loos/srDFT_GW
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

minor corrections in theory section

Browse Source
This commit is contained in:
Pierre-Francois Loos 2019-10-22 22:36:23 +02:00
parent 7c7a2fe812
commit d6ab5a604f
1 changed files with 8 additions and 7 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

15
Manuscript/GW-srDFT.tex
View File

@ -337,7 +337,7 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
In this subsection, we provide the minimal set of equations required to describe {\GOWO}.
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
For sake of generality, we consider closed-shell systems with a $\KS$ reference.
For the sake of simplicity, we consider closed-shell systems with a $\KS$ single-particle reference.
The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals.
Within the {\GW} approximation, the correlation part of the self-energy reads
@ -359,9 +359,9 @@ The screened two-electron integrals
\end{equation}
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}
\begin{equation}
(pq|rs) = \iint \MO{p}(\br{}) \MO{q}(\br{}) \frac{1}{r_{12}} \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}',
(pq|rs) = \iint \MO{p}(\br{1}) \MO{q}(\br{1}) \frac{1}{r_{12}} \MO{r}(\br{2}) \MO{s}(\br{2}) \dbr{1} \dbr{2},
\end{equation}
and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a (direct) random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
and the transition densities $(\bX+\bY)_{ia}^{m}$ originating from a (direct) random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
\begin{equation}
\label{eq:LR}
\begin{pmatrix}
@ -387,7 +387,7 @@ with
B_{ia,jb} & = 2 (ia|jb),
\end{align}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which represent the poles of the screened Coulomb potential $\W{}{}$.
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which represent the poles of the screened Coulomb potential $\W{}{}$.
The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
\begin{equation}
@ -418,9 +418,10 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
%%%%%%%%%%%%%%%%%%%%%%%%
The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{TouGorSav-TCA-05,PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{FerGinTou-JCP-19,LooPraSceTouGin-JPCL-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior~\cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} using the on-top pair density from the uniform-electron gas~\cite{LooPraSceTouGin-JPCL-19}.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{TouGorSav-TCA-05,PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{FerGinTou-JCP-19,LooPraSceTouGin-JPCL-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior~\cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} using the on-top pair density from the uniform-electron gas. \cite{LooPraSceTouGin-JPCL-19}
Additionally to the one-electron density, these RS-DFT functionals requires a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error and is computed using the opposite-spin on-top pair density.
We refer the interested reader to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided.
The basis set corrected {\GOWO} quasiparticle energies are thus given by
\begin{equation}
\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas}
@ -435,8 +436,8 @@ with
& = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{}.
\end{split}
\end{equation}
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-GW correction.
Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent GW calculations.
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-{\GW} correction.
Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent {\GW} calculations.
%%% TABLE I %%%
\begin{squeezetable}