Interpolated energy densities, correlation indicators and lower bounds from approximations to the strong coupling limit of DFT

Abstract

We investigate the construction of approximated exchange-correlation functionals by interpolating locally along the adiabatic connection between the weak- and the strong-coupling regimes, focussing on the effect of using approximate functionals for the strong-coupling energy densities. The gauge problem is avoided by dealing with quantities that are all locally defined in the same way. Using exact ingredients at weak coupling we are able to isolate the error coming from the approximations at strong coupling only. We find that the nonlocal radius model, which retains some of the non-locality of the exact strong-coupling regime, yields very satisfactory results. We also use interpolation models and quantities from the weak- and strong-coupling regimes to define a correlation-type indicator and a lower bound to the exact exchange-correlation energy. Open problems, related to the nature of the local and global slope of the adiabatic connection at weak coupling, are also discussed.

Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy

“…This has been recently used for constructing XC functionals from a local interpolation along the adiabatic connection. 14,27−29 Although this approach is promising for treating strong correlation within the realm of DFT, 14 it can still easily overcorrelate (for example for stretched bonds it overcorrelates the fragments), again because the SCE (exact or approximate) quantities are often far from the physical ones. 29 …”

“…The XC energy densities along the adiabatic connection (i.e., position-dependent quantities w λ ( r ) that integrate to W λ [ρ] when multiplied by the density) are not uniquely defined and therefore we have to be specific on their gauge . 14,29,43−45 A physically sound gauge often considered in DFT is the one of the electrostatic potential of the XC hole. 14,34,35,46 Within this gauge we can express the λ-dependent energy density in terms of the corresponding spherically averaged XC hole 14,34,35,47 h xc λ ( r , u ) obtained from |Ψ λ [ρ]| 2 ,where u = | r – r ′| is the distance from a reference electron at r .…”

“…While in the stretched H 2 molecule the static correlation effects are dominant, at intermediate bond lengths, around L b ∼ 5.0 au, there is a subtle interplay between dynamic and static correlation effects. 29 This region can be even more challenging for DFAs than the stretched case, given that certain DFAs that dissociate H 2 correctly fail in this scenario yielding a positive “bump” (see, e.g., refs (14 and 65−67)). We can see that the MRF-1 energy densities are very accurate at L b = 5.0 au, hardly distinguishable from the reference ones.…”

“…To recover the missing T c [ρ] component, one can combine W 1 MRF [ρ] with the quantities from the weak coupling limit, namely, W 0 [ρ] and W 0 ′ [ρ], to interpolate W λ [ρ] and thus obtain E xc [ρ]. For this purpose, we employ a very simple interpolation form, the two-legged representation, 14,29,68 which has recently been used to construct a tight lower bound to correlation energies. 29 This form reads asAs in this work, we use W 1 MRF [ρ] as an approximation to W 1 [ρ], we call this approach the “2-leg MRF”, and from Figure 4 we can see that it substantially improves the MRF-1 energies.…”

Assessment of interaction-strength interpolation formulas for gold and silver clusters

“…It has been found that the local interpolations are in general more accurate than their global counterpart. 89 Thus, the study carried out here provides also a very useful first idea of what could be achieved with these higher-level approaches.…”