Symmetry of the One-electron Approximation Hamiltonian

The set of one-electron functions transforming according to the n j-dimensional irrep d( ) is called the shell. For molecules, these shells are connected with irreps of the point-symmetry group. For a crystal, f3 = ( fe,7) - full irreducible representation of space group G, defined by the star of wavevector k and irrep 7 of the point group of this vector. Taking into consideration the spin states a a) we have 2n/ one-electron states in the shell. The functions ( )Q ( r) span the space of the rep 

Now we check that for the closed-shell systems the group G is a symmetry group of the Hartree-Fock equations (4.21). First we note that the sum 

The invariance of the operator H is already proved. For Coulomb J and exchange K operators and for an arbitrary function (p(r) we have 

the operator F(F(p)) in the electron equation (4.21) has the symmetry group G of the equilibrium nuclear configuration if the electron-density matrix p r, r ) p P r, r ) is invariant under G. The eigenfunctions of the operator F(F p ) form the bases of irreps of G. The invariance of p r, r ) p r, r )] is assured for the system with closed shells. 

taking as the initial approximation that the electron density p ° r, r ) is invariant under the group G we have, during any iteration step and in the self-consistent limit, the one-electron functions classified according to the irreps of the group G -the symmetry group of the equilibrium nuclear con%uration. 

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