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Clotilde Marut 2019-11-25 11:12:27 +01:00
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Manuscript/FarDFT.tex
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@ -480,9 +480,9 @@ We select STO-3G as minimal basis, and study the behaviour of the total energy o
The bonding and antibonding orbitals of the \ce{H2} molecule are given by The bonding and antibonding orbitals of the \ce{H2} molecule are given by
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\MO{1}{}(\br{}) & = \qty[ \AO{A}(\br{}) + \AO{B}(\br{}) ]/\sqrt{2 + S_{AB}}, \MO{1}{}(\br{}) & = \qty[ \AO{A}(\br{}) + \AO{B}(\br{}) ]/\sqrt{2(1 + S_{AB})},
\\ \\
\MO{2}{}(\br{}) & = \qty[ \AO{A}(\br{}) - \AO{B}(\br{}) ]/\sqrt{2 - S_{AB}}, \MO{2}{}(\br{}) & = \qty[ \AO{A}(\br{}) - \AO{B}(\br{}) ]/\sqrt{2(1 - S_{AB})},
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$. where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$.
@ -501,10 +501,10 @@ with
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:HF0} \label{eq:HF0}
\E{\HF}{(0)} & = \eHc{1} + 2 \eJ{11} - \eK{11}, \E{\HF}{(0)} & = 2 \eHc{1} + 2 \eJ{11} - \eK{11},
\\ \\
\label{eq:HF1} \label{eq:HF1}
\E{\HF}{(1)} & = \eHc{2} + 2 \eJ{22} - \eK{22}, \E{\HF}{(1)} & = 2 \eHc{2} + 2 \eJ{22} - \eK{22},
\end{align} \end{align}
\end{subequations} \end{subequations}
and and
@ -536,10 +536,10 @@ At the (ground-state) LDA level (\ie, we only consider ground-state functionals)
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LDA0} \label{eq:LDA0}
\E{\LDA}{(0)} & = \eHc{1} + 2 \eJ{11} + \int \e{\xc}{\LDA}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{}, \E{\LDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \e{\xc}{\LDA}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{},
\\ \\
\label{eq:LDA1} \label{eq:LDA1}
\E{\LDA}{(1)} & = \eHc{2} + 2 \eJ{22} + \int \e{\xc}{\LDA}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}, \E{\LDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \e{\xc}{\LDA}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{},
\end{align} \end{align}
\end{subequations} \end{subequations}
with with
@ -554,13 +554,13 @@ At the eLDA, we have
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:eLDA0} \label{eq:eLDA0}
\E{\eLDA}{(0)} & = \eHc{1} + 2 \eJ{11} + \int \be{\xc}{(0)}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{}, \E{\eLDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \be{\xc}{(0)}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{},
\\ \\
\label{eq:eLDA1} \label{eq:eLDA1}
\E{\eLDA}{(1)} & = \eHc{2} + 2 \eJ{22} + \int \be{\xc}{(1)}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}, \E{\eLDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \be{\xc}{(1)}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{},
\end{align} \end{align}
\end{subequations} \end{subequations}
with $\be{\xc}{(0)} \equiv \e{\xc}{\LDA}$ and $\be{\xc}{(1)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{}) + \e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})$. with $\be{\xc}{(0)}(\n{}{}) \equiv \e{\xc}{\LDA}(\n{}{})$ and $\be{\xc}{(1)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{}) + \e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})$.
%\titou{Note that we do not consider symmetry-broken solutions.} %\titou{Note that we do not consider symmetry-broken solutions.}
@ -592,19 +592,19 @@ For HF, we have
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\bE{\HF}{\ew{}} \bE{\HF}{\ew{}}
& = (1-\ew{}) \eHc{1} + \ew{} \eHc{2} & = (1-\ew{}) 2 \eHc{1} + \ew{} 2 \eHc{2}
+ \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
\\ \\
& = \ldots & = \ldots
\end{split} \end{split}
\end{equation} \end{equation}
which is clearly quadratic with respect to $\ew{}$. which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term.
For LDA, we have For LDA, we have
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\bE{\LDA}{\ew{}} \bE{\LDA}{\ew{}}
& = (1-\ew{}) \eHc{1} + \ew{} \eHc{2} & = (1-\ew{}) 2 \eHc{1} + \ew{} 2 \eHc{2}
+ \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{} & + \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{}
@ -618,7 +618,7 @@ For eLDA, we have
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\bE{\eLDA}{\ew{}} \bE{\eLDA}{\ew{}}
& = (1-\ew{}) \eHc{1} + \ew{} \eHc{2} & = (1-\ew{}) 2 \eHc{1} + \ew{} 2 \eHc{2}
+ \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{} & + \int \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{}