Molecular

Orbital
Theory
Dr. Ashima Srivastava,
Dept of Chemistry, JSSATE, NOIDA
SALIENT FEATURES OF MOLECULAR ORBITAL THEORY (MOT)
1. When atoms combine to form molecules, their individual atomic orbitals
lose their identity and forms new orbitals called molecular orbitals.
2. The shapes of molecular orbitals depend upon the shapes of combining
atomic orbitals.
3. The number of molecular orbitals formed is the same as the number of
combining atomic orbitals. Half the number of molecular orbitals formed will
have lower energy than the corresponding atomic orbital, while the
remaining molecular orbitals will have higher energy. The molecular orbital
with lower energy is called bonding molecular orbital and the one with
higher energy is called anti-bonding molecular orbital. The bonding
molecular orbitals are represented as σ (Sigma), π (pi) and the corresponding
antibonding orbitals are denoted as σ*, π*.
4. The electrons in a molecule are accommodated in the newly formed
molecular orbitals. The filling of electrons in these orbitals follows Aufbau's
principle, Pauli's exclusion principle and Hund's rule as in the case of filling of
electrons in atomic orbitals.
5. Bond order gives the number of covalent bonds between the two
combining atoms. The bond order of a molecule can be calculated using the
following equation,

Where
Nb = Total number of electrons present in the bonding molecular orbitals
Na = Total number of electrons present in the antibonding molecular orbitals and
A bond order of zero value indicates that the molecule doesn't exist.
CONDITIONS FOR LINEAR COMBINATION OF ATOMIC ORBITALS

1. Same Energy of Combining Orbitals

2. Same Symmetry about Molecular Axis
3. Effective Overlap between Atomic Orbitals
Linear Combination of Atomic Orbitals (LCAO)
Molecular orbitals are formed by the combination of atomic orbitals of bonded
atoms.
In wave mechanics atomic orbitals are expressed by wave functions (ψ).
The wave functions are obtained as the solutions of Schrodinger wave equation.
Just like atomic orbitals, Schrodinger wave equation can be written to describe
the behaviour of the electron in molecules also. However, because of the
complex nature of the Schrodinger wave equation, it may not be easy to solve it
for molecules. Thus, in view of it, for the sake of convenience an approximate
technique to obtain the wave functions for molecular orbitals was applied. This
approximate method is known as Linear Combination of Atomic Orbitals Method
(LCAO method).
Application LCAO method to homonuclear diatomic molecules

Let us consider two atoms of hydrogen in the molecule as A and B. Each

hydrogen atom has one electron in 1s orbital in the ground state. These
atomic orbitals may be represented by wave functions ψA and ψB .Then
according to LCAO method, the molecular orbitals in H2 molecule are given
by linear combination (addition or subtraction of wave functions of
individual atoms) of ψA and ψB as shown below :
ψMO = ψA ± ψB
ψb = ψA + ψB
ψa = ψA − ψB
The molecular orbital ψb formed by the addition overlap (constructive
interference of waves) of atomic orbitals is called bonding molecular orbitals
and the molecular orbital ψa formed by subtraction overlap (destructive
interference of waves) of atomic orbitals is called antibonding molecular
orbital.
LCAO of Hydrogen Molecule (1s+1s combination)
Combination of s-atomic orbitals
The combination of 1s orbitals of hydrogen atoms to form molecular orbitals
has been shown in Fig
Combination of p-atomic orbitals
Overlapping of two pz atomic orbitals

by the linear additive and subtractive combination to form an σ2pz BMO and σ*2pz ABMO
Combination of px and px orbitals

Overlapping of two px AO’s

by linear additive and subtractive combination to give π2px BMO and π*2px ABMO.
Difference between Bonding and Antibonding MO’s
 Energy Level Diagram for Molecular Orbitals
Energy levels of such molecular orbitals have been found out experimentally
by spectroscopic studies. The order of rising energy in case of the diatomic
homo-nuclear molecules of first and second period of the periodic table is
as shown below:
Molecular orbital
energy level diagram
for diatomic
homo-nuclear
molecules of first and
second period (i.e.,
except O2, F2 and so
on.)
Molecular orbital
energy level diagram
for diatomic
homo-nuclear
molecules of O2
and other heavier
elements
Why the order of energy of molecular orbitals is different for molecules lighter than O2 , F2
(N2, C2, B2).

