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JCTC_revision/GW-srDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-12-05 10:37:28 +0100 %% Created for Pierre-Francois Loos at 2019-12-09 09:35:19 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -17,7 +17,8 @@
Pages = {084302}, Pages = {084302},
Title = {Ab Initio Determination of The Ionization Potentials of {{DNA}} And {{RNA}} Nucleobases}, Title = {Ab Initio Determination of The Ionization Potentials of {{DNA}} And {{RNA}} Nucleobases},
Volume = {125}, Volume = {125},
Year = {2006}} Year = {2006},
Bdsk-Url-1 = {https://doi.org/10.1063/1.2336217}}
@article{Govoni_2018, @article{Govoni_2018,
Author = {Marco Govoni and Giulia Galli}, Author = {Marco Govoni and Giulia Galli},
@ -2371,7 +2372,7 @@
@article{Loos_2016, @article{Loos_2016,
Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.}, Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
Date-Added = {2018-02-24 12:51:10 +0000}, Date-Added = {2018-02-24 12:51:10 +0000},
Date-Modified = {2018-02-24 12:51:10 +0000}, Date-Modified = {2019-12-09 09:35:15 +0100},
Doi = {10.1002/wcms.1257}, Doi = {10.1002/wcms.1257},
File = {/Users/loos/Zotero/storage/HEXYAMEN/50.pdf}, File = {/Users/loos/Zotero/storage/HEXYAMEN/50.pdf},
Issn = {17590876}, Issn = {17590876},
@ -2381,7 +2382,7 @@
Number = {4}, Number = {4},
Pages = {410--429}, Pages = {410--429},
Shorttitle = {The Uniform Electron Gas}, Shorttitle = {The Uniform Electron Gas},
Title = {The Uniform Electron Gas: {{The}} Uniform Electron Gas}, Title = {{{The}} Uniform Electron Gas},
Volume = {6}, Volume = {6},
Year = {2016}, Year = {2016},
Bdsk-Url-1 = {https://dx.doi.org/10.1002/wcms.1257}} Bdsk-Url-1 = {https://dx.doi.org/10.1002/wcms.1257}}

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JCTC_revision/GW-srDFT.tex
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@ -328,7 +328,7 @@ and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the
with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
This is found from Eq.~\eqref{eq:stat} by using the chain rule, This is found from Eq.~\eqref{eq:stat} by using the chain rule,
\begin{equation} \begin{equation}
\frac{\delta \bE{}{\Bas}[\n{}{}]}{\delta \G{}{}(\br{},\br{}',\omega)} = \int \frac{\delta \bE{}{\Bas}[\n{}{}]}{\delta \n{}{}(\br{}'')} \frac{\delta \n{}{}(\br{}'')}{\delta \G{}{}(\br{},\br{}',\omega)} \dbr{}'', \fdv{\bE{}{\Bas}[\n{}{}]}{\G{}{}(\br{},\br{}',\omega)} = \int \fdv{\bE{}{\Bas}[\n{}{}]}{\n{}{}(\br{}'')} \fdv{\n{}{}(\br{}'')}{\G{}{}(\br{},\br{}',\omega)} \dbr{}'',
\end{equation} \end{equation}
and and
\begin{equation} \begin{equation}
@ -352,7 +352,7 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
\titou{Note that the present basis-set correction is applicable to any approximation of the self-energy (irrespectively of the diagrams included) without altering the CBS limit of such methods. \titou{Note that the present basis-set correction is applicable to any approximation of the self-energy (irrespectively of the diagrams included) without altering the CBS limit of such methods.
Consequently, it can be applied, for example, to GF2 methods (also known as second Born approximation \cite{Stefanucci_2013} in the condensed-matter community) or higher orders. \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} Consequently, it can be applied, for example, to GF2 methods (also known as second Born approximation \cite{Stefanucci_2013} in the condensed-matter community) or higher orders. \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018}
Note, however, that the basis-set correction is optimal for the \textit{exact} self-energy within a given basis set, since it corrects only for the basis-set errors and not for the chosen approximate form of the self-energy within the basis set.} Note, however, that the basis-set correction is optimal for the \textit{exact} self-energy within a given basis set, since it corrects only for the basis-set error and not for the chosen approximate form of the self-energy within the basis set.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The {\GW} Approximation} \subsection{The {\GW} Approximation}
@ -361,7 +361,7 @@ Note, however, that the basis-set correction is optimal for the \textit{exact} s
In this subsection, we provide the minimal set of equations required to describe {\GOWO}. 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}. More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
For the sake of simplicity, we only give the equations for closed-shell systems with a $\KS$ single-particle reference (with a local potential). For the sake of simplicity, we only give the equations for closed-shell systems with a $\KS$ single-particle reference (with a local potential).
