1. MO Theory
H2+ and H2 solns

2. Solutions to Hydrogen Molecule Ion
Y2, E2 = eV (for H2 )
Y1, E1 = 1.37 eV (for H2)

3. Solutions to Hydrogen Molecule
MOs created from combinations of p-orbitals
pxA + pxB, pyA + pyB
pxA - pxB, pyA - pyB
pzA - pzB
pzA + pzB

4. Solutions to Hydrogen Molecule
px+ px OR py + py
px - px OR py - py
pz - pz
pz + pz

5. Gerade = symmetric with inversion
Parity
represents center of inversion
Gerade = symmetric with inversion
inversion
Ungerade = antisymmetric with inversion
inversion

6. MO Energy Level Diagram for Homonuclear Diatomics
lone atom s*
lone atom p* 2p 2p p s s* 2s 2s s s* 1s 1s s

7. Molecular Term Symbols
ML = S (over all e-) l
l identifies “z-component” of angular momentum of an e-
Symbols used to id l
| l | 1 2 3 4 s p d f g

8. Molecular Term Symbols
Angular momentum about “z-axis” for all electrons is
L = |ML|
Symbol used to id L L 1 2 3 4 S P D F G

9. Molecular Term Symbols
Symbol is 2S + 1 L g/u
2S + 1 is multiplicity as already used for atomic term symbols
g or u identifies overall parity
To determine overall parity, make use of multiplication of symmetric and antisymmetric functions
If the term is a S term, a right superscript of + or – is added to indicate whether the wavefunction is symmetric or antisymmetric with respect to reflection through a plane containing the two nuclei

10. Molecular Term Symbols
Remember sigma orbs:
From s orbs
From pz orbs
Remember pi orbs:
From px orbs
From py orbs

11. Molecular Term Symbols
Remember sigma-star orbs:
From s orbitals
From p orbitals
Remember pi-star orbs:
From px orbitals
From py orbitals

12. Spectroscopy – Selection Rules
DL = 0, +1, -1
DS = 0
note S = Ms
note W refers to spin-orbit coupling and
W = L + S
DW = 0, +1, -1

13. Molecular Term Symbols
Molecular Orbitals not always so “clear-cut”
Remember how orbitals change energy as go across PT
Can affect MO energy pattern too

14. MO Energy Level Diagram for Homonuclear Diatomics
Essentially LCAOs involving four orbitals are made. The sigma orbitals that we thought of as being made by the 2s orbitals are lowered in E while the sigma orbitals that we thought of as being made by the 2pz orbitals are raised in E.
As you move to the right on PT, 2s and 2p energy gap increases. Early, in the period, then, this permits mixing of 2s and 2pz orbitals.
Atkins, Fig 14.30

15. MO Energy Level Diagram for Homonuclear Diatomics (N2 and “before”)
lone atom
lone atom p* s 2p 2p
Use this diagram for N2 and earlier in PT p s* 2s 2s s s* 1s 1s s

16. Taking a look at heteronuclear diatomic molecules

17. Taking a look at heteronuclear diatomic molecules

18. MOs of HF Unoccupied, E = -0.124 eV Occupied, E = -0.3523 au

19. MOs of HF
H atom
F atom
H – F molecule s 1s p 2p s 2s 1s

20. Computational Chemistry
Considering complexity of the calculations we’ve been doing, certainly, using computers to do these calcs should be useful  Computational Chemistry
For polyatomic molecules can make LCAOs yMO = S ciyi
- Yi constitute basis set (computational forms of atomic orbitals)
Use variation theory to find ci
To find structure of molecule, must move nuclei and find MOs  find structure with lowest overall energy

21. Computational Chemistry
May “solve” for MOs using ab initio or semi-empirical methods
Semi-empirical methods: empirical parameters substituted for some “integrals” to save time in calculations
Ab initio methods: supposedly make no assumptions
NOTE: computational chemistry may determine Energy and some other properties without using quantum chemistry
Such calculations are referred to as molecular mechanics calculations

