Position Representation of Effective Electron-Electron Interactions in Solids

An essential ingredient in many model Hamiltonians, such as the Hubbard model, is the effective electron-electron interaction $U$, which enters as matrix elements in some localized basis. These matrix elements provide the necessary information in the model, but the localized basis is incomplete for describing $U$. We present a systematic scheme for computing the manifestly basis-independent dynamical interaction in position representation, $U({\bf r},{\bf r}';\omega)$, and its Fourier transform to time domain, $U({\bf r},{\bf r}';\tau)$. These functions can serve as an unbiased tool for the construction of model Hamiltonians. For illustration we apply the scheme within the constrained random-phase approximation to the cuprate parent compounds La$_2$CuO$_4$ and HgBa$_2$CuO$_4$ within the commonly used 1- and 3-band models, and to non-superconducting SrVO$_{3}$ within the $t_{2g}$ model. Our method is used to investigate the shape and strength of screening channels in the compounds. We show that the O 2$p_{x,y}-$Cu 3$d_{x^2-y^2}$ screening gives rise to regions with strong attractive static interaction in the minimal (1-band) model in both cuprates. On the other hand, in the minimal ($t_{2g}$) model of SrVO$_3$ only regions with a minute attractive interaction are found. The temporal interaction exhibits generic damped oscillations in all compounds, and its time-integral is shown to be the potential caused by inserting a frozen point charge at $\tau=0$. When studying the latter within the three-band model for the cuprates, short time intervals are found to produce a negative potential.