A general approach to the realistic electronic structure
calculations of non-local correlated systems, which combines the
Local Density Approximation with the cluster Dynamical Mean-Field
Theory (LDA+CDMFT) is discussed. We present a quantum theory of
the Mott transition in Ti2O3  and VO2 . Strong competition
between the local Coulomb interaction and chemical bonding in a
metal-metal pair results in a small insulating gap of low
temperature phase. The cluster LDA+DMFT scheme successfully
describe the non-local spin and orbital fluctuation effects and
Mott transition in these compounds. The interplay of strong
electronic Coulomb interactions and structural distortions, in
particular the dimerization of metal atoms in the low temperature
phase, plays a crucial role. We find these oxides are not a
conventional Mott insulator, but that the formation of dynamical
meta-metal singlet pairs due to strong Coulomb correlations is
necessary to trigger the opening of a Peierls gap. The
possibilities to includes the general interaction vertex in the
cluster DMFT calculations within the continuous time QMC-scheme
as well as k-dependence of self-energy  will be discussed.
 A. Poteryaev, A. Lichtenstein, and G. Kotliar "Nonlocal Coulomb Interactions and Metal-Insulator Transition in Ti2O3: A Cluster LDA + DMFT Approach ", Phys. Rev. Lett. 93, 086401 (2004).
 S. Biermann, A. Poteryaev, A. I. Lichtenstein, and A. Georges "Dynamical Singlets and Correlation-Assisted Peierls Transition in VO2", Phys. Rev. Lett. 94, 026404 (2005).
 V. Savkin, A. Rubtsov, M. Katsnelson, and A. Lichtenstein "Correlated Adatom Trimer on a Metal Surface: A Continuous-Time Quantum Monte Carlo Study", Phys. Rev. Lett. 94, 026402 (2005).
 A.N. Rubtsov, M.I. Katsnelson and A.I. Lichtenstein, "Dual fermion approach to nonlocal correlations in the Hubbard model", cond-mat/0612196.
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