In case of homonuclear diatomic molecules of second row elements, if the

energy difference between 2s and 2p atomic orbital is large, then the 2s
orbital of one atom interacts very slightly with the 2pz orbital of other atom.
Though the symmetry is proper for overlap, the energy difference is very
high and the overlap is not effective. This is true for diatomic molecules such
as O2 and F2. Thus the MO’s are composed of two 2s atomic orbitals and the
order of energy is as shown in Fig.
Why the order of energy of molecular orbitals is different for molecules lighter than O2 , F2
(N2, C2, B2).

In case of elements upto nitrogen, in the second row, the energy difference
between 2s and 2p atomic orbitals is small. As the two orbitals are of same
symmetry their overlap cannot be ignored. As a result of interaction between s
orbital of one atom with the p orbital of the other and vice versa, s2s and s*2s
do not retain their pure s character and s2pz and s*2pz do not retain their pure p
character.
Why the order of energy of molecular orbitals is different for molecules lighter than O2 , F2
(N2, C2, B2).

Due to this mixing, the energies of all the four MO’s change in such a way that
resulting MOs s2s and s*2s which also contain some p character, become more
stable and thus lowered in energy.
Similarly MOs s2pz and s*2pz which also contain some s character become less
stable and their energy is raised. Thus there is a upward displacement of s2pz so
that it lies above π2px and π2py.
Since π2p orbitals are not involved in mixing, the energies of π2px and π2py
remain unchanged
Change in
energies of
MOs due to the
mixing of
orbitals of same
symmetry
HOMONUCLEAR DIATOMIC MOLECULES
Bond Order= 2-
Hydrogen molecule, H2
0/2=1
Does molecule He molecule exist
The electronic configuration of helium atom is 1s2. In helium molecule, there are four electrons
which are arranged as s1s2, s*1s2. Thus, stabilizing effect of bonding orbitals is cancelled out by
the destabilizing effect of antibonding orbitals and the molecule does not exist. .
Helium molecule ion, He2+

Bond order =
2 -1 / 2= 0.5
Lithium molecule, Li2

Bond order =
4 -2 / 2= 1
Beryllium molecule, Be2

Electronic configuration of beryllium atom: 1s2, 2s2

Total number of electrons in beryllium molecule: 8

MO configuration of beryllium molecule: s1s2, s*1s2, s2s2, s*2s2

. This molecule does not exist. Bond order =

4 -4 / 2= 0
Boron molecule, B2

Electronic configuration of boron atom: 1s2, 2s2, 2p1

Total number of electrons in boron molecule: 10

MO configuration of boron molecule: s1s2, s*1s2, s2s2, s*2s2, π2px1 = π2py1 Bond order =
. It is paramagnetic in nature as it has two unpaired electrons in molecular orbitals
6 -4 / 2= 1
MO diagram
of Boron
molecule
 MO Diagram
of Nitrogen
Molecule

Bond order =
10 -4 / 2= 3

Diamagnetic
N2+ ion: N2+ has one electron less than N2 molecule. This electron will be
lost from s2pz orbital.
Hence the MO configuration will be: s1s2, s*1s2, s2s2, s*2s2, π2px2 = π2py2,
s2pz1
The ion is paramagnetic due to the presence of unpaired electron and is
less stable than N2 molecule.
N2- ion: N2- has one electron more than N2 molecule. This electron will be
added in π*2px orbital.
Hence the MO configuration will be: s1s2, s*1s2, s2s2, s*2s2, π2px2 = π2py2,
s2pz2, π*2px1
 MO
Diagram of
Oxygen
Molecule
Bond order =
10 -6 / 2= 2

