theory
This commit is contained in:
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@ -78,6 +78,7 @@
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\newcommand{\eSat}[2]{\epsilon_{#1,#2}}
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\teHF}[1]{\Tilde{\epsilon}^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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@ -94,6 +95,8 @@
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% Matrix elements
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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\newcommand{\tA}{\Tilde{A}}
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\newcommand{\tB}{\Tilde{B}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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@ -102,6 +105,7 @@
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
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\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
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\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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@ -131,6 +135,14 @@
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\bZ}{\boldsymbol{Z}}
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\newcommand{\fc}{f_\text{c}}
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\newcommand{\Vc}{V_\text{c}}
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\newcommand{\MO}[1]{\phi_{#1}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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@ -170,6 +182,8 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The GW Approximation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
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More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
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@ -259,7 +273,30 @@ Unless otherwise stated, in the remaining of this paper, the {\GOWO} QP energies
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In the case of {\evGW}, the QP energy, $\eGW{p}$, are obtained via Eq.~\eqref{eq:QP-G0W0}, which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref{eq:SigC}]. \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
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At least in the weakly correlated regime where a clear QP solution exists, we believe that, within {\evGW}, the self-consistent algorithm should select the solution of the QP equation \eqref{eq:QP-G0W0} with the largest renormalization weight $\Z{p}(\eGW{p})$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Basis Set Correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The present basis set correction is a two-level correction.
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First, one has to correct the neutral excitations $\Om{x}$ from the RPA calculation.
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The corrected matrix elements read
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\begin{align}
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\label{eq:RPA}
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\tA_{ia,jb} & = \A{ia,jb} + (ia|\fc|jb),
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&
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\tB_{ia,jb} & = \B{ia,jb} + (ia|\fc|bj),
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\end{align}
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where the elements $\A{ia,jb}$ and $\B{ia,jb}$ are given by Eq.~\eqref{eq:RPA}.
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\begin{equation}
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\fc(\br{1},\br{2})= \frac{\delta^2 \Ec}{\delta n(\br{1})\delta n(\br{2})}
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\end{equation}
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In a second time, we correct the GW energy
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\begin{equation}
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\tSigC{p} = \SigC{p} + (p|\Vc|p)
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\end{equation}
