Outline
Molecular Orbitals in
Inorganic Chemistry
L2 build a MO diagram to show you the process
quick revision
stage 1: basic MO diagram for H2O
stage 2: include MO mixing
Dr. P. Hunt
[email protected]
Rm 167 (Chemistry)
http://www.ch.ic.ac.uk/hunt/
Socrative:
WHZ9KBWC3
real MOs (how good are qualitative diagrams)
Walsh or correlation diagrams
L3 come back and look closely at the details
@quantumchemist
#MO_lectures
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Revision
Setting Up
MO are combinations of AOs
determine the shape of the molecule
Fig. 1
details:
Lecture 1
find the point group of the molecule: C2v
Maths lectures with
Kim Jelfs
↓
ψ 1s
atomic orbitals
define the axial system
find all of the symmetry elements
↓
R1sY1s
✦ radial and angular components
✦ radial: as quantum shell increases
x
↓
‣ inner nodes
‣ radius max density increases
✦ cartoons represent outer portion
✦ shaded part represents negative part of function
✦ angular nature represented by the lobes
θ
θ
positive
the “Orbitron”
C2v
Y1s
z
y=ax2+bx+c
x
negative
sign change with θ
✦ link from my website
Fig. 2
3
C2(z)
σv(yz)
y
⎡⎣ 2Z 3 2 e− ρ 2 ⎤⎦ ⎡⎣(1 4π )1 2 ⎤⎦
!#"#$ !#"#$
R1s
z
E
A1
1
A2
B1
B2
σv(xz) σv'(yz)
C2
h=4
z
1
1
1
1
1
-1
-1
1
-1
1
-1
x
1
-1
-1
1
y
O
H1
H2
σv(xz)
Fig. 4
Fig. 3
4
Fragments
Fragments
map onto each other under the symmetry operations
✦ H atoms map onto each other
✦ O atom maps onto its self
✦ H2 and O atom are the fragments
z
x
O
H
place holder
y
H
use place holders!
place holder
O
H
H
H2 fragment
map onto each other under the symmetry operations
H
O
✦ H atoms map onto each other
✦ O atom maps onto its self
✦ H2 and O atom are the fragments
H
whole molecule
z
place holder
x
O
H
O fragment
place holder
y
O
H
H
use place holders!
H
H2 fragment
H
O
H
whole molecule
O fragment
Fig. 5
Fig. 5
5-1
5-2
Fragments
map onto each other under the symmetry operations
✦ H atoms map onto each other
✦ O atom maps onto its self
✦ H2 and O atom are the fragments
z
O
H
y
place holder
x
H
use place holders!
place holder
O
H
H
H2 fragment
H
O
H
whole molecule
O fragment
reproduce the whole of the
molecular structure in the MO
fragment orbitals
✦ H2 orbitals
✦ O atom orbitals
Set Up MO
Diagram
vertical axis: Energy
tant!
Impor
H 1s AO
position AOs
✦ O more electronegative than
H so valence orbitals lie
below H 1s reference orbital
✦ H atoms are further apart
than in H2 so stablilsation
and destabilsation are less ie
splitting energy is less
py
H
x
y
s
H
O
H
H
O
H
H2O
O (s and p) orbitals
Fig. 6
6
pz
z
O
H
H2 orbitals
px
???
Fig. 6
7
Fragment Orbital Symmetry
Short Cuts!
How does each orbital transform under the symmetry
operations of the group?
σv
Lecture 1
C2v
No change under σv
1
1
χ =1
t!
u
Short-C
rtant!
Impo
a1
1
C2v
✦ sometimes also written as just x, y, z
✦ gives you the axis symmetry label
C2 σv(xz) σv'(yz)
E
1
Γ
look in the last columns of the
character table
ut!
Short-C
✦ find Tx, Ty, Tz
σv(yz)
✦ orbital is unchanged => character = 1
✦ a sign change => character = -1
✦ check against the character table
E
C2 σv(xz) σv'(yz) h=4
A1
1
1
1
1
A2
1
1
-1
-1
B1
1
-1
1
-1
Tx
B2
1
-1
-1
1
Ty
Tz
Fig. 7
Fig. 7
totally bonding orbitals are always totally symmetric
✦ which is the first symmetry label listed for all point groups
✦ for C2v this is the a1 irreducible representation
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Fragment Orbitals
Fragment Orbitals
look for similarities in the phase of orbitals
and the cartesian axes
C2 v
E
ut!
