Working on paper
This commit is contained in:
parent
41b18cade3
commit
753392c87d
@ 626,4 +626,63 @@ note={Gaussian Inc. Wallingford CT}


issn = {00219606},


publisher = {American Institute of Physics},


doi = {10.1063/1.1487829}


}




@article{Ludovicy_2019,


author = {Ludovicy, Jil and Mood, Kaveh Haghighi and


Lüchow, Arne},


title = {{Full Wave Function Optimization with Quantum Monte Carlo{}A


Study of the Dissociation Energies of ZnO, FeO, FeH, and CrS}},


journal = {J. Chem. Theory Comput.},


volume = {15},


number = {10},


pages = {52215229},


year = {2019},


month = {Oct},


issn = {15499618},


publisher = {American Chemical Society},


doi = {10.1021/acs.jctc.9b00241}


}




@article{HaghighiMood_2017,


author = {Haghighi Mood, Kaveh and Lüchow, Arne},


title = {{Full Wave Function Optimization with Quantum Monte Carlo and Its Effect on


the Dissociation Energy of FeS}},


journal = {J. Phys. Chem. A},


volume = {121},


number = {32},


pages = {61656171},


year = {2017},


month = {Aug},


issn = {10895639},


publisher = {American Chemical Society},


doi = {10.1021/acs.jpca.7b05798}


}




@article{Scemama_2006,


author = {Scemama, Anthony and Filippi, Claudia},


title = {{Simple and efficient approach to the optimization of correlated wave functions}},


journal = {Phys. Rev. B},


volume = {73},


number = {24},


pages = {241101},


year = {2006},


month = {Jun},


issn = {24699969},


publisher = {American Physical Society},


doi = {10.1103/PhysRevB.73.241101}


}




@article{Filippi_2000,


author = {Filippi, Claudia and Fahy, Stephen},


title = {{Optimal orbitals from energy fluctuations in correlated wave functions}},


journal = {J. Chem. Phys.},


volume = {112},


number = {8},


pages = {35233531},


year = {2000},


month = {Feb},


issn = {00219606},


publisher = {American Institute of Physics},


doi = {10.1063/1.480507}


}

@ 103,14 +103,6 @@ The favorable scaling of QMC, its very low memory requirements and


its adequation with massively parallel architectures make it a


serious alternative for highaccuracy simulations on large systems.




Different molecular orbitals can be chosen to build the single Slater


determinant. Three main options are commonly chosen: HartreeFock (HF),


KohnSham (KS) or natural (NO) orbitals of a correlated wave function.


The nodal surfaces obtained with the KS determinant are in general


better than those obtained with the HF determinant,\cite{Per_2012} and


of comparable quality to those obtained with a Slater determinant


built with NOs.\cite{Wang_2019}




As it is not possible to minimize directly the DMC energy with respect


to the variational parameters of the trial wave function, the


fixednode approximation is much more difficult to control than the


@ 118,10 +110,11 @@ finitebasis approximation.


The conventional approach consists in multiplying the trial wave


function by a positive function, the \emph{Jastrow factor}, taking


account of the electronelectron cusp and the shortrange correlation


effects. The wave function is then reoptimized in the presence of the


Jastrow factor and the nodal surface is expected to be improved.


Using this technique, it has been shown that the chemical accuracy


could be reached within DMC.\cite{Petruzielo_2012}


effects. The wave function is then reoptimized within Variational


Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal


surface is expected to be improved. Using this technique, it has been


shown that the chemical accuracy could be reached within


DMC.\cite{Petruzielo_2012}




Another approach consists in considering the DMC method as a


\emph{postFCI method}. The trial wave function is obtained by


@ 138,8 +131,8 @@ exponential scaling of the size of the trial wave function.


Extrapolation techniques have been used to estimate the DMC energies


obtained with FCI wave functions,\cite{Scemama_2018} and other authors


have used a combination of the two approaches where highly truncated


CIPSI trial wave functions are reoptimized under the presence of a


Jastrow factor to keep the number of determinants


CIPSI trial wave functions are reoptimized in VMC under the presence


of a Jastrow factor to keep the number of determinants


small,\cite{Giner_2016} and where the consistency between the


different wave functions is kept by imposing a constant energy


difference between the estimated FCI energy and the variational energy


@ 189,41 +182,76 @@ within this context.


\section{Combining rangeseparated DFT with CIPSI}


\label{sec:rsdftcipsi}




Starting from a HartreeFock determinant in a small basis set,


we have seen that we can systematically improve the trial wave


function in two directions. The first one is by increasing the


size of the atomic basis set, and the second one is by


increasing the determinant expansion towards the FCI limit.


