work done on algorithm
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%% This BibTeX bibliography file was created using BibDesk.
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%% Created for Pierre-Francois Loos at 2020-08-07 20:17:07 +0200
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%% Created for Pierre-Francois Loos at 2020-08-08 08:15:47 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Pulay_1980,
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Author = {Pulay, P{\'e}ter},
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Date-Added = {2020-08-08 08:15:42 +0200},
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Date-Modified = {2020-08-08 08:15:42 +0200},
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Doi = {10.1016/0009-2614(80)80396-4},
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Issn = {00092614},
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Journal = {Chem. Phys. Lett.},
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Language = {en},
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Month = jul,
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Number = {2},
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Pages = {393--398},
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Title = {Convergence Acceleration of Iterative Sequences. the Case of Scf Iteration},
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Volume = {73},
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Year = {1980},
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Bdsk-Url-1 = {https://dx.doi.org/10.1016/0009-2614(80)80396-4}}
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@article{Pulay_1982,
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Author = {Pulay, P.},
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Date-Added = {2020-08-08 08:15:42 +0200},
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Date-Modified = {2020-08-08 08:15:42 +0200},
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Doi = {10.1002/jcc.540030413},
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Issn = {0192-8651, 1096-987X},
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Journal = {J. Comput. Chem.},
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Language = {en},
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Number = {4},
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Pages = {556--560},
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Title = {{{ImprovedSCF}} Convergence Acceleration},
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Volume = {3},
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Year = {1982},
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Bdsk-Url-1 = {https://dx.doi.org/10.1002/jcc.540030413}}
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@article{Assaraf_2000,
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Author = {Assaraf, Roland and Caffarel, Michel and Khelif, Anatole},
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Date-Added = {2020-08-07 20:12:45 +0200},
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@ -334,7 +334,10 @@ the FCI wave function.
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\centering
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\includegraphics[width=0.7\linewidth]{algorithm.pdf}
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\caption{Algorithm showing the generation of the RS-DFT wave
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function $\Psi^{\mu}$ starting from the .}
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function $\Psi^{\mu}$ starting from $\Psi^{(0)}$.
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The outer (macro-iteration) and inner (micro-iteration) loops are represented in red and blue, respectively.
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The steps common to both loops are represented in purple.
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DIIS extrapolations of the one-electron density are introduced in both the outer and inner loops in order to speed up convergence of the self-consistent process.}
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\label{fig:algo}
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\end{figure*}
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@ -347,23 +350,28 @@ connecting HF to FCI which consists in increasing the number of determinants.
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We follow the KS-to-FCI path by performing FCI calculations using the
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RS-DFT Hamiltonian with different values of $\mu$.
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Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a
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single- and multi-determinant wave function $\Psi^{(0)}$.
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single- or multi-determinant wave function $\Psi^{(0)}$ which can
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be obtained in many different ways depending on the system that one considers.
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One of the particularity of the present work is that we
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have used the CIPSI algorithm to perform approximate FCI calculations
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with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
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In the outer loop (red), a CIPSI selection is performed with a RS Hamiltonian
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parameterized using the current one-electron density.
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An inner loop (blue) is introduced to accelerate the
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use the CIPSI algorithm to perform approximate FCI calculations
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with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
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This provides a multi-determinant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.
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In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selection is performed
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to obtain $\Psi^{\mu\,(k)}$ with the RS Hamiltonian $\hat{H}^{\mu\,(k)}$
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parameterized using the current one-electron density $n^{(k)}$.
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At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
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One exits the macro-iteration loop when the absolute energy difference between two successive iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to \titou{???} in the present study.
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An inner (micro-iteration) loop (blue) is introduced to accelerate the
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convergence of the self-consistent calculation, in which the set of
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determinants is kept fixed, and only the diagonalization of the
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RS-Hamiltonian is performed iteratively with the updated density.
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determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
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$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
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The micro-iteration loop is exited when the absolute energy difference between two successive iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been set to \titou{???} in the present study.
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The convergence of the algorithm was further improved
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by introducing a direct inversion in the iterative subspace (DIIS)
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step to extrapolate the density both in the outer and inner loops.
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step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
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Note that any range-separated post-HF method can be
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implemented using this scheme by just replacing the CIPSI step by the
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post-HF method of interest.
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\titou{T2: introduce $\tau_1$ and $\tau_2$. More description of the algorithm needed.}
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Note that, thanks to the self-consistent nature of the algorithm,
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the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.
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