Quantum Mechanical Treatment of Valence Bond Theory

Valence Bond (V.B.) or Heitler-London (H.L.) Theory

Let us consider the formation of H₂ molecule from two isolated H-atoms : H_A & H_B. Electron -1 of one H-atom having the probability of occurring around the nucleus is shown by ψ_HA1, and similarly another by ψ_HB2. These two H-atoms interact during the formation of molecule when both the electrons have probability of occurring around both nuclei. It is shown by two wave functions

Read Valence Bond Theory in details

φ₁ = ψ_HA1 · ψ_HB2  &  φ₂ = ψ_HA2 · ψ_HB1

φ₁ & φ₂ are linear combinations of two possible wave functions

ψ = C₁ φ₁ + C₂ φ₂

where coefficients C₁ & C₂ should be such that the resulting wave function has the minimum energy. This necessitates C₁² = C₂². i.e. C₁ = ± C₂. Hence we have two wave functions—

Symmetrical = C₁ (φ₁ + φ₂) ;      Asymmetrical = C₂ (φ₁ – φ₂).

In symmetrical interaction, electron waves are in the same phase i.e. electron spins are opposite. This gives lower energy state (E_s) than the interacting atoms. This results in covalent bond formations—

E_s = H₁₁ + H₁₂S₁₁ + S₁₂

where,    H₁₁ = ∫ φ₁ H φ₁ dτ
              H₁₂ = ∫ φ₁ H φ₂ dτ
              S₁₁ = ∫ φ₁ φ₁ dτ
              S₁₂ = ∫ φ₁ φ₂ dτ

Various parameters in the energy equation can be substituted and the distance corresponding to the minimum energy has been worked out. This corresponds to bond distance between two H-atoms. However, an asymmetrical interaction gives a higher energy state (E_as) and there is a constant increase in P.E. when atoms come closer and hence no molecule is formed—

E_as = H₁₁ – H₁₂S₁₁ – S₁₂

Thus a covalent bond is formed by the neutralisation of opposite spins of two atomic orbitals. Spin neutralisation of one, two and three pairs give single, double and triple bonds respectively. Three pairs of electrons pull nuclei more than two, hence a triple bond is stronger than a double bond. Similarly double bond is stronger than a single bond. s–s or s–p_x interaction gives to cylindrical symmetry with no nodal plane along the internuclear axis. Such a covalent bond is called σ-bond. p_y–p_y or p_z–p_z internuclear gives a bond with nodal plane. Such a covalent bond is called π-bond. Pauling & Slater suggested that the covalent bond formation occurs in the direction where there is maximum overlap. VBT considers the valence orbitals of atoms participating in bond formation. Though it explains stereochemistry but fails to explain paramagnetic property of O₂ and electronic spectra.

Hybridisation

In case when pure atomic orbitals cannot affect good overlap for a covalent bond formation, then atomic orbitals of same or similar energy combine to form equal number of equivalent orbitals called hybrid orbitals. If φ₁ & φ₂ atomic orbitals combine to form two hybrid orbitals ψ_h1 & ψ_h2—

ψ_h1 = C₁ φ₁ + C₂ φ₂
ψ_h2 = C₃ φ₁ + C₄ φ₂

Values of co-efficients are such that hybrid orbitals are ortho-normal i.e.—

∫ (C₁ φ₁ + C₂ φ₂)² dτ = 1

or,

C₁² ∫ φ₁² dτ + C₂² ∫ φ₂² dτ + 2 C₁ C₂ ∫ φ₁ φ₂ dτ = 1

or,

C₁² + C₂² + 0 = 1;          or,    C₁ = ± C₂

Similarly C₃ = ± C₄

Further, ∫ (C₁ φ₁ + C₂ φ₂)(C₃ φ₁ + C₄ φ₂) dτ = 0

or,

C₁ C₃ ∫ φ₁² dτ + C₂ C₄ ∫ φ₂² dτ + C₁ C₄ ∫ φ₁ φ₂ dτ + C₂ C₃ ∫ φ₁ φ₂ dτ = 0

or,

C₁ C₃ + C₂ C₄ = 0

Hybrid orbitals provide better overlap and result in more stable bonds and a state of low energy. The energy released called hybridisation energy is partly used in the excitation of electrons from lower to higher orbitals in atom.

Related Topics
Valence Bond Theory
Hybridization and Exceptions
Quantum Mechanical Treatment of Molecular Orbital Theory.
Quantum Mechanical Treatment of a Rigid Rotator