mirror of https://gitlab.com/scemama/eplf
Improved gamma parameter
This commit is contained in:
Anthony Scemama
parent
cc7163ebd3
commit
a0eeaf84d3
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\input texinfo @c -*-texinfo-*-
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@c %**start of header (This is for running texinfo on a region.)
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@setfilename eplf.info
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@settitle Electron Pair Localization Function
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@afourpaper
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@c %**end of header (This is for running texinfo on a region.)
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@macro exe
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{@acronym{EPLF} }
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@end macro
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@ignore
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@ifinfo
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@format
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START-INFO-DIR-ENTRY
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* EPLF: (eplf.info). Electron Pair Localization Function
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END-INFO-DIR-ENTRY
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@end format
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@end ifinfo
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@end ignore
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@ifinfo
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This file documents @exe, a code to compute the Electron Pair
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Localization Function.@refill
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@end ifinfo
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@titlepage
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@title @exe
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@subtitle Electron Pair Localization Function: User's Guide
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@subtitle May, 2010
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@author @email{scemama@@irsamc.ups-tlse.fr,Anthony Scemama}
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@end titlepage
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@ifinfo
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@node Top, (dir), (dir)
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@top General Introduction
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@c Preface or Licensing nodes should come right
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@c after the Top node, in `unnumbered' sections,
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@c then the chapter, `What is foogol'.
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This file documents @exe, a code to compute the Electron Pair
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Localization Function.@refill
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@end ifinfo
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@menu
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* Introduction:: Definition the Electron Pair Localization Function
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* Using EPLF :: A basic introduction to using @exe.
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@end menu
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@node Introduction
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@chapter Introduction
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@include intro.texi
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@node Using EPLF
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@chapter Using the @exe program
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@include wf.texi
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@include ezfio.texi
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@bye
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@ -0,0 +1,23 @@
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@macro ezfio
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{@acronym{EZFIO} }
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@end macro
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@section Preparing the @ezfio file
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The @exe program uses the @ezfio (Easy Fortran Input/Output) library
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generator@footnote{@url{http://ezfio.sourceforge.net}} to handle data persistence.
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An @ezfio ``file'' is a directory which contains all the needed input/output data
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needed by the code. These files are accessible via a simple @acronym{API} both
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in Fortran and Python.
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The output file containing the computed wave function has to be transformed
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into an @ezfio file in order to be used by @exe. This step is realized by
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calling the @command{to_ezfio.py} Python script followed by the name of the
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file containing the wave function:
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@example
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to_ezfio.py test.out
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@end example
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An @ezfio file is produced, named @file{test.out.ezfio}.
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@section QMC Formulation
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The EPLF@footnote{``Electron pair localization function, a practical tool to visualize electron
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localization in molecules from quantum Monte Carlo data'', A. Scemama, P. Chaquin and M. Caffarel,
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@i{J. Chem. Phys.} @b{121}, 1725-1735 (2004).
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}
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has been designed to describe local electron pairing in molecular
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systems. It is defined as a scalar function defined in the three-dimensional
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space and taking its values in the [-1,1] range. It is defined as follows:
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@tex
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$$
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{\rm EPLF}(\vec{r}) = { d_{\sigma \sigma} (\vec{r}) - d_{\sigma {\bar \sigma}} (\vec{r}) \over d_{\sigma \sigma} (\vec{r}) + d_{\sigma {\bar \sigma}}
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(\vec{r})}
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$$
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@end tex
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where the quantity @math{ d_{\sigma \sigma} (\vec{r}) } [@math{resp. d_{\sigma
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\bar \sigma} (\vec{r})} ] denotes the quantum-mechanical average of the
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distance between an electron of spin @math{\sigma} located at @math{\vec r} and the closest
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electron of same spin (resp., of opposite spin @math{\bar\sigma}).
