INTRODUCTION :
The molecular orbital theory was proposed and developed mainly by Hund, Mulliken, Hückel and Lennard- Jones in between 1929-1932 A.D. This theory was mainly used first to explain the paramagnetic behavior of oxygen (as per VBT, oxygen molecule is diamagnetic), but later on it was found to be very useful regarding explanation of - relative bond energies , magnetic behavior(diamagnetic/paramagnetic), Ionization energies, reactivity etc. for different molecular species. In this approach, we have to develop molecular orbitals from the combining atomic orbitals. There are two well-known methods available for such combination of monocentric atomic orbitals to produce multicentric molecular orbitals. These are- LCAO- Linear Combination Of Atomic Orbitals and United Atom Method. In this context, we mainly discuss on LCAO method as it is simpler and has many similarities with our Valence Bond method.
LCAO –LINEAR COMBINATION OF ATOMIC ORBITALS:
Salient features of LCAO- Linear Combination Of Atomic Orbitals :
1. When two (or more) Atomic Orbitals (AO) combine, they lose their identity and form Molecular Orbitals.(MO). Thus MOs are polycentric but AOs are monocentric. 2. The AOs which can combine to form MOs must satisfy the following three conditions: 1. ENERGY : The combining AOs(wave functions) should have comparable energy. 2. OVERLAP : The AOs must have proper orientation to undergo appreciable overlapping between them. 3. SYMMETRY : The lobes of combining AOs must have the same symmetry with respect to the bond axis. 3. The number of MOs formed is equal to the total number of combining AOs. 4. Depending upon the overlap integral(S), the MOs are classified as: 1. Bonding Molecular Orbital (BMO) → S > 0 2. Non-Bonding Molecular Orbital (NBMO) → S = 0 3. Antibonding Molecular Orbital (ABMO) → S < 0 5. After construction of MOs, the electrons are filled in these MOs as per Aufbau principle, Hund’s rule and Pauli’s exclusion principle. 6. Just like AOs, the MOs are also defined by four quantum numbers. The principal quantum number(n) and azimuthal quantum number(l) are retained from the combining AOs. The magnetic quantum number(λ) are designated as follows:
Value of λ Name of molecular orbital (MO) 0 + 1 Π + 2 Δ The spin quantum number can have two values- (+1/2) or (-1/2) depending upon the direction of spinning motion of electrons. 7. The bonding MOs concentrate the electron density between the nuclei to stabilize the system and antibonding MOs remove the electron density from the space between the nuclei to stabilize the system.
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DIFFERENCES BETWEEN ATOMIC AND MOLECULAR ORBITALS :
Sl Atomic Orbital Molecular Orbital 1 They are monocentric(under the influence They are polycentric(under the influence of of one nucleus). more than one nucleus). 2 They are represented by s,p,d,f etc. They are represented by , π, δ etc. They may be bonding or antibonding in nature. 3 They have simple shapes They have complex shape.
FORMATION OF MOLECULAR ORBITALS BY LCAO :
The formation of MOs can be explained on the basis of LCAO keeping in mind that an atomic orbital is an electron –wave having wave function Ψ. Suppose ΨA and ΨB represents the amplitudes of the two atoms A and B respectively. When these two atomic orbitals of atom A and B approach each other, following two situations may appear-
Case-I : The two waves may undergo constructive interference(or they are in-phase). In that case the amplitude of resulting wave(Φ1) can be determined by the addition of the individual wave amplitudes.
Φ1 = ΨA + ΨB
Case-II : The two waves may undergo destructive interference(or they are out-of-phase). In that case the amplitude of resulting wave(Φ2) can be determined by the substraction of the individual wave amplitudes.
Φ2 = ΨA - ΨB
We know, the probability of finding electron is proportional to the square of amplitude of the wave. Therefore,
2 2 2 2 2 2 2 Φ1 = (ΨA + ΨB) = ΨA + ΨB + 2ΨA ΨB (i.e. Φ1 > ΨA + ΨB )
2 2 2 2 2 2 2 Φ2 = (ΨA - ΨB) = ΨA + ΨB - 2ΨA ΨB (i.e. Φ2 < ΨA + ΨB )
Thus, by combination of two atomic orbitals, two new molecular orbitals are formed. The molecular orbital formed by constructive interference is known as bonding molecular orbital and the molecular orbital formed by destructive interference is known as antibonding molecular orbital.
