Insights from exact exchange-correlation kernels

The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness ${f}_{\text{xc}}$ are set out. frequency dependence found be largely due its analytic structure, i.e., singularities at certain frequencies, which required in order capture particular transitions, including those double excitation character. However, within first few interacting excitations, approximately $\ensuremath{\omega}$ independent, meaning adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies failings local and random phase these lowest transitions. key differences between common approximations analyzed, cannot eliminated by exploiting limited gauge freedom ${f}_{\text{xc}}$. optical spectrum benefits from using as accurate possible an ground-state xc potential, while maintaining compatibility two less importance.