loos/srDFT_SC
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

changes in SI

Browse Source
This commit is contained in:
Julien Toulouse 2020-01-20 23:34:35 +01:00
parent 24be5f2ed0
commit 5d1afbdc56
1 changed files with 6 additions and 18 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

24
Manuscript/SI/srDFT_SC-SI.tex
View File

@ -286,16 +286,16 @@
\section{Size consistency of the basis-set correction}
\subsection{General considerations}
\subsection{Sufficient condition for size consistency}
The following paragraph gives a proof of the size consistency of the basis-set correction. The basis-set correction is expressed as an integral in real space
The basis-set correction is expressed as an integral in real space
\begin{equation}
\label{eq:def_ecmdpbebasis}
\begin{aligned}
& \efuncdenpbe{\argebasis} = \\ & \int \text{d}\br{} \,\denr \ecmd(\argrebasis),
\end{aligned}
\end{equation}
where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$
where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$
\begin{equation}
\label{eq:def_ecmdpbebasis}
\begin{aligned}
@ -329,25 +329,13 @@ where the left-hand-side quantities are for the supersystem and the right-hand-s
\begin{equation}
\ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}},
\end{equation}
where $\otimes$ is the antisymmetric tensor product. In this case, it is well known that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid.
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 explicity 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 $S_z$, i.e. 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 (e.g., coming from atomic p states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragement. This applies to the systems treated in this work.
For spatial degeneracies such as different p states, by selected the same member of the ensemble in the supersystem and the isolated fragements.
applies to the systems treated in this work.
Even though this does deal with the case of degeneracies,
To avoid difficulties arising from spin-multiplet degeneracy, we can use the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ depending only on $n(\br{})$ and $n_{2}(\br{})$ or a zero spin polarization $\zeta(\br{}) = 0$.
Regarding the density and its gradients, these are necessary intensive quantities. The remaining questions are therefore the local range-separation parameter $\murpsi$ and the on-top pair density.
\subsection{Property of the on-top pair density}
\subsection{Intensivity of the on-top pair density}
A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as
\begin{equation}
\label{eq:def_n2}
n_{2,\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
n_{2{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$.
Assume now that the wave function $\wf{A+B}{}$ of the super system $A+B$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$