The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then the $\KS$ orbitals and orbital energies. The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then the $\KS$ orbitals and their orbital energies.
Within the {\GW} approximation, the correlation part of the self-energy reads Within the {\GW} approximation, the correlation part of the self-energy reads
\begin{equation} \begin{equation}
@ -475,7 +475,8 @@ Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
\infty, & \text{otherwise}, \\ \infty, & \text{otherwise}, \\
\end{cases} \end{cases}
\end{equation} \end{equation}
where, in this work, $f^{\Bas}(\br{},\br{}')$ and $\n{2}{\Bas}(\br{},\br{}')$ are calculated using the opposite-spin two-electron density matrix of a spin-restricted single determinant (such as HF and KS), for a closed-shell system, where, in this work, $f^{\Bas}(\br{},\br{}')$ and $\n{2}{\Bas}(\br{},\br{}')$ are calculated using the opposite-spin two-electron density matrix of a spin-restricted single determinant (such as HF and KS).
For a closed-shell system, we have
\begin{equation} \begin{equation}
\label{fBsum} \label{fBsum}
f^{\Bas}(\br{},\br{}') = 2 \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'), f^{\Bas}(\br{},\br{}') = 2 \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'),
@ -505,24 +506,26 @@ Since the present basis-set correction employs complementary short-range correla
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 $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. 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 $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
\jt{In this work, we have tested two complementary density functionals coming from two approximations to the short-range correlation functional with multideterminant (md) reference of RS-DFT~\cite{Toulouse_2005}. The first one is a short-range local-density approximation ($\srLDA$)~\cite{Toulouse_2005,Paziani_2006} \titou{In this work, we have tested two complementary density functionals coming from two approximations to the short-range correlation functional with multideterminant (md) reference of RS-DFT. \cite{Toulouse_2005}
The first one is a short-range local-density approximation ($\srLDA$) \cite{Toulouse_2005,Paziani_2006}
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\srLDA}{\Bas}[\n{}{}] = \bE{\srLDA}{\Bas}[\n{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\srLDA}\qty(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \int \n{}{}(\br{}) \be{\text{c,md}}{\srLDA}\qty(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation} \end{equation}
where the correlation energy per particle $\be{\text{c,md}}{\srLDA}\qty(\n{}{},\rsmu{}{})$ has been parametrized from uniform-electron gas calculations in Ref.~\onlinecite{Paziani_2006}. The second one is a short-range Perdew-Burke-Ernzerhof ($\srPBE$) approximation \cite{Ferte_2019, Loos_2019} where the correlation energy per particle $\be{\text{c,md}}{\srLDA}\qty(\n{}{},\rsmu{}{})$ has been parametrized from calculations on the uniform electron gas \cite{Loos_2016} reported in Ref.~\onlinecite{Paziani_2006}.
The second one is a short-range Perdew-Burke-Ernzerhof ($\srPBE$) approximation \cite{Ferte_2019, Loos_2019}
\begin{equation} \begin{equation}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\srPBE}{\Bas}[\n{}{}] = \bE{\srPBE}{\Bas}[\n{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\srPBE}\qty(\n{}{}(\br{}),s(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \int \n{}{}(\br{}) \be{\text{c,md}}{\srPBE}\qty(\n{}{}(\br{}),s(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation} \end{equation}
where $s(\br{})=\nabla n(\br{})/n(\br{})^{4/3}$ is the reduced density gradient and the correlation energy per particle $\be{\text{c,md}}{\srPBE}\qty(\n{}{},s,\rsmu{}{})$ interpolates between the usual PBE correlation energy per particle \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density of the Coulomb uniform-electron gas (see Ref.~\onlinecite{Loos_2019}). Note that the information on the local basis-set incompleteness error is provided to these RS-DFT functionals through the range-separation function $\rsmu{}{\Bas}(\br{})$. where $s(\br{})=\nabla n(\br{})/n(\br{})^{4/3}$ is the reduced density gradient and the correlation energy per particle $\be{\text{c,md}}{\srPBE}\qty(\n{}{},s,\rsmu{}{})$ interpolates between the usual PBE correlation energy per particle \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density of the Coulombic uniform electron gas (see Ref.~\onlinecite{Loos_2019}). Note that the information on the local basis-set incompleteness error is provided to these RS-DFT functionals through the range-separation function $\rsmu{}{\Bas}(\br{})$.
