One-Electron Molecules and Orbitals

If a molecule has certain symmetry properties, important predictions about the solutions of the electronic Schrodinger equation can be made without having to solve the equation itself. Consider the case of a planar molecule, i.e. of a molecule whose nuclei lie in a plane. This plane is a symmetry plane for the molecule, and it can be shown that any eigenfunction is either symmetric or antisymmetric with respect to this plane. If one chooses the plane of the nuclei as the (y, z) plane of a Cartesian coordinate system, this means that 

If there is more than one symmetry element (symmetry plane, axis etc.), relations similar to Eq. (2.5) hold for every element, and the wave functions can be classified according to group-theoretical symbols (see e.g. ). 

Let us consider more specifically wave functions depending on the coordinates (and possibly on the spin) of a single electron. Such functions are called orbitals (or, if spin is explicitly included, spin orbitals ). According to their behaviour under a reflection with respect to the nuclear plane of a planar molecule, they are classified as o or n orbitals one has 

The eigenfunctions of the Schrodinger equation for planar one-electron systems, e.g. those of the HS+ ion in a non-linear configuration, must be either a or n orbitals the former are symmetric about the molecular plane, the latter antisymmetric. The a orbitals have in general maximum values close to the nuclei, whereas the n orbitals have a nodal surface on the nuclear plane and different signs on the two sides of this plane, as can be seen from Eq. (2.6) by letting xi 0. 

Orbitals can also be defined for many-electron systems, and the molecular orbital theory mentioned in the previous section is indeed based on this possibility. In order to assess the significance and limitations of the molecular orbital scheme and the meaning of a and n orbitals, we have to discuss the definition and the determination of orbitals in a many-electron system at some length. 

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