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. 23, 2009 Reality and the Orbital Approximation For copyright notice see final page of this file

Exam 1 - Friday, Sept. 25 ! Session 1 10:15-11:15 Room 111 SCL Session 2 10:30-11:30 Room 160 SCL Review/Help Sessions Tonight 8:00-10:00 pm Room 207 WLH (McBride) and Tomorrow Night (Thursday) 7-10 pm Room 211 WLH (Eugene then Peer Tutors)

The angular part of a p orbital Polar Plot of cos(  ) [radius] vs.  [angle]  = 0°  = ±30°  = ±60°  = ±45°  = ±90° + 2()2()

  e -  /2 cos(  ) Find Max:  = 0 d  e -  /2 )/d  -  e -  /2 / 2 + e -  /2  (-  / 2 + 1) e -  /2    Polar 2p Contour Plot    ? ? (max for C +6 = 14 e/Å 3 )

Atom-in-a-Box Shape of H-like  Special thanks to Dean Dauger (physicist/juggler)

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

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) 1 eV ~ 23 kcal/mole Formula for 

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?  r2 r2

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.

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

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

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

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

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

Scaling e-Density with Z Normalization:     d  = 1 (reason for most constants)  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)

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

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

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 E=0 4 n =

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

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

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!

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

Change Shape by Hybridization sp n = a 2  + b 2p x (sp n ) 2 = a 2 2  2 + b 2 2p x 2 + 2ab 2  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?

 (Pure 2p) Change Shape by Hybridization sp n = a 2  + b 2p x (sp n ) 2 = a 2 2  2 + b 2 2p x 2 + 2ab 2  2p x b2b2 a2a2 n n  (a weighted sum) E

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.

 (.. 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

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

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

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.

End of Lecture 10 Sept 23, 2009 Good luck on the exam. Copyright © J. M. McBride 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