Molecular orbital theory

Molecular orbital theory (MOT) is an essential topic in quantum chemistry, particularly in understanding bond formation between atoms in a molecule. In simple terms, molecular orbital theory proposes that atomic orbitals combine to form molecular orbitals, which are spread across multiple atoms in a molecule. These molecular orbitals can accommodate electrons resulting in the formation of chemical bonds.

To understand MOT, it is sensible to compare it with valence bond theory (VBT). While VBT emphasizes localized electrons between bonded atoms, molecular orbital theory describes electrons in molecules as delocalized, providing a more holistic view of bond formation, especially in molecules with resonance or complex three-dimensional structures.

Fundamentals of molecular orbital theory

The major principles of molecular orbital theory can be summarized as follows:

• Linear combination of atomic orbitals (LCAO): This is the fundamental approach used in the construction of molecular orbitals. According to LCAO, atomic orbitals having similar energy and symmetry combine linearly to form molecular orbitals. For example, consider two hydrogen atoms. They have 1s atomic orbitals that can combine linearly to form bonding and restricting molecular orbitals.
• Conservation of orbitals: The total number of molecular orbitals formed in a molecule is equal to the number of atomic orbitals used to form them.
• Bonding and antibonding orbitals: When atomic orbitals interact, they form bonding orbitals (low energy) and antibonding orbitals (high energy). Bonding orbitals arise from constructive interference of atomic orbital wave functions, which increases the electron density between atoms, leading to the formation of a bond. In contrast, antibonding orbitals arise from destructive interference, which decreases the electron density between atoms and weakens the bond.
• Population of molecular orbitals: Electrons reside in molecular orbitals according to the Auf Bau principle, Pauli exclusion principle and Hund's rule.

  Example:
  Keeping the order of increasing energy levels in mind, 1σ g is filled before 1σ u *.
              

• Bond order: Bond order is a numerical value that describes the strength and stability of a bond. It is determined using the formula:

  Bond order = (number of electrons in bonding orbitals - number of electrons in blocking orbitals) / 2
              

  Higher bond orders correlate with stronger and shorter bonds.

Construction of molecular orbitals

Homodiatomic molecule example: _H2

Consider the simplest molecule, H _2. Each hydrogen atom contributes a 1s orbital that can combine linearly to form two molecular orbitals: a low energy bonding orbital (σ 1s) and a higher energy antibonding orbital (σ * 1s).

Let us illustrate this interaction with simple wave interaction diagrams:

Atomic Orbital Combination:

H (1s) + H (1s) → Bonding molecular orbital (σ 1s)
                  → Antibonding molecular orbital (σ * 1s)
                              
Energy diagram:

H(1s) H(1s)
 ,
  ,
  ,
  |------σ 1s ------| (low energy)
  |-------σ * 1s ------|(high energy)
    

The bonding molecular orbital (σ 1s) arises from constructive interference of 1s atomic orbitals, in which there is increased electron density between the nuclei. The antibonding orbital (σ * 1s) arises from destructive interference, in which there is a node where the electron density is minimum.

In _H2, both electrons occupy the low energy bonding molecular orbital. Therefore, the bond order is:

Bond order = (2 bonds – 0 valence bonds) / 2 = 1
    

This indicates a stable single bond between the two hydrogen atoms.

Heterodiatomic molecule example: HF

For heterogeneous diatomic molecules such as HF, the approach is a little more complicated because the atomic orbitals have different energies and shapes. In HF, the 1s orbital of hydrogen overlaps with the 2p orbitals of fluorine.

interaction:

H(1s) + F( 2pz ) → σ
                        → σ*
    

The higher electronegativities of fluorine attract electrons more effectively, resulting in polar covalent bonds. The molecular orbitals may be tilted toward fluorine because of its higher electronegativities.

Delocalisation and resonance

One of the greatest strengths of molecular orbital theory is its ability to describe delocalized electrons, which cannot be explained well with valence bond theory. In molecules such as benzene, the electrons are not localized. Instead, they are delocalized in pi orbitals, giving benzene its characteristic stability and symmetry.

The structure of benzene (C ₆ H ₆) appears to have alternating single and double bonds, but in fact, all the C-C bonds are identical. Molecular orbital theory describes this by the displacement of the six pi electrons over the entire ring, which is often represented in a resonance hybrid structure.

Molecular orbital representation for benzene:

1) Sigma bond framework (localized):
   CC
   Chowdhary
        
2) Pi molecular orbitals (delocalized):

   π₁ (bonding, all in-phase)
   π₂ and π₃ (sets of non-bonding)
   π₄ (restriction, charges cancel out)
    

These details explain the resonance energy and increased stability of aromatic systems.

Ideas of symmetry

Molecular orbital theory makes extensive use of symmetry considerations. Molecular orbitals themselves are solutions of the Schrödinger equation that are inherently symmetric or antisymmetric. Symmetry operations help determine whether orbitals can combine:

• Only orbitals with the same symmetry about the molecular axis will combine.
• The concept of symmetry optimized linear combinations (SALCs) helps to make the process of construction of molecular orbitals in large systems economical.

Extended molecular systems

Molecular orbital theory is equally capable of dealing with larger, more complex molecules. Molecular orbital diagrams can become complex as the number of atoms increases, but the same principles apply. For example, in a polyene chain such as butadiene, the overlap of p orbitals can be expanded into long conjugated systems, such as in catalytic polymerization processes or electronic properties in organic solar cells, further proving the flexibility and utility of MOT.

Example: butadiene
CH₂=CH-CH=CH₂
    
Orbital Overlap:
    
    σ-bond: localized, involves sigma (σ) orbitals
    π-bond: Extended conjugation results in delocalized electrons which contribute to the formation of π molecular orbital.
    

Benefits and limitations

Molecular orbital theory offers several advantages over simpler models:

• It can explain magnetic properties like paramagnetism and diamagnetism due to the presence of unpaired electrons in molecular orbitals.
• It provides an explanation for molecular stability using concepts such as bond order and bond energy.
• It is efficient in understanding electron displacement in conjugated and aromatic systems.

However, it has some limitations:

• MOT can be computationally challenging, especially for larger molecules.
• It may be less intuitive than valence bond theory for viewing electron sharing and local interactions.
• Some approximations in LCAO do not fully capture electron correlation effects and require more advanced techniques such as configuration interactions or density functional theory.

Conclusion

Molecular orbital theory stands as a foundational pillar in modern chemistry, providing a complex view of the electronic structure and bonding of molecules. By conceptualizing electrons spread throughout molecules in a molecular orbital, it revolutionized our understanding of chemical behavior, stability, and color. Despite its complexity, the principles of molecular orbital theory provide a powerful framework for predicting and rationalizing the properties of both simple diatomic molecules and sophisticated extended systems.

As we advance our explorations and applications in the field of chemistry, concepts arising from molecular orbital theory continue to be integral to the pursuit of knowledge in areas such as quantum chemistry and beyond.

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