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13
docs/source/_static/links.rst
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@ -40,3 +40,16 @@
.. |qp| replace:: *Quantum Package*
.. |SLURM| replace:: `SLURM`_
.. |ZeroMQ| replace:: `ZeroMQ`_
.. |CAS| replace:: :abbr:`CAS (Complete Active Space)`
.. |CAS-SD| replace:: :abbr:`CAS-SD (Complete Active Space plus all Singles and Doubles)`
.. |CCD| replace:: :abbr:`CCD (Coupled Cluster with Double Excitations)`
.. |CCSD| replace:: :abbr:`CCSD (Coupled Cluster with Single and Double Excitations)`
.. |CID| replace:: :abbr:`CID (Configuration Interaction with Double Excitations)`
.. |CIPSI| replace:: :abbr:`CIPSI (Configuration Interaction using a Perturbative Selection)`
.. |CI| replace:: :abbr:`CI (Configuration Interaction)`
.. |CISD| replace:: :abbr:`CISD (Configuration Interaction with Single and Double Excitations)`
.. |CIS| replace:: :abbr:`CIS (Configuration Interaction with Single Excitations)`
.. |DDCI| replace:: :abbr:`DDCI (Difference Dedicated Configuration Interaction)`
.. |FCI| replace:: :abbr:`FCI (Full Configuration Interaction)`
.. |HF| replace:: :abbr:`HF (Hartree-Fock)`

6
docs/source/conf.py
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@ -47,9 +47,9 @@ suppress_warnings = [
# ones.
extensions = [
'sphinx.ext.mathjax',
# 'sphinx.ext.imgmath',
'sphinx.ext.githubpages',
'sphinxcontrib.bibtex'
]
# Add any paths that contain templates here, relative to this directory.
@ -134,13 +134,15 @@ latex_elements = {
# Additional stuff for the LaTeX preamble.
#
# 'preamble': '',
# 'preamble': '\usepackage{physics}\n',
# Latex figure (float) alignment
#
# 'figure_align': 'htbp',
}
# Grouping the document tree into LaTeX files. List of tuples
# (source start file, target name, title,
# author, documentclass [howto, manual, or own class]).

182
docs/source/intro/selected.bib Normal file
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@ -0,0 +1,182 @@
@article{Bytautas_2009,
doi = {10.1016/j.chemphys.2008.11.021},
url = {https://doi.org/10.1016%2Fj.chemphys.2008.11.021},
year = 2009,
month = {feb},
publisher = {Elsevier {BV}},
volume = {356},
number = {1-3},
pages = {64--75},
author = {Laimutis Bytautas and Klaus Ruedenberg},
title = {A priori identification of configurational deadwood},
journal = {Chemical Physics}
}
@article{Anderson_2018,
doi = {10.1016/j.comptc.2018.08.017},
url = {https://doi.org/10.1016%2Fj.comptc.2018.08.017},
year = 2018,
month = {oct},
publisher = {Elsevier {BV}},
volume = {1142},
pages = {66--77},
author = {James S.M. Anderson and Farnaz Heidar-Zadeh and Paul W. Ayers},
title = {Breaking the curse of dimension for the electronic Schrodinger equation with functional analysis},
journal = {Computational and Theoretical Chemistry}
}
@article{Bender_1969,
doi = {10.1103/physrev.183.23},
url = {http://dx.doi.org/10.1103/PhysRev.183.23},
year = 1969,
month = {jul},
publisher = {American Physical Society ({APS})},
volume = {183},
number = {1},
pages = {23--30},
author = {Charles F. Bender and Ernest R. Davidson},
title = {Studies in Configuration Interaction: The First-Row Diatomic Hydrides},
journal = {Phys. Rev.}
}
@article{Whitten_1969,
doi = {10.1063/1.1671985},
url = {https://doi.org/10.1063%2F1.1671985},
year = 1969,
month = {dec},
publisher = {{AIP} Publishing},
volume = {51},
number = {12},
pages = {5584--5596},
author = {J. L. Whitten and Melvyn Hackmeyer},
title = {Configuration Interaction Studies of Ground and Excited States of Polyatomic Molecules. I. The {CI} Formulation and Studies of Formaldehyde},
journal = {The Journal of Chemical Physics}
}
@article{Huron_1973,
doi = {10.1063/1.1679199},
url = {https://doi.org/10.1063%2F1.1679199},
year = 1973,
month = {jun},
publisher = {{AIP} Publishing},
volume = {58},
number = {12},
pages = {5745--5759},
author = {B. Huron and J. P. Malrieu and P. Rancurel},
title = {Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions},
journal = {The Journal of Chemical Physics}
}
@article{Knowles_1984,
author="Peter J. Knowles and Nicholas C Handy",
year=1984,
journal={Chem. Phys. Letters},
volume=111,
pages="315--321",
title="A New Determinant-based Full Configuration Interaction Method"
