1. Chem 125 Lecture 13 10/3/08 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. Overlap & Energy-Match

3. Consider how the Overlap Integral (the “sum” of A x B over all space) depends on the Distance between two Carbon Atoms and on Hybridization of their Atomic Orbitals

4. 2s C Overlap Scale Diameter of node for 2s C is 0.7 Å Sliding together to 1.4Å (~CC bond distance) superimposes the two 'X's xx

5. 2s x C Overlap Scale 2s x x x x x x x Sliding together to 1.4Å superimposes the two 'X's Overlap Integral = 0.41 ! Guess the overlap integral,  A  B (remember that  A  A = 1)

6. C Overlap 1.0 0.8 0.6 0.4 0.2 0.0 Overlap Integral 1.21.31.41.5Å s-p  2s2p  2s2p    + x - + x + 2p  xx s-s p-p  CCCCCC  and  are “orthogonal” (net overlap = 0) to -1 at D = 0 to 0 at D = 0 to 1 at D = 0 p-p   (sigma) is Greek “s” MO analogue of s AO. (no node through nuclei)  (pi) is Greek “p” MO analogue of p AO. (nodal plane through nuclei)

7. Curiosity: Over most of this range 2s overlaps with 2p  better than either 2s with 2s or 2p  with 2p  1.0 0.8 0.6 0.4 0.2 0.0 Overlap Integral 1.21.31.41.5Å s-p  p-p  s-s p-p  sp 3 -sp 3 s 2 p-s 2 p CCCCCC sp 3 -sp 3 sp 2 -sp 2 sp-sp xx sp 2 -sp 2 sp-sp

8. 1.0 0.8 0.6 0.4 0.2 0.0 Overlap Integral 1.21.31.41.5Å s-p  p-p  s-s p-p  sp 3 -sp 3 s 2 p-s 2 p CCCCCC sp 2 -sp 2 sp-sp Hybrids overlap about twice as much as pure atomic orbitals. sp gives best overlap, but only allows two orbitals (50% s in each) sp 3 gives four orbitals with nearly as much overlap (25% s in each) (because they allow nearly full measure of s with p overlap plus s with s, and p with p.)

9. Influence of Overlap on “MO” Energy of a One-Dimensional Double Minimum Case I: Perfect Energy Match

10. Degenerate Energy Rising Energy Falling Increasing Overlap No Significant Energy Difference Creates Splitting

11. Overlap Holds Atoms Together A B Electron Energy separate 1/√2 (A+B) 1/√2 (A-B) together < > with greater overlap

12. Electron Count and Bond Strength A B Electron Energy separate together # Effect 1Bonding 2Strongly Bonding 3Weakly Bonding 4Antibonding

13. Why Doesn’t Increasing Overlap Make Molecular Plum Puddings Collapse? H2H2 He ? Electrons do become 55% more stable (~650 kcal/mole) But proton-proton repulsion increases much more dramatically (  1/r) (already increases by 650 kcal/mole from H-H to 0.3 Å) Unless one uses neutron “glue” D 2  He fusion fuels the Sun (200 million kcal/mole)

14. Finally we understand the atom-atom …. force law! … …. Bonding Potential Electron pair becomes more stable with increasing overlap. Nuclear repulsion becomes dominant All from Coulomb’s Law and Schr ö dinger Kinetic Energy of Electrons (This curve provides the potential for studying molecular vibration.) Atom-Atom Distance Energy

15. Newton Opticks (1717) Query 31 There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the business of experimental Philosophy to find them out. shop.rpg.net

16. Overlap & Energy-Match

17. What if partner is lower in energy than A? A B Electron Energy separate 1/√2 (A+B) 1/√2 (A-B) together < > “Splitting”  Overlap ? B * *) approximately

18. Why use any of an“Inferior” Orbital? The 1s “core” AOs did indeed remain pure and unmixed during creation of molecular orbitals for CH 3 CHFOH :

