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Spin invariance

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Anthony Scemama 2020-08-02 16:25:57 +02:00
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Manuscript/rsdft-cipsi-qmc.tex
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@ -434,7 +434,7 @@ than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
0.3$~m\hartree{} at the triple-zeta level are obtained for water, and
a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, using the
RSDFT-CIPSI trial wave function with a range-separation parameter
RS-DFT-CIPSI trial wave function with a range-separation parameter
$\mu=1.75$~bohr$^{-1}$ with the double-zeta basis one can obtain for
water a FN-DMC energy $2.6 \pm 0.7$~m\hartree{} lower than the energy
obtained with the FCI trial wave function. This can be explained by
@ -463,12 +463,12 @@ $\mu=\infty$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{overlap.pdf}
\caption{Overlap of the RSDFT CI expansion with the
\caption{Overlap of the RS-DFT CI expansion with the
CI expansion optimized in the presence of a Jastrow factor.}
\label{fig:overlap}
\end{figure}
This data confirms that RSDFT-CIPSI can give improved CI coefficients
This data confirms that RS-DFT-CIPSI can give improved CI coefficients
with small basis sets, similarly to the common practice of
re-optimizing the trial wave function in the presence of the Jastrow
factor. To confirm that the introduction of sr-DFT has an impact on
@ -476,7 +476,7 @@ the CI coefficients similar to the Jastrow factor, we have made the
following numerical experiment. First, we extract the 200 determinants
with the largest weights in the FCI wave function out of a large CIPSI
calculation. Within this set of determinants, we diagonalize
self-consistently the RSDFT Hamiltonian with different values of
self-consistently the RS-DFT Hamiltonian with different values of
$\mu$. This gives the CI expansions $\Psi^\mu$. Then, within the same
set of determinants we optimize the CI coefficients in the presence of
a simple one- and two-body Jastrow factor. This gives the CI expansion
@ -504,7 +504,7 @@ atoms than molecules and atomization energies usually tend to be
underestimated with variational methods.
In the context of FN-DMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded on an atom-centered basis
set. So we expect the fixed-node error to be also tightly related to
set, so we expect the fixed-node error to be also tightly related to
the basis set incompleteness error.
Increasing the size of the basis set improves the description of
the density and of electron correlation, but also reduces the
@ -552,7 +552,7 @@ one-electron, two-electron and one-nucleus-two-electron terms.
The problematic part is the two-electron term, whose simplest form can
be expressed as
\begin{equation}
J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.
J_\text{ee} = \sum_i \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
\end{equation}
The parameter
$a$ is determined by cusp conditions, and $b$ is obtained by energy
@ -606,12 +606,13 @@ Ref.~\onlinecite{Scemama_2015}).
In this section, we make a numerical verification that the produced
wave functions are size-consistent for a given range-separation
parameter.
We have computed the energy of the dissocited fluorine dimer, where
We have computed the energy of the dissociated fluorine dimer, where
the two atoms are at a distance of 50~\AA. We expect that the energy
of this system is equal to twice the energy of the fluorine atom.
The data in table~\ref{tab:size-cons} shows that it is indeed the
case, so we can conclude that the proposed scheme provides
size-consistent FN-DMC energies for all values of $\mu$.
size-consistent FN-DMC energies for all values of $\mu$ (within
$2\times$ statistical error bars).
\subsection{Spin-invariance}
@ -621,8 +622,8 @@ fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for open-shell and
closed-shell systems. A good test is to check that all the components
of a spin multiplet are degenerate.
FCI wave functions are invariant with respect to the spin quantum
number $m_s$, but the introduction of a
FCI wave functions have this property and give degenrate energies with
respect to the spin quantum number $m_s$, but the multiplication by a
Jastrow factor introduces spin contamination if the parameters
for the same-spin electron pairs are different from those
for the opposite-spin pairs.\cite{Tenno_2004}
@ -632,9 +633,9 @@ is used.
Within DFT, the common density functionals make a difference for
same-spin and opposite-spin interactions. As DFT is a
single-determinant theory, the functionals are designed to work in
with the highest value of $m_s$, and therefore different
values of $m_s$ lead to different energies.
single-determinant theory, the density functionals are designed to be
used with the highest value of $m_s$, and therefore different values
of $m_s$ lead to different energies.
So in the context of RS-DFT, the determinantal expansions will be
impacted by this spurious effect, as opposed to FCI.
@ -671,7 +672,7 @@ bias is relatively small, more than one order of magnitude smaller
than the energy gained by reducing the fixed-node error going from the single
determinant to the FCI trial wave function. The highest bias, close to
2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly
below 1~m\hartree when $\mu$ increases. As expected, with $\mu=\infty$
below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
there is no bias (within the error bars), and the bias is not
noticeable with $\mu=5$~bohr$^{-1}$.
@ -682,7 +683,7 @@ noticeable with $\mu=5$~bohr$^{-1}$.
The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
of the RSDFT-CIPSI trial wave functions for energy differences.
of the RS-DFT-CIPSI trial wave functions for energy differences.
@ -704,7 +705,7 @@ B3LYP & 0 & 11.27 & -10.98 & 9.59
\hline
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
\hline
RSDFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
@ -712,7 +713,7 @@ RSDFT-CIPSI & 0 & 4.53 & -1.66 & 5.91
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
\phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
\hline
DMC@RSDFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30 & 3.52(19) & -1.03(19) & 4.39 & 3.16(26) & -0.12(26) & 4.12 \\
DMC@RS-DFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30 & 3.52(19) & -1.03(19) & 4.39 & 3.16(26) & -0.12(26) & 4.12 \\
\phantom{} & 1/4 & 4.04(37) & -3.13(\phantom{0.}37) & 4.88 & 3.39(77) & -0.59(77) & 4.44 & 2.90(25) & 0.25(25) & 3.74 \\
\phantom{} & 1/2 & 3.74(35) & -3.53(\phantom{0.}35) & 4.03 & 2.46(18) & -1.72(18) & 3.02 & 2.06(35) & -0.44(35) & 2.74 \\
\phantom{} & 1 & 5.42(29) & -5.14(\phantom{0.}29) & 4.55 & 4.38(94) & -4.24(94) & 5.11 & --- & --- & --- \\
@ -820,7 +821,7 @@ Simulation of Functional Materials.
\item Exponential increase of number of Slater determinants
\item Cite papiers RS-DFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)
\item Question: does such a scheme provide better nodal quality ?
\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC
\item In RS-DFT we cannot optimize energy with $\mu$ , but in FNDMC
\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI
\begin{itemize}
\item less determinants $\Rightarrow$ large systems