Molecular orbital theory describes how atomic orbitals of metal ions and ligand group orbitals interact to form molecular orbitals. For interaction to occur, the atomic orbitals must have similar energies, symmetry, and overlap appreciably. Molecular orbitals are formed through sigma and pi bonding. Sigma bonding results from the combination of metal orbitals along the bonding axis with ligand group orbitals of the same symmetry. Pi bonding occurs through overlap of metal ion t2g orbitals with ligand pi orbitals. The type of pi bonding that results depends on whether the ligand pi orbitals are filled or empty.

1. MOLECULAR ORBITAL THEORY
Dr Sakina.Z.Bootwala
Wilson College
Mumbai ,India
4/9/2020
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Dr.S.Z.Bootwala,WilsonCollege

2. MOLECULAR ORBITAL THEORY
On the basis of MOT, for the atomic orbitals to interact
to form molecular orbitals, the metal atomic orbitals
and ligand group orbitals must have
 i) similar energies
 ii) similar symmetry
 iii) appreciable overlapping.
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3. ASSUMPTIONS:
 The central metal ion provides 9 atomic orbitals ( one s
three p and five d ). These orbitals are divided into 2
groups.
 Atomic orbitals having lobes along the axis. These are s, px,
py, pz, dx
2-dy
2, dz
2
 Atomic orbitals having lobes in between the axis. These
include dxy, dyz, dzx. These orbitals are suitable for π bonding
as they can provide lateral overlap.
 Each ligand is assumed to possess atleast one filled σ
orbital.
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4. The six available σ ligand orbitals combine or
unite to form a set of six ligand group orbitals
(LGO)
To form the complex ML6 ,the six ligands
orbital approach the metal in direction of x,y
and z axes.hence ligands orditals are
represented as σx , σ-x, σy, σ-y, σz and σ-z
respectively.
Metal orbitals having lobes along the axis are
s, px, py, pz, dx
2-y
2, dz
2 are available for sigma
bonding.
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5. CONSTRUCTION OF LIGANDS GROUP ORBITALS(LGOS)
 The six σ ligands orbitals
combine linearly to form a set of
six matching symmetry orbital
like central metal orbitals, so to
overlap effectively. These six
ligand orbitals are called LGOs or
LSOs
 Algebrically these are the
combination of individual σ
ligands orbitals and are named
as Ʃa, Ʃz2, Ʃx2-y2, Ʃx, Ʃy, Ʃz
according to the symmetryof
metal d-orbitals as
a1g , eg , t1u
Metal ion LSO
a1g Σa .
t1u Σx, Σy, Σz
eg Σz
2 .Σx
2-y
2
g -gerade (centrosymmetric/even)
u - ungerade(non-centrosymmetric/uneven)
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6.  4s orbital of metal has same sign in all direction ,the linear
combination of σ ligand oritals that can overlap effectively
to s-orbital is therefore shown as
Ʃa = 1/√6 (σx + σ –x+ σy+ σ-y+ σz+ σ-z )-----a1g
 4pz orbital has one positive lobe sign and other negative
,thus σ ligands orbital generated by linear combination like
4pz orbital and it is represented as
Ʃz = 1/√2 (σz - σ-z )
 Similarly the linear combination of ligands σ orbitals to
forms LGOs, that can effectively overlap 4px & 4py metal
orbitals can be shown as above .these are represented as
t1u
Ʃz = 1/√2 (σz - σ-z ) ,
Ʃy = 1/√2 (σy- σ-y ), --------- t1u
Ʃx = 1/√2 (σx - σ-x )
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7. Ʃa = 1/√6 (σx + σ –x+ σy+ σ-y+ σz+ σ-z )----------------a1g
Ʃz = 1/√2 (σz - σ-z )
Ʃy = 1/√2 (σy- σ-y ) ------ t1u
Ʃx = 1/√2 (σx - σ-x )
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pzpypx