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with
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\begin{equation}
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\Vc(\br{}) = \fdv{\Ec}{n(\br{})}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdetails}
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232
srLDA.nb
232
srLDA.nb
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|
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- (-0.5 + (1 + 0.020711000000000035*z + 0.0819306*z**2 - \
|
||||
0.0127713*z**3 + 0.00185898*z**4)/(2.*E**(0.752411*z))))/z**3) +
|
||||
- mu**3*((0.1932056675843198*rs**3*(1 + 0.020711000000000035*rs + \
|
||||
0.0819306*rs**2 - 0.0127713*rs**3 + 0.00185898*rs**4)*(-1 + z**2))/
|
||||
- E**(0.752411*rs) - 0.013547702504838225*rs**5*(((-4.95*rs + \
|
||||
rs**2)*(1 - z**2))/(E**(0.31*rs)*rs**3) +
|
||||
- (3*1.5**0.3333333333333333*Pi**0.6666666666666666*(1 - \
|
||||
0.028562410201116776*rs*(1/(1 - z))**0.3333333333333333)*(1 - z**2))/
|
||||
- (10.*rs**2*(1 + 0.5441599014495957*rs*(1/(1 - \
|
||||
z))**0.3333333333333333 + 0.06349604207872797*rs**2*(1/(1 - \
|
||||
z))**0.6666666666666666)*
|
||||
- (1/(1 - z))**0.6666666666666666) + \
|
||||
(3*1.5**0.3333333333333333*Pi**0.6666666666666666*(1 + z**2)*
|
||||
- (1 - 0.028562410201116776*rs*(1/(1 + \
|
||||
z))**0.3333333333333333))/
|
||||
- (10.*rs**2*(1/(1 + z))**0.6666666666666666*(1 + \
|
||||
0.5441599014495957*rs*(1/(1 + z))**0.3333333333333333 +
|
||||
- 0.06349604207872797*rs**2*(1/(1 + \
|
||||
z))**0.6666666666666666)))) +
|
||||
- mu**4*(2.330975903948103*ec*rs**4 - (0.3632210918893606*rs**6*(1 - \
|
||||
rs**2)*
|
||||
- (-0.5 + (1 + 0.020711000000000035*z + 0.0819306*z**2 - \
|
||||
0.0127713*z**3 + 0.00185898*z**4)/(2.*E**(0.752411*z))))/z**3 -
|
||||
- 0.02122440884682295*rs**5*(((-0.388*rs + 0.676*rs**2)*(1 - \
|
||||
z**2))/(E**(0.547*rs)*rs**2) +
|
||||
- (3*1.5**0.3333333333333333*Pi**0.6666666666666666*(1 - \
|
||||
0.028562410201116776*rs*(1/(1 - z))**0.3333333333333333)*(1 - z**2))/
|
||||
- (10.*rs**2*(1 + 0.5441599014495957*rs*(1/(1 - \
|
||||
z))**0.3333333333333333 + 0.06349604207872797*rs**2*(1/(1 - \
|
||||
z))**0.6666666666666666)*
|
||||
- (1/(1 - z))**0.6666666666666666) + \
|
||||
(3*1.5**0.3333333333333333*Pi**0.6666666666666666*(1 + z**2)*
|
||||
- (1 - 0.028562410201116776*rs*(1/(1 + \
|
||||
z))**0.3333333333333333))/
|
||||
- (10.*rs**2*(1/(1 + z))**0.6666666666666666*(1 + \
|
||||
0.5441599014495957*rs*(1/(1 + z))**0.3333333333333333 +
|
||||
- 0.06349604207872797*rs**2*(1/(1 + \
|
||||
z))**0.6666666666666666)) -
|
||||
- (3*1.5**0.3333333333333333*Pi**0.6666666666666666*((1 - \
|
||||
z)**2.6666666666666665 + (1 + z)**2.6666666666666665))/(20.*rs**2))) +
|
||||
- (((1 - z)**0.6666666666666666 + (1 + z)**0.6666666666666666)**3*(-2 \
|
||||
+ 2*Log(2))*
|
||||
- Log((1 + (31.33952*mu**3*rs**1.5)/((1 - z)**0.6666666666666666 + \
|
||||
(1 + z)**0.6666666666666666)**3 +
|
||||
- (29.798101530536215*mu**2*rs)/((1 - z)**0.6666666666666666 + \
|
||||
(1 + z)**0.6666666666666666)**2 +
|
||||
- (11.6921*mu*Sqrt(rs))/((1 - z)**0.6666666666666666 + (1 + \
|
||||
z)**0.6666666666666666))/
|
||||
- (1 + (13.79404*mu**2*rs)/((1 - z)**0.6666666666666666 + (1 + \
|
||||
z)**0.6666666666666666)**2 +
|
||||
- (11.6921*mu*Sqrt(rs))/((1 - z)**0.6666666666666666 + (1 + \
|
||||
z)**0.6666666666666666))))/(8.*Pi**2))/(1 + 0.6232944601*mu**2*rs**2)**4\
|
||||
\>", "Output",
|
||||
CellChangeTimes->{{3.771781666009734*^9, 3.7717816755992613`*^9},
|
||||
3.771781811362009*^9, 3.7717821333483133`*^9},
|
||||
3.771781811362009*^9, 3.7717821333483133`*^9, 3.771836600647401*^9},
|
||||
CellLabel->
|
||||
"Out[209]//FortranForm=",ExpressionUUID->"c4c0b3fa-e41e-4c64-937f-\
|
||||
db62244b9041"]
|
||||
"Out[20]//FortranForm=",ExpressionUUID->"e8c89351-e3be-455d-b993-\
|
||||
40a6da00af8b"]
|
||||
}, Open ]]
|
||||
}, Open ]],
|
||||
|
||||
@ -901,30 +883,30 @@ Cell[2584, 84, 1207, 40, 52, "Input",ExpressionUUID->"d1a8b6ea-55ae-4a5e-8673-b9
|
||||
