Insights from exact exchange-correlation kernels

Abstract

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

Double and Charge-Transfer Excitations in Time-Dependent Density Functional Theory

“…An explicit computation of the frequency-dependence of the exact xc kernel is quite a computational feat, given that it is effectively solving an inverse problem which is very sensitive to small errors. Yet it has been achieved (32,33,34) on model systems, and results verify the simple pole structure of the xc kernel near double or multiple excitations that was postulated in a simple model in Ref. (29) (Sec.…”

“…Ref. (32) performed real-time calculations of a kick-perturbation that is localized in space and time, to find effectively the functional derivatives in χ(r, r , t − t ) and χS(r, r , t − t ), then Fourier-transforming to the frequency-domain, to reveal a full spatial and frequency-dependency of the kernel, (33,34) worked directly in the frequency domain to construct the true and KS response functions. In both approaches, regions of small density need much care; a thorough analysis together with different ways to deal with this can be found in Ref.…”

“…In both approaches, regions of small density need much care; a thorough analysis together with different ways to deal with this can be found in Ref. (34). An interesting issue that arose is the "gauge-freedom" of the xc kernel (34,35,36): adding functions g(r, ω) (independent of r ), h(r , ω) (independent of r), or a spatiallyindependent term, to fXC(r, r , ω) has no effect in the Dyson equation Eq.…”

“…(34). An interesting issue that arose is the "gauge-freedom" of the xc kernel (34,35,36): adding functions g(r, ω) (independent of r ), h(r , ω) (independent of r), or a spatiallyindependent term, to fXC(r, r , ω) has no effect in the Dyson equation Eq. 4.…”

Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals

“…Post-SCF RPA methods can describe noncovalent interactions very accurately without the need of semiempirical corrections. ,, However, they also suffer from unphysical self-interaction errors due to the neglect of the exchange kernel. , This results in favorable error cancellation in some cases, for instance in the description of bond dissociation or the calculation of barrier heights, but also in overestimated magnitudes of total correlation energies and a poor description of ionization or atomization energies. ,,, In order to overcome these shortcomings, several first-principles approaches to go beyond the RPA have been developed, which often yield major improvements over RPA correlation energies. Those approaches are typically either derived within the ACFD theorem ,,− or within MBPT. ,− However, most of them come with an increased computational cost.…”