A third direction of improvement exists. It is the path which connects


the HartreeFock determinant to the KohnSham determinant, which


usually has a better nodal surface. This third path is obtained by


the socalled \emph{rangeseparated} DFT framework which splits the


electronelectron interaction in a longrange part and a shortrange


part. HELP MANU!




In singledeterminant DMC calculations, the degrees of freedom used to


reduce the fixednode error are the molecular orbitals on which the


Slater determinant is built.


Different molecular orbitals can be chosen:


HartreeFock (HF), KohnSham (KS), natural (NO) orbitals of a


correlated wave function, or orbitals optimized under the


presence of a Jastrow factor.


The nodal surfaces obtained with the KS determinant are in general


better than those obtained with the HF determinant,\cite{Per_2012} and


of comparable quality to those obtained with a Slater determinant


built with NOs.\cite{Wang_2019} Orbitals obtained in the presence


of a Jastrow factor are generally superior to KS


orbitals.\cite{Filippi_2000,Scemama_2006,HaghighiMood_2017,Ludovicy_2019}




The description of electron correlation within DFT is very different


from correlated methods.


In DFT, one solves a mean field problem with a modified potential


incorporating the effects of electron correlation, whereas in


correlated methods the real Hamiltonian is used and the


electronelectron interactions are considered.


Nevertheless, as the orbitals are oneelectron functions,


the procedure of orbital optimization in the presence of the


Jastrow factor can be interpreted as a selfconsistent field procedure


with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.


So DFT can be viewed as a very cheap way of introducing the effect of


correlation in the parameters determining the nodal surface. But in


the general case, even at the complete basis set limit a fixednode


error will remain because the singledeterminant ans\"atz does not


have enough freedom to describe the exact nodal surface.


If one wants to have to exact CBS limit, a multideterminant


parameterization of the wave functions is required.




\subsection{CIPSI}




\emph{Configuration interaction using a perturbative selection made


iteratively} (CIPSI)\cite{Huron_1973} belongs to the class of selected


CI methods, whose goal is to build compressed configuration interaction


(CI) wave functions with a controlled accuracy, together with an estimate


of the associated eigenvalue obtained with perturbation theory.


Beyond the singledeterminant representation, the optimal wave


function is obtained with FCI. FCI is often presented as a


\emph{postHartreeFock} method, which implies that there is a


continuous connection between the HartreeFock and FCI wave functions.




Selected CI methods take a very short path between the HartreeFock


determinant and the FCI wave function by increasing iteratively the


number of determinants on which the wave function is expanded and


producing relatively compact wave functions.




Using a conventional iterative diagonalization method such as


Davidson's,\cite{Davidson_1975} the size of the CI space is


predefined, and at each iteration the length of the vectors is equal


to the size of the CI space. Within selected CI methods, at each


iteration the size of the determinant space is increased such that


one keeps only the most important determinants that will lead to an


accurate description of the state of interest. The lowest eigenpairs


are extracted from the CI matrix expressed in this determinant


subspace, and the CI energy can be estimated by computing a


secondorder perturbative correction (PT2) to the variational energy


computed in the complete CI space. The magnitude of the PT2 correction


is a measure of the distance to the exact eigenvalue, and is an


adjustable parameter to control the quality of the wave function.


Davidson's,\cite{Davidson_1975} the dimension of the configuration


interaction (CI) space is predefined, so the dimensions of the CI


vectors grow rapidly towards the size of the CI space. Within selected


CI methods, at each iteration the size of the determinant space is


increased such that one keeps only the most important determinants


that will lead to an accurate description of the state of


interest. The lowest eigenpairs are extracted from the CI matrix


expressed in this determinant subspace, and the CI energy can be


estimated by computing a secondorder perturbative correction (PT2) to


the variational energy computed in the complete CI space. The


magnitude of the PT2 correction is a measure of the distance to the


exact eigenvalue, and is an adjustable parameter to control the


quality of the wave function.


Within the \emph{Configuration interaction using a perturbative


selection made iteratively} (CIPSI)\cite{Huron_1973} method, the PT2


correction is computed along with the determinant selection. So the


magnitude of the PT2 correction can be made the only parameter of the


algorithm, and we choose this parameter as the convergence criterion.




Considering that the perturbatively corrected energy is a reliable


estimate of the FCI energy, using a fixed value of the PT2 as a stopping


criterion enforces a constant distance of all the calculations to the


FCI energy. In this work, we target the chemical accuracy so all the


CIPSI selections were made such that $\EPT < 1$~mE$_h$.








\subsection{Rangeseparated DFT in a nutshell}



Loading…
Reference in New Issue
Block a user