On the Kohn--Sham density response in a localized basis set

We construct the Kohn--Sham density response function $\chi_{0}$ in a previously described basis of the space of orbital products. The calculational complexity of our construction is $O(N^{2}N_{\omega})$ for a molecule of $N$ atoms and in a spectroscopic window of $N_{\omega}$ frequency points. As a first application, we use $\chi_{0}$ to calculate molecular spectra from the Petersilka--Gossmann--Gross equation. With $\chi_{0}$ as input, we obtain correct spectra with an extra computational effort that grows also as $O(N^2 N_{\omega})$ and, therefore, less steeply in $N$ than the $O(N^{3})$ complexity of solving Casida's equations. Our construction should be useful for the study of excitons in molecular physics and in related areas where $\chi_{0}$ is a crucial ingredient.Comment: 20 pages, 11 figure

Fast construction of the Kohn--Sham response function for molecules

The use of the LCAO (Linear Combination of Atomic Orbitals) method for excited states involves products of orbitals that are known to be linearly dependent. We identify a basis in the space of orbital products that is local for orbitals of finite support and with a residual error that vanishes exponentially with its dimension. As an application of our previously reported technique we compute the Kohn--Sham density response function $\chi_{0}$ for a molecule consisting of $N$ atoms in $N^{2}N_{\omega}$ operations, with $N_{\omega}$ the number of frequency points. We test our construction of $\chi_{0}$ by computing molecular spectra directly from the equations of Petersilka--Gossmann--Gross in $N^{2}N_{\omega}$ operations rather than from Casida's equations which takes $N^{3}$ operations. We consider the good agreement with previously calculated molecular spectra as a validation of our construction of $\chi_{0}$. Ongoing work indicates that our method is well suited for the computation of the GW self-energy $\Sigma=\mathrm{i}GW$ and we expect it to be useful in the analysis of exitonic effects in molecules

Adaptively truncated Hilbert space based impurity solver for dynamical mean-field theory

We present an impurity solver based on adaptively truncated Hilbert spaces. The solver is particularly suitable for dynamical mean-field theory in circumstances where quantum Monte Carlo approaches are ineffective. It exploits the sparsity structure of quantum impurity models, in which the interactions couple only a small subset of the degrees of freedom. We further introduce an adaptive truncation of the particle or hole excited spaces, which enables computations of Green functions with an accuracy needed to avoid unphysical (sign change of imaginary part) self-energies. The method is benchmarked on the one-dimensional Hubbard model.Comment: 10 pages, 7 figure