Short-C
look for similarities in the phase of orbitals
and the cartesian axes
C2 σv(xz) σv'(yz)
E
C2 σv(xz) σv'(yz)
Γ
1
-1
-1
1
py
b2
Γ
1
-1
1
-1
px
b1
Γ
1
1
1
1
pz
C2v
x
Γ
O
a1
1
Γ
1
Γ
1
-1
-1
1
-1
1
1
1
Ty
b2
-1
Tx
b1
1
Tz
a1
O
H
H
O
H
H
O
a1
Fig. 7
H2
Fig. 8
σv(xz) σv'(yz)
E
C2
Γ
1
1
1
1
a1
Γ
1
-1
1
-1
b1
C2v
Fig. 7
10
11
ut!
Short-C
Form the MOs
work out MOs first
then the splitting
tant!
Impor
Only fragment orbitals (FOs) of
the same symmetry can combine
bels to
try la
ymme
Add s O diagram
M
+
✦ for water: a1 and b1
+
FO can only combine ONCE
a1
✦ more than one FO of same symmetry?
✦ then combine the lowest energy two
✦ leave the last one non-bonding (for now!)
Form MOs by
✦ “adding” FOs together “as is”
✦ “adding” FOs with ONE FO phase inverted
b1
Fig 10
Fig. 9
12
13
Stage One MO Diagram
Stage One
MO Diagram
Evaluate splitting
✦ FOs far apart interact only weakly
✦ in-phase interactions are bonding
✦ destabilisation is always larger
✦ number within symmetry label
✦ count the core orbitals
t!
tan
2b1
4a1
b1
a1
Annotate your
diagrams!
MO labeling
or
Imp
antibonding MO is always
destabilised more than the
bonding MO is stabilised
b2
1b 2
a1 FO left nonbonding in the first
stage diagram
a1
3a1
FOs are closer in energy
and so the interaction
between the orbitals is
larger for the b1 MO
1b 1
Annotate your
diagrams!
b1
Figure 11
a1
2a1
add the electrons!
Figure 12
✦ fill sequentially
1a 1 is the O 1sAO which
is not shown because this
is a valence MO diagram
z
O
H
y
x
H
O
H
H
2 valence e
14
15
H
O
H
6 valence e
Fig 11/12
Stage 2: MO Mixing
a1 MO Mixing
mixing orbitals
ONLY MOs of the same symmetry can mix
✦ “add” MOs together “as is”
0
✦ “add” MOs with ONE MO phase inverted
✦ inspect to determine which is the bonding mixed MO
to occur mixing MUST stabilise the total energy
mixing tends to be large when:
✦ MOs are close in energy
✦ one MO is non-bonding
✦ one MO is unoccupied
✦ orbitals are in HOMO-LUMO region
tant!
occupied
Impor
antibonding due to
increased out-ofphase interaction
unoccupied
+
ψ (3a1 )
ψ (4a1 )
ψ 1 = ψ (3a1 ) + ψ (4a1 )
antibonding MO
Fig 13
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In-Class Activity
4a1
Mixing
mixing orbitals
✦ “add” MOs together “as is”
✦ “add” MOs with ONE MO phase inverted
0
✦ inspect to determine which is the bonding mixed MO
2b1
b1
strong mixing:
a1
✦ 4a1 MO unoccupied
✦ 3a1 MO non-bonding
✦ in HOMO/LUMO region
✦ close in energy
✦ occupied MO is stabilised
form (4a 1)
)+ψ
3a 1
-ψ(
b2
1b 2
a1
b1
significant mixing
drives 3a1 MO
down in energy
and chanes the
shape of the MO
3a1
1b 1
a1
extra lines drawn
to show the main
mixing interaction
2a1
Fig 14
z
O
H
y
x
H
O
H
H
2 valence e
18
19
H
O
H
H2O
6 valence e
Fig 15
4a1
Final MO
Diagram of
H2O
Annotate your
diagrams!
antibonding MO is always
destabilised more than the
bonding MO is stabilised
4a1
Final MO
Diagram of
H2O
2b1
the 4a1 lies above the 2b1 MO because
there is strong directional antibonding
overlap in the 4a1 MO and weaker less
directional overlap in the 2b1 MO
b1
a1
b2
1b 2
b2 FO left nonbonding as there is
no other FO of this
symmetry for it to
interact with
3a1
FOs are closer in energy and
so the interaction between the
orbitals is larger for the 1b 1
than the 2a1 MO
O
H
y
H
2 valence e
H
do not repeat
information
tant!