The probability of finding electron in bonding MO increases (by an amount of 2ΨA ΨB) i.e. most of the electron density is located in between the nuclei of the two bonded atoms(A and B). The probability of finding electron in antibonding MO decreases (by an amount of 2ΨA ΨB) i.e. most of the electron density is located away from the space between the nuclei of the two bonded atoms(A and B).
constructive destructive
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COMBINATION OF ATOMIC ORBITALS :
Combination of different AOs produces different molecular orbitals. It is clear that overlapping with same signed lobes will produce a bonding interaction and overlapping with oppositely signed lobes will produce anti-bonding interaction.
Sl no Type of interaction/overlapping Nature of molecular orbital formed 1 1 lobe + 1 lobe interaction Sigma ( ) 2 2 lobes + 2 lobes interactions Pi (π) 3 4 lobes + 4 lobes interactions Delta (δ) 4 6 lobes + 6 lobes interactions Phi (Φ)
1. The s orbitals can show only -type interaction. 2. The p-orbitals can show –type and π-type interactions. 3. The d orbitals can show , π, and δ-type interactions 4. The f orbitals can show ,π,, δ and Φ-type interactions. Some combinations of atomic orbitals are given below :
ΨA ΨB Type of MO produced s S s P s D p P , π p D , π , δ
SYMMETRY OF MOLECULAR ORBITALS :
If any molecular orbital possseses centre of symmetry/centre of inversion, then that MO is labelled as ‘g’ or ‘gerade’.For such MOs, the wave functions are even-type and follow the condition:
Ψ(x,y,z) = Ψ(-x,-y,-z)
If any molecular orbital does not possses centre of symmetry/centre of inversion, then that MO is labelled as ‘u’ or ‘ungerade’.For such MOs, the wave functions are odd-type and follow the condition: Ψ(x,y,z) = -Ψ(- x,-y,-z)
Ψ(x,y,z) = - Ψ(-x,-y,-z)
It can be noted that for homonuclear diatomic molecules,following designations can be acceptable-
Types of MO Centre of inversion Designation of MO - bonding Present g –antibonding Absent *u π –bonding Absent πu * π -antibonding Present π g
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Non-bonding molecular orbitals (NBMO) :If the combining AOs much differ in energy, or don’t have proper symmetry/orientation for overlapping, then those AOs will not undergo combination but these AOs will experience the nuclear charge of more than one nuclei. As a result they will remain as non-bonding MOs and these are also polycentric in nature.
Sigma- MO Pi-MO It is formed by overlapping of AOs along the It is formed by the lateral/sideways overlapping of internuclear axis. AOs along perpendicular to the internuclear axis. Due to head-on overlapping, the extent of Due to lateral overlapping, the extent of overlapping overlapping is maximum. is comparatively lower. Can not participate in delocalization. Can participate in delocalization. Magnetic quantum number, λ = 0 Magnetic quantum number, λ = + 1
DIAGRAMMATIC REPRESENTATION OF FORMATION OF MOLECULAR ORBITALS :
Formation of sigma-MOs Formation of pi-MOs
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Formation of non-bonding MOs (with zero overlapping/interactions) Formation of delta-MO
ENERGY OF MOLECULAR ORBITALS :
It is important to note that compared with the energy of the combining AOs, the energy of the ABMO is raised by an amount greater than the amount by which the energy of the BMO gets lowered, although the summation of the energies of the MOs remain same with the total energy of the combining AOs. This is because, in bonding MO, the electron density in the internuclear region is high. As a result the nuclei are shielded are from each other. Therefore the repulsions between the nuclei are small. In contrast, for antibonding MOs, the electron density between the nuclei is very low and strong repulsion operates between the nuclei. Thus the energy of ABMOs rise slightly. Hence we can conclude that, the destabilization caused by ABMOs are relatively higher than the stabilization caused by the BMOs.
