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Molecular Orbitals
Atomic orbitals interact to form molecular orbitals
Electrons are placed in molecular orbitalsfollowing the same rules as for atomic orbitals
In terms of approximate solutions to the Scrödinger equationMolecular Orbitals are linear combinations of atomic orbitals (LCAO)
caacbb (for diatomic molecules)
Interactions depend on the symmetry propertiesand the relative energies of the atomic orbitals
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 overlapthe energy of the orbitals must be similarthe interatomic distance must be short enough but not too short
If the total energy of the electrons in the molecular orbitalsis less than in the atomic orbitals, the molecule is stable compared with the atoms
Combinations of two s orbitals (e.g. H2)
Antibonding
Bonding
More generally:ca(1sa)cb(1sb)]
n A.O.’s n M.O.’s
Electrons in bonding orbitals concentrate between the nuclei and hold the nuclei together(total energy is lowered)
Electrons in antibonding orbitals cause mutual repulsion between the atoms(total energy is raised)
Bothand notation means symmetric/antisymmetric with respect to rotation
zC2
zC2 zC2
zC2Not
Combinations of two p orbitals (e.g. H2)
and notation meanschange of sign upon C2 rotation
and notation means nochange of sign upon rotation
Combinations of two p orbitals
zC2
zC2
Combinations of two sets of p orbitals
Combinations of s and p orbitals
Combinations of d orbitals
No interaction – different symmetry means change of sign upon C4
NO NOYES
Is there a net interaction?
Relative energies of interacting orbitals must be similar
Strong interaction Weak interaction
Molecular orbitalsfor diatomic molecules
From H2 to Ne2
Electrons are placedin molecular orbitals
following the same rulesas for atomic orbitals:
Fill from lowest to highestMaximum spin multiplicity
Electrons have different quantum numbers including spin (+ ½, -
½)
Bond order = # of electrons
in bonding MO's# of electrons in antibonding MO's
12
-
O2 (2 x 8e)
1/2 (10 - 6) = 2A double bond
Or counting onlyvalence electrons:
1/2 (8 - 4) = 2
Note subscriptsg and u
symmetric/antisymmetricupon i
Place labels g or u in this diagram
g
g
u
u
g
u
g
u
u
g
g or u?
Orbital mixing
Same symmetry and similar energies !shouldn’t they interact?
orbital mixing
When two MO’s of the same symmetry mixthe one with higher energy moves higher and the one with lower energy moves lower
H2 g2 (single bond)
He2 g2 u
2 (no bond)
Molecular orbitalsfor diatomic molecules
From H2 to Ne2
E (Z*)
E > E Paramagneticdue to mixing
C2 u2 u
2 (double bond)
C22- u
2 u2 g
2(triple bond)
O2 u2 u
2 g1 g
1 (double bond)paramagneticO2
2- u2 u
2 g2 g
2 (single bond)diamagnetic
Bond lengths in diatomic molecules
Filling bonding orbitals
Filling antibonding orbitals
Photoelectron Spectroscopy
h(UV o X rays) e-
Ionization energy
hphotons
kinetic energy of expelled electron= -
N2O2
*u (2s)
u (2p)
g (2p)
*u (2s)
g (2p)u (2p)
u (2p)
(Energy required to remove electron, lower energy for higher orbitals)
Very involved in bonding(vibrational fine structure)
Simple Molecular Orbital TheoryA 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
)...3()2(1...)3,2,1( 211 fffF
electrons
iihH ...)3,2,1(
Where hi is the energy operator for electron i and involves only electron i
AO
llluaf
MO Theory - 2Seek F such that
...)3,2,1(...)3,2,1(...)3,2,1( EFFH
i
i ffEffffhFH )...3()2()1()...)3()2()1((...)3,2,1(...)3,2,1( 211211
Divide by F(1,2,3…) recognizing that hi works only on electron i.
electrons
i j
ji Eif
ifh
)(
)(
jjj fefh
Since each term in the summation depends on the coordinates of a different electron then each term must equal a constant.
MO Theory - 3
jjj fefh
Multiply by uk and integrate.
dvhuuh jkjk ,
dvfuedvhfu jkjk
AO
llluaf
Recall the expansion of a molecular orbital in terms of the atomic orbitals.
Define
dvuuS jkjk ,
Substituting the expansion for f
0)( ,, AO
llklkl eSha
These integrals are fixed numerical values.
MO Theory - 40)( ,,
AO
llklkl eSha For k = 1 to AO
These are the secular equations. The number of such equations is equal to the number of atomic orbitals, AO.
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
There are AO equations with AO unknowns, the al.
0
)()(
)()(
,,1,1,
,1,11,11,1
AOAOAOAOAOAO
AOAO
eSheSh
eSheSh
MO Theory 6
0
)()(
)()(
,,1,1,
,1,11,11,1
AOAOAOAOAOAO
AOAO
eSheSh
eSheSh
Drastic assumptions can now be made. We will use the simple Huckle approximations.
hi,i = 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.
ExampleThe allyl pi system. 1
2 3
The secular equations:
(-e)a1 + a2 + 0 a3 = 0
a1 + (-e) a2 + a3 = 0
0 a1 + a2 + (-e) a3 = 0
Simplify by dividing every element by and setting (-e)/ = x
0)0(1)1(
10
11
012
xxx
x
x
x
2,0 x
For x = -sqrt(2)
e = sqrt(2)
0210
0121
0012
321
321
321
aaa
aaa
aaa
321 2 uuuf normalized
)2()121(
1321
5.0222uuuf
12 3
For x = 0
2/1
2/1
2/1
)2(
2/1
2/1
2/1
0
0
Verify that
h f = e f
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.
Corrections
Energy
Zero order (no correction): ei0
First Order correction: 1,
010iiii HdvfHf
Wave functions corrections to f0i
Zero order (no correction): f0i
First order correction: 000
1,
jij ji
ij fee
H
ExamplePi system only:
Perturbed system: allyl system
Unperturbed system: ethylene + methyl radical
12 3
12 3
2/)( 210
3 uuf 03e
30
2 uf 02e
2/)( 210
1 uuf 01e
00
0
00H
H
0
0
H
00
00
00001
HHH
0
0
21
21
00
00
000
02
1
2
111
e
03
02
03
2
03
03
01
11,3
02
02
01
11,2
11
02
1
))()/((
0
21
21
00
00
000
02
12
1
))/((
0
21
21
00
00
000
100
)/()/(
ff
f
f
feehfeehf
+
0
0
21
21
00
00
000
02
1
2
111
e
03
02
03
2
03
03
01
11,3
02
02
01
11,2
11
02
1
))()/((
0
21
21
00
00
000
02
12
1
))/((
0
21
21
00
00
000
100
)/()/(
ff
f
f
feehfeehf
+
Mixes in bonding
Mixes in bonding
Mixes in anti-bonding
Mixes in anti-bonding
Projection OperatorAlgorithm of creating an object forming a basis for an irreducible rep from an arbitrary function.
^^
RRh
lP
jj
jj
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.
1sA 1sB
z
y
Starting with the 1sA create a function of A1 sym
¼(E1sA + C21sA + v1sA + v’1sA) = ½(1sA + 1sB)