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Physical chemistry 272

Related topics Thewavenature of matter Elementary valence theory (H1)

Related topics Valence bond theory (H2)

The Born-Oppenheimer approximation
Many-electron atoms (G6)
In contrast to valence bond theory (see Topic H2), molecular orbital theory attempts to
describe the bonding orbitals in a molecule in their entirety, as opposed to focusing on the
creation of each bond individually. Just as the explicit calculation of atomic orbitals is
only possible for the hydrogen atom, direct calculation of the molecular orbitals is
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possible only in the simplest possible molecule, H 2 . Unlike atomic wavefunctions, the
molecular wavefunction must describe the relative motion of nuclei in addition to the
motion of the electrons. Primarily because of their relative masses, the electrons move
3
some 10 times faster than the nucleus, and the Born-Oppenheimer approximation
simplifies the calculation of molecular orbitals by assuming that the nuclei are stationary
relative to the motion of the electron. This approximation allows the internuclear
repulsion terms to be treated completely separately from the electrostatic behavior of the
electrons.
The potential energy of an electron in the electric field resulting from two protons is
readily calculated using:

where r H1 and r H2 are the distances of the electron from each proton. Incorporation of this
potential energy expression into the Schrödinger equation (see Topic G4) yields the exact
solutions for the hydrogen molecular ion. As with atomic orbitals, the impossibility of
calculating orbitals for systems with three or more bodies in relative motion makes
mathematical solutions for the molecular orbitals impossible. Molecular orbital theory
therefore makes the approximation that the molecular orbitals may be formed by the
linear combination of atomic orbitals (LCAO).

The LCAO approximation
The total wavefunction for a molecule is given by:

where ψ n represents the wavefunction for each electron in the molecule.
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Explicit calculations for the molecular orbitals H 2 show that the lowest energy
solution of the Schrödinger equation is given by the addition of the two 1s orbitals. In all
other cases, it is necessary to make the approximation that molecular orbitals may be
calculated from a linear combination of atomic orbitals. This linear combination
generates molecular orbitals by direct addition of atomic orbitals on the bonding atoms.