3 Molecular Orbital (MO) Theory

3 Molecular Orbital (MO) Theory In MO theory electrons are not regarded as belonging to particular bonds, but should be treated as spreading throughou...
Author: Allen Eaton
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3 Molecular Orbital (MO) Theory In MO theory electrons are not regarded as belonging to particular bonds, but should be treated as spreading throughout the entire molecule.

MO theory is more fully developed than VB theory and is widely used in modern discussions of bonding.

3.1 The hydrogen molecule ion H2+

rA1, rB1 are the distances of the electron from the two nuclei and R is the distance between the two nuclei.

SE:

2 2  1  V 2 me

V 

e2 1 1 1 (   ) 4 0 rA1 rB1 R

Solving the SE: H=E results in one-electron wavefunctions, called molecular orbitals (MO). The SE can be solved for H2+ but the wavefunctions are very complicated functions. The solution cannot be extended to polyatomic systems.

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3.1 Linear combinations of atomic orbitals (LCAO) If an electron can be found in an atomic orbital (AO) belonging to atom A and also in an AO belonging to atom B, then the overall wavefunction is a superposition of the atomic orbitals: =N(AB) where for H2+ A denotes H1sA, B denotes H1sB and N is a normalization factor. An approximate MO formed from a linear combination of AOs is called an LCAO-MO. An MO with cylindrical symmetry around the internuclear axis is called a s orbital (because it has zero orbital angular momentum around the internuclear axis).

3.2 Shape of s orbitals

A general indication of the shape of the boundary surface of a s orbital.

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3.3 Bonding orbitals Probability density of the electron in H2+: +2=N2 (A2 + 2AB+B2) A2: probability density if the electron were confined to the atomic orbital A B2: probability density if the electron were confined to the atomic orbital B 2AB: an extra contribution to the density (‘overlap density’) The electron density calculated from the linear combination of atomic orbitals. Note the accumulation of electron density in the internuclear region.

3.3 Bonding orbitals

A representation of the constructive interference that occurs when two H1s orbitals overlap and form a bonding s orbital.

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3.4 Antibonding orbitals -2=N2 (A2 - 2AB+B2) - corresponds to a higher energy than +. It is also a s orbital. This 2s orbital is an example of an antibonding orbital. If occupied it contributes to a reduction in the cohesion between the two atoms.

A representation of the destructive interference that occurs when two H1s orbitals overlap and form an antibonding s* orbital.

3.4 Antibonding orbitals

The electron density calculated from the linear combination of atomic orbitals. Note the elimination of electron density from the internuclear region.

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3.4.1 Molecular potential energy curves

The calculated and experimental molecular potential energy curves for a hydrogen moleculeion.

3.5 Bonding and antibonding effects

A partial explanation of the origin of bonding and antibonding effects. (a) In a bonding orbital, the nuclei are attracted to the accumulation of electron density in the internuclear region. (b) In an antibonding orbital the nuclei are attracted to an accumulation of electron density outside the internuclear region

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3.6 The structures of diatomic molecules 3.6.1 Molecular orbital energy level diagram of H2

E

A molecular orbital energy level diagram constructed from the overlap of H1s orbitals; the separation of the levels corresponds to that found at the equilibrium bond length. The ground electronic configuration of H2 is obtained by accommodating the two electrons in the lowest available orbital (the bonding orbital).

3.6.2 Molecular orbital energy level diagram of He2

E

The ground electronic configuration of the hypothetical four-electron molecule He2 has two bonding electrons and two antibonding electrons. It has a higher energy than the separated atoms, and so is unstable.

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3.6.3 Period 2 diatomic molecules Elementary concepts only include the orbitals of the valence shell to form molecular orbitals (such as VB theory). A general principle of MO theory is that all orbitals of the appropriate symmetry contribute to a molecular orbital. To form s orbitals, for example, all orbitals with cylindrical symmetry about the internuclear axis are taken into account.

From the 2s and 2pz orbitals four molecular orbitals can be built.

The general form of the s orbitals that may be formed is

  c A2 s A2 s  cB 2 s B 2 s  c A2 pz A2 p z  cB 2 p z B 2 p z From the four atomic orbitals four molecular orbitals of s symmetry can be formed by an appropriate choice of the coefficients c.

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3.6.4 s orbitals built from the overlap of p orbitals If 2s and 2pz orbitals have distinctively different energies they may be treated separately. The four s orbitals then fall into two sets.

A schematic representation of the composition of bonding and antibonding s orbitals built from the overlap of p orbitals.

3.6.5 p orbitals The 2px and 2py orbitals of each atom are perpendicular to the internuclear axis giving rise to a bonding and an antibonding p orbital.

A schematic representation of the structure of p bonding and antibonding molecular orbitals

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3.6.6 Molecular orbital energy level diagram for homonuclear diatomic molecules

E

In some cases, p orbitals are less strongly bonding than s orbitals because their maximum overlap occurs off-axis, suggesting that the molecular orbital diagram should be as shown on the left (O2 and F2).

3.6.7 Orbital energies of period 2 diatomics The order of energies of the s and p orbitals varies along Period 2.

The variation of the orbital energies of Period 2 homonuclear diatomics.

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3.6.8 Orbital energies of period 2 diatomics

The order shown is appropriate as far as N2. The relative order is controlled by the separation of the 2s and 2p orbitals in the atoms, which increases across the group. The consequent switch in order occurs at about N2.

3.6.9 Bond order

A measure of the net bonding in a diatomic molecule is its bond order, b:

b

1 (n  n*) 2

Where n is the number of electrons in bonding orbitals and n* is the number in antibonding orbitals. The bond order is useful for discussing the characteristics of bonds.

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3.6.10 The structures of homonuclear diatomic molecules Ground state electron configuration of N2 is 1s2 2s*2 1p4 3s2 and bond order is ½ (8-2)=3. O2: 1s2 2s*2 3s2 1p4 2p*2 bond order: ½ (8-4)=2. F2: 1s2 2s*2 3s2 1p4 2p*4 bond order: ½ (8-6)=1. O2: The 2p*2 electrons occupy different orbitals and have parallel spins! The net spin angular momentum is S=1(triplet state) and oxygen should be paramagnetic (not revealed by VB theory).

3.6.11 Bond order and bond length

Typical bond lengths in diatomic and polyatomic molecules. For bonds of a given pair of elements: The greater the bond order, the shorter the bond. Similarly, the greater the bond order, the greater the bond strength.

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3.7 Heteronuclear diatomic molecules

The electron distribution in the covalent bond between the atoms is not evenly shared resulting in a polar bond. In HF, for example, the F atom has a net negative charge and the H atom a partial positive charge.

3.7.1 Electronegativity The charge distribution in bonds is commonly discussed in terms of the eletronegativity, , of the elements involved. It is a measure of the power of an atom to attract electrons to itself when it is part of a compound. The greater the difference in electronegativities, the greater the polar character of the bond. Examples: HF: 1.78; CH (commonly regarded as almost nonpolar): 0.51

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3.7.1 Electronegativity

The charge distribution in bonds is commonly discussed in terms of the eletronegativity, , of the elements involved.

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