Chem 125 Lecture 10 9/24/08 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.

Exam 1 - Friday, Sept. 26 ! 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 116 WLH Tomorrow Night (Thursday) 7-10 pm Rooms 113 and 116 WLH

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

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.

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 ≈ 300 kcal/mole 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 24, 2008 Good luck on the exam.