19.  1  1s (F) Core 1

20.  2  1s(O) Core 2

21.  3  1s(C 1 ) Core 3

22.  4  1s(C 2 ) Core 4

23. Why use any of an“Inferior” Orbital? but the valence-level AOs were heavily mixed. The compact 1s “core” AOs did indeed remain pure and unmixed during creation of molecular orbitals for CH 3 CHFOH,

24.  5 “1s(valence)” 2s of F 2sp hybrid of O 2s of C

25. (aA + bB) 2 = a 2 A 2 + b 2 B 2 + 2abAB Why use any of an“Inferior” Orbital? Suppose the energy of the A orbital is much higher (less favorable) than that of the B orbital. Can one profit from shifting electron density toward the internuclear AB region (from the “outside” region) without paying too much of the high-energy“cost” of A? Yes, because for a small amount ( a ) of A in the MO, the amount of A 2 probability density ( a 2 ) is REALLY small, while the amount of AB shifting (2ab) is much larger. e.g. a = 0.03, b = 0.98 means a 2 = 0.001, b 2 = 0.96, 2ab = 0.06 (Incidentally, this is normalized, since the integral of AB is ~0.6, and 0.6 x 0.06 is ~0.04 = 1 - 0.96)

26. Influence of Overlap on “MO” Energy of a One-Dimensional Double Minimum Case II: Poor Energy Match

27. Degenerate Energy Rising Energy Falling Increasing Overlap Splitting due only to Original Well Offset Fights Well Difference Note Small Energy Mismatch still Mixing non-degenerate AOs Negligible Mixing Still Biased

28. What if partner is lower in energy than A? What are the ultimate energies? A B Electron Energy separate 1/√2 (A+B) 1/√2 (A-B) together < > ? C A-CA-C A+CA+C larger energy shifts smaller energy shifts looks mostly like C in shape & energy looks mostly like A

29. B A given overlap yields this splitting for perfect E-match How much smaller is the bonding shift when energy is mismatched? C A Electron Energy separate together Average of A and C Energy- mismatch

30. B How much smaller is the bonding shift when energy is mismatched? C A Electron Energy separate together With E-mismatch larger splitting for same overlap A given overlap yields this splitting for perfect E-match Energy- mismatch

31. B How much smaller is the bonding shift when energy is mismatched? C A-CA-C A+CA+C A Electron Energy separate together (shift up a bit for >,< normalization) Splitting is less sensitive to lesser contributor of mismatch / overlap For a given overlap, bonding shift is reduced by energy mismatch. (Still A+ C ends lower than A+B, because C starts lower.) e.g. when mismatch is relatively large, a given amount of overlap doesn’t make much difference

32. Important Generalizations Mixing two overlapping orbitals gives one composite orbital that is lower in energy than either parent and one that is higher in energy than either parent. The lower-energy combination looks more like the lower-energy parent, both in shape and in energy (ditto for higher-). For a given overlap, increasing energy mismatch decreases the amount of mixing and decreases the magnitude of energy shifts.

33. Which Bond is Stronger A-B or A-C? A B Electron Energy separate C Compared to What? A-B stronger if forming Ions (A + B - ) together A-C electrons clearly lower in energy, but…

34. Which Bond is Stronger A-B or A-C? A B Electron Energy separate C Compared to What? A-B stronger if forming Ions (A + B - ) A-C stronger if forming Atoms (A C) together mismatch aids Heterolysis mismatch hinders Homolysis

35. Experimental Evidence Is All This True?

36. H-H vs. H-F  ** Homolysis to A B kcal/mole 136104 HF Bond is Stronger Heterolysis to A + B - kcal/mole (gas phase) 400373 HF Bond is Weaker Big on F Big on H "Hydrofluoric Acid " antibonding molecular orbital : empty (match)(mismatch)

37. End of Lecture 13 Oct. 3, 2008