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Molecular Orbital Theory


Introduction
One of the methods of constructing molecular orbitals is the method of linear combination of atomic orbitals (the abbreviation is LCAO). This method may be illustrated by taking an example of hydrogen molecule, written as HA—HB.
Consider the formation and breaking of hydrogen molecule as shown:


 

Making and Breaking of a Hydrogen Molecule

From the figure, it follows that the two monocentric atomic orbitals, 1s(HA) and 1s(HB), transform into the bicentric molecular orbital ψ as the two atoms HA and HB are brought close to each other to form hydrogen molecule. Alternatively, the molecular orbital ψ degenerates into the two atomic orbitals 1s(HA) and 1s(HB) as the two atoms in hydrogen molecule recede from each other to give two hydrogen atoms.

Talking in terms of probability distribution of electron, it can be said that on increasing the distance between the two atoms, the probability distribution obtained from molecular orbital must resemble those obtained from atomic orbitals and in the extreme case it should be reducible to those of atomic orbitals. In case of H2+ species where only one electron is present in the lowest molecular orbital, the probability distribution in H2+ will be reducible to that of one of the atoms as the two nuclei recede from each other.

To satisfy the above criterion, the molecular orbital is written as the linear combination of the appropriate atomic orbitals. Taking an example where the molecular orbital is reduced to 1s orbital of the two hydrogen atoms, we can write the molecular orbital as
ψMO=C1ψ 1s(HA)+C2ψ is(HB) .............................(1.27)
In the extreme case, we will have
ψMO= ψ Is (HA), i.e. C1 = 1 and c2 = 0 when the electron is near 1s(HA)
ψMO= ψ Is (HB), i.e. C1 = 0 and c2 = 1 when the electron is near 1s(HB)

In any other situation C1 and C2 may have identical or different values depending upon the involved atomic orbitals in the linear combination.

There are a few guidelines about the type of orbitals which can mix up to form a molecular orbital and also the number of molecular orbitals that can be formed by the linear combination of the given atomic orbitals. These are described in the following:
  1. The energies of atomic orbitals taken in the linear combination must be comparable.
Illustrations
  1. The atomic orbitals 1s(HA) and 1s(HB) have comparable (in fact, identical) energies, therefore, there is a 'good mixing' of these atomic orbitals in the linear combination.
    ψ MO= C1 ψ Is (HA)+C2ψ Is (HB)
    The meaning of 'good mixing' is that the values of coefficients C1 and C2 are significant. In the present case, both C1 and C2 will have identical values because of the symmetry requirements.*
  2. The atomic orbitals 1s(HA) and 2s(HB) or vice versa do not have comparable energies, therefore, there is 'poor mixing' of these atomic orbitals in the linear combination
    ψ MO=C1 ψ Is(HA)+C2ψ 1S (HB)
    The meaning of 'poor mixing' is that the values of coefficients C1 and C2 will be such that either

    C1 >> C2 or C2 >> C1.
Comment
With a very large difference in energies of atomic orbitals, we will have either C1 = 1 and C2 = 0 or C1 = 0 and C2 = 1. This implies that there is no mixing between the two involved atomic orbitals.
  1. There should exist a positive overlap between the atomic orbitals at the stable internuclear distance.
llustrations
  1. The mixing between 1s(HA) and 1s(HB) is effective as there is a positive overlap betwen these atomic orbitals when the two atoms are placed at the stable internuclear distance of hydrogen molecule.
  2. The mixing between 2s(A) and 2px(B) (or 2py(B)) or vice versa is not effective as there is a zero overlap between these atomic orbitals when the two atoms A and B are placed at the stable internuclear distance of AB molecule.
Note: The positive and negative signs are those of wave functions.

  1. The scheme of linear combination of atomic orbitals yields as many molecular orbitals as the number of atomic orbitals included in the combination.
Illustration
  1. In hydrogen molecule, the combination of two atomic orbitals 1s(HA) and 1s(HB) gives two molecular orbitals; one involving positive combination and the other involving negative combination.
    ψ 1=C1Is (HA)+ ψ Is (HB)].............................(1.28)
    ψ 1=C2[ψ Is (HA) - ψ Is (HB)].............................(1.29)
    It may be noted that the contributions of atomic orbitals 1s(HA) and 1s(HB) in both the molecular orbitals Ψ1 and Ψ2 are equal and have values C1 and C2, respectively.
  2. In oxygen molecule, the combination of four atomic orbitals 2s(OA), 2pz(OA), 2s(OB) and 2pz(OB) gives four molecular orbitals differing in the values of coefficients C1, C2, C3, and C4.
  1. In the combination of two atomic orbitals, one of the molecular orbitals is more stable than the more stable of the two atomic orbitals and the second one is less stable than the lesser stable atomic orbital
    The term 'more stable' implies that the molecular orbital has lesser energy than the energies of the two involved atomic orbitals and the term 'less stable' implies that the molecular orbital has larger energy than the energies of the two involved atomic orbitals.
Illustration  
In the combination
ψ = C1[ψ Is (HA) + C2ψ MO = ψ Is (HB)]

the energies of the two resultant molecular orbitals relative to those of the two atomic orbitals are shown.