}
@article{Scemama_2013,
author = {{Scemama}, A. and {Giner}, E.},
title = "{An efficient implementation of Slater-Condon rules}",
journal = {ArXiv [physics.comp-ph]},
pages = {1311.6244},
year = 2013,
month = nov,
url = {https://arxiv.org/abs/1311.6244}
}
@article{Sharma_2017,
doi = {10.1021/acs.jctc.6b01028},
url = {https://doi.org/10.1021%2Facs.jctc.6b01028},
year = 2017,
month = {mar},
publisher = {American Chemical Society ({ACS})},
volume = {13},
number = {4},
pages = {1595--1604},
author = {Sandeep Sharma and Adam A. Holmes and Guillaume Jeanmairet and Ali Alavi and C. J. Umrigar},
title = {Semistochastic Heat-Bath Configuration Interaction Method: Selected Configuration Interaction with Semistochastic Perturbation Theory},
journal = {Journal of Chemical Theory and Computation}
}
@article{Holmes_2016,
doi = {10.1021/acs.jctc.6b00407},
url = {https://doi.org/10.1021%2Facs.jctc.6b00407},
year = 2016,
month = {aug},
publisher = {American Chemical Society ({ACS})},
volume = {12},
number = {8},
pages = {3674--3680},
author = {Adam A. Holmes and Norm M. Tubman and C. J. Umrigar},
title = {Heat-Bath Configuration Interaction: An Efficient Selected Configuration Interaction Algorithm Inspired by Heat-Bath Sampling},
journal = {Journal of Chemical Theory and Computation}
}
@article{Evangelisti_1983,
doi = {10.1016/0301-0104(83)85011-3},
url = {https://doi.org/10.1016%2F0301-0104%2883%2985011-3},
year = 1983,
month = {feb},
publisher = {Elsevier {BV}},
volume = {75},
number = {1},
pages = {91--102},
author = {Stefano Evangelisti and Jean-Pierre Daudey and Jean-Paul Malrieu},
title = {Convergence of an improved {CIPSI} algorithm},
journal = {Chemical Physics}
}
@article{Booth_2009,
doi = {10.1063/1.3193710},
url = {https://doi.org/10.1063%2F1.3193710},
year = 2009,
publisher = {{AIP} Publishing},
volume = {131},
number = {5},
pages = {054106},
author = {George H. Booth and Alex J. W. Thom and Ali Alavi},
title = {Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space},
journal = {The Journal of Chemical Physics}
}
@article{Booth_2010,
doi = {10.1063/1.3407895},
url = {https://doi.org/10.1063%2F1.3407895},
year = 2010,
month = {may},
publisher = {{AIP} Publishing},
volume = {132},
number = {17},
pages = {174104},
author = {George H. Booth and Ali Alavi},
title = {Approaching chemical accuracy using full configuration-interaction quantum Monte Carlo: A study of ionization potentials},
journal = {The Journal of Chemical Physics}
}
@article{Cleland_2010,
doi = {10.1063/1.3302277},
url = {https://doi.org/10.1063%2F1.3302277},
year = 2010,
month = {jan},
publisher = {{AIP} Publishing},
volume = {132},
number = {4},
pages = {041103},
author = {Deidre Cleland and George H. Booth and Ali Alavi},
title = {Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo},
journal = {The Journal of Chemical Physics}
}
@article{Garniron_2017b,
doi = {10.1063/1.4992127},
url = {https://doi.org/10.1063%2F1.4992127},
year = 2017,
month = {jul},
publisher = {{AIP} Publishing},
volume = {147},
number = {3},
pages = {034101},
author = {Yann Garniron and Anthony Scemama and Pierre-Fran{\c{c}}ois Loos and Michel Caffarel},
title = {Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory},
journal = {The Journal of Chemical Physics}
}

205
docs/source/intro/selected_ci.rst
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@ -1,90 +1,155 @@
Selected Configuration Interaction
==================================
These methods rely on the same principle as the usual CI approaches, except
.. default-role:: cite
These methods rely on the same principle as the usual |CI| approaches, except
that determinants aren't chosen *a priori* based on an occupation or
excitation criterion, but selected *on the fly* among the entire set of
determinants based on their estimated contribution to the Full-CI wave function.
Conventional CI methods can be seen as an exact resolution of Schrödinger's
equation for a complete, well-defined subset of determinants (and for a given
atomic basis set), while selected CI methods are closer to a truncation of the
Full-CI.
determinants based on their estimated contribution to the |FCI| wave function.
