Done with effective operator
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@ -390,7 +390,7 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
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However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
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First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br{})$ varying in space. %(see Sec.~\ref{sec:weff}).
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Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05}, that we evaluate at $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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%=================================================================
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@ -400,24 +400,35 @@ Then, we choose a specific class of short-range density functionals, namely the
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One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent Coulomb interaction.
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Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
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The present paragraph briefly describes how to obtain an effective interaction $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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%----------------------------------------------------------------
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%\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
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\subsection{Effective Coulomb operator}
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%----------------------------------------------------------------
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Consider the Coulomb operator projected in $\Bas$
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\begin{equation}
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\begin{aligned}
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\weeopbasis = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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\end{aligned}
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\end{equation}
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where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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Consider now the expectation value of $\weeopbasis$ over a general wave function $\wf{}{\Bas}$ belonging to the $\Nel$-electron Hilbert space spanned by $\Bas$.
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One can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
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In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that
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\begin{equation}
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\label{eq:expectweeb}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2}
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\label{eq:int_eq_wee}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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(where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), it is key to realise that
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\begin{equation}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
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\end{equation}
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which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$
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\begin{equation}
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\hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
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\end{equation}
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over the same wave function $\wf{}{\Bas}$, where the indices run over all \alert{occupied} spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{subequations}
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\begin{align}
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\label{eq:expectweeb}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} & = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\\
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\label{eq:expectwee}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} & = \frac{1}{2} \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{align}
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\end{subequations}
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where
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\begin{multline}
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\label{eq:fbasis}
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@ -425,37 +436,14 @@ where
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\\
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= \sum_{ijklmn \in \Bas} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1} \SO{i}{1} \SO{j}{2},
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\end{multline}
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and
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and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that
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\begin{equation}
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\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}},
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}.
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\end{equation}
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is the two-body density tensor of $\wf{}{\Bas}$, $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$.
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Then, consider the expectation value of the exact Coulomb operator over $\wf{}{\Bas}$
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\begin{equation}
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\label{eq:expectwee}
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\mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
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\end{equation}
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where $\n{\wf{}{\Bas}}{(2)}$ is the two-body density associated with $\wf{}{\Bas}$.
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Because $\wf{}{\Bas}$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
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\begin{equation}
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\mel*{\wf{}{\Bas}}{\weeopbasis}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}},
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\end{equation}
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which can be rewritten as:
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\begin{multline}
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\label{eq:int_eq_wee}
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\iint \wbasis \twodmrdiagpsi \dr{1} \dr{2}
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\\
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= \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}.
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\end{multline}
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where we introduced $\wbasis$
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})},
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\end{equation}
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which is the effective interaction in the basis set $\Bas$.
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As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\Bas$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\Bas$.
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Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ tends to the regular coulomb interaction $r_{12}^{-1}$ for all points $(\bx{1},\bx{2})$ and any choice of $\wf{}{\Bas}$ in the limit of a complete basis set.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set $\Bas$.
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Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$.
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%----------------------------------------------------------------
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@ -568,7 +556,7 @@ where $\weeopmu$ is the long-range electron-electron interaction operator
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with
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\begin{equation}
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\label{eq:erf}
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w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) = \frac{\text{erf}(\mu |{\bf r}_1 - {\bf r}_2|)}{|{\bf r}_1 - {\bf r}_2|},
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w^{\text{lr},\mu}(r_{12}) = \frac{\text{erf}(\mu r_{12})}{r_{12}},
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\end{equation}
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and the pair-density operator $\hat{n}^{(2)}({\bf r}_1,{\bf r}_2) =\hat{n}({\bf r}_1) \hat{n}({\bf r}_2) - \delta ({\bf r}_1-{\bf r}_2) \hat{n}({\bf r}_1)$.
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The ECMD functionals admit two limits as function of $\mu$
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