# KS-DFT versus G0W0

• The central equation that KS-DFT will solve is:

$$\tag{1}\left[ -\dfrac{1}{2}\nabla^2+V_{ext}+V_{hartree}+V_{xc} \right]\psi_i(\vec{r})=E_i\psi_i(\vec{r})$$

Here $$V_{xc}$$ is approximated with LDA or GGA.

• The central equation that GW will solve is:

$$\tag{2}\left[ -\dfrac{1}{2}\nabla^2+V_{ext}+V_{hartree}+\Sigma \right]\psi_i(\vec{r})=E_i\psi_i(\vec{r})$$

Here the self-energy operator $$\Sigma$$ is calculated with GW approximation.

One can see the GW method is the same as KFT except for replacing $$V_{xc}$$ by $$\Sigma$$. KS-DFT usually underestimates the bandgap of semiconducting materials while GW can give a satisfactory description for that. So, what's the difference between $$V_{xc}$$ and $$\Sigma$$?

I am going to try to go systematically through your question and answer the critical bits. I may edit this answer going forward as I'm able to express my thoughts better or people point out important subtleties that I may be missing.

First, in KS-DFT (without any Generalized KS (GKS) or w/e), $$V_{xc}$$ is explicitly multiplicative and local. So it is necessarily always something of the form $$V_{xc}(\vec{r}) \psi(\vec{r})$$. Also, $$V_{xc}$$ is static, meaning that you cannot get quasiparticle lifetimes from it. Which isn't a problem since KS states are not quasiparticle states anyway (except some more recent GKS formulations). But the primary message right now is: KS DFT with necessarily local functionals does not give you QP states. There is some fantastic work by Baerends and co. out there that goes into the details of the relations between KS states and QP states.

Now, onto GW. The GW self-energy is frequency-dependent, non-Hermitian, and nonlocal (although short-ranged, it exponentially decays away for gapped systems, or finite-temperature metals, see ref 1, p 257). Given that $$\Sigma(1, 2)=iG(1, 2)W(1^+, 2)$$, where $$1\equiv(\vec{r}_1, t_1)$$ and similarly for $$2$$, you see that $$\Sigma$$ acts between two points. It's a way (roughly) of describing 'ah yes, the electron (or more commonly, test charge) here is feeling this potential due to the rest of the material'.

The GW approximation (with whatever approximation you want for $$W$$, typically the random phase approximation is taken for the $$\epsilon$$ used to construct $$W$$) is rigorously formulated so that, given a noninteracting set of wavefunctions to act as a basis (the DFT wavefunctions, in this case), the energies it gives you are rigorously quasiparticle energies. Meaning that it will actually model (obviously to within the limitations of the approximation) addition-removal energies that you would measure using photoemission/inverse photoemission.

I've really shortened many things here, and I'm sure some people here may even think I've shortened it so far as to be misleading (which I've tried not to be, but we are all human).

So if you want to look more into this, here are a few of my favourite references on the matter:

1. Interacting Electrons Theory and Computational Approaches: Martin, Reining, Ceperley. This is probably the one-stop text for an introduction to GW and the general ideas behind it, and even the specifics. The presentation is extremely pedagogical, and I personally find this text positively fantastic. Chapter 9+ has what you're looking for.
2. From the Kohn–Sham band gap to the fundamental gap in solids. An integer electron approach: Pedagogical review by Baerends on what the different gaps in KSDFT mean, and hopw they are related to each other. https://pubs.rsc.org/en/content/articlelanding/2017/cp/c7cp02123b#!divAbstract
3. The Kohn–Sham gap, the fundamental gap and the optical gap: the physical meaning of occupied and virtual Kohn–Sham orbital energies: Another review by Baerends and colleagues. Very illuminating, and just generally a great read. https://pubs.rsc.org/en/content/articlelanding/2013/cp/c3cp52547c#!divAbstract

All three of the above resources have numerous 'aha!' moments, where connections about the physics and the formalism click, and I have personally found them extremely helpful in my journey through electronic structure.

• So say your density is represented on a 3D grid of points. So is your $V_{xc}$. Both are 3D grids. To get the energy contribution from $V_{xc}$, you pointwise multiply these two arrays, and sum up/integrate. So to get $E_{xc}$, you've done a element-by-element multiplication of the density and the potential, followed by a sum (you've done a dot product). May 20 at 14:48