Paramagnetic
O2+ ion: O2+ has one electron less than O2 molecule. This electron will be
lost from π*2p orbital.
Hence the MO configuration will be: s1s2, s*1s2, s2s2, s*2s2, s2pz2, π2px2
= π2py2 π*2px1
The ion is paramagnetic due to the presence of unpaired electron and is
more stable than O2 molecule.
O2- ion: O2- has one electron more than O2 molecule. This electron will
be added in π*2px orbital.
Hence the MO configuration will be: s1s2, s*1s2, s2s2, s*2s2, s2pz2, π2px2
= π2py2 π*2px2 = π*2py1
.
HETERONUCLEAR DIATOMIC MOLECULES
ISSUE for drawing the MO diagrams of heteronuclear diatomic molecules:
Where do we place the atomic orbitals of an atom relative to atomic orbitals
on other atom?

The answer comes from our understanding of electronegativity

The more electronegative element's orbitals are placed lower on the MO
diagram than those of the more electropositive element.
Examples: NO, CO, HF
Explanation
Electronegative atom is the one that can attract the electrons (negative
charge) towards itself more efficiently (i.e., closer to the positive charge
(nuclei of it's own). Therefore comparatively on an average the
electrons of an electronegative atoms are more closer to it's own
nuclei than the one which is less electronegative, so the electrons in
electronegative atoms are at lower energy than when compared with
the electrons in the less electronegative atom, so electronegative
atoms are kept at lower position in MO diagram, showing that they have
lower potential energy than the electropositive atom.
Energy level diagram for Heteronuclear molecule

Since there is energy difference between the two set of atomic orbitals, they
interact less strongly and the energy lowering as a result of overlap is less
pronounced in heteronuclear molecules than in homonuclear molecules
Nitric oxide molecule, NO:
The electronic configurations of N and O atoms are:
N = 1s22s22p3
O = 1s22s22p4
There are total of 15 electrons to be accommodated in the MO’s of the
molecule. The bonding MO’s are closer to oxygen and antibonding MO’s
are closer to nitrogen because oxygen is more electronegative than
nitrogen.
MO configuration of NO:

s 1s2, s*1s2, s 2s2, s* 2s2, s 2pz2, p2px2 = p2py2, p*2px1 = p*2py0

MO Diagram of NO
Electronic configuration, Bond Order & Magnetic
Property of NO
MO configuration of NO:
s 1s2, s*1s2, s 2s2, s* 2s2, s 2pz2, p2px2 = p2py2, p*2px1 = p*2py0
Bond Order 10-5 /2= 2.5
Magnetic Property: Paramagnetic since it contains one unpaired electron
Carbon Monoxide molecule, CO

The electronic configurations of C and O atoms are:

C = 1s22s22p2
O = 1s22s22p4
There are total of 14 electrons to be accommodated in the MO’s of the
molecule. The bonding MO’s are closer to oxygen and antibonding MO’s are
closer to carbon because oxygen is more electronegative than carbon.
MO configuration of CO may be written as:
s 1s2, s*1s2, s 2s2, s* 2s2, p2px2 = p2py2 , s 2pz2 Bond order:
The above configuration is incorrect 10-4 /2=3
If the MO configuration for of CO is correct, then bond order of CO+ should be reduced
to 2.5 and bond length should be increased.
Bond Length of CO: 1.128 Å Bond Length of CO+: 1.115 Å
The most likely explanation for the decrease in bond length (increase in bond order) is
that the electron must have been removed from antibonding molecular orbital. This
can be explained from the MO configuration and molecular orbital diagram given
below:
s 1s2, s*1s2, s 2s2, , p2px2 = p2py2, s 2pz2, s* 2s2
Hydrogen fluoride molecule, HF
The electronic configurations of hydrogen and fluorine atoms are:
H= 1s1
F= 1s22s22p5
The formation of HF molecule takes place by linear combination of H (1s)
atomic orbital with 2pz orbital of fluorine. Spectroscopic evidence shows
that the energies of 1s and 2s electrons of fluorine are very low and they
do not take part in bonding. Out of the three 2p orbitals, only 2pz orbital is
able to combine with s-orbital because of symmetry reasons
Energy level diagram for HF
MO configuration of HF:
1s2, 2s2, s spz2, 2px2, 2py2, s*spz0
Note that sigma spz (BMO) is the only MO that contains two
electrons. sigma*spz (ABMO)is empty.
Bond order: 2-0/2 = 1
 Metallic Bonding
Metallic bonding is the a type of chemical bonding between
atoms within metals. It is the electrostatic attractive forces
between the delocalized electrons, called conduction
electrons and the positively charged metal ions. Metallic
bonding accounts for many physical properties of metals, such
as strength, malleability, ductility, thermal and electrical
conductivity, opacity and lustre.
Why is metallic bonding different from ionic or covalent bonding?

