1. Chem 125 Lecture 15 10/5/2005 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 gives Orbitals their Shape? Potential Energy Kinetic Energy 4d 2s

3. e-density contours of H 2 Single “United Atom” distorted by a fragmented nucleus Which contour should we use? The Plum-Pudding View of Molecular Orbitals Shows Generality of Kinetic-Energy-Based Clouds Atom-Pair Bonding But One Must Probe Harder to Gain a Qualitative Understanding of Chemical Bonds

4. Reality: Structure (Nuclear Arrangement ) Stability/Reactivity (Energy) Total Electron Density LCAO Atom-Pair Localized Bond Orbitals (cf. H 2 When necessary, mix Localized Bond Orbitals to give MOs) MO Plum Pudding (MO-to-Atom analogy ; useful for one-electron phenomena) NOT a "sphere of uniform density” à la J.J. Thomson Appearance depends on chosen contour. Models help Understand Electrons: Orbitals Simplify        x i,y i,z i ) (  2 for total e-density) i=1 n i i

5. Molecules from Atoms: LCAO MO 1 √2 ( AO a + AO b )  (x 1,y 1,z 1 ) = SUM of AOs (like “hybridization” but with two atoms) Why is this sensible?

6. H 2 at Great Distance 1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = H 2 at Bonding Distance

7. Overlap Creates Bonding If we approximate a molecular orbital as a sum of atomic orbitals: and square to find electron density: then subtract the average of the atom electron densities: we find bonding, the difference electron density due to overlap: < (normalization)

8. AA 2 BB 2 Where is  A   B significant?Where is  A significant? no yesa littleno!  b small yes! At the center  A  B is as large as  A 2 ! “Overlap Integral” is   A  B Region of Significant Overlap

9. 92.9% of Total Electronic Energy (almost all of which was already present in the atoms) Great accuracy required to calculate correct value of bond energy (a difference). (Cf. X-ray difference density) Total e-Density Difference Density 1s (atomic) 52% Bond Energy 0.02 e/a o 3 Coutoured at 0.025 e/a o 3 Coutoured at 0.004 e/a o 3 State-of-the-art 40 years ago

10. Total e-Density Difference Density 1s (atomic) 52% 0.02 1s (optimize exponent) 73% 0.04 Bond Energy

11. Total e-Density Difference Density Hybridized + SCF (96.7% 1s; 0.6% 2s; 2.7% 2p) 76% Bond Energy 0.11 1s (expanded) 73% 0.04 100% 1sHybrid: 96.7% 1s 0.6% 2s 2.7%2p Helps overlap but at the cost of 3% n=2 character

12. Total e-Density Difference Density Hybridized + SCF (96.7% 1s; 0.6% 2s; 2.7% 2p) 76% Bond Energy 0.11 + some correlation 90% 0.11

13. LCAO-MO Looks like atoms (especially near nuclei) (the Main Event for electrons; ~100  bond) <1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = Virtues: Builds up e-density between nuclei (through Overlap - the source of Bonding) Hybridizing AOs provides flexibility (unlimited if you use all H-like AOs in hybrid) Easy to formulate and understand (but keep it simple - valence shell is fairly good)

14. LCAO-MO <1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = <1 2 (AO A 2 + AO B 2 + 2 AO A AO B ) == Atoms Bond (overlap / product) >1>1 >1>1 Anti

15. Overlap & Energy-Match

16. 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

17. 2s C Overlap Scale SCALE: At node of 2s orbital  = 2 2 =  = r * 2Z/na o or r node = na o /Z n for 2s is 2 a o = 0.58Å Z eff for C 2s is 3.2 Diameter of node is 0.7 Å 0.7 Å node diameter Sliding together to 1.4Å (~CC bond distance) superimposes the two 'X's xx

18. 2s x C Overlap Scale 2s x x x x x x x Sliding together to 1.4Å (~CC bond distance) superimposes the two 'X's Overlap Integral = 0.41

19. C Overlap 1.0 0.8 0.6 0.4 0.2 0.0 Overlap Integral 1.21.31.41.5Å s-p  p-p  2s2p  2s2p  + x - + x +  and  are orthogonal 2p   + x - + x +  and  are orthogonal 2p  xx s-s p-p  CCCCCC

20. Curiosity: Over most of this range 2s overlaps with 2p  better than 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 sp 2 -sp 2 sp-sp s 2 p-s 2 p CCCCCC sp 3 -sp 3 sp 2 -sp 2 sp-sp xx

21. 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 sp 2 -sp 2 sp-sp s 2 p-s 2 p CCCCCC 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)

22. Influence of Overlap on “MO” Energy of a Double Minimum Case I: Perfect Energy Match

23. Degenerate Energy Rising Energy Falling Increasing Overlap

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

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

26. 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 by much more (  1/r) (increases by 650 kcal/mole already by 0.3 Å) Unless one uses neutron “glue” (200 million kcal/mole; D 2  He fusion fuels the Sun)

27. Overlap & Energy-Match

28. What if partner is lower in energy than A? A B Electron Energy separate 1/√2 (A+B) 1/√2 (A-B) together < > ? B

29. 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 you 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 orbital, 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)

30. Influence of Overlap on “MO” Energy of a Double Minimum Case II: Poor Energy Match

31. Energy Mismatch Note Energy Mismatch Increasing Overlap Tiny Energy Shifts Mixing non-degenerate AOs

32. What if partner is lower in energy than A? A B Electron Energy separate 1/√2 (A+B) 1/√2 (A-B) together < > ? B A-BA-B A+BA+B larger energy shifts smaller energy shifts

33. B How much smaller is the bonding shift when energy is mismatched? C A Electron Energy separate together Splitting for perfect match mismatch

34. B How much smaller is the bonding shift when energy is mismatched? C A Electron Energy separate together Splitting for perfect match mismatch Splitting with mismatch (shift up for >,< normalization)

35. B How much smaller is the bonding shift when energy is mismatched? C A-CA-C A+CA+C A Electron Energy separate together Splitting with mismatch Splitting not very sensitive to lesser contributor of mismatch / overlap (shift up for >,< normalization)

36. Important Generalizations Mixing two orbitals gives one new orbital lower in energy than either parent and one higher in energy than either parent. The lower-energy combination looks mostly like the lower-energy parent, both in shape and in energy (and vice versa). For a given overlap, increasing energy mismatch decreases the amount of mixing and decreases the magnitude of energy shifts.

37. 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

38. 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 Heterolysis Homolysis