The filling of electrons in the MOs are done by considering-
1. Aufbau principle. 2. Hund’s rule of maximum multiplicity. 3. Pauli’s exclusion principle.
i.e. maximum number of electron that can be accommodated in each MO = 2.
If total number of electrons in BMOs= NB
If total number of electrons in ABMOs = NA
Then Bond Order (B.O) =(NB – NA)/2
Higher the bond order, higher will be the Note: stability of the molecule.
1. If the number of electrons in BMOs are greater than that of ABMOs, the molecule will be stable. 2. If the number of electrons in BMOs are lower than that of ABMOs, the molecule will be unstable and will not exist. 3. If the number of electrons in BMOs are equal to that of ABMOs, then the molecule will be unstable and it will not exist. This is due to the fact that the destabilization caused by ABMOs are relatively higher than the stabilization caused by the BMOs.
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4. Bond order signifies the number of bonds formed between two atoms. Fractional bond order indicates delocalization.
Bond order ∝ bond energy or bond dissociation energy ∝ bond strength ∝ 1/bond length
5. If the molecule contains unpaired electron in its MOs,it will be paramagnetic. If it has no unpaired electron in any of its MOs, it will be diamagnetic.
Q. Sketch the overelap of the following orientation of orbitals(z-axis taken as inter nuclear axis) and comment on the nature of the MO formed:
1. s + pz 2 2. pz + dz 3. px + dzx 4. py + dyz 5. dxz + dxz 6. dyz + dyz
7. dxz + pz 2 8. s + dz
9. s + dyz
CONCEPT OF FRONTIER MOLECULAR ORBITALS(FMO):
During filling of electrons in the MO diagram obeying Hunds rule, Aufbau principle and Pauli’s exclusion principle, the MO in which the last electron enters in known as Highest Occupied Molecular Orbital (HOMO) and the MO which remains vacant and addition of one more electron will lead to occupancy of that MO, is known as Lowest Unoccupied Molecular Orbital(LUMO). Sometimes a single MO can behave as both HOMO and LUMO. Such an MO is known as Singly Occupied Molecular Orbital(SOMO).
These HOMO and LUMO (or SOMO) are collectively known as Frontier Molecular Orbitals(FMO). These are very much important in predicting the reactivity and chemical nature of the corresponding molecule.
MO DIAGRAM FOR HOMONUCLEAR DIATOMIC MOLECULES :
1. H2 molecule: Hydrogen molecule is formed by combination of two hydrogen atoms. As each hydrogen atom
contains one electron,total number of electrons present in H2 molecule is 2. These two 1s orbitals
from each hyderogen atom combine to form two molecular orbitals : g(1s) which is a bonding MO * (of lower energy) and u (1s), which is an antibonding MO(of higher energy). As each molecular orbital can accommodate two electrons(in accordance with Pauli’s exclusion
principle), the two electrons of H2 molecule will occupy the lowest energy g(1s) MO. 2 The electronic configuration of H2 molecule is - [ g(1s)]
Bond order of H2 molecule – (2-0)/2 = 1 (i.e one single bond is present between two H atoms in H2 molecule)
As no unpaired electron is present in the MOs of H2, it is a diamagnetic molecule.
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Species Electronic configuration Bond order Magnetic nature 2 H2 [ g(1s)] 1 Diamagnetic + 1 H2 [ g(1s)] 0.5 Paramagnetic - 2 * 1 H2 [ g(1s)] [ u (1s)] 0.5 Paramagnetic
+ - + From the bond order data, it is clear that, H2 is more stable compared to H2 and H2 . In between H2 - - + - and H2 , H2 is slightly less stable than H2 . This is because, H2 contains one electron in antibonding orbital and the destabilization caused by antibonding electron is more than the stabilization caused + - the bonding electrons. Hence the order of stability runs as : H2 > H2 > H2
2. He2 molecule :
He2 molecule does not exist due to following reasons:
The bond order of the molecule is zero.