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The mathematical definition of these quantities can be written as
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@tex
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$$
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d_{\sigma \sigma}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r_N}) \left[
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\sum_{i=1}^N \delta(\vec{r}-\vec{r}_i) \min_{j\neq
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i;\sigma_j=\sigma_i}|\vec{r}_i - \vec{r}_j| \right] d\vec{r}_1 \dots d\vec{r}_N
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$$
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$$
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d_{\sigma {\bar \sigma}}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N)
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\left[ \sum_{i=1}^N \delta(\vec{r}-\vec{r}_i) \min_{j\ne
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i;\sigma_j\neq\sigma_i}|\vec{r}_i - \vec{r}_j| \right] d\vec{r}_1 \dots
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d\vec{r}_N
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$$
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@end tex
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where @math{\sigma} is the spin (\alpha or \beta), @math{\bar \sigma} is the
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spin opposite to @math{\sigma} @math{\Psi(\vec{r}_1,\dots,\vec{r}_N)} is the
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wave function, and @math{N} is the number of electrons.
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In a region of space, if the shortest distance separating anti-parallel
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electrons is smaller than the shortest distance separating electrons of same
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spin, the EPLF takes positive values and indicates pairing of anti-parallel
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electrons. In contrast, if the shortest distance separating anti-parallel
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electrons is larger than the shortest distance separating electrons of same
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spin, the EPLF takes negative values and indicates pairing of parallel
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electrons (which in practice never happens). If the shortest distance
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separating anti-parallel electrons is equivalent to the shortest distance
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separating electrons of same spin, the EPLF takes values close to zero and
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indicates no electron pairing.
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The original formulation of EPLF is extremely easy to compute in the quantum
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Monte Carlo framework. However, it is not possible to compute it analytically
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due to the presence of the min function in the definitions of @math{d_{\sigma
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\sigma}} and @math{d_{\sigma \bar \sigma}} .
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@section Analytical Formulation
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@subsection The EPLF operator
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To evaluate exactly the average of the min function is not possible. We
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proposed@footnote{``Structural and optical properties of a neutral Nickel bisdithiolene
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complex. Density Functional versus ab-initio methods.'', F. Alary, J.-L.
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Heully, A. Scemama, B. Garreau-de-Bonneval, K. I. Chane-Ching and M. Caffarel,
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@i{Theoretical Chemistry Accounts}, DOI: 10.1007/s00214-009-0679-9}
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to introduce a representation of this function in terms of gaussians
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and to construct our @math{d}-functions in a slightly different way so that the
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gaussian contributions can be exactly integrated out. The representation of the
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min function we are considering here is written as
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@tex
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$$
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\min_{j\neq i} |\vec{r}_i - \vec{r}_j| = \lim_{\gamma \rightarrow \infty}
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\sqrt{ -{1 \over \gamma} \ln ( \sum_{j \neq i} e^{ -\gamma |\vec{r}_i -
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\vec{r}_j|^2 } ) }
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$$
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@end tex
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and we propose to define the new modified average-distances as
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@tex
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$$
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d_{\sigma \sigma}(\vec{r}) \sim \lim_{\gamma \rightarrow \infty}
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\sqrt{-{1 \over \gamma} \ln \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N)
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\sum_{i=1}^{N} \delta(\vec{r}-\vec{r}_i) \sum_{j \neq i ; \sigma_i =
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\sigma_j}^{N} e^{ -\gamma |\vec{r}_i - \vec{r}_j|^2 } d\vec{r}_1 \dots
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d\vec{r}_N}
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$$
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$$
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d_{\sigma \bar{\sigma}} (\vec{r}) \sim \lim_{\gamma \rightarrow \infty}
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\sqrt{-{1\over \gamma} \ln \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N)
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\sum_{i=1}^{N} \delta(\vec{r}-\vec{r}_i) \sum_{j\neq i ; \sigma_i \neq
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\sigma_j}^{N} e^{ -\gamma |\vec{r}_i - \vec{r}_j|^2 } d\vec{r}_1 \dots
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d\vec{r}_N} $$
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@end tex
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Note that these new definitions of the average-distances are different from the
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previous ones essentially because @math{\ln \langle X \rangle \neq \langle \ln
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X\rangle} where @math{X} denotes the sum @math{\sum_{j} e^{ -\gamma |\vec{r} -
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\vec{r}_j|^2 }} and the symbol @math{\langle\dots\rangle} denotes the quantum
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average. Remark that the equality is almost reached when the
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fluctuations of @math{X} are small. In our applications it seems that we are
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not too far from this regime and we have thus systematically observed that both
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definitions of the @math{d}-functions lead to similar EPLF landscapes.