It has been noticed long ago that, even inside a predefined subspace of
determinants, only a small number significantly contributes to the wave
function.:cite:`Bytautas_2009,Anderson_2018` Therefore, an *on the fly*
function. `Bytautas_2009,Anderson_2018` Therefore, an *on the fly*
selection of determinants is a rather natural idea that has been proposed
in the late 60's by Bender and Davidson :cite:`Bender_1969` as well as Whitten
and Hackmeyer :cite:`Whitten_1969`.
in the late 60's by Bender and Davidson `Bender_1969` as well as Whitten
and Hackmeyer. `Whitten_1969`
The approach we are using in the |qp| is based on :abbr:`CIPSI (Configuration
Interaction using a Perturbative Selection` developed by Huron, Rancurel and
Malrieu,:cite:`Huron_1973` that iteratively selects *external* determinants
|kalpha| (determinants which are not present in the wave
function :math:`|\Psi \rangle = \sum_i c_i |i \rangle`) using a perturbative
criterion
The approach we are using in the |qp| is based on |CIPSI| developed by Huron,
Rancurel and Malrieu, `Huron_1973` that iteratively selects *external*
determinants (determinants which are not present in the variational space)
using a perturbative criterion.
.. math::
e_\alpha = \frac{\langle \Psi |{\hat H}| \alpha \rangle^2 }{E - H_{\alpha\alpha}}
with |kalpha| the external determinant being considered, and |ealpha| the
estimated gain in correlation energy that would be brought by the inclusion of
|kalpha| in the wave function. |EPT| is an estimation of the total missing
correlation energy:
.. math::
\begin{align}
E_\text{PT2} & = \sum_{\alpha} e_\alpha \\
E_\text{FCI} & \approx E + E_\text{PT2}
\end{align}
There is however a computational downside. In \textit{a priori} selected
methods, the rule by which determinants are selected is known \textit{a
priori}, and therefore, one can map a particular determinant to some row or
column index.\cite{Knowles_1984} As a consequence, it can be systematically
determined to which matrix element of $\widehat{H}$ a two-electron integral
There is however a computational downside. In *a priori* selected
methods, the rule by which determinants are selected is known *a
priori*, and therefore, one can map a particular determinant to some row or
column index. `Knowles_1984` As a consequence, it can be systematically
determined to which matrix element of :math:`\hat H` a two-electron integral
contributes. This allows for the implementation of so-called
\emph{integral-driven} methods, that work essentially by iterating over
integrals.
*integral-driven* methods, that work essentially by iterating over
integrals and are very fast.
On the contrary, in selected methods an explicit list has to be maintained, and
there is no immediate way to know whether a determinant has been selected, or
what its index is. Consequently, so-called \emph{determinant-driven} approaches
will be used, in which iteration is done over determinants rather than
integrals. This can be a lot more expensive, since the number of determinants
is typically much larger than the number of integrals. The number of
determinants scales as $\order{\Norb!}$ while the number of integrals scales as
$\order{\Norb^4}$ with the number of MOs.
On the contrary, in selected methods an explicit list of determinants has to be
kept, and there is no immediate way to know whether a determinant has been
selected, or what its index is in the list. Consequently, a
*determinant-driven* approach will be used, in which the loops run over
determinants rather than integrals. This can be a lot more computationally
expensive since the number of determinants is typically much larger than the
number of integrals.
Furthermore, determinant-driven methods require an effective way to compare
determinants in order to extract the corresponding excitation operators, and a
way to rapidly fetch the associated integrals involved
section~\ref{sec:meth_mel}.
What makes *determinant-driven* approaches possible here is:
Because of this high computational cost, approximations have been
proposed.\cite{Evangelisti_1983} And recently, the \emph{Heat-Bath
Configuration Interaction (HCI)} algorithm has taken farther the idea of a more
approximate but extremely cheap selection.\cite{Holmes_2016, Sharma_2017}
Compared to CIPSI, the selection criterion is simplified to
- the fact that selected |CI| methods will keep the number of determinants small
enough, orders of magnitude smaller than in *a priori* selected methods for
wave functions with equal energies,
- an efficient way to compare determinants in order to extract the
corresponding excitation operators `Scemama_2013`,
- an intense filtering of the internal space to avoid as much as possible
determinant comparisons of disconnected determinants,
- a fast retrieval of the corresponding two-electron integrals in memory.
Simple Algorithm
----------------
.. default-role:: math
.. |SetDI| replace:: `\{|D_I\rangle\}^{(n)}`
.. |Psi_n| replace:: `|\Psi^{(n)}\rangle`
.. |H| replace:: `\hat H`
.. |kalpha| replace:: `|\alpha\rangle`
.. |kalpha_star| replace:: `\{ |\alpha \rangle \}_\star ^{(n)}`
.. |ealpha| replace:: `e_\alpha`
.. |EPT| replace:: `E_\text{PT2}`
The variational wave function |Psi_n| is defined over a set of determinants
|SetDI| in which we diagonalize |H|.
