MO Theory
H2+ and H2 solns
Solutions to Hydrogen Molecule Ion
2, E2 = -10.16 eV (for H2 )
1, E1 = 1.37 eV (for H2)
Solutions to Hydrogen Molecule
MOs created from combinations of p-orbitals
pzA - pzB
pxA + pxB, pyA + pyB pxA - pxB, pyA - pyB
pzA + pzB
Solutions to Hydrogen Molecule
px+ px OR py + py
px - px OR py - py
pz - pz
pz + pz
Parity
Gerade = symmetric
with inversion
Ungerade = antisymmetric with inversion
represents center of inversion
inversion
inversion
MO Energy Level Diagram for Homonuclear Diatomics
*
*
*
lone atom lone atom
1s 1s
2s 2s
2p 2p
Molecular Term Symbols
• ML = (over all e-)
• identifies “z-component” of angular momentum of an e-
• Symbols used to id
| | 0 1 2 3 4
Molecular Term Symbols
• Angular momentum about “z-axis” for all electrons is LM
= |ML|
Symbol used to id
0 1 2 3 4
Molecular Term Symbols
• Symbol is 2S + 1 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 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
Molecular Term SymbolsRemember sigma orbs:
Remember pi orbs:
From s orbs From pz orbs
From px orbsFrom py orbs
Molecular Term SymbolsRemember sigma-star orbs:
Remember pi-star orbs:
From s orbitals From p orbitals
From px orbitals From py orbitals
Spectroscopy – Selection Rules
= 0, +1, -1
S = 0
note = Ms
= 0
note refers to spin-orbit coupling and
= 0, +1, -1
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
MO Energy Level Diagram for Homonuclear Diatomics
Atkins, Fig 14.30
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.
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.
MO Energy Level Diagram for Homonuclear Diatomics (N2 and “before”)
*
*
*
lone atom lone atom
1s 1s
2s 2s
2p 2pUse this diagram for N2 and earlier in PT
Taking a look at
heteronuclear diatomic molecules
Taking a look at
heteronuclear diatomic molecules
MOs of HF
E = -0.491 au
Unoccupied, E = -0.124 eV
Occupied, E = -0.3523 au
E = -1.086 au
MOs of HF
H atom F atomH – F molecule
1s
1s
2s
2p
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 MO = cii
i 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
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
Valence Bond Theory
• H2
• Initial approx is = 1sA(1) 1sB(2)
– But, is this a symm or antisymm wavefxn?
• So, make LCs– = 1sA(1) 1sB(2) + 1sB(1) 1sA(2)
– = 1sA(1) 1sB(2) - 1sB(1) 1sA(2)
• In this case, turns out that + is lower E
Valence Bond Theory
• Ground state wavefunction would bebond = [1sA(1) 1sB(2) + 1sB(1) 1sA(2)][(1)(2) – (2)(1)]
• 2 electrons in overlapping orbitals – with spins paired
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)
Hybrid Orbitals
• Valence s and p orbitals on C hybrids1 = a12s + a22px + a32py + a42pz
= b12s + b22px + b32py + b42pz
= c12s + c22px + c32py + c42pz
= d12s + d22px + d32py + d42pz
• Consider ethyne– Only two hybrids
1 = s + pz and y2 = s – pz
– Leftover px and py on one C overlap with px and py on other C
Simplification to MO Approach
Huckel Approach
Symmetry of Molecules
Determining Point Groups
Special Group?
No
Cn
YesC∞v , D∞h , Td , Oh , Ih , Th
No
h
Yes
S2n or S2n and i only, collinearwith highest order Cn
YesSn
Cs
Yes
YesCiNo
C1
No
nC2 perpendicular to Cn
Yes h
DnhYesNo
n dNoDn
No
h
YesCnh
Non v
CnvYes
NoCn
DndYes
iNo
C2v Character Table
C2v E C2 v(xz) v’(yz)
A1 1 1 1 1
A2 1 1 -1 -1
B1 1 -1 1 -1
B2 1 -1 -1 1
Now go practice!!!
Applying Symmetry to MOs
Water
MOs of Water
HOMO-4
Looks like s orbital on O, nbo
E = -18.6035 au
a1
MOs of Water
HOMO-3 from two viewpoints
Looks like s orbital on O with constructive interference with 1 - bo E = -0.9127
au
a1
MOs of Water
HOMO-1
Looks like combination of p on O along C2 with constructive interference with bo (close to nbo)
E = -0.3356 au
HOMO-2
Looks like combination of p on O (perp to C2, but in plane of molecule) with constructive interference with bo
E = -0.4778 au
a1b2
MOs of Water
HOMO from two viewpoints
Looks like p orbital on O, perpendicular to plane of molecule - nbo
E = -0.2603 au
b1
MOs of Water
LUMO
Looks like combination of p on O along C2 with destructive interference with abo
E = -0.0059 au
LUMO +1
Looks like combination of p on O (perp to C2, but in plane of molecule) with destructive interference with abo
E = 0.0828 au
a1b2
Filling Pattern for Water
1a1 (nbo)
2a1 (bo)
1b2 (bo)
3a1 (bo/nbo)
1b1 (nbo)
4a1 (abo)
2b2 (abo)
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
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