Unrestricted and Restricted Open-shell Hartree-Fock Methods for Crystals

The unrestricted and restricted open-sheU Hartree-Fock Methods (UHF and ROHF) for crystals use a single-determinant wavefunction of type (4.40) introduced for molecules. The differences appearing are common with those examined for the RHF LCAO method use of Bloch functions for crystalline orbitals, the dependence of the Fock matrix elements on the lattice sums over the direct lattice and the Brillouin-zone summation in the density matrix calculation. The use of one-determinant approaches is the only possibility of the first-principles wavefunction-based calculations for crystals as the many-determinant wavefunction approach (used for molecules) is practically unrealizable for the periodic systems. The UHF LCAO method allowed calculation of the bulk properties of different transition-metal compounds (oxides, perovskites) the qrstems with open shells due to the transition-metal atom. We discuss the results of these calculations in Chap. 9. The point defects in crystals in many cases form the open-sheU systems and also are interesting objects for UHF LCAO calculations (see Chap. 10). 

In the UHF or ROHF cases two sets of matrix equations are solved self-consistently for a and / spins [69]  

The Fourier components of the density matrix for a and / spins are different as they are obtained by using the C (fe) and C (fe) eigenvectors obtained from (4.67) and (4.68), respectively  

The F Rm) matrix is defined as in (4.59), where the total density matrix F Rm) defined in (4.71) is used in the Coulomb and exchange terms X (Rm) is defined as Xfj,v Rm) in (4.63), where, however, the spin density matrix P P (Rm) is used instead of the total density matrix F Rm)- 

The ROHF LCAO method for crystals differs from the UHF method in the equations defining the CO. Let n = na + n electrons per primitive unit cell be considered (ria nfj). For the Ufj CO the closed-shell RHF LCAO equations are solved, the CO of ria + rifj electrons with a spin are found from the equation analogous to (4.57). In the ROHF method the total density P(i m) and spin density P P (iim) matrices are defined as 

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