Metallic bonds are formed by attraction between metal ions

and delocalized or free electrons
Whereas
ionic bond are formed by transfer of electrons
from one atom to another and
in covalent, sharing of electrons takes place
Metallic Bonding is non-directional !!
Bonds in metals are non-directional because the electrons are NOT shared with one
atom in one direction; however, they are shared with many other neighbouring
atoms in all directions.
Covalent bonds are directional since the atoms prefer specific orientations in space
relative to one another. As a result, molecules in which atoms are bonded covalently
have definite shapes.
Ionic bonding is non-directional because an ion has the same attraction from all
directions for an ion of opposite charge.
and the number of anions surrounding a cation is limited by the charges of the ions,
their sizes, and the efficiency of the lattice packing.
Theories of Metallic Bonding
There are theories to explain most of the properties of metals.
These theories do not involve directional bond formation.
1. Free electron theory or Electron Sea model for metallic
bonding
2. Band theory for metallic bonding
Free electron theory or Electron sea model
The main features of this model are:
1. A metal atom is supposed to consist of two parts, valence electrons and the
remaining part (nucleus and inner shells) which is called kernel.
2. The metallic crystal consists of closely packed metal atoms in three
dimensions. The kernels of the metal atoms occupy the fixed positions
called lattice sites while the space between the kernels is occupied by
valence electrons. The arrangement of kernels and valence electrons is
shown in the Fig
 Band theory for metallic bonding
Based on molecular orbital theory:
Overlapping of atomic orbitals of metal atoms give rise to molecular
orbitals.
It can be explained by taking an example of a sodium metal. Sodium atom
has electronic configuration 1s22s22p63s1 and a sodium crystal is formed
by gradual addition of sodium atoms.
Now let us see the Molecular Orbital model of sodium
For Nan, n/2 bonding and n/2 antibonding molecular orbitals are formed.
All n-valence electrons of sodium (Nan) are accommodated in bonding
molecular orbitals while n/2 antibonding molecular orbitals are vacant.
It has been observed that the separation of molecular orbitals decreases
as the number of sodium atoms increases which are seen as closely
spaced bands. So the theory is called the band theory.
Molecular Orbital model of sodium (Band Theory)

The band formed by valence electrons (i.e. 3s1) is called valence band and that formed by 3p (vacant)
of sodium is called conduction band. There is no energy gap between valence
band and conduction band (overlapped).
Molecular Orbital model of Magnesium
Molecular Orbital model of Magnesium
Molecular Orbital model of Magnesium
The band formed by valence electrons (i.e. 3s2) is called valence band and is
completely filled and that formed by 3p (vacant) of Mg is called conduction
band. There is no enegy gap between the two as they are overlapping.
THINK ABOUT THE MOLECULAR ORBITAL MODEL OF
ALUMINIUM !!!!!!!
Band Theory in Solids
Conductors
Metals are conductors. There is no band gap between their valence and
conduction bands, since they overlap. There is a continuous availability of
electrons in these closely spaced orbitals. Hence, electrons are easily
transferred from valence band to conduction band
Semiconductors
Semiconductors have a small energy gap (0.7-1.0 eV) between the valence
band and the conduction band. Electrons can make the jump up to the
conduction band, but not with the same ease as they do in conductors. This
energy gap leads to insulation at ordinary conditions, but on heating or by
doping with certain elements (B or As) it functions as conductor
Insulators
In insulators, the band gap (larger than 3 eV) between the valence band
the conduction band is so large that electrons cannot make the energy
jump from the valence band to the conduction band.
Types of Semiconductors