The Schrodinger equation is a 2nd order differential equation that can be solved . Atomic Orbitals . Molecular Orbitals.

1. Molecular Orbital Theory
Dr. K. Shahzad Baig
Memorial University of Newfoundland (MUN)
Canada
Petrucci, et al. 2011. General Chemistry: Principles and Modern Applications. Pearson Canada Inc., Toronto, Ontario.
Tro, N.J. 2010. Principles of Chemistry. : a molecular approach. Pearson Education, Inc.

2. Molecular Orbital Theory
Molecular orbital (MO) theory is a more sophisticated
quantum mechanical model of bonding in molecules that
can be applied successfully to both simple and complex
molecules

3. Atomic Orbitals
Heisenberg Uncertainty Principle states that it is impossible to define what time and
where an electron is and where is it going next.
Since it is impossible to know where an electron is at a certain time, a series of
calculations are used to approximate the volume and time in which the electron can be
located. These regions are called Atomic Orbitals.
These are also known as the quantum states of the electrons.
Only two electrons can occupy one orbital and they must have different spin states, ½
spin and – ½ spin (easily visualized as opposite spin states).

4. Atomic Subshells
These are some examples of atomic orbitals:
s Orbirals:
(Spherical shape) There is one S orbital in an s subshell. The electrons can be located
anywhere within the sphere centered at the atom’s nucleus.
p Orbitals:
(Shaped like two balloons tied together) There are 3 orbitals in a p subshell that are denoted
as p x , p y , and p z orbitals. These are higher in energy than the corresponding s orbitals.
d Orbitals:
The d subshell is divided into 5 orbitals (d xy , d xz , d yz , d z 2 and d (x2 -y2 ) . These
orbitals have a very complex shape and are higher in energy than the s and p orbitals.

5. Wave particle duality describes electromagnetic radiation and matter (such as electrons,
protons...
Schrodinger equation:
Accounts for wave particle duality, the motion of electrons in an atom, and quantized
nature of atomic structure
𝑇𝑜𝑡𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝐾𝐸 + 𝑃𝐸
𝐸 𝜑 = 𝐾𝐸𝜑 + 𝑃𝐸 𝜑
𝐸 𝜑 = −
ℎ3
2𝜇
𝛻3 𝜑 −
𝑍𝑒2
4𝜋𝜖0 𝑟
𝜑

6. The Schrodinger equation is a 2nd order differential equation that can be solved exactly
for H2, but only numerically for many electron atoms and molecules. Only certain
solutions produce physically acceptable results quantization
Wave function:
Solution to the Schrodinger equation, describes the behavior of an electron moving in
x, y, and z directions –where it is and what it is doing.
Each wave function that is a solution to the Schrodinger equation is an atomic orbital.
Quantum numbers uniquely label each orbital
The probability of finding an electron at a particular point in space is proportional to the
square of the wave function at that point (x, y, z)
Nodes: Regions where wave functions pass through zero

7. 1𝑠 𝐴 + 1 𝑠 𝐵
1𝑠 𝐴 − 1 𝑠 𝐵
Constructive interference corresponds to adding the two mathematical functions (the
positive sign puts the waves in phase), while
Destructive interference corresponds to subtracting the two mathematical functions
(the minus sign puts the waves out of phase).
As the atoms approach, the two wave functions combine; they do this by interfering
constructively or destructively.

8. Electron probability or electron charge density
The electron probability or electron charge density in the 𝜎1𝑠orbital is
1𝑠 𝐴 + 1𝑠 𝐵
2
The square has an extra term [2 x 1𝑠 𝐴 1𝑠 𝐵] which is the extra charge density between the
nuclei.
A high electron charge density between atomic nuclei reduces repulsions between the
positively charged nuclei and promotes a strong bond. This bonding molecular orbital,
designated 𝜎1𝑠, is at a s lower energy than the 1s atomic orbitals.
The electron charge density in the molecular orbital formed by destructive interference of
the 2, 𝜎1𝑠
∗
orbital is reduced 1𝑠 𝐴 − 1𝑠 𝐵
2
The square has minus term [2 x 1𝑠 𝐴 1𝑠 𝐵] which reduces the charge density between the
nuclei.
A low electron charge density between atomic nuclei promotes repulsions between the
positively charged nuclei. This bonding molecular orbital, designated 𝜎 ∗1𝑠, is at the higher
energy than the 1s atomic orbitals.