22. Valence Bond Theory H2 Initial approx is y = y1sA(1) y1sB(2)
But, is this a symm or antisymm wavefxn?
So, make LCs
y+ = y1sA(1) y1sB(2) + y1sB(1) y1sA(2)
y- = y1sA(1) y1sB(2) - y1sB(1) y1sA(2)
In this case, turns out that y+ is lower E

23. ybond = [y1sA(1) y1sB(2) + y1sB(1) y1sA(2)][a(1)b(2) – a(2)b(1)]
Valence Bond Theory
Ground state wavefunction would be
ybond = [y1sA(1) y1sB(2) + y1sB(1) y1sA(2)][a(1)b(2) – a(2)b(1)]
2 electrons in overlapping orbitals – with spins paired

24. Remember CH4
If try to make combinations of the valence s of C with s of H, will be different type of wavefxn, hence diff’t kind of bond than when make combination of a p of C with an s of H
DON’T see any diff in bonding of 4 H’s
Make LCs of valence orbitals on central atom
Call these LCs hybrid orbitals
Use these hybrid orbitals to make sigma bonds with H
Atomic orbitals NOT used to make sigma bonds used to make pi bonds (Huckel method for conjugated)

25. Hybrid Orbitals Valence s and p orbitals on C  hybrids
y1 = a12s + a22px + a32py + a42pz
y2 = b12s + b22px + b32py + b42pz
y3 = c12s + c22px + c32py + c42pz
y4 = d12s + d22px + d32py + d42pz
Consider ethyne
Only two hybrids
y1 = s + pz and y2 = s – pz
Leftover px and py on one C overlap with px and py on other C

26. Simplification to MO Approach
Huckel Approach

27. Symmetry of Molecules

28. Determining Point Groups

29. Special Group?
Yes
C∞v , D∞h , Td , Oh , Ih , Th No Cn No sh
Yes
S2n or S2n and i only, collinear
with highest order Cn Cs
Yes i No No
nC2 perpendicular to Cn No C1
Yes Ci
Yes Sn No sh
Yes sh No
n sv No
n sd
Yes
Cnh
Dnh
Yes
Cnv
Yes No Dn
Dnd
Yes No Cn

30. C2v Character Table
C2v E C2
sv(xz)
sv’(yz) A1 1 A2 -1 B1 B2

31. Now go practice!!!

32. Applying Symmetry to MOs
Water

33. MOs of Water
HOMO-4 a1
Looks like s orbital on O, nbo
E = au

34. MOs of Water HOMO-3 from two viewpoints a1
Looks like s orbital on O with constructive interference with c1 - bo
E = au

35. MOs of Water HOMO-1 HOMO-2 b2 a1
Looks like combination of p on O (perp to C2, but in plane of molecule) with constructive interference with c2, bo
Looks like combination of p on O along C2 with constructive interference with c1, bo (close to nbo)
E = au
E = au

36. MOs of Water HOMO from two viewpoints b1
Looks like p orbital on O, perpendicular to plane of molecule - nbo
E = au

37. MOs of Water LUMO LUMO +1 b2 a1
Looks like combination of p on O (perp to C2, but in plane of molecule) with destructive interference with c2, abo
Looks like combination of p on O along C2 with destructive interference with c1, abo
E = au
E = au

38. Filling Pattern for Water
2b2 (abo)
4a1 (abo)
1b1 (nbo)
3a1 (bo/nbo)
1b2 (bo)
2a1 (bo)
1a1 (nbo)

39. Molecular Spectroscopy
Molecule has a number of motions
Translational, vibrational, rotational, electronic
Sum them to get total energy of molecule
Changes may occur in any of these modes through absorption or emission of energy
Vibrational: IR
Rotational: Microwave
Electronic: UV-Vis
CHP 16, 17, 18 of text

40. Statistical Mechanics
Quantum gives you possible energy levels (states)
In a real sample, not all molecules in the same energy level
With statistics and total energy, can predict (on average) how many molecules in each state
Dynamic Equilibrium
Role of Temperature
Can predict macroscopic properties/behavior
Heat capacity, pressure, etc.
CHP 19, 20 of text