} }
\jt{From these energy functionals, we generate the potentials $\bpot{\srLDA}{\Bas}[\n{}{}](\br{}) = \delta \bE{\srLDA}{\Bas}[\n{}{}]/\delta \n{}{}(\br{})$ and $\bpot{\srPBE}{\Bas}[\n{}{}](\br{}) = \delta \bE{\srPBE}{\Bas}[\n{}{}]/\delta \n{}{}(\br{})$ (considering $\rsmu{}{\Bas}(\br{})$ as being fixed) which are then used to obtain the basis-set corrected {\GOWO} quasiparticle energies \titou{From these energy functionals, we generate the potentials $\bpot{\srLDA}{\Bas}[\n{}{}](\br{}) = \delta \bE{\srLDA}{\Bas}[\n{}{}]/\delta \n{}{}(\br{})$ and $\bpot{\srPBE}{\Bas}[\n{}{}](\br{}) = \delta \bE{\srPBE}{\Bas}[\n{}{}]/\delta \n{}{}(\br{})$ (considering $\rsmu{}{\Bas}(\br{})$ as being fixed) which are then used to obtain the basis-set corrected {\GOWO} quasiparticle energies
\begin{equation} \begin{equation}
\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas} \beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas},
\label{eq:QP-corrected} \label{eq:QP-corrected}
\end{equation} \end{equation}
with with
@ -533,8 +536,7 @@ with
\end{split} \end{split}
\end{equation} \end{equation}
where $\bpot{}{\Bas}[\n{}{}](\br{})=\bpot{\srLDA}{\Bas}[\n{}{}](\br{})$ or $\bpot{\srPBE}{\Bas}[\n{}{}](\br{})$ and the density is calculated from the HF or KS orbitals. where $\bpot{}{\Bas}[\n{}{}](\br{})=\bpot{\srLDA}{\Bas}[\n{}{}](\br{})$ or $\bpot{\srPBE}{\Bas}[\n{}{}](\br{})$ and the density is calculated from the HF or KS orbitals.
} The explicit expressions of these srLDA and srPBE correlation potentials are provided in the {\SI}.}
\titou{The explicit expressions of these srLDA and srPBE correlation potentials are provided in the {\SI}.}
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis-set correction is a non-self-consistent, \textit{post}-{\GW} correction. 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. Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent {\GW} calculations.
@ -637,7 +639,7 @@ The FC density-based basis-set correction~\cite{Loos_2019} is used consistently
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \ie, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization). The {\GOWO} quasiparticle energies have been obtained ``graphically'', \ie, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization).
Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero. Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
Compared to the conventional $\order*{\Nocc^3 \Nvirt^3}$ computational cost of {\GW}, the present basis-set correction represents a marginal $\order*{\Nocc^2 \Nbas^2 \Ngrid}$ additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}. \titou{Compared to the conventional $\order*{\Nocc^3 \Nvirt^3}$ computational cost of {\GW}, the present basis-set correction represents a marginal $\order*{\Nocc^2 \Nbas^2 \Ngrid}$ additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.}
Note, however, that the formal $\order*{\Nocc^3 \Nvirt^3}$ computational scaling of {\GW} can be significantly reduced thanks to resolution-of-the-identity techniques \cite{vanSetten_2013, Bruneval_2016, Duchemin_2017} and other tricks. \cite{Rojas_1995, Duchemin_2019} Note, however, that the formal $\order*{\Nocc^3 \Nvirt^3}$ computational scaling of {\GW} can be significantly reduced thanks to resolution-of-the-identity techniques \cite{vanSetten_2013, Bruneval_2016, Duchemin_2017} and other tricks. \cite{Rojas_1995, Duchemin_2019}
%%% FIG 1 %%% %%% FIG 1 %%%