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In the context of the Holstein polaron problem it is shown that the dynamicalmean field theory (DMFT) corresponds to the summation of a special class oflocal diagrams in the skeleton expansion of the self-energy. In the real spacerepresentation, these local diagrams are characterized by the absence of vertexcorrections involving phonons at different lattice sites. Such correctionsvanish in the limit of infinite dimensions, for which the DMFT provides theexact solution of the Holstein polaron problem. However, for finite dimensionalsystems the accuracy of the DMFT is limited. In particular, it cannot describecorrectly the adiabatic spreading of the polaron over multiple lattice sites.Arguments are given that the DMFT limitations on vertex corrections found forthe Holstein polaron problem persist for finite electron densities andarbitrary phonon dispersion.

Diagrammatic content of the DMFT for the Holstein polaron problem in finite dimensions