Molecular orbitals of

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Lecture 6
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Molecular orbitals of heteronuclear diatomic
molecules
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The general principle of molecular orbital
theory Interactions of orbitals (or groups of
orbitals) occur when the interacting orbitals
overlap. the energy of the orbitals must be
similar the interatomic distance must be short
enough but not too short A bonding interaction
takes place when regions of the same sign
overlap An antibonding interaction takes place
when regions of opposite sign overlap
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Combinations of two s orbitals in a homonuclear
molecule (e.g. H2)
In this case, the energies of the A.O.s are
identical
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More generally Y NcaY(1sa) cbY (1sb)
n A.O.s
n M.O.s
The same principle is applied to heteronuclear
diatomic molecules But the atomic energy levels
are lower for the heavier atom
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Orbital potential energies (see also Table 5-1 in
p. 134 of textbook)
Average energies for all electrons in the same
level, e.g., 3p (use to estimate which orbitals
may interact)
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The molecular orbitals of carbon monoxide
Y NccY(C) coY (O)
E(eV)
2s 2p
C -19.43 -10.66
O -32.38 -15.85
Each MO receives unequal contributions from C and
O (cc ? co)
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Group theory is used in building molecular
orbitals
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Bond order 3
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A related example HF
s (A1) 2p(A1, B1, B2)
H -13.61 (1s)
F -40.17 (2s) -18.65
No s-s int. (DE gt 13 eV)
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Extreme cases ionic compounds (LiF)
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Molecular orbitals for larger molecules
1. Determine point group of molecule (if linear,
use D2h and C2v instead of D8h or C8v)
2. Assign x, y, z coordinates (z axis is
higher rotation axis if non-linear y axis in
outer atoms point to central atom)
3. Find the characters of the representation for
the combination of 2s orbitals on the outer
atoms, then for px, py, pz. (as for vibrations,
orbitals that change position 0, orbitals
that do not change 1 and orbitals that remain
in the same position but change sign -1)
4. Find the reducible representations (they
correspond to the symmetry of group orbitals,
also called Symmetry Adapted Linear
Combinations SALCs of the orbitals).
5. Find AOs in central atom with the same
symmetry
6. Combine AOs from central atom with those
group orbitals of same symmetry and similar E
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F-H-F- D8h, use D2h
1st consider combinations of 2s and 2p orbitals
from F atoms
8 GROUP ORBITALS DEFINED
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Group orbitals can now be treated as atomic
orbitals and combined with appropriate AOs from H
1s(H) is Ag so it matches two group orbitals 1
and 3
Both interactions are symmetry allowed, how about
energies?
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-13.6 eV
-13.6 eV
-40.2 eV
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CO2 D8h, use D2h
(O O) group orbitals the same as for F F
But C has more AOs to be considered than H !
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CO2 D8h, use D2h
Carbon orbitals
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All four are symmetry allowed
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Group orbitals 3(Ag) -4 (B1u)
Group orbitals 1(Ag) and 2 (B1u)
2s(C) (Ag) and Groups 1 3 (Ag)
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LUMO
The frontier orbitals of CO2
HOMO
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Molecular orbitals for larger molecules H2O
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2
0
0
2
For H H group orbitals
G A1 B1
E two orbitals unchanged
C2 two orbitals interchanged
sv two orbitals unchanged
sv two orbitals interchanged
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Molecular orbitals for NH3
Find reducible representation for 3Hs
G
1
0
3
Irreducible representations G A1 E
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Projection Operator
Algorithm of creating an object forming a basis
for an irreducible rep from an arbitrary function.
Where the projection operator sums the results of
using the symmetry operations multiplied by
characters of the irreducible rep. j indicates
the desired symmetry. lj is the dimension of the
irreducible rep. h the order order of the group.
Starting with the 1sA create a function of A1 sym
¼(E1sA C21sA sv1sA sv1sA) ¼ (1sA
1sB 1sB 1sA)
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Consider the bonding in NF3
GA 3 0 -1
GA A2 E
GB 3 0 1
GB GC GD A1 E
GC 3 0 1
GD 3 0 1
2
3
1
A B
C D
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Now construct SALC
GA A2 E
PA2(p1)
1/6 (p1 p2 p3 (-1)(-p)1
(-1)(-1p3) (-1)(-p2)
No AO on N is A2
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E
Apply projection operator to p1
PA2(p1) 2/6 (2p1 - p2 - p3) E1
But since it is two dimensional, E, there should
be another SALC
PA2(p2) 2/6 (2p2 - p3 - p1) E
But E1 and E should be orthogonal want sum of
products of coefficients to be zero. E2 E k
E1. (-1 k2) p1 (2 k(-1)) p2 (-1
k(-1)) 0 Have to choose k such that they are
orthogonal. 0 (2/6)2 (2(-1 k2) -1 (2
k(-1)) -1 (-1 k(-1)) k ½ E2 2/6 (3/2 p2
- 3/2 p3)
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Molecular shapes When we discussed VSEPR theory
Can this be described in terms of MOs?
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Hybrid orbitals