Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revisionNext revision | Previous revision |
| ericmarkusnotes:start [2017/04/13 16:08] – [Kohn-Sham equations] estachura | ericmarkusnotes:start [2017/04/13 19:10] (current) – [Kohn-Sham equations] estachura |
|---|
| * There is also the previous result of Jerome here [[https://arxiv.org/pdf/1309.3587.pdf|Jerome's previous paper]] which is more numerical in nature. Here bounded domains with homogeneous B.C. are also studied. | * There is also the previous result of Jerome here [[https://arxiv.org/pdf/1309.3587.pdf|Jerome's previous paper]] which is more numerical in nature. Here bounded domains with homogeneous B.C. are also studied. |
| |
| * 1D Hartree potential for the electron charge density $\rho = |\psi|^2$: $$V_H(x,t) = 1/|x| \ast \rho (x,t) = \int_a^b \dfrac{\rho(y,t)}{|x-y|} dy$$ and Hilbert transform for $f \in L^2 (\mathbb{R})$ is $$H f (y) = \lim_{\epsilon \to 0} \dfrac{1}{\pi} \int_{|x| > \epsilon} \dfrac{f(y-x)}{x} dx$$ Recall that $H: L^2 (\mathbb{R}) \to L^2 (\mathbb{R})$ is an isometry and for $1<p < \infty$, $H: W^{s,p}(\mathbb{R}) \to W^{s,p}(\mathbb{R})$ is an isomorphism. In the 1D case Jerome suggests considering the softened potential given (e.g. for Helium atom) by $$V(x_1, x_2) = \dfrac{1}{\sqrt{(x_1-x_2)^2+1}}$$ subject to a softened external potential $$V(x) = \dfrac{-2}{\sqrt{x^2+1}}$$ Jerome suggests Hilbert transform is needed for this, but maybe if another type of softened potential is considered, e.g. Eric's $V(x) = \dfrac{e^{-C/|x|}}{|x|}$ no Hilbert transform is needed? | * 1D Hartree potential for the electron charge density $\rho = |\psi|^2$: $$V_H(x,t) = 1/|x| \ast \rho (x,t) = \int_a^b \dfrac{\rho(y,t)}{|x-y|} dy$$ and Hilbert transform for $f \in L^2 (\mathbb{R})$ is $$H f (y) = \lim_{\epsilon \to 0} \dfrac{1}{\pi} \int_{|x| > \epsilon} \dfrac{f(y-x)}{x} dx$$ Recall that $H: L^2 (\mathbb{R}) \to L^2 (\mathbb{R})$ is an isometry and for $1<p < \infty$, $H: W^{s,p}(\mathbb{R}) \to W^{s,p}(\mathbb{R})$ is an isomorphism. In the 1D case Jerome suggests considering the softened potential given (e.g. for Helium atom) by $$V(x_1, x_2) = \dfrac{1}{\sqrt{(x_1-x_2)^2+1}}$$ subject to a softened external potential $$V_{\text{ext}}(x) = \dfrac{-2}{\sqrt{x^2+1}}$$ Jerome suggests Hilbert transform is needed for this, but maybe if another type of softened potential is considered, e.g. Eric's $V(x) = \dfrac{e^{-C/|x|}}{|x|}$ no Hilbert transform is needed? |
| | |
| | * Ionic potentials not included in Jerome's analysis, can we include them? What general form do they take? |
| |
| |