8.  In dx
2-y
2 metal orbital ,one opposite pair of lobes has
positive sign and other has negative sign ,the liner
combination of ligands orbital to form LGO is shown as
Ʃx2-y2 = 1/2 [( σx + σ –x) – ( σy+ σ-y)]
 The formation of LGOs which can effectively overlap dz
2
metal orbital which is located on z-axis can be shown as
Ʃz2
Ʃz2 = 1/ √12 ( 2 σz+ 2σ-z - σx - σ –x- σy- σ-y)
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9. CONSTRUCTION OF Σ MOLECULAR ORBITALS
 MO’s are formed between atomic orbitals of metal ion
and LGO/LSOs
 Each metal orbitals s, px, py, pz, dx
2-dy
2, dz
2 is then
combined with its matching symmetry LGOs to form one
bonding and one antibonding molecular orbitals
 Metal orbitals used for this purpose are;
a1g (ns), eg (n-1) dx
2-dy
2, dz
2, t1u (np) px, py, pz
t2g (n-1) (dxy, dyz, dzx) remain non bonding.
 Ligand orbitals used for this purpose are
Σa -a1g . Σz
2 .Σx
2-y
2 - eg Σx, Σy, Σz - t1u
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10. Metal orbitals Ligand group
orbitals
BMO AMO
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BMO
AMO