The Energies of Molecular Orbitals Relative to
the Energies of Involved Atomic Orbitals

  1. Only the valence atomic orbitals are considered for the construction of molecular orbitals. The inner core atomic orbitals are strongly bound to their respective nuclei and thus are considered nonbonding orbitals.
Illustration  
For homonuclear diatomic molecules involving atoms of the second period, only 2s and 2p orbitals are considered for constructing the molecular orbitals.
  1. Constructive and Destructive Interactions
    The combination of atomic orbitals leading to the formation of bonding molecular orbital is said to be constructive interaction between the involved atomic orbitals.
    The combination of atomic orbitals leading to the formation of antibonding molecular orbital is said to be destructive interaction between the involved atomic orbitals.
  2. Sigma and Pi Molecular Orbitals for simple molecules, two types of molecular orbitals are formed. These are:
    1. Sigma molecular orbital is an orbital in which electron density is concentrated symmetrically around the line joining the two nuclei of the bonding atoms. It is represented by the symbol σ.
  1. Pi molecular orbital is an orbital in which electron density is concentrated above and below the line joining the two nuclei of the bonding atoms. It is represented by the symbol π.

Bonding in Some Homonuclear Diatomic Molecules
In t his section we shall discuss bonding in some homonuclear diatomic molecules.
  1. Hydrogen Molecule (H2): It is formed by the combination of two hydrogen atoms. Each hydrogen atom has one electron in 1s orbital.
    Therefore, in all there are two electrons in hydrogen molecule which are present in σ 1s molecular orbital. So electronic configuration of hydrogen molecule is H2: (σ 1s)2
    The bond order of H2 molecule can be calculated as given below:
    Bond order
    This means that the two hydrogen atoms are bonded together by a single covalent bond. The bond dissociation energy of hydrogen molecule has been found to be 438 kJ mol-1 and bond length equal to 74 pm. Since no unpaired electron is present in hydrogen molecule, therefore, it is diamagnetic.
  2. Helium Molecule (He2): The electronic configuration of helium atom is 1s2. Each helium atom contains 2 electrons, therefore, in He2 molecule there would be 4 electrons. These electrons will be accommodated in σ 1s and σ *1s molecular orbitals leading to electronic configuration:
    He2: (σ 1s)2 (σ *1s)2
    Bond order of He2 is €(2 - 2) = 0
    He2 molecule is therefore unstable and does not exist.
    Similarly, it can be shown that Be2 molecule
    (σ 1s)2 (σ *1s)2 (σ 2s)2 (σ *2s)2also does not exist.
  3. Lithium Molecule (Li2): The electronic configuration of lithium is 1s2, 2s1 . There are six electrons in Li2. The electronic configuration of Li2 molecule, therefore, is Li2: (σ 1s)2 (σ *1s)2 (σ 2s)2 The above configuration is also written as KK(σ 2s)2 where KK represents the closed K shell structure (σ 1s)2 (σ *1s)2.

    From the electronic configuration of Li2 molecule it is clear that there are four electrons present in bonding molecular orbitals and two electrons present in antibonding molecular orbitals. Its bond order, therefore, is € (4 - 2) = 1. It means that Li2 molecule is stable and since it has no unpaired electrons it should be diamagnetic. Indeed diamagnetic Li2 molecules are known to exist in the vapour phase.
  4. Carbon Molecule (C2): The electronic configuration of carbon is 1s22s22p2. There are twelve electrons in C2. The electronic configuration of C2 molecule, therefore, is
    C2: (σ s1 )2 (σ *1 )2 (σ 2s )2 (σ *2s)2
    or
    The bond order of C2 is € (8 - 4) = 2 and C2 should be diamagnetic. Diamagnetic C2 molecules have indeed been detected in vapour phase. It is important to note that double bond in C2 consists of both pi bonds because of the presence of four electrons in two pi molecular orbitals. In most of the other molecules a double bond is made up of a sigma bond and a pi bond. In a similar fashion the bonding in N2 molecule can be discussed.
  5. Oxygen Molecule (O2): The electronic configuration of oxygen atom is 1s22s22p4. Each oxygen atom has 8 electrons, hence, in O2 molecule there are 16 electrons. The electronic configuration of O2 molecule, therefore, is
    From the electronic configuration of O2 molecule it is clear that ten electrons are present in bonding molecular orbitals and six electrons are present in antibonding molecular orbitals. Its bond order, therefore, is
    Bond order
    So in oxygen molecule, atoms are held by a double bond. Moreover, it may be noted that it contains two unpaired electrons in is π *2px  and π *2py molecular orbitals, therefore, O2 molecule should be paramagnetic, a prediction that corresponds to experimental observation. In this way, the theory successfully explains the paramagnetic nature of oxygen.
    Similarly, the electronic configurations of other homonuclear diatomic molecules of the second row of the periodic table can be written. In are given the molecular orbital occupancy and molecular properties for B2 through Ne2. The sequence of MOs and their electron population are shown. The bond energy, bond length, bond order, magnetic properties and valence electron configuration appear below the orbital diagrams.





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