Ch 5 Lecture 1 Molecular Orbitals for Diatomics
I.
Introduction
A.
B.
II.
Molecular Orbital (MO) Theory uses Group Theory to describe bonding
1) Simple Bonding Theories don’t explain all molecular properties
2) MO Theory often better explains some properties (ex: magnetism)
How MO Theory Works
1) Atomic orbitals of like symmetry interact to form molecular orbitals
2) Electrons fill the MO’s and determine bonding, stability, and properties
3) A simple pictorial approach is often adequate to find MO’s
4) More rigorous Group Theory approaches are sometimes needed
Forming Molecular Orbitals
A.
Linear Combinations of Atomic Orbitals (LCAO’s)
1) Atomic Orbitals (s, p, d, f…) are pictorial representations of solutions to
the quantum mechanical equations describing electron locations in atoms
a) A wave function is a solution to the quantum mechanical equation
b) We use Y to represent a given wave function
c) The orbitals are actually pictures of Y2 functions
2)
B.
Molecular Orbitals are formed by combining AO’s
a.
Y = caYa + cbYb
b. Y = a molecular wave function
c.
Ya = an atomic wave function from atom A
d. c = constant that can be + or –
Bonding in Diatomic Molecules
1) As 2 atoms approach each other, their atomic orbitals overlap in space
2) Electrons from both atoms occupy the same space in this overlap
3) MO’s form from the interaction of these electrons in the same space
4)
Bonding MO’s occupy the space between the nuclei
a) In a bonding interaction, the overall electronic energy is lowered
b) Electronic energy is lowered when an electron can occupy more space
5)
Conditions for bonding
a) Symmetry of AO’s must allow overlap of like signed areas (+/+)
b) Energies of the AO’s must be similar
c) A—B distance must be appropriate for overlap
C.
Bonding in H2 (Ha—Hb)
1) Only the occupied AO’s are important in forming MO’s
2) For H only the 1s orbital is occupied
3) There are two possible LCAO’s for H2
1
Y1s a   Y 1s b 
Ψσ   Nc a Y 1s a   c b Y 1s b  
2
1
Y 1s a   Y 1s b 
Ψσ *  Nc a Y 1s a   c b Y 1s b  
2
a)
b)
N = normalizing factor to keep total probability = 1
ca = cb = 1 since the two H atoms are identical
D.
4
s signifies that the MO is symmetric to rotation about internuclear axis
a.
s means the lower energy bonding MO
b. s* means the higher energy antibonding MO
5)
N Atomic Orbitals (2) produce N Molecular Orbitals (2)
6)
Nonbonding orbitals are also possible, although none form in diatomics
a. A nonbonding MO has the same energy as its substituent AO’s
b. If symmetry doesn’t match, the MO is made up of only 1 atom’s AO’s
c. Sometimes the AO/MO energies are identical by coincidence
MO’s produced from p-orbitals
1) p-orbitals are more complex than s-orbitals due to +/- lobes and more
complex symmetry
a) A + lobe only means the sign of the amplitude is positive, not charge
b) A – lobe only means the sign of the amplitude is negative, not charge
2) We must choose a common z-axis connecting the 2 nuclei
a) 1 s MO and 1 s* MO result from pz/pz overlap
b) 2 p MO’s and 2 p* MO’s result from px/px and py/py overlap
p signifies sign change upon C2 rotation about the internuclear axis
3)
Because of differing symmetries, no overlap is possible between s/p or
between px/py/pz orbitals (equal overlap with +/- lobes cancels out)
4)
Energies Match and Molecular Orbitals
a. Overlap of AO’s with the same energy leads to strong interaction (greatly
stabilized bonding MO and greatly destabilized antibonding MO)
b. Overlap of AO’s with differing energies progressively weakens
interaction
a. 1s/2s overlap can’t occur for a diatomic even though symmetry ok
b. Heteronuclear diatomics must use these types of MO’s
III. Homonuclear Diatomics
A.
N N
O O
B B
B.
Why do we need MO’s
1) The N2 Lewis structure explains
bonding and reactivity quite well
2) Problem: The O2 Lewis structure
doesn’t explain why it is
paramagnetic
3) Problem: The B2 Lewis structure
doesn’t have an octet
4) MO descriptions are better
MO’s of the First Row Elements
1) Correct ordering only if identical
orbitals interact
2) Relative energy depend on element
3) Electrons fill MO’s from bottom up
4) Bond Order
a) BO = ½[bonding-antibonding]
b) O2: BO = 2
5) MO symmetry labels
a) g = gerade = symmetric to i
b) u = ungerade = antisymmetric
to i
C.
