1. Chem 125 Lecture 9 9/22/08 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.

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

3. Single- vs. Double Minimum For Hooke's Law the Blue Energy is too Low and the Red Energy is too High. The Correct Lowest Energy must lie between these values. Single-Mimimum Actually this is a Double- Minimum. The Blue and Red  s are correct! What if the wells were further apart? Closer wells give lowered minimum energy and higher next energy ~ same as single-minimum solution “ Splitting ”

4. in Ain B Wells far apart Wells far apart Total Energy of Particle "Mixing" localized   s for double minimum Wells close together in AB Antibonding Holds A & B together Black line is energy Blue line is  Bonding! Stabilzation of Particle

5. Dynamics: Tunneling

6. The word "Tunneling" is one of my pet peeves: It is misleading and mischievous because it suggests that there is something weird about the potential energy in a double minimum.. In fact it simply involves the same negative kinetic energy that one sees in the tails of EVERY bounded wavefunction. The word reveals naiveté about quantum mechanics.

7. 1.4 kcal/mole splitting  ~4  10 -14 sec to get from well to well. Well-to-Well time  5  10 -14 sec  Energy (kcal/mole) Assertion from time-dependent q. mech. Dynamics: Tunneling

8. Reward for Finding  Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity Coming soon After Exam, Atoms, Molecules

9. Morse Quantization "Erwin" can find  s for any complicated V(x) 7 Å and rank them by energy / "curvature" / # of nodes Don’t cross 0 in “forbidden” continuum. Don’t slope out and away in “forbidden” continuum. What’s wrong with this picture?

10. Even Multiple Minima

11. This curve-tracing recipe won't work in more dimensions (e.g. 3N). But Schrödinger had no trouble finding solutions for the 3-dimensional H atom, because they were familiar from a long tradition of physicists studying waves. When there are many curvatures, it is not clear how to partition the kinetic energy among the different (d 2  / dx i 2 ) /  contributions to E total.

12. E. F. F. Chladni (1756-1827) Acoustics (1803) e.g. Chladni Figures in 2 Dimensions

13. Sand Collects in Nodes Touch in Different Places Bow in Different Places

14. Click for Short Chladni Movie (3MB) Click for Longer Chladni Movie (9.5MB)

15. Crude Chladni Figures 3 Diameters / 1 Circle3 Circles 1 Diameter / 2 Circles 4 Diameters / 1 Circle from in-class demo

16. Chladni’s Nodal Figures for a Thin Disk Portion inside outer circular node Cf. http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html (1,2)

17. Chladni’s Nodal Figures for a Thin Disk

18. Number of Diametrical Nodes Number of Circular Nodes PITCH 47 Patterns!

19. "These pitch relationships agree approximately with the squares of the following numbers:" Frequency ≈ (Diametrical Nodes + 2  Circular Nodes) 2 Note: Increasing number of ways to get a higher frequency by mixing different numbers of circles and lines 8 Lines 4 Circles 2 Circles 4 Lines 3 Circles 2 Lines 1 Circle 6 Lines Number of Circles Number of Diameters 1 Circle 2 Lines

20. Great Mathematicians Worked on Chladni’s 2-D Problems: e.g. Daniel Bernoulli

21.  s for one-electron atoms involve “Spherical Harmonics” (3D-Analogues of Chladni Figures)

22. 3-Dimensional H-Atom Wavefunctions   ( , ,  ) = R(r)   (   )   (   ) Adrien-Marie Legendre (1752 -1833)  (  ) is the normalized “Associated Legendre Polynomial” Edmond Laguerre (1834-1886) R(r) is the normalized “Associated Laguerre Function” Available from other old-time mathematicians

23.  Table for H-like Atoms V( x,y,z ) = sqrt(x 2 + y 2 + z 2 ) 1 simplifies V( r, ,  ) = r c Name  by quantum numbers (n > l ≥ m) or by nickname (1s, etc.)  = R nl (r)   lm (  )   m (  ) product of simple functions of only one variable each and  (x,y,z) is very complicated change coordinate system: x,y,z  r  x y z n e r  

24.  Table for H-like Atoms  = R(r)   (  )   (  ) 1s  r 2Z2Z na o Why  instead of r? Allows using the same e  2 for any nuclear charge (Z) and any n. = K e -  /2 N.B. No surprise for Coulombic Potential x y z n e r   Note: all contain (Z / a o ) 3/2 Squaring gives a number, Z 3 per unit volume (units of probability density)

25.   r 2Z2Z na o exp -  r =  2Z2Z na o r 1H =  2 0.53Å r 1C =  12 0.53Å All-Purpose Curve shrunk by Z; expanded by n Å (1s H ) (0.26  Å ) 0.51.0 Increasing nuclear charge sucks standard 1s function toward the nucleus 0.1 Å (1s C ) 0.2 (renormalization keeps probability density constant)  1/6  6 (0.044  Å) Å (1s C )0.10.2 (0.044  Å) Different Å scales Common Å scale

26. H 1s C 1s Relative Electron Density +5 0.51.0 Increasing nuclear charge sucks standard 1s function toward the nucleus 0.10.2 (renormalization keeps total probability constant) Common Å scale Summary  r 2Z2Z na o What would the exponential part of……. look like? C2sC2s +5

27. For Wednesday: 1) Why are there no Chladni Figures with an odd number of radial nodes? (e.g. 3 or 5 radii) 2) Why are the first two cells [(0,0) and (1,0)] in Chladni's tables vacant? 3) Compare 1s H with 2s C +5 in Energy 4) Do the 6 atomic orbital problems Click Here Click Here

28. 2 2 2  Table for H-like Atoms 1s = K e -  /2 2s = K'(2-  ) e -  /2 Shape of H-like  = K'''(  cos(  )) e -  /2 2p z z Guess what 2p x and 2p y look like. Simpler (!) than Erwin 1-D Coulombic x y z n e r  

29. The angular part of a p orbital Polar Plot of cos(  ) vs.  0.5- 0.5- 101  = 0° 0.86 0.71 0.5  = ±30°  = ±60°  = ±45°  = ±90° + cos 2 (  ) vs. 

30.   e -  /2 cos(  ) Find Max:  = 0 d  e -  /2 )/d  -  e -  /2 / 2 + e -  /2  (-  / 2 + 1) e -  /2    Polar 2p Contour Plot  

31. Atom-in-a-Box Shape of H-like  Special thanks to Dean Dauger (physicist/juggler) http://dauger.com

32. Dean at Apple World Wide Developers Conference 2003 permission D. Dauger Link to Video of Dauger Juggling at Apple Developer's Conference

34. Information from Atom-in-a-Box r 2   R(r)  2 Probability Density Surface Weighting Where is the density highest? What is the most likely distance? n,l,m (nickname) Schr ö dinger Equation Energy (ev) Formula

35. Which shell (1 or 2) has higher density? 1 2 Which shell contains more stuff (probability) ? 2 has ~ 3  the radius ~9  the volume of 1.

36. Information from Atom-in-a-Box Single Slice 3D2D at different levels near far

37. Information from Atom-in-a-Box Nodes (Shape & Energy) ?3d4d Cf.

38. End of Lecture 9 Sept 23, 2008