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Now, introducing
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@tex
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$$
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f_{\sigma\sigma}^\gamma(\vec{r}) = \sum_{i=1}^{N} \delta(\vec{r}-\vec{r}_i)
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\sum_{j\ne i ; \sigma_i = \sigma_j}^{N} e^{ -\gamma |\vec{r}_i - \vec{r}_j|^2 }
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$$
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$$
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f_{\sigma\bar{\sigma}}^\gamma(\vec{r}) = \sum_{i=1}^{N}
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\delta(\vec{r}-\vec{r}_i) \sum_{j ; \sigma_i \neq \sigma_j}^{N} e^{ -\gamma
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|\vec{r}_i - \vec{r}_j|^2 } $$
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@end tex
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we obtain
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@tex
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$$
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{\rm EPLF}(\vec{r}) = {\sqrt{-{1\over \gamma} \ln \langle \Psi |
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f_{\sigma\sigma}^\gamma(\vec{r}) | \Psi \rangle} - \sqrt{-{1\over \gamma} \ln
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\langle \Psi | f_{\sigma\bar{\sigma}}^\gamma(\vec{r}) | \Psi \rangle} \over
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\sqrt{-{1\over \gamma} \ln \langle \Psi | f_{\sigma\sigma}^\gamma(\vec{r}) |
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\Psi \rangle} + \sqrt{-{1\over \gamma} \ln \langle \Psi |
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f_{\sigma\bar{\sigma}}^\gamma(\vec{r}) | \Psi \rangle}}
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$$
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@end tex
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@subsection Single determinant wave functions
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@tex
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$$
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\Psi(r_1,\dots,r_N) = D_0 = |\phi_1,\dots,\phi_N|
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$$
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$$
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\langle \Psi | f_{\sigma\sigma}^\gamma(\vec{r}) | \Psi \rangle = \sum_{\sigma =
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\alpha,\beta} \sum_{i=1}^{N_\sigma} \sum_{j\ne i}^{N_\sigma} \int
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\phi_i(\vec{r}) \phi_j(\vec{r'}) e^{-\gamma |\vec{r'}-\vec{r}|^2}
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\phi_i(\vec{r}) \phi_j(\vec{r'}) d\vec{r'}
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$$
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$$
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- \int \phi_i(\vec{r})
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\phi_j(\vec{r'}) e^{-\gamma |\vec{r'}-\vec{r}|^2} \phi_j(\vec{r})
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\phi_i(\vec{r'}) d\vec{r'}
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$$
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$$
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\langle \Psi | f_{\sigma\bar{\sigma}}^\gamma(\vec{r}) | \Psi \rangle =
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\sum_{\sigma=\alpha,\beta} \sum_{i=1}^{N_\sigma} \sum_{j=1}^{N_{\bar \sigma}}
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\int \phi_i(\vec{r}) \phi_j(\vec{r'}) e^{-\gamma |\vec{r'}-\vec{r}|^2}
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\phi_i(\vec{r}) \phi_j(\vec{r'}) d\vec{r'}
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$$
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@end tex
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@section Multi-determinant wave functions
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@tex
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$$
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\Psi = \sum_k c_k D_k
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$$
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$$
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\langle \Psi | f(\vec{r}) | \Psi \rangle = \sum_k \sum_l c_k c_l \langle D_k |
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f(\vec{r}) | D_l \rangle $$
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$$
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\langle D_k | f_{\sigma\sigma}^\gamma(\vec{r}) | D_l \rangle = \sum_{\sigma =
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\alpha,\beta} \sum_{i=1}^{N_\sigma} \sum_{j\ne i}^{N_\sigma} \int
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\phi^k_i(\vec{r}) \phi^k_j(\vec{r'}) e^{-\gamma |\vec{r'}-\vec{r}|^2}
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\phi^l_i(\vec{r}) \phi^