1. Chem 125 Lecture 12 10/4/2005 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.

2. What's Coming for Next Exam? Atoms 3-Dimensional Reality (H-like Atoms) Hybridization Orbitals for Many-Electron Atoms Payoff for Organic Chemistry! Molecules Plum-Pudding Molecules ("United Atom" Limit) Understanding Bonds (Pairwise LCAO) "Energy-Match & Overlap" (Wrong!) Recovering from the Orbital Approximation Reactivity HOMOs and LUMOs Recognizing Functional Groups Structure (and Dynamics) of XH 3 Molecules

3. 2 2 2  Table for H-like Atoms r   x y z e n 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

4. 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. 

5. 2p2p

6. Atom-in-a-Box http://www.dauger.com/hobbies/ Shape of H-like  Thanks to Dean Dauger (physicist)

7. Information from Atom-in-a-Box r2r2  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

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

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

10. Scaling H-like  for Changing Nuclear Charge (Z) Size e-Density Energy

11. Scaling Size with Z  r 2Z2Z na o Increasing Z shrinks wave function (makes r smaller for same  ) H + : C +6 : K +19 = 1 : 1/6 : 1/19

12. Scaling Size with Z : 1s H + : C +6 : K +19 = 1 : 1/6 : 1/19

13. Scaling e-Density with Z Normalization:     d  = 1 (why most constants are there)  Table for H-like Atoms Note:     Z 3 H + : C +6 : K +19 = 1 : 216 : 6859 (Helps X-ray find heavier atoms more easily; H very difficult)

14. Scaling Kinetic Energy with Z F(Zr)  Z F'(Zr) '' Z 2 F"(Zr) "" ""   Z 2

15. Scaling Potential Energy with Z Distance Shrinkage  1/Z (thus 1/r  Z ) V at fixed distance  Z Coulomb's Law V  Ze r V  Z 2

16. Scaling Total Energy with Z (and n) E = -RZ 2 n 2 Independent of l, m (e.g. 3s = 3p = 3d) for 1-electron atoms R ≈ 300 kcal/mole As we saw for 1-D Coulomb 1 2 3 5 E=0 4 n =

17. Scaling H-like  for Changing Nuclear Charge (Z) Size e-Density Energy  1/Z  Z 3  Z 2 (n/Z) /n 2

18. Multiplying and Adding Wave Functions Multiply “pieces” to create 1-electron wave function for atom:   ( , ,  ) = R(r)   (   )   (   ) “ORBITAL” Add orbitals of an atom to create a “hybrid” atomic orbital: 2p y + 2p z = hybrid orbital Function of what? Position of one electron!

19. Physicist’s 2p (m=1) with “orbital angular momentum” Information from Atom-in-a-Box Superposition (a kind of hybridization) Chemist’s 2p y

20. End