Orbital Mixing
1) Orbitals of similar energies can interact if symmetry allows
2) This leads to adjustments in the energy ordering of diatomic MO’s
a) Example: sg(2s) and sg(2p) can mix lowering sg(2s) and raising sg(2p)
b) Example: su(2s) and su(2p) can mix lowering su(2s) and raising su(2p)
3) Effects of mixing
a) Energy ordering changes as we move across the first row elements
b) Bonding/Antibonding nature of a given MO may change
D.
MO Theory and Properties of Homonuclear Diatomics
1) Magnetism
a) Paramagnetic = strongly attracted by magnetic field due to upaired eb) Diamagnetic = weakly repelled by magnetic field by all paired e2)
H2 [sg2(1s)]
a) BO = 1 and BL = 74 pm
b) H2+ has BO = ½ and is less stable with BL = 106 pm
3)
4)
5)
He2 [sg2(1s)su*2(1s)] BO = 0, Noble gas that does not exist as a diatomic
Li2 [sg2(2s)] BO = 1
Be2 [sg2(2s)su*2(2s)] BO = 0, not stable
6)
B2 [pu1pu1(2p)]
a) Paramagnetism of B2 is not explained by Lewis structure B B
b) BO = 1
c) Without mixing, all e- paired = diamagnetic
d) Mixing makes degenerate pu(2p) lower than sg(2p)
e) Filling degenerate orbitals equally (Hund’s Rule) makes it paramagnetic
7)
C2 [pu2pu2(2p)]
a. BO = 2, mixing suggest that both are p-bonds with no s-bond?
b. C2 is rare, but C22- (with a s-bond as well BO = 3) is more stable
8)
N2 [sg2pu2pu2(2p)]
a. BO = 3 agrees with short strong bond (BL = 109.8 pm, D = 945 kJ/mol)
b. After nitrogen the “normal” energy ordering initially predicted returns
i. Nuclear charge gets higher as we go across first row
ii. Mixing becomes less important because the energy difference between
2s and 2p orbitals increases (s closer to nucleus, p more diffuse)
iii. Large energy difference means mixing is less important
9)
O2 [sg2pu2pu2pg*1pg*1 (2p)]
a. BO = 2, paramagnetic (not explained by Lewis structure)
b. O2 ions correlate BO and BL very well
O O
E.
10) F2 [sg2pu2pu2pg*2pg*2 (2p)] BO = 1, diamagnetic
11) Ne2 BO = 0 Noble gas, diatomic not stable
Photoelectron Spectroscopy: Where Orbital Energies Come From
1) Radiation can dislodge an electron from a molecule
a) UV radiation removes outer eb) X-Ray radiation can remove inner e2) The KE of the expelled e- tells us the energy of the orbital it came from
a) Ionization energy is equivalent to the orbital energy
b) IE = hn – KE
3)
a)
N2 spectrum
Lower E at top (outer orbital)
b) Fine structure is due to vibrational
Energy levels within electronic levels
i. Many levels = bonding orbital
ii. Few levels = less bonding
4)
Correlating Spectrum with Orbitals
a. Sometimes spectrum has orbitals out of order, because energies shift
as electrons are removed from molecule
b. Higher level calculations can correct for these “errors”
c. N2 levels pu(2p) and sg(2p) are very close; some debate as to order
5) O2 Spectrum: ordering is
correct as shown in the spectrum
F.
Correlation Diagrams
1) Calculated effect of bringing atoms together to a merged nucleus
2) Example: 2 H atoms approaching and merging into 1 He
a) Won’t happen chemically
b) Useful to see what happens to orbitals
c) MO’s for a homonuclear diatomics are somewhere in the middle,
depending on inherent energy of the element
Larger F2 farther to right, Smaller B2 farther to left (closer together)
3) As Nuclei Approach
a) Bonding MO’s decrease in energy
b) Antibonding MO’s increase in energy
c) Atomic Orbitals are connected to MO’s by symmetry
The 2 1s orbitals become 1sg and 1su*, then 1s and 2pz
4)
The Noncrossing Rule: orbitals of the same symmetry interact so that
their energies never cross on a correlation diagram
a) Helpful in assigning correlations
b) If 2 orbital sets of the same symmetry cross, you must change the
grouping to avoid it