reformating equations
This commit is contained in:
parent
a65b8a8a19
commit
fab902e121
@ -178,7 +178,7 @@ The present eLDA functional is specifically designed to compute double excitatio
|
||||
The paper is organised as follows.
|
||||
In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
|
||||
Section \ref{sec:func} provides details about the construction of the weight-dependent xc LDA functional.
|
||||
The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}.
|
||||
The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:resdis}.
|
||||
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
|
||||
Unless otherwise stated, atomic units are used throughout.
|
||||
|
||||
@ -418,13 +418,13 @@ Combining these, we build a two-state weight-dependent correlation functional:
|
||||
Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}.
|
||||
The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ldd}
|
||||
& \tabc{Ground state} & \tabc{Doubly-excited state} \\
|
||||
& \tabc{$I=0$} & \tabc{$I=1$} \\
|
||||
\begin{tabular}{lll}
|
||||
& \tabc{Ground state} & \tabc{Doubly-excited state} \\
|
||||
& \tabc{$I=0$} & \tabc{$I=1$} \\
|
||||
\hline
|
||||
$a_1$ & -0.023\,818\,4 & -0.014\,463\,3 \\
|
||||
$a_2$ & +0.005\,409\,94 & -0.050\,601\,9 \\
|
||||
$a_3$ & +0.083\,076\,6 & +0.033\,141\,7 \\
|
||||
$a_1$ & $-0.023\,818\,4$ & $-0.014\,463\,3$ \\
|
||||
$a_2$ & $+0.005\,409\,94$ & $-0.050\,601\,9$ \\
|
||||
$a_3$ & $+0.083\,076\,6$ & $+0.033\,141\,7$ \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
@ -552,8 +552,8 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o
|
||||
%%%%%%%%%%%%%%%
|
||||
%%% RESULTS %%%
|
||||
%%%%%%%%%%%%%%%
|
||||
\section{Results}
|
||||
\label{sec:res}
|
||||
\section{Results and discussion}
|
||||
\label{sec:resdis}
|
||||
Here, we consider as testing ground the minimal-basis \ce{H2} molecule.
|
||||
We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr).
|
||||
This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry.
|
||||
@ -561,6 +561,11 @@ We have then access to the individual densities of the ground and doubly-excited
|
||||
Moreover, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities.
|
||||
In other words, there is no density-driven error and the only error that we are going to observe is the functional-driven error (and this is what we want to study).
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Results}
|
||||
\label{sec:res}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The bonding and antibonding orbitals of the \ce{H2} molecule are given by
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
@ -703,7 +708,6 @@ Alternatively to Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}, o
|
||||
\end{equation}
|
||||
(This is what one would do in practice, \ie, by performing a KS ensemble calculation.)
|
||||
We will label these energies as $\tE{}{\ew{}}$ to avoid confusion with the expressions reported in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}.
|
||||
\begin{widetext}
|
||||
For HF, we have
|
||||
\begin{equation}
|
||||
\label{eq:bEwHF}
|
||||
@ -711,10 +715,12 @@ For HF, we have
|
||||
\tE{\HF}{\ew{}}
|
||||
& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
|
||||
+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
|
||||
+ \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
|
||||
\\
|
||||
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
|
||||
+ (1-\ew{})^2 (2\eJ{11}- \eK{11}) + \ew{}^2 (2\eJ{22}- \eK{22}) + 2 (1-\ew{})\ew{} (2 \eJ{12} - \eK{12}),
|
||||
& + \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
|
||||
\\
|
||||
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} + (1-\ew{})^2 (2\eJ{11}- \eK{11})
|
||||
\\
|
||||
& + \ew{}^2 (2\eJ{22}- \eK{22}) + 2 (1-\ew{})\ew{} (2 \eJ{12} - \eK{12}),
|
||||
\end{split}
|
||||
\end{equation}
|
||||
which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term.
|
||||
@ -725,14 +731,17 @@ In the case of the LDA, it reads
|
||||
\tE{\LDA}{\ew{}}
|
||||
& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
|
||||
+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
|
||||
+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
|
||||
\\
|
||||
& + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
|
||||
+ \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
|
||||
\\
|
||||
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
|
||||
+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
|
||||
\\
|
||||
& + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
|
||||
\\
|
||||
& + (1-\ew{}) \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
|
||||
+ \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
|
||||
\end{split}
|
||||
\end{equation}
|
||||
which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term.
|
||||
@ -743,25 +752,33 @@ For eLDA, the ensemble energy can be decomposed as
|
||||
\tE{\eLDA}{\ew{}}
|
||||
& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
|
||||
+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
|
||||
+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
|
||||
\\
|
||||
& + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
|
||||
+ \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
|
||||
\\
|
||||
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
|
||||
+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
|
||||
\\
|
||||
& + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
|
||||
+ \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
& + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
|
||||
\\
|
||||
& + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
+ \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
|
||||
\\
|
||||
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
|
||||
+ (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ]
|
||||
+ \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ]
|
||||
\\
|
||||
& + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12}
|
||||
+ \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
+ \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ],
|
||||
& + \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
|
||||
% & + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
|
||||
% \\
|
||||
% & + \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
% \\
|
||||
% & + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
% \\
|
||||
% & + \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
|
||||
% \\
|
||||
% & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
|
||||
% \\
|
||||
% &
|
||||
% + (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ]
|
||||
% \\
|
||||
% & + \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ]
|
||||
% \\
|
||||
% & + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12}
|
||||
% + \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
% & + \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ],
|
||||
\end{split}
|
||||
\end{equation}
|
||||
which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term.
|
||||
@ -782,36 +799,55 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
\eps{1}{\ew{},\LDA}
|
||||
& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \frac{1}{2} \int \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(0)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \frac{1}{2} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{},
|
||||
\end{split}
|
||||
\\
|
||||
\begin{split}
|
||||
\eps{2}{\ew{},\LDA}
|
||||
& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \frac{1}{2} \int \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(1)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \frac{1}{2} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
|
||||
\end{split}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
\eps{1}{\ew{},\eLDA}
|
||||
& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \frac{1}{2} \int \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(0)}(\br{}) d\br{},
|
||||
\\
|
||||
& + \frac{1}{2} \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{},
|
||||
\end{split}
|
||||
\\
|
||||
\begin{split}
|
||||
\eps{2}{\ew{},\eLDA}
|
||||
& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2\ew{} \eJ{22}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{}.
|
||||
\\
|
||||
& + \frac{1}{2} \int \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(1)}(\br{}) d\br{}.
|
||||
\\
|
||||
& + \frac{1}{2} \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}.
|
||||
\end{split}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
\end{widetext}
|
||||
|
||||
The derivative discontinuity is modelled by the last term of the right-hand-side of Eq.~\eqref{eq:dEdw}.
|
||||
Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Discussion}
|
||||
\label{sec:dis}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Numerical results are reported in Table \ref{tab:Energies}.
|
||||
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user