Impor
O
O
y
a1
b1
significant mixing
drives 3a1 MO
down in energy
and chanes the
shape of the MO
a1
2a1
1a 1 is the O 1sAO which
is not shown because this
is a valence MO diagram
z
H
6 valence e
3a1
FOs are closer in energy and
so the interaction between the
orbitals is larger for the 1b 1
than the 2a1 MO
H
H2O
b2
1b 2
1b 1
2a1
O
H
b1
a1
a1
x
H
the 4a1 lies above the 2b1 MO because
there is strong directional antibonding
overlap in the 4a1 MO and weaker less
directional overlap in the 2b1 MO
b2 FO left nonbonding as there is
no other FO of this
symmetry for it to
interact with
Annotate your
diagrams!
1a 1 is the O 1sAO which
is not shown because this
is a valence MO diagram
z
2b1
a1
b1
significant mixing
drives 3a1 MO
down in energy
and chanes the
shape of the MO
1b 1
antibonding MO is always
destabilised more than the
bonding MO is stabilised
x
H
O
H
H
2 valence e
Fig 15
H
O
H
H2O
6 valence e
20
21
“Real” MOs
Experimental Evidence
Fig 15
Photoelectron spectrum
✦ energy required to eject an
electron from its orbital
computed
molecular
orbitals
2a1
“traditional” theory:
1b1
✦ expect 2 equivalent bonds and 2
equivalent lone pairs for water = 2
lines in photo-electron spectrum
we have
“solved” the
Schrödinger
equation!!!
3a1
BUT
O
b2
H
H
Fig 17
Fig 16
22
✦ have 3 lines in photo-electron
spectrum which relate to
delocalised 1b1, 3a1 and 1b2 MOs
Fig 5.13
from
“Structure
and
Bonding" by
J. Barrett
Fig 18
23
Delocalisation
Analysis
4a1
MO diagrams are
transferable
MOs are delocalised
✦ not 2 center 2 electron
✦ most of MOs extend over ALL atoms in molecule
✦ there are no “bonds”
2b1
✦ one diagram is good for many
molecules or fragments!
✦ molecules BeH2 (homework), H2S
✦ fragments CH2 (lecture 4), NH2
✦ we can even treat metal fragments: MH2
Bonds represent a build up of the TOTAL
electron density
b1
a1
1b1
✦ A=main group element, M=metal
✦ slight modifications:
✦ different numbers of electrons
✦ slightly different position of the
fragment orbitals
a1
2a1
z
O
H
y
x
H
O
H
H
2 valence e
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b1
3a1
general formula AH2
We keep ideas of hybridisation and 2c-2e
bonds because they are USEFULL
b2
1b2
H
O
H
H2O
6 valence e
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Correlation Diagram
Walsh Diagram
Walsh diagram: change one geometric
parameter and examine changes in MOs
and energies
typically start from the
highest symmetry
✦ I’ve constructed the MO diagram for
linear H2O for you (Self-study for
you to reproduce)
✦ normally a bond distance or angle
✦ link the MOs for two extreme geometries
example: Why is H2O bent?
start with high
symmetry
Fig 20
≈105º
C2v
180º
O
H
end with lower
symmetry
O
H
H
H
H
O
H
D∞h
Prof. A.D Walsh
University of Dundee
from: http://
www.dundee.ac.uk/museum/
scientists.htm
H-O-H angle
Fig 21
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27
a1
Walsh Diagram
Energy of MOs
then examine how MOs
change under geometric
distortion
examine how MOs change
under geometric
distortion
2σ g+ (2a1) stabilised
✦ ↑ H...H through space bonding
overlap
✦ ↓ O-H bonding overlap
✦ net result small stabilisation
Fig 21
Fig 21
28
29
Energy of MOs
Energy of MOs
then examine how MOs
change under geometric
distortion
then examine how MOs
change under geometric
distortion
1π
1σ u+ (1b1) destabilised
u (1b2) no change
✦ non-bonding orbital
✦ ↑ H...H through space antibonding
✦ ↓ O-H bonding overlap
✦ net result large destabilisation
Fig 21
30
Fig 21
31
Energy of MOs
Energy of MOs
then examine how MOs
change under geometric
distortion
then examine how MOs
change under geometric
distortion
1π u / 3σ g+ or 3a1/4a1 special
✦ as planar molecule 1πu and 3σg+
cannot mix (not same symmetry)
✦ when molecule distorts they become
the same symmetry: mixing occurs
✦ 1πu goes to 3a1
✦ 3σg+ goes to 4a1
✦ on mixing the 1πu ( 3a1) is stabilised
2σ u+(2b1) stabilised
socrative quiz!