11. Metal
orbitals
Ligand group
orbitals
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BMO
AMO

12. CONSTRUCTION OF LIGANDS GROUP
ORBITALS(LGOS)
 Each metal orbital combines with its matching
LGO to produce bonding and antibonding
molecular orbitals. In these orbitals, the electrons
are filled on the basis of
a) Hund’s rule of maximum multiplicity and
b) Pauli exclusion principle.
 If the ligands possess π orbitals along with σ
orbitals, then the π orbitals are also considered to
provide π bonding with metal orbitals to
produce bonding as well as antibonding M.O.’s.
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13. MOLECULAR ORBITAL DIAGRAM
 a1g and t1u MO’s occupy the lower part of MO energy
level diagram. This indicates high stability for MO and
appreciable overlap between metal ion orbitals and LGO’s
 eg MO’s occupies higher level, indicate poor overlap
between eg orbitals of metal and its corresponding
counterpart ligand orbitals
 t2g orbitals do not undergo any change in the energy
hence remain non bonding.
 e*g (antibonding MO’s) remains slightly above t2g non
bonding MO’s, energy of e*g is in between (n-1)d metal
orbitals and ns metal orbitals.
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14. MOLECULAR ORBITAL DIAGRAM
 Non bonding t2g MO and e*g antibonding MO’s are at the
centre of MO energy level diagram. Both of them have metal
ion character.
 a*1g and t*1u MO’s possess higher energy above e*g
antibonding MO’s. These orbitals have higher energy
with respect to ns np atomic orbitals of metal ions.
 The energy difference Δo varies from nature of ligands
involved in the formation of metal ion complexes.
 For weak ligand field, the difference is small while in strong
ligand field the difference is large
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16. HEXAAQUO TITANIUM (III) ION [TI(H2O)6]3+
Ti (At. No. 22) -3d2 4s2 ,Ti3+ 3d1 4s0
 Number of electrons in d orbital of Ti3+ = 1
 Number of electrons of 6 ligands = 12
 Total electrons to be filled = 13
 Out of the 13 electrons, the first 12
electrons will be placed in the lower
energy six bonding MO’s. a1g t1u and eg.
These orbitals have more ligand
character.
 The remaining 1 electron occupies non
bonding t2g orbital. which explains their
weak paramagnetic nature of the
complex.
 A single band in the absorption spectrum
of this complex is attributed to a transition
from triply degenerate t2g ground state to
doubly degenerate excited e*g state.
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17. HEXAFLUORO FERRATE (III) ION [FE (F)6]3-
 Fe (At. No. 26) 3d6 4s2
 Fe3+ 3d5 4s0 4p0
 Number of electrons in d orbital of Fe3+=
5
 Number of electrons of 6 ligands = 12
 Total electrons to be filled = 17
 Out of the 17 electrons, the first 12
electrons will be placed in the lower
energy six bonding MO’s. a1g t1u and eg.
These orbitals have more ligand
character.
 The remaining 5 electrons occupies non
bonding t2g orbital and antibonding e*g
orbitals.
 Thus in this complex, 5 unpaired
electrons are present. Therefore this
complex is highly paramagnetic and it’s
a high spin complex.
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18. HEXACYANO FERRATE (III) ION [FE(CN)6]4-
 Fe (At. No. 26) 3d6 4s2
 Fe2+ -3d6 4s0 4p0
 Number of electrons in d orbital of Fe2+ = 6
 Number of electrons of 6 ligands = 12
 Total electrons to be filled = 18
 Out of the 18 electrons, the first 12 electrons
will be placed in the lower energy six
bonding MO’s. a1g t1u and eg. These orbitals
have more ligand character. The remaining 6
electrons occupy non bonding t2g orbital.
 The electron prefer to pair up in t2g rather
than occupying e*g orbital, as the energy gap
between t2g and e*g orbital is large. This
higher value of Δo is credited to the greater
extent of overlap between eg orbitals of
metal and ligand orbitals.
 Thus in this complex all electrons are
paired. Hence it is diamagnetic and a low
spin complex.
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19. MOLECULAR DIAGRAM OF THE FOLLOWING
 Hexacyano ferrate (III) ion [Fe(CN)6]3-
 Hexafluoro ferrate (II) ion [Fe(F)6]4-
 Hexafluoro cobaltate (III) ion [Co(F)6]3-
 Hexammine cobalt (III) ion [Co(NH3)6]3+
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20. Π- BONDING FOR METAL ION COMPLEXES
 If the ligands have π orbitals which may be filled or unfilled, it is
necessary to consider their interaction with non bonding t2g orbitals
of metal ion (dxy, dyz, dxz,).
 If we consider px, py, pz, orbitals with each ligand, then total number
of 18 orbitals are available, out of which 6 orbitals have been used
for σ bonding.
 The remaining 12 orbitals are grouped into 4 triply degenerate set of
orbitals belonging to the symmetry class. They are t1g, t2g, t1u, t2u.
 t1g and t2u will be the new class of ligand orbitals. These orbitals
do not have any matching orbitals for metal ions and therefore
they remain non bonding for the formation of π bonding.
 t1u set can interact with metal ion π orbitals but these have
already taken part in σ bonding. Hence they remain non
bonding.
 This leaves t2g orbitals of ligands to overlap with metal ion t2g
orbitals to establish π molecular orbitals of the complex.
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21. Three types of ligand orbitals can be
used for π bonding with metal ion
orbitals
 Simple filled p π orbitals as in
F-, Cl- , Br-
 Simple empty d π orbitals as
in phosphine and arsine
 Empty π molecular orbitals of
polyatomic ligands like CN- and
CO
 When metal orbitals combine
with ligand orbitals for the
formation of π bonding, the
overlap depends on the energy
of ligand π orbitals the energy
of metal t2g orbitals whether the
orbital is filled/not
Types of π interactions:
Effect of filled π orbitals ( L → M interaction)
Effect of empty π orbitals ( M → L interaction)
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22. A)EFFECT OF FILLED Π ORBITALS ( L → MΠ-INTERACTION)
 In an octahedral complex like
[Fe(F)6]3- the F- ion have π
orbitals as well as σ orbitals.
The σ interaction takes place as
shown earlier.
 As fluorine atom is
electronegative, the π orbitals
will be at a low energy level
than the corresponding t2g metal
orbitals.
 Under these conditions, the
bonding π orbitals ( t2g
(b) ) will
be nearer to ligand symmetry
orbitals used for π bonding.
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23. A)EFFECT OF FILLED Π ORBITALS ( L → MΠ INTERACTION)
Metal- d Ligand-
L
p(t2g)
M
Ligand p (full)
e.g. halide ion, X-
RO-
The anti bonding π orbitals ( t2g
* ) which are formed are slightly
above the t2g non bonding orbitals for metal.
 e*
g Molecular orbital remains at the same energy level as in the case
of σ bonding and the difference between t2g
* and e*
g (Δo) is
reduced
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24. B) EFFECT OF EMPTY Π ORBITALS ( M → LΠ INTERACTION)
 In an octahedral complex like
[Fe(CN)6]3- the cyanide ions have
vacant π orbitals in addition to σ
orbitals.
 As the π orbitals of ligands are
empty, they are comparatively at
higher energy level than t2g orbitals
of metal ion.
 The bonding and antibonding π
molecular orbitals are formed due to
π interactions, out of which t2g
*
molecular orbital occupies higher
energy level than e*
g , t2g bonding
molecular orbital however goes to
lower energy level at much more
stabilised Δo.
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25. B) EFFECT OF EMPTY Π ORBITALS ( M → LΠ INTERACTION
CFSE is measured between t2g bonding molecular orbital and e*g
molecular orbital due to unfilled nature of π ligand orbitals.
In general, ligands which can form π bonds in addition to σ bonds,
cause more splitting of d orbitals, i.e. More Δo than those
which form only σ bonds. Thus ligands like CN and CO
cause more splitting of d orbitals and are strong field ligands
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