1. Atomic Orbitals

2. Atomic Orbitals S

3. Atomic Orbitals pz

4. Atomic Orbitals py

5. Atomic Orbitals py

6. Atomic Orbitals
for a Carbon Atom

7. Atomic Orbitals
for a Carbon Atom
Carbon has 6 electrons
but only 4 valence electrons

8. Ĥ is the Hamiltonian operator that operates on
Where do these orbitals come from?
The orbitals are actually mathematical functions
that satisfy Schrödinger’s equation.
Ĥ Y = E Y
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.

9. Ĥ Y = E Y Ĥ F2s = E F2s Where do these orbitals come from?
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

10. F2s describes where we can find the electron
Ĥ F2s = E F2s
F2s2 gives the probability of finding the electron.
If you integrate over
all space the probability
is unity.
F2s2 = 1
Functions that behave this way
are said to be normalized.

11. E gives the energy of the electron
Ĥ F2s = E F2s
You can equate the
energy E of the electron
to its ionization energy.
This is the energy needed
for the electron to escape
from the atom.

12. Fa Ĥ Fa = Fa E Fa = E Fa2 Fa Ĥ Fa = E
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
This is a Coulomb integral
Gives E of function F
Fa Ĥ Fa = E

13. Fa Ĥ Fb = Eint Another important integral is called an
interaction integral. It gives the energy
of “interaction” of two different functions.
Fa Ĥ Fb = Eint

14. Y = S ci Fi 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 = S ci Fi
The coefficients, ci, of each orbital
must be determined

15. Y = S ci Fi Y = c1F1 + c2F2 Let us start with 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 = c1F1 + c2F2

16. Y = c1F1 + c2F2 Y1 = c1F1 + c1F2 Y2 = c1F1 - c1F2 The H2 molecule has
symmetry, this means that
the electron density on one end must be the
same as the electron density on the other end.
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 = c1F1 + c1F2
Y2 = c1F1 - c1F2 or

17. Y1 = c1F1 + c1F2 Fa Ĥ Fa = E = a = 0 Fa Ĥ Fb = Eint = b
How do we determine the Energy?
First we need to assume some integral values.
Let us assume:
Fa Ĥ Fa = E = a = 0
Coulomb integral
Fa Ĥ Fb = Eint = b
Interaction integral

18. Y1 = c1F1 + c1F2 Y1 Ĥ Y1 = E1 (c1F1 + c1F2 ) Ĥ (c1F1 + c1F2 ) = E1
( 2 c12F1 Ĥ F1 + 2 c12F1 Ĥ F2 ) = E1
2 c12 x c12 x b = 2 c12 b = E1

19. Y1 = c1F1 + c1F2 Y1Y1 = 1 (c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1 Fa2 = 1
But what is c1 ?
Y1 = c1F1 + c1F2
Y1Y1 = 1
(c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1
define
Fa2 = 1
FaFb = Sab
Overlap integral

20. Y1 Y1 = 1 Y1 = c1F1 + c1F2 (c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1
But what is c1 ?
Y1 Y1 = 1
Y1 = c1F1 + c1F2
(c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1
C ( 2 F1 F F1 F2 ) = 1
2 + 2 S12
c1 = 1
c12 ( S12 ) = 1

21. E1 = b Y1 = c1F1 + c1F2 E2 = - b Y2 = c1F1 - c1F2 1 c1 = 2 c12 b = E1
2 + 2 S12
c1 = 1
2 c12 b = E1
1 + S12 1
E1 = b
Y1 = c1F1 + c1F2
for
on your own show:
1 - S12 1
E2 = - b
Y2 = c1F1 - c1F2
for

22. Y2 = c1F1 - c1F2 E2 = - b Y1 = c1F1 + c1F2 E1 = b 1 1 - S12 1 1 + S12
antibonding
1 - S12 1
E2 = - b
Y1 = c1F1 + c1F2
bonding
1 + S12 1
E1 = b

23. 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