Question

In addition to the molecular orbital treatment of H2, the ground state can be treated by the so-called valence bond (Heitler-London) approach, where individual covalent bonds are combined to form an antisymmetric wave function with spins up and down. Look up this approach and compare the results with the simple MO treatment in terms of bond energy (kJ/mol) and bond distance (pm).

Answer 1

Heitler-London model for the hydrogen molecule.is an important model for the understanding of magnetism since it introduces the concept of an exchange energy very nicely for a two-electron wave function in H2.

We assume that we have two hydrogen atoms A and B. We know the hamiltonian for these as

H_A = - (h²∇²₁ / 2 m_e) − (e² / 4π₀ ) (1 / [R_A − r₁])...................................... (1)

and

H_B = − (h²∇²₂ / 2 m_e) − (e² / 4π₀) (1 / [R_B − r₂]) ...................................... (2)

where R_A,B are the coordinates for the two nuclei, r_1,2 are the coordinates for the electron belonging to atom A and atom B, respectively and ∇²_1,2 acts on electron 1 and 2, respectively. We also know the ground state solutions for the hydrogen atom hamiltonians. We call the ground state energy

E_A = E_B = E₀ and the wave function Ψ_A(r₁) and Ψ_B(r₂).

A practical calculations proceeds using the variational principle of quantum mechanics. This tells us that the energy calculated by

E = Ψ(r₁, r₂) HΨ(r₁, r₂) dr₁dr₂ / Ψ(r₁, r₂)Ψ(r₁, r₂)dr₁dr₂

using a trial wave function Ψ(r₁, r₂) is always greater or equal than the ground state energy of the system. For a Ψ(r₁, r₂) that closely approximates the true ground wave wave function, we can hope to get an accurate solution for the energy. In the present case, using the atomic wave function for the construction of Ψ(r₁, r₂) might not be entirely correct because the presence of the second atom can be expected to affect these. The simple approach might, however, still be a good guess. The resulting energies from the explicit calculation can be written as

E = 2E₀ + ∆E,

where E₀ is the ground state energy of a hydrogen atom and the second term is the energy change due the interaction between the two atoms.

Following the MO treatment of H2+, assume the (normalized) ground electronic state wavefunction wavefunction is given by:

Ψ_gs= ψ+1 x ψ+2 [α₁β₂- β₂α₁] / 2

Evaluate the ground state electronic energy based on this approximate eigenfunction

E_gs = dr₁dr₂ψ (_{1 2}) H_el ψ (_{1 2})

= 2 E_{1s +} (J₀ / R) - (2 j¹ +2K / 1+ S) + (J+2K+M+4l / 2(1+S)²)