The valence bond (VB) model of bonding in H₂

Valence bond theory considers the interactions between separate atoms as they are brought together to form molecules. We begin by considering the formation of H₂ from two H atoms, the nuclei of which are labelled H_A and HB, and the electrons of which are 1 and 2, respectively.

  When the atoms are so far apart that there is no interaction between them, electron 1 is exclusively associated with HA, while electron 2 resides with nucleus H_B. Let this state be described by a wavefunction ψ¹.

  When the H atoms are close together, we cannot tell which electron is associated with which nucleus since, although we gave them labels, the two nuclei are actually indistinguishable, as are the two electrons. Thus, electron 2 could be with H_A and electron 1 with HB. Let this be described by the wavefunction ψ².

Equation 1.1 gives an overall description of the covalently bonded H₂  molecule; ψ_covalent is a linear combination of wavefunctions ψ¹ and ψ².  The equation contains a normalization factor, N (see Box 1.4). In the general case where:

                         (1.1)

  (1.2)

In terms of the spins of electrons 1 and 2  ,ψ+‏ corresponds to spin-pairing, and ψ- corresponds to parallel spins (nonspin- paired). Calculations of the energies associated with these states as a function of the inter nuclear separation of H_A and H_B show that, while -  represents a repulsive state (high energy), the energy curve for ‏ψ +‏  reaches a minimum value when the inter nuclear separation, d, is 87pm and this corresponds to an H-H bond dissociation energy, ΔU, of 303 kJ mol ^-1. While these are near enough to the experimental values of d = 74pm and ΔU = 458 kJ mol^-1 to suggest that the model has some validity, they are far enough away from them to indicate that the expression for  ψ+‏ needs refining.

  The bond dissociation energy (ΔU) and enthalpy (ΔH) values for H₂ are defined for the process:

H₂(g) → 2H(g)

Improvements to equation 1.1 can be made by:

•  allowing for the fact that each electron screens the other from the nuclei to some extent;
•   taking into account the possibility that both electrons 1 and 2 may be associated with either H_A or H_B, i.e. allowing for the transfer of one electron from one nuclear centre to the other to form a pair of ions H_A⁺ H_B^- or H_A^-H_B⁺.

The latter modification is dealt with by writing two additional wavefunctions, ψ₃ and ψ₄ (one for each ionic form), and so equation 1.1 can be rewritten in the form of equation 1.3. The coefficient c indicates the relative contributions made by the two sets of wavefunctions. For a homonuclear diatomic such as H₂, the situations described by ψ ₁ and ₂ are equally probable, as are those described by ψ₃

and ψ₄.

            (1.3)

Since the wavefunctions ψ₁ and ψ₂ arise from an internuclear interaction involving the sharing of electrons among nuclei, and ₃ and ψ₄ arise from electron transfer, we can simplify equation 1.3 to 1.4 in which the overall wavefunction, ψ _molecule, is composed of covalent and ionic terms.

       (1.4)

Based on this model of H₂, calculations with c ≈ 0.25 give values of 75pm for d(H–H) and 398 kJ mol^-1 for the bond dissociation energy.

Now consider the physical significance of equations 1.3 and 1.4. The wavefunctions ψ₁ and ψ₂ represent the structures shown in 1.7 and 1.8, while  ψ₃ and ψ ₄ represent the ionic forms 1.9 and 1.10. The notation H_A(1) stands for ‘nucleus H_A with electron (1)’, and so on.

Dihydrogen is described as a resonance hybrid of these contributing resonance or canonical structures. In the case of H₂ , an example of a homonuclear diatomic molecule which is necessarily symmetrical, we simplify the picture to 1.11. Each of structures 1.11a, 1.11b and 1.11c is a resonance structure and the double-headed arrows indicate the resonance between them. The contributions made by 1.11b and 1.11c are equal. The term ‘resonance hybrid’ is somewhat unfortunate but is too firmly established to be eradicated.

A crucial point about resonance structures is that they do not exist as separate species. Rather, they indicate extreme bonding pictures, the combination of which gives a description of the molecule overall. In the case of H₂, the contribution made by resonance structure 1.11a is significantly greater than that of 1.11b or 1.11c. Notice that 1.11a describes the bonding in H₂ in terms of a localized two-centre two-electron, 2c-2e, covalent bond. A particular resonance structure will always indicate a localized bonding picture, although the combination of several resonance structures may result in the description of the bonding in the species as a whole as being delocalized .