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small changes in appendices

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Julien Toulouse 2020-01-26 15:22:56 +01:00
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Manuscript/srDFT_SC.tex
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@ -834,7 +834,7 @@ where all the local quantities $\argrebasis$ are obtained from the same wave fun
\\
+ \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis).
\end{multline}
Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, \ie, they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we must have
Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, \ie, they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, we must have, for $\br{} \in \Omega_\text{X}$,
\begin{subequations}
\begin{gather}
n_\text{A+B}(\br{}) = n_\text{X}(\br{}),
@ -858,11 +858,11 @@ where the left-hand-side quantities are for the supersystem and the right-hand-s
\ket*{\wf{\text{A}+\text{B}}{}} = \ket*{\wf{\text{A}}{}} \otimes \ket*{\wf{\text{B}}{}},
\label{PsiAB}
\end{equation}
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency (which could arise from spin-multiplet degeneracies) can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency, which could arise from spatial degeneracies (\eg, coming from atomic $p$ states), can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work.
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (\eg, coming from atomic $p$ states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work.
\subsection{Intensivity of the on-top pair density and the local range-separation function}
The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as
The on-top pair density can be written in an orthonormal spatial orbital basis set $\{\SO{p}{}\}$ as
\begin{equation}
\label{eq:def_n2}
n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
@ -877,7 +877,7 @@ where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X
\end{equation}
in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [see Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity.
The local range-separation function is defined as
The local range-separation function is defined as, for $n_{2}(\br{}) \not=0$,
\begin{equation}
\label{eq:def_murAnnex}
\mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})},
@ -887,7 +887,7 @@ where
\label{eq:def_f}
f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation}
Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both $A$ and $B$ vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable
Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both A and B vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation}
\label{eq:def_fa+b}
f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}),
@ -901,14 +901,15 @@ So, $f(\bfr{},\bfr{})$ is a local intensive quantity.
As a consequence, the local range-separation function of the supersystem $\text{A}+\text{B}$ is
\begin{equation}
\label{eq:def_murAB}
\alert{\mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})},}
\mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})},
\end{equation}
which gives
which implies
\begin{equation}
\label{eq:def_murABsum}
\mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{A}}(\bfr{}) + \mu_{\text{B}}(\bfr{}),
\mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{X}}(\bfr{}) \;\; \text{if} \;\; \bfr{} \in \Omega_\text{X},
\end{equation}
with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. The local range-separation function is thus a local intensive quantity.
where $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$.
The local range-separation function is thus a local intensive quantity.
We can therefore conclude that, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent.
@ -930,10 +931,10 @@ where
\label{def_n2_act}
n_{2,\mathcal{A}}(\br{}) & = \sum_{pqrs \in \mathcal{A}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\\
n_{\mathcal{A}}(\br{}) & = \sum_{pq\, \in \mathcal{A}} \phi_p (\br{}) \phi_q (\br{})
n_{\mathcal{A}}(\br{}) & = \sum_{pq \in\mathcal{A}} \phi_p (\br{}) \phi_q (\br{})
\mel*{\wf{}{}}{ \aic{p_\uparrow}\ai{q_\uparrow} + \aic{p_\downarrow}\ai{q_\downarrow} }{\wf{}{}},
\\
n_{\mathcal{I}}(\br{}) & = 2 \sum_{p\, \in \mathcal{I}} \phi_p (\br{})^2
n_{\mathcal{I}}(\br{}) & = 2 \sum_{p \in \mathcal{I}} \phi_p (\br{})^2
\end{align}
\end{subequations}
are the purely active part of the on-top pair density, the active part of the density, and the inactive part of the density, respectively.