Symmetry of Molecular and Crystalline Orbitals

Let the wavefunction p(r) depend on the coordinates of only one electron, H is a Hamiltonian operator and E the energy of the eigenstate p(r). The function p(r) satisfies the time-independent equation 

The transformation g is said to be a symmetry operation of the Hamiltonian H if H(p r) = H(pr), i.e. if H is invariant under g. Symmetry operations g that leave H invariant form a group. This group is called the symmetry group of the Hamiltonian or of the Schrodinger equation (3.1). 

We introduce in the linear vector space L of functions v (r) the operators g = D(g) as 

This relation holds for any eigenfunction of H and for all linear combinations of the eigenfunctions, i.e. for any function in the linear vector space L. Therefore, the following operator relation may be written  

This expresses in mathematical form the symmetry properties of the Hamiltonian of a quantum mechanical system. 

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