10. Basic Ideas Concerning Molecular Orbitals
1. The number of molecular orbitals (MOs) formed is equal to the number of atomic
orbitals (AOs) combined.
2. Of the two MOs formed when two atomic orbitals are combined, one is a bonding
MO at a lower energy than the original atomic orbitals. The other is an antibonding
MO at a higher energy
3. e- fill the lowest energy MO first (aufbau process)
4. Maximum 2 e- per orbital (Pauli Exclusion Principle)
5. In ground-state configurations, electrons enter MOs of identical energies fill singly
before they pair up (Hund’s Rule).
𝐵𝑜𝑛𝑑 𝑂𝑟𝑑𝑒𝑟 =
𝑛𝑜. 𝑜𝑓 𝑒−
𝑖𝑛 𝑏𝑜𝑛𝑑𝑖𝑛𝑔 𝑀𝑂𝑠 − 𝑛𝑜. 𝑜𝑓 𝑒−
𝑖𝑛 𝑎𝑛𝑡𝑖𝑏𝑜𝑛𝑑𝑖𝑛𝑔 𝑀𝑂𝑠
2

11. Applying Bond Order

12. Molecular Orbitals of the Second-Period Elements
First period use only 1s orbitals.
Second period have 2s and 2p orbitals available.
p orbital overlap:
End-on overlap is best – sigma bond (𝜎2𝑝 𝑎𝑛𝑑 𝜎2𝑝
∗
).
Side-on overlap is good – pi bond (𝜋2𝑝 𝑎𝑛𝑑 𝜋2𝑝
∗
)
2𝑝 𝑥
2𝑝 𝑦
2𝑝 𝑧
𝜋2𝑝𝑦
𝜋2𝑝
𝜎2𝑝

14. Formation of bonding and antibonding orbitals from 2p orbitals

15. the second-shell molecular orbitals of a diatomic molecule in the order of increasing energy
the orbitals are at a lower energy than𝜋2𝑝 𝜎2𝑝
If we assign the eight valence electrons of the molecule C2. We start with the lowest energy
orbitals filled. Then we add electrons, in order of increasing energy, to the available molecular
orbitals of the second principal shell.
Experiment shows that the
molecule is diamagnetic, not
paramagnetic, and the
configuration just described is
incorrect.

17. The molecular orbital diagram of CO
As a result of 2s and 2p orbital mixing,
the 3rd σ –orbital, 3σ , has an orbital
energy greater than that of the π 2p
orbitals
therefore, the highest occupied molecular orbital (HOMO) is 3σ
The Zeff of C is much lower than that of O,
and so the 2s and 2p orbitals of C are
higher in energy than those of O.
The 2s-2p separation in C is much lower
than in O, and so we expect 2s-2p mixing
to occur in the σ-orbitals.

18. 1𝜎2𝑠
2
𝜎2𝑠
∗2
1𝜋4 𝜎2𝑝
2
1𝜎2 2𝜎2 1𝜋4 3𝜎2 2𝜋1
As we have lost the designations, such as σ2s through 2s and 2p mixing. The
configuration of CO (ignoring the orbitals) can therefore be written as follows:
1𝜎2 2𝜎2 1𝜋4 3𝜎2
As an approximation, not ignoring the orbitals, the configuration would be :
Considering the free radical NO, the extra electron is in the 2π-orbital; thus, the
configuration is

19. Example :
Write the molecular orbital for the cyanide ion, CN- and determine the bond order for this
ion.
Solution
The number of valence electrons to be assigned to the molecular orbitals is 10
i.e. 4 +5 + 1 = 10
Because of the 2s-2p mixing we use the modified order of molecular orbitals and write
the configuration of CN- as
1𝜎2 2𝜎2 1𝜋4 3𝜎2
Or, as an approximation,
2𝜎2𝑠
2
𝜎2𝑠
∗2
1𝜋4 𝜎2𝑝
2
which gives a bond order of (2 + 4 + 2 – 2) / 2 = 3
𝜋4
∗
𝜋5
∗
𝜋6
∗
𝜋1
𝜋2 𝜋3