Cell[3794, 126, 1033, 35, 52, "Input",ExpressionUUID->"8257931c-d5a2-4fa5-bfcf-242ed662fad5"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[4852, 165, 249, 4, 67, "Section",ExpressionUUID->"da7db347-895a-4445-adf0-185e3e7ae4bc"],
|
||||
Cell[5104, 171, 2181, 58, 69, "Input",ExpressionUUID->"2424c22e-2944-42fb-a926-6931d6b950db"],
|
||||
Cell[7288, 231, 632, 17, 50, "Input",ExpressionUUID->"5e7071ef-4b10-4195-9ab5-72086ae9360d"],
|
||||
Cell[7923, 250, 1502, 41, 53, "Input",ExpressionUUID->"431fbccd-665c-479c-825d-74e37905a953"],
|
||||
Cell[9428, 293, 2440, 73, 115, "Input",ExpressionUUID->"68450462-8882-467d-85de-668cca5b67df"],
|
||||
Cell[11871, 368, 2918, 88, 210, "Input",ExpressionUUID->"76293f49-0f9c-448d-9fa4-7a7821619364"],
|
||||
Cell[14792, 458, 1627, 46, 53, "Input",ExpressionUUID->"7b438198-7fff-4469-8902-d63008ab1edd"],
|
||||
Cell[16422, 506, 1439, 41, 53, "Input",ExpressionUUID->"bb62c6dd-19f1-49b0-93ed-7a72089471a5"],
|
||||
Cell[17864, 549, 1518, 44, 112, "Input",ExpressionUUID->"1b3202c0-1ab4-429c-8dba-a60935c07597"],
|
||||
Cell[19385, 595, 883, 23, 52, "Input",ExpressionUUID->"30aeeec3-f7a5-4138-bd8c-548696d2f752"],
|
||||
Cell[5104, 171, 2180, 58, 69, "Input",ExpressionUUID->"2424c22e-2944-42fb-a926-6931d6b950db"],
|
||||
Cell[7287, 231, 631, 17, 50, "Input",ExpressionUUID->"5e7071ef-4b10-4195-9ab5-72086ae9360d"],
|
||||
Cell[7921, 250, 1501, 41, 53, "Input",ExpressionUUID->"431fbccd-665c-479c-825d-74e37905a953"],
|
||||
Cell[9425, 293, 2439, 73, 115, "Input",ExpressionUUID->"68450462-8882-467d-85de-668cca5b67df"],
|
||||
Cell[11867, 368, 2917, 88, 210, "Input",ExpressionUUID->"76293f49-0f9c-448d-9fa4-7a7821619364"],
|
||||
Cell[14787, 458, 1627, 46, 53, "Input",ExpressionUUID->"7b438198-7fff-4469-8902-d63008ab1edd"],
|
||||
Cell[16417, 506, 1439, 41, 53, "Input",ExpressionUUID->"bb62c6dd-19f1-49b0-93ed-7a72089471a5"],
|
||||
Cell[17859, 549, 1518, 44, 112, "Input",ExpressionUUID->"1b3202c0-1ab4-429c-8dba-a60935c07597"],
|
||||
Cell[19380, 595, 883, 23, 52, "Input",ExpressionUUID->"30aeeec3-f7a5-4138-bd8c-548696d2f752"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[20293, 622, 357, 7, 30, "Input",ExpressionUUID->"408e19ad-23db-4b1a-8d84-8acca9835fae"],
|
||||
Cell[20653, 631, 265, 5, 34, "Output",ExpressionUUID->"a3d7ec91-f8e5-4d9d-8b72-6c30f73a7867"]
|
||||
Cell[20288, 622, 563, 12, 30, "Input",ExpressionUUID->"408e19ad-23db-4b1a-8d84-8acca9835fae"],
|
||||
Cell[20854, 636, 386, 6, 34, "Output",ExpressionUUID->"0b40f4ad-6233-4b3d-8a80-07aca06e731a"]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[20955, 641, 286, 7, 30, "Input",ExpressionUUID->"7d031781-f5af-40e6-b079-1bc0fdb02f2b"],
|
||||
Cell[21244, 650, 4505, 87, 1485, "Output",ExpressionUUID->"c4c0b3fa-e41e-4c64-937f-db62244b9041"]
|
||||
Cell[21277, 647, 282, 6, 30, "Input",ExpressionUUID->"7d031781-f5af-40e6-b079-1bc0fdb02f2b"],
|
||||
Cell[21562, 655, 3550, 64, 531, "Output",ExpressionUUID->"e8c89351-e3be-455d-b993-40a6da00af8b"]
|
||||
}, Open ]]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[25798, 743, 156, 3, 67, "Section",ExpressionUUID->"1774903d-cd4a-427c-913b-6d03354a574f"],
|
||||
Cell[25957, 748, 968, 30, 57, "Input",ExpressionUUID->"a08e663f-6de1-46b7-a344-1b47fcd2e96e"],
|
||||
Cell[26928, 780, 975, 31, 33, "Input",ExpressionUUID->"760031e2-bb91-4ce7-8b6e-8566604d6f6c"],
|
||||
Cell[27906, 813, 1090, 39, 65, "Input",ExpressionUUID->"be645cf4-80f8-46a4-a854-7e7f3cf3adaa"],
|
||||
Cell[28999, 854, 640, 20, 33, "Input",ExpressionUUID->"2bcddbd2-7cce-438d-a1b4-20e94d4eb82b"]
|
||||
Cell[25161, 725, 156, 3, 67, "Section",ExpressionUUID->"1774903d-cd4a-427c-913b-6d03354a574f"],
|
||||
Cell[25320, 730, 968, 30, 57, "Input",ExpressionUUID->"a08e663f-6de1-46b7-a344-1b47fcd2e96e"],
|
||||
Cell[26291, 762, 975, 31, 33, "Input",ExpressionUUID->"760031e2-bb91-4ce7-8b6e-8566604d6f6c"],
|
||||
Cell[27269, 795, 1090, 39, 65, "Input",ExpressionUUID->"be645cf4-80f8-46a4-a854-7e7f3cf3adaa"],
|
||||
Cell[28362, 836, 640, 20, 33, "Input",ExpressionUUID->"2bcddbd2-7cce-438d-a1b4-20e94d4eb82b"]
|
||||
}, Open ]]
|
||||
}, Open ]]
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user