.. math::
e^{\text{HCI}}_\alpha = \max \qty( \qty| c_I \Hij{D_I}{\alpha} | )
This algorithmically allows for an extremely fast selection of doubly
excited determinants by an integral-driven approach.
Full Configuration Interaction Quantum Monte Carlo (FCI-QMC) is an alternate
approach to selection recently proposed in 2009 by Alavi \textit{et
al.},\cite{Booth_2009,Booth_2010,Cleland_2010} where signed walkers spawn from
one determinant to connected ones, with a probability that is a function of the
associated matrix element. The average proportion of walkers on a determinant
converges to its coefficient in the FCI wave function.
|\Psi^{(n)}\rangle = \sum_{I} c_I^{(n)} |D_I\rangle
.. |kalpha| replace:: :math:`| \alpha \rangle`
.. |ealpha| replace:: :math:`e_\alpha `
.. |EPT| replace:: :math:`E_{\text PT2}`
The determinants in |SetDI| will be characterized as **internal**.
#. For all **external** determinants |kalpha| `\notin` |SetDI|, compute the
Epstein-Nesbet second-order perturbative contribution to the energy
.. math::
e_\alpha = \frac{ \langle \Psi^{(n)}| {\hat H} | \alpha \rangle^2 }{E^{(n)} - \langle \alpha | {\hat H} | \alpha \rangle }.
`E^{(n)}` is the variational energy of the wave function at the current
iteration. Note that another perturbation theory could be used to estimate
|ealpha|.
#. An estimate of the total missing correlation energy can be computed
by summing all the |ealpha| contributions
.. math::
E_\text{PT2} & = \sum_{\alpha} e_\alpha \\
E_\text{FCI} & \approx E + E_\text{PT2}
#. |kalpha_star|, the subset of determinants |kalpha| with the largest
contributions |ealpha|, is added to the variational space
.. math::
\{ |D_I \rangle \}^{(n+1)} = \{|D_I\rangle\}^{(n)} \cup \{ |\alpha\rangle \}_\star^{(n)}
#. Go to iteration n+1, or exit on some criterion (number of determinants in
the wave function, low |EPT|, ...).
Approximations to accelerate the selection
------------------------------------------
The simple algorithm would be too slow to make calculations possible. Instead,
we use the 3-class |CIPSI| approximation to accelerate the selection,
:cite:`Evangelisti_1983` where instead of generating all possible |kalpha|,
we only generate a subset which are likely to be selected.
We define **generator** determinants, as determinants of the internal space
from which the |kalpha| are generated.
We then define **selector** determinants, a truncated wave function
used in the computation of |ealpha|.
For calculations in the |FCI| space, the determinants are sorted by
decreasing `|c_I|^2`, and thresholds are used on the squared norm of the wave
function. The default is to use :ref:`threshold_generators` = 0.99 for the
generators, and :ref:`threshold_selectors` = 0.999 for the selectors.
The computation of |EPT| using a truncated wave function is biased,
so if an accurate estimate of the |FCI| energy is desired, it is preferable
to recompute |EPT| with the hybrid deterministic/stochastic algorithm
:cite:`Garniron_2017b` which is unbiased (this is the default).
Modifying the selection space
-----------------------------
By changing the definition of generators, and the rules for the generation of
the |kalpha|, it is easy to define selected variants of traditional |CI| methods.
For example, if one defines the |HF| determinant as the only generator,
one will produce a selected |CISD|. If one also changes the rules for the generation
to generate only the double excitations, one will have a selected |CID|.
The generators can also be chosen as determinants belonging to a |CAS|. If the
rules allow only for excitations inside the |CAS|, we obtain a selected
|CAS| |CI|. If the rules allow for excitations in the |FCI| space, we obtain
a selected |CAS-SD|. And if one add the rule to prevent for doing double
excitations with two holes and two particles outside of the active space, one
obtains a selected |DDCI| method.
All such things can be done very easily when programming the the |qp|.
-----------------------------------
.. bibliography:: selected.bib
:style: unsrt
:labelprefix: A

4
src/FCI/FCI.irp.f
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@ -20,10 +20,10 @@ program fci_zmq
threshold_davidson = threshold_davidson_in * 100.d0
SOFT_TOUCH threshold_davidson
if (N_det < N_states) then
! if (N_det < N_states) then
call diagonalize_CI
call save_wavefunction
endif
! endif
call ezfio_has_hartree_fock_energy(has)
if (has) then