The presence of antibonding electrons causes more destabilization than the stabilsation caused by the bonding electrons.
He is a noble gas having highly stable full filled electronic configuration.
The dissociation of He molecule is entropically favored. 2
He2 → 2 He
Species Electronic configuration Bond order Magnetic nature 2 2 He2 [ g(1s)] [ *u(1s)] 0 ------+ 2 1 He2 [ g(1s)] [ *u(1s)] 0.5 Paramagnetic - 2 * 2 1 He2 [ g(1s)] [ u(1s)] [ g(2s)] 0.5 Paramagnetic
3. Li2 molecule :
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2 The electronic configuration of Li2 molecule is- [ g(1s)] 2 2 [ *u(1s)] [ g(2s)]
Here, the inner closed shell do not take part in bonding and hence the above configuration can be written as :
2 KK[ g(2s)]
2 Where KK represents the closed shell structure [ g(1s)] 2 [ *u(1s)] . As these shells do not take part in bonding, they are known as non bonding molecular orbitals. (NBMOs)
Note : In the vapour phase, lithium exists as discrete diatomic molecules and hence is diamagnetic in nature.
4. Be2 molecule : 2 2 2 2 This molecule will have electronic configuration [ g(1s)] [ *u(1s)] [ g(2s)] [ *u(2s)] . Just like
He2 molecule, Be2 molecule does not exists.
Note :
For the homonuclear diatomic molecules like B2 , C2 , N2 , O2 , F2 , Ne2 , the 2p orbitals also are involved the formation of molecular orbitals. The combination of two p orbitals leads to the formation of both and π MOs. Since bonds are stronger than π bonds, we expect that the energy of bonding MOs must be lower than that of π bonding MOs.
This expectation fits well for the molecules like O2 , F2 but for the lighter elements like N2 , C2 , B2 this becomes invalid. That’s why we can find two different energy level digrams as follows :
(a) MO energy level diagram for O2 , F2 and their ions.
(b) MO energy level diagram for B2 , C2 , N2 and their ions. This energy level diagram can be represented as :
(a) [ g(1s)] < [ *u(1s)] < [ g(2s)] <
[ *u(2s)] < [ g(2pz)] < [πu(2px)] = * * [πu(2py)] < [πg (2px)] = [πg (2py)] <
[ *u(2pz)]
(b) [ g(1s)] < [ *u(1s)] < [ g(2s)] <
[ *u(2s)] < [πu(2px)] = [πu(2py)] * * <[ g(2pz)] < [πg (2px)] = [πg (2py)]
< [ *u(2pz)]
In the diagram the old convention for designating the MOs are used.
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Reason For The Difference :
1. Symmetry Interaction : According to the non-crossing rule, orbitals of same symmetry mutually repel each other. This effect is more prominent for MOs compared to the π MOs. As a consequence, the energy of
g(2s) (shown as 1 g) goes down and the energy of g(2pz) (shown as 2 g) rises and ultimately exceeds the energy level of the πu(2px) = πu(2py) (shown as 1πu). In this way, fig (a) changes to fig(b).
2. s-p mixing :
If the energy difference between 2s and 2pz orbitals of each atom are small, there is a possibility of mixing of these AOs to produce two mixed spz hybridised orbitals on each atom. These two hydbridised orbitals of each atom will be of unequal energy. Since the 2s AO was of lower
energy than the 2pz AO, the lower energy mixed spz orbital thus formed on each atom will have more contribution from 2s AO and less contribution from 2pz AO. Similarly, the higher energy mixed spz orbital formed on each atom will have less contribution from 2s AO and more
contribution from 2pz AO.Since all the 4 mixed spz AOs of both the atoms are of the similar symmetry, they will undergo linear combination to produce 4 MOs.
1. The 4 MOs are arranged as per following energy level :
1 < 2 < 3 < 4
2. In the atoms where the difference between 2s and 2p orbitals are high(highly electronegative O and F atom),
then the mixing of 2s and 2pz orbitals are insignificant.
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