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The detailed calculation of the electronic structure of the spin-Peierls CuGeO 3 crystal was made within the limits of a spin-unrestricted Hartree-Fock method, simplified by means of a complete neglecting of difference overlap approximation. The crystal lattice polarization due to one-particle excitations has been also taken into account to improve the description of the insulating gap and vacant crystal states. This calculation has given the local magnetic moments, ionic charges, crystal valences, and partial densities of states. The correct insulating gap value of about 3 eV was obtained. It was shown that this gap corresponds to the charge transfer from 2 p oxygen to 3d copper atomic orbitals. The nature of several first optical transitions was considered. We have compared our results with those results which have been given by local-density approximation-linear augmented plane-wave and local-density approximation-linear muffin-tin orbital methods. We considered the eight different Néel magnetic states using the expansion of the unit cell. We calculated the total energies of this states and employed them to obtain the values of exchange integrals J c ϭ11.5Ϯ0.3 meV, J b ϭ0.45 Ϯ0.20 meV, J a ϭ0.05Ϯ0.05 meV, which are in good agreement with experimental data on magnetic neutron scattering and x-ray diffraction and antiferromagnetic EPR for this compound. We discuss possible schemes of exchange interactions between spins localized on neighboring Cu atoms. The electronic structure of the same Néel states of CuGeO 3 in a dimerized configuration was determined. We did not find any changes of the band structure for all considered Néel spin states in the dimerization. The dependence of J c as a function of Cu-Cu distance along the c axis was obtained, which enabled us to determine the alternation parameter ␦ϭ0.007 in the linearized model of the dimerized state ͓J c 1,2 ϭJ c (1Ϯ␦)͔. The values of ⌬ SP (␦ϭ0.007)ϭ0.43 meV and ⌬E 0 (␦ϭ0.007)ϭ5ϫ10 Ϫ3 meV were calculated in terms of Bulaevskii's Hartree-Fock analysis.

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ABSTRACT:The spin structure of the first order reduced density matrix (RDM-1) for an arbitrary many-electron state with zero z-projection of the total spin is examined. It is well known that for the state S0 (r 1 σ 1 , . . . , r N σ N ), which is an eigenstate of operators S 2 and S z with quantum numbers S and M = 0, the matrix elements for spins α and β are equal for any r and r : ρ α S0 (r|r ) = ρ β S0 (r|r ). In the present article, it is shown that the same is true for any state M=0 (r 1 σ 1 , . . . , r N σ N ) with indefinite total spin if in the expansion M=0 = S D S S0 only spins S with the same parity are present. To prove the statement, it is shown that the wave function S0 acquires the phase factor (−1) N/2−S when all spin functions α(σ i ) are changed for β(σ i ) and vice versa. In the developed proof, the Hamiltonian was not used at all and it was not even assumed that the wave function S0 is an eigenfunction of some Hamiltonian. Therefore the obtained result is valid for the stationary and non-stationary states, ground and excited states, with and without homogeneous magnetic field imposed, exact and approximate wave functions. From the result obtained it follows, in particular, that for the stationary state to be spin-polarized (ρ α 0 (r|r) = ρ β 0 (r|r)) it is necessary for the Hamiltonian to mix states with different parity spins. The consequences from the proved statement for the antiferromagnetic state are discussed.