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.

reward For Finding   kids