3
KBWC
9
Z
H
W
nt!
Importa
Fig 21
Fig
Fig 21
18
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33
Change of Axes
When the symmetry point
group changes the axial
definition changes!
z
z
y
Walsh Diagram
x
✦ z-axis reorientated
✦ orbital remains the same
✦ only labels change
✦ then follow with mixing
x
x
O
H
H
O
H
y
z
y
2σ g+
4a1
H
H
y
H
H
3a1
z
✦ AOs move with the atoms
✦ form or shape of AOs remains
constant
✦ !! except for MOs which undergo
mixing
Molecular stability
H
✦ examine how occupied MOs
change under geometric distortion
✦ look for occupied MOs which show
a large change in energy
✦ these orbitals drive the change in
shape
1π u
mixing
+
4a1
Orbital changes
x
3a1
Fig 22
Fig 21
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Symmetry Breaking
Symmetry Breaking
But how does the drop in symmetry start?
But how does the drop in symmetry start?
nuclear vibrations provide infinitesimal
distortion required for MO mixing
nuclear vibrations provide infinitesimal
distortion required for MO mixing
vibronic coupling = coupling of electronic
and nuclear motions
vibronic coupling = coupling of electronic
and nuclear motions
breakdown of the Born-Oppenheimer
approximation!
breakdown of the Born-Oppenheimer
approximation!
✦ collapse of a VERY fundamental approximation
✦ more common than you think!
✦ of which Jahn-Teller theorem is a special case
✦ collapse of a VERY fundamental approximation
✦ more common than you think!
✦ of which Jahn-Teller theorem is a special case
36-1
36-2
Symmetry Breaking
MO checklist
Steps to construct a MO diagram
symmetry & symmetry breaking underlies
many theories in physics and chemistry
Noether’s
Theorem
Shows that a conservation
law can be derived from any
continuous symmetry.
VERY
nt!
Importa
• invariance with respect to translation gives the law
of conservation of linear momentum
• invariance with respect to time translation gives the
law of conservation of energy
Emmy Noether
ity
magnetism
nductiv
superco
Heisen
Serious
b
ity
Uncerta erg
l relativ
a
r
e
n
e
in
g
Stuff!
Princip ty
le
ce of
standard model of
existen ticle
ar
particle physics
eory
Higgs p
field th
37
source: http://en.wikipedia.org/wiki/
Image:Noether.jpg accessed
17/08/07
✦ determine the molecular shape and identify the point group
✦ define the axial system and all of the symmetry operations
✦ identify the chemical fragments, put them on the bottom of
the diagram
✦ determine the energy levels and symmetry labels of the
fragment orbitals (use H1s as a reference)
✦ combine fragment orbitals of the same symmetry, determine
the MOs and then estimate the splitting energy; draw in the
MO energy levels and MOs (in pencil!)
✦ determine the number of electrons in each fragment and
hence the central MO region, add them to the diagram
✦ identify if any MO mixing occurs, determine the mixed
orbitals and redraw the MO diagram with shifted energy
levels and the mixed MOs
✦ annotate your diagram
✦ use the MO diagram to understand the structure, bonding
and chemistry of the molecule
38
Finally
Key Points
be able to form MO diagrams for molecules with the general
formula AH2 and AH3 (tutorial) where A= main group element or a
metal
be able to explain and illustrate MO mixing
be able to critically evaluate VSEPR theory, localised 2c-2e bonding
and the delocalised MO picture of bonding
be able to describe how a PES is formed and be able to relate a
spectrum to the MOs, and MO diagram of a molecule
be able to form correlation diagrams and explain why a particular
geometry is more stable than another with reference to the
stability of the MOs
be able to discuss symmetry breaking and vibronic coupling
http://www.huntresearchgroup.org.uk/
See my web-site
✦ notes AND slides
✦ link to panopto when it becomes available
✦ optional background support for beginners
✦ optional material to take you a little further
✦ links to interesting people and web-sites
✦ links to relevant research papers on MOs
✦ model answers!!
Tweet!
@quantumchemist
#MO_lectures
◆ share questions
◆ let me know what is easy or hard
◆ I can adjust the problems classes & workshops
(even lectures) based on your feedback!
be able to describe the process of forming a MO diagram (the MO
checklist)
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