Table 1 Summary of the general COT approach and its declinations.
From: Connector theory for reusing model results to determine materials properties
Kohn–Sham | General | 1D potential | Band structure | |
|---|---|---|---|---|
Object of interest | vxc(r; [n]) | O(x; [Q]) | E(j; ω0, L) | GKK(k, ω; [V]) |
Model | HEG | t.b. chosen | 1D square well | HEG |
Object in model | \({v}_{{{{\rm{xc}}}}}^{h}({n}^{h})\) | \({{{{\mathcal{O}}}}}_{x}({{{\mathcal{Q}}}})\) | \({{{{\mathcal{E}}}}}_{j}(L)={\pi }^{2}{j}^{2}/(2{L}^{2})\) | \({{{{\mathcal{G}}}}}_{{{{\bf{k}}}}\omega {{{\bf{K}}}}}({{{\mathcal{V}}}})\) |
Connector | \({n}_{{{{\bf{r}}}}}^{c}[n]\) | \({{{{\mathcal{Q}}}}}_{x}^{c}[Q]\) | \({L}_{j{\omega }_{0}L}^{c}=\frac{\pi j}{\sqrt{2E(j;{\omega }_{0},L)}}\) | \({{{{\mathcal{V}}}}}_{{{{\bf{k}}}}\omega {{{\bf{K}}}}}^{c}\) |
Approximation | 1st order in densitya | t.b. chosen | 1st order in curvature | transferability |
Appr. real | \({v}_{{{{\rm{xc}}}}}^{h}({n}_{0})+\int d{{{\bf{r}}}}^{\prime} (n({{{\bf{r}}}}^{\prime})-{n}_{0}){f}_{{{{\rm{xc}}}}}(| {{{\bf{r}}}}-{{{\bf{r}}}}^{\prime} | ;{n}_{0})\) | Oapprox(x; [Q]) | Eapprox(j; ω0, L) | G(k) ≈ G(k0) |
Appr. model | \({v}_{{{{\rm{xc}}}}}^{h}({n}_{0})+({n}^{h}-{n}_{0}){f}_{{{{\rm{xc}}}}}^{h}({n}_{0})\) | \({{{{\mathcal{O}}}}}_{x}^{{{{\rm{approx}}}}}({{{\mathcal{Q}}}})\) | \({{{{\mathcal{E}}}}}_{j}^{{{{\rm{approx}}}}}(\tilde{L})\,\,\,\,\,\) | \({{{{\mathcal{G}}}}}_{{{{\bf{k}}}}}\approx {{{{\mathcal{G}}}}}_{{{{{\bf{k}}}}}_{0}}\) |
Appr. connector | \({n}_{{{{\bf{r}}}}}^{c,{{{\rm{approx}}}}}[n]=\frac{1}{{f}_{{{{\rm{xc}}}}}^{h}({n}_{0})}\int d{{{\bf{r}}}}^{\prime} \,n({{{\bf{r}}}}^{\prime}){f}_{{{{\rm{xc}}}}}(| {{{\bf{r}}}}-{{{\bf{r}}}}^{\prime} | ;{n}_{0})\) | \({{{{\mathcal{Q}}}}}_{x}^{c,{{{\rm{approx}}}}}\) | \({L}_{{\omega }_{0}L}^{c,{{{\rm{approx}}}}}\,\,\,\,\,\,\,\,\,\,\) | \({{{{\mathcal{V}}}}}_{{{{\bf{k}}}}\omega {{{\bf{K}}}}}^{c,{{{\rm{approx}}}}}={{{{\mathcal{V}}}}}_{{{{{\bf{k}}}}}_{0}\omega {{{\bf{K}}}}}^{c}\) |
Appr. result | \({v}_{{{{\rm{xc}}}}}^{h}({n}_{{{{\bf{r}}}}}^{c,{{{\rm{approx}}}}}[n])\) | \({{{{\mathcal{O}}}}}_{x}({{{{\mathcal{Q}}}}}_{x}^{c,{{{\rm{approx}}}}})\) | \({{{{\mathcal{E}}}}}_{j}({L}_{{\omega }_{0}L}^{c,{{{\rm{approx}}}}})\) | \({{{{\mathcal{G}}}}}_{{{{\bf{k}}}}\omega {{{\bf{K}}}}}({{{{\mathcal{V}}}}}_{{{{\bf{k}}}}\omega {{{\bf{K}}}}}^{c,{{{\rm{approx}}}}})\) |
