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Atomic Orbitals
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Atomic Orbitals
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Atomic Orbitals
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Atomic Orbitals
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Atomic Orbitals
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Atomic Orbitals
for a Carbon Atom
Energy
2p
2s
1s
Atomic Orbitals
for a Carbon Atom
Energy
2p
Carbon has 6 electrons
but only 4 valence electrons
2s
1s
Where do these orbitals come from?
The orbitals are actually mathematical functions
that satisfy Schrödinger’s equation.
ĤY=EY
Ĥ is the Hamiltonian operator that operates on
the wave funtions Y to give the Energy E.
This is a special form of what is called a wave equation.
Y is a wave function, a function that fully describes the
electrons of a system in a particular state.
Where do these orbitals come from?
ĤY=EY
The atomic orbitals 1s, 2s, 2p etc are wave
functions that describe different states of an
electron of a atom with one electron.
Ĥ F2s = E
F2s
F2s describes where we can find the electron
Ĥ F2s = E
F2s
2
F2s gives the probability of finding the electron.
2
F2s
= 1
If you integrate over
all space the probability
is unity.
Functions that behave this way
are said to be normalized.
E gives the energy of the electron
Ĥ F2s = E
F2s
E=0
You can equate the
energy E of the electron
to its ionization energy.
Energy
This is the energy needed
for the electron to escape
from the atom.
2p
2s
1s
How do you actually calculate the Energy of
some wave function Fa ?
Start with
Ĥ Fa = E Fa
Multiply by Fa and integrate
Fa Ĥ Fa = Fa E Fa = E Fa2
Fa Ĥ Fa = E
This is a Coulomb integral
Gives E of function F
Another important integral is called an
interaction integral. It gives the energy
of “interaction” of two different functions.
Fa Ĥ Fb = Eint
Now we have the tools we need
to describe Molecular Orbitals.
We are going to use an approximate
method called the LCAO approach.
Molecular orbitals, Y, are expressed as
Linear Combinations of Atomic Orbitals, F.
Y
=
Sc F
i
i
The coefficients, ci, of each orbital
must be determined
Y
=
Let us start with H2
Sc F
i
i
H1
H2
Each hydrogen has a 1s orbital
The MOs must be a linear combination
of these 1s orbitals. We will call the two
1s orbitals F1 and F2.
Y
= c 1F 1 + c 2F 2
Y
= c 1F 1 + c 2F 2
The H2 molecule has
1
2
symmetry, this means that
the electron density on one end must be the
same as the electron density on the other end.
H
H
This can only happen if the coefficients c1 and c2
are of equal magnitude. There are two ways to
do this. Either c2 = c1 or c2 = - c1 .
Y1
= c 1F 1 + c 1F 2
or
Y2
= c 1F 1 - c 1F 2
Y1
= c 1F 1 + c 1F 2
How do we determine the Energy?
First we need to assume some integral values.
Let us assume:
Coulomb integral
Fa Ĥ Fa = E = a = 0
Interaction integral
Fa Ĥ Fb = Eint = b
Y1
= c 1F 1 + c 1F 2
Y1 Ĥ Y1 = E1
(c1F1 + c1F2 ) Ĥ (c1F1 + c1F2 ) = E1
( 2 c 12F 1 Ĥ F 1 + 2 c 12F 1 Ĥ F 2 ) = E 1
2 c 12 x 0
+
2 c12 x
b = 2 c 12 b = E 1
But what is c1 ?
Y1
= c 1F 1 + c 1F 2
Y 1Y 1 = 1
(c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1
define
Fa = 1
2
FaFb = Sab
Overlap integral
But what is c1 ?
Y1
= c 1F 1 + c 1F 2
Y1 Y1 = 1
(c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1
C12
( 2 F1 F1 + 2 F1 F2 ) = 1
1
c 12
( 2 + 2 S12 ) = 1
c1 =
2 + 2 S12
2 c 12
E1 =
1
b = E1
1
1 + S12
c1 =
b
for
2 + 2 S12
Y1
= c 1F 1 + c 1F 2
on your own show:
E2 = -
1
1 - S12
b
for
Y2
= c 1F 1 - c 1F 2
Y2
= c 1F 1 - c 1 F 2
E2 = -
Y1
antibonding
1
b
1 - S12
= c 1F 1 + c 1 F 2
E1 =
1
1 + S12
b
bonding
This means
that He2 would
be unstable,
because two He
atoms, a four
electron system
would repel
each other
Antibonding
goes up
more than:
Bonding
goes down