1. After discussion of how increased nuclear charge affects the energies of one-electron atoms and discussion of hybridization, this lecture finally addresses the simple, inconvenient truth that multi-electron systems cannot be properly described in terms of one-electron orbitals. Chemistry 125: Lecture 10 Sept. 22, 2010 Reality and the Orbital Approximation For copyright notice see final page of this file Preliminary

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

3. Dean at Apple World Wide Developers Conference 2003 permission D. Dauger

5. 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) Schrödinger Equation Energy (eV) 1 eV ~ 23 kcal/mole Formula for  Two Questions

6. Radius = 122 mm Shell thickness ~1.6 mm Rubber Basketball density ~0.9 g/ml Lead Sinker density 13.4 g/ml Surface Weighting Which is heavier, Rubber or Lead?  r2 r2

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

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 Cf. 2D

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 (the reason for most constants)  Table for H-like Atoms Note:     Z 3 At the same value of (scaled)  H + : C +6 : K +19 = 1 : 216 : 6859 X-ray finds heavy atoms easily (more electrons and more concentrated) H very difficult to locate

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 for corresponding positions (same 

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 ~ 314 kcal/mole (ionization energy of H) 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. Physicist’s 2p (m=1) with “orbital angular momentum” Information from Atom-in-a-Box Superposition (a kind of hybridization) Chemist’s 2p y complex numbers

19. 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!

20. Change Orientation by Hybridization a 2p y + b 2p z (a weighted sum) 2pz2pz 2py2py 25%50% 75% 50:50 mixture of p z and p y ? Other mixtures of p z and p y ? Orientation 0%

21. 0.00 0.020.040.060.090.110.180.250.330.501.00 Change Shape by Hybridization sp n = a 2s + b 2p x (sp n ) 2 = a 2 2s 2 + b 2 2p x 2 + 2ab 2s × 2p x b2b2 a2a2 n n  (a weighted sum) Maximum extension for sp 1 hybrid (see Web & A-i-B) E Shape What would happen to 2s in an electric field?

22. 1.00 4 2 3 9 24  (Pure 2p) Change Shape by Hybridization sp n = a 2s + b 2p x (sp n ) 2 = a 2 2s 2 + b 2 2p x 2 + 2ab 2s × 2p x b2b2 a2a2 n n  (a weighted sum) E

23. Multiplying and Adding Wave Functions Multiply “pieces” to create 1-electron wave function for atom:   ( , ,  ) = R(r)   (   )   (   ) “ORBITAL” Add orbitals of one atom to create a “hybrid” atomic orbital: 2s + 2p z = hybrid orbital Allows adjusting to new situations (e.g. electric field) while preserving the virtues of real solutions for the nuclear potential.

24.  (.. function of what? ) r1,1,1,r2,2,2r1,1,1,r2,2,2 2-e Wave Function “An Orbital is... a One-Electron Wave Function”  a (r 1,  1,  1 )   b (r 2,  2,  2 ) = ? Multiply 1-e Wave Functions 2 22

25. If so - Orbital Paradise Total e-density (x,y, z) =  1 2 (x, y, z) +  2 2 (x, y, z) + … Total e-Energy =  1 + E 2 + … e.g. Ne (1s) 2 (2s) 2 (2p x ) 2 (2p y ) 2 (2p z ) 2  (3N e variables) =  1 (x 1, y 1, z 1 )   2 (x 2, y 2, z 2 )  … Whole = Sum of Parts

26. Two (or more) Electrons: a Problem in Joint Probability Prob (A and B) = Prob (A)  Prob (B) like tossing two coins for two heads IF the events are independent

27. 2-e Wave Function (r1,1,1,r2,2,2)(r1,1,1,r2,2,2)  a (r 1,  1,  1 )   b (r 2,  2,  2 ) = Multiply 1-e Wave Functions 2 22 No way can electrons be independent! They repel one another. ?

28. 2-e Wave Function (r1,1,1,r2,2,2)(r1,1,1,r2,2,2)  a (r 1,  1,  1 )   b (r 2,  2,  2 ) = Multiply 1-e Wave Functions 2 22 Orbitals can’t work!

29. Exam 1 - Friday, Sept. 24 ! Session 1 10:15-11:15 Room 111 SCL Session 2 10:30-11:30 Room 160 SCL Extra Review/Help Session Wednesday 8:00-10:00 pm Room 119 WLH (McBride)

30. End of Lecture 10 Sept 22, 2010 Good luck on the exam. 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).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