The hydrogen molecule

Key points: In valence bond theory, the wavefunction of an electron pair is formed by superimposing the wavefunctions for the separated fragments of the molecule; a molecular potential energy curve shows the variation of the molecular energy with internuclear separation. The two-electron wavefunction for two widely separated H atoms is , where A and B are H1s orbitals on atoms A and B. (Although, chi, is also used for electronegativity, the context makes it unlikely that the two usages will be confused: is commonly used to denote an atomic orbital in computational chemistry.) When the atoms are close, it is not possible to know whether it is electron 1 that is on A or electron 2. An equally valid description is therefore , in which electron 2 is on A and electron 1 is on B. When two outcomes are equally probable, quantum mechanics instructs us to describe the true state of the system as a superposition of the wavefunctions for each possibility, so a better description of the molecule than either wavefunction alone is the linear combination of the two possibilities.

This function is the (unnormalized) VB wavefunction for an HH bond. The formation of the bond can be pictured as being due to the high probability that the two electrons will be found between the two nuclei and hence will bind them together. More formally, the wave pattern represented by the term X_A (1) X_B (2) interferes constructively with the wave pattern represented by the contribution X_A (2) X_B (1) and there is an enhancement in the amplitude of the wavefunction in the internuclear region. For technical reasons stemming from the Pauli principle, only electrons with paired spins can be described by a wavefunction of the type written in eqn 2.1, so only paired electrons can contribute to a bond in VB theory. We say, therefore, that a VB wavefunction is formed by spin pairing of the electrons in the two contributing atomic orbitals.

The electron distribution described by the wavefunction in eqn 2.1 is called a (2.1) bond. As shown in Fig. 2.2, a bond has cylindrical symmetry around the internuclear axis, and the electrons in it have zero orbital angular momentum about that axis. The molecular potential energy curve for H₂ , a graph showing the variation of the energy of the molecule with internuclear separation, is calculated by changing the in ternuclear separation R and evaluating the energy at each selected separation (Fig. 2.3). The energy is found to fall below that of two separated H atoms as the two atoms are brought within bonding distance and each electron becomes free to migrate to the other atom. However, the resulting lowering of energy is counteracted by an increase in energy from the Coulombic (electrostatic) repulsion between the two positively charged nuclei. This positive contribution to the energy becomes large as R becomes small. Consequently, the total potential energy curve passes through a minimum and then climbs

Fig. 2.2 The formation of a bond from (a) s orbital overlap, (b) p orbital overlap. A bond has cylindrical symmetry around the internuclear axis.

to a strongly positive value at small internuclear separations. The depth of the minimum of the curve is denoted De . The deeper the minimum, the more strongly the atoms are bonded together. The steepness of the well shows how rapidly the energy of the molecule rises as the bond is stretched or compressed. The steepness of the curve, an indication of the stiffness of the bond, therefore governs the vibrational frequency of the molecule (Section 8.4).

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Polyatomic molecules

Homonuclear diatomic molecules

The basic shapes

Resonance

Polarizability

Electronegativity

Electron affinity

Atomic and ionic radii

The format of the periodic table

The periodic table

Penetration and shielding

The angular variation of atomic orbitals