Molecular Orbitals

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Molecular Orbitals
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Atomic orbitals interact to form molecular
orbitals Electrons are placed in molecular
orbitals following the same rules as for atomic
orbitals
In terms of approximate solutions to the
Scrödinger equation Molecular Orbitals are linear
combinations of atomic orbitals (LCAO) Y caya
cbyb (for diatomic molecules)
Interactions depend on the symmetry
properties and the relative energies of the
atomic orbitals
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As the distance between atoms decreases
Atomic orbitals overlap
Bonding takes place if the orbital symmetry
must be such that regions of the same sign
overlap the energy of the orbitals must be
similar the interatomic distance must be short
enough but not too short
If the total energy of the electrons in the
molecular orbitals is less than in the atomic
orbitals, the molecule is stable compared with
the atoms
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Combinations of two s orbitals (e.g. H2)
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Both s (and s) notation means symmetric/antisymme
tric with respect to rotation
s
s
s
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Combinations of two p orbitals (e.g. H2)
s (and s) notation means no change of sign upon
rotation
p (and p) notation means change of sign upon C2
rotation
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Combinations of two p orbitals
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Combinations of two sets of p orbitals
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Combinations of s and p orbitals
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Combinations of d orbitals
No interaction different symmetry
d means change of sign upon C4
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Is there a net interaction?
NO
NO
YES
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Relative energies of interacting orbitals must be
similar
Weak interaction
Strong interaction
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Molecular orbitals for diatomic molecules From H2
to Ne2
Electrons are placed in molecular
orbitals following the same rules as for atomic
orbitals Fill from lowest to highest Maximum
spin multiplicity Electrons have different
quantum numbers including spin ( ½, - ½)
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O2 (2 x 8e)
1/2 (10 - 6) 2 A double bond
Or counting only valence electrons 1/2 (8 - 4)
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Note subscripts g and u symmetric/antisymmetric up
on i
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Place labels g or u in this diagram
su
pg
pu
sg
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su
sg
g or u?
pg
pu
du
dg
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Orbital mixing
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s orbital mixing
When two MOs of the same symmetry mix the one
with higher energy moves higher and the one with
lower energy moves lower
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Molecular orbitals for diatomic molecules From H2
to Ne2
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Bond lengths in diatomic molecules
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Photoelectron Spectroscopy
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O2
N2
sg (2p)
pu (2p)
pu (2p)
sg (2p)
pu (2p)
Very involved in bonding (vibrational fine
structure)
su (2s)
su (2s)
(Energy required to remove electron, lower energy
for higher orbitals)
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Simple Molecular Orbital Theory

• A molecular orbital, f, is expressed as a linear
  combination of atomic orbitals, holding two
  electrons.

The multi-electron wavefunction and the
multi-electron Hamiltonian are
Where hi is the energy operator for electron i
and involves only electron i
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MO Theory - 2
Seek F such that
Divide by F(1,2,3) recognizing that hi works
only on electron i.
Since each term in the summation depends on the
coordinates of a different electron then each
term must equal a constant.
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MO Theory - 3
Recall the expansion of a molecular orbital in
terms of the atomic orbitals.
Multiply by uk and integrate.
Define
These integrals are fixed numerical values.
Substituting the expansion for f
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MO Theory - 4
For k 1 to AO
These are the secular equations. The number of
such equations is equal to the number of atomic
orbitals, AO.
There are AO equations with AO unknowns, the al.
For there to be a nontrivial (all al equal to
zero) solution to the set of secular equations
then the determinant below must equal zero
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MO Theory 6
Drastic assumptions can now be made. We will use
the simple Huckle approximations.
hi,i a, if orbital i is on a carbon atom. Si,i
1, normalized atomic orbitals hi,j b, if atom
i bonded to atom j, zero otherwise
Expand the secular determinant into a polynomial
of degree AO in e. Obtain the allowed values of
e by finding the roots of the polynomial.
Choose one particular value of e, substitute into
the secular equations and obtain the coefficients
of the atomic orbitals within the molecular
orbital.
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Example
The allyl pi system.
The secular equations (a-e)a1 b a2 0 a3
0 ba1 (a-e) a2 b a3 0 0 a1 b a2
(a-e) a3 0
Simplify by dividing every element by b and
setting (a-e)/b x
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For x -sqrt(2) e a sqrt(2) b
For x 0
normalized
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Verify that h f e f
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Perturbation Theory
The Hamiltonian is divided into two parts H0 and
H1 H0 is the Hamiltonian of for a known system
for which we have the solutions the energies,
e0, and the wavefunctions, f0. H0f0 e0f0 H1
is a change to the system and the Hamiltonian
which renders approximation desirable. The
change to the energies and the wavefunctions are
expressed as a summation.
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Corrections
Energy Zero order (no correction) ei0 First
Order correction
Wave functions corrections to f0i Zero order (no
correction) f0i First order correction
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Example
Pi system only Perturbed system allyl
system Unperturbed system ethylene methyl
radical
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Mixes in anti-bonding
Mixes in bonding
Mixes in anti-bonding
Mixes in bonding
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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 results from using
the symmetry operations multiplied by characters
of the irreducible reps. j indicates the desired
symmetry. lj is the dimension of the irreducible
rep.
Starting with the 1sA create a function of A1 sym
¼(E1sA C21sA sv1sA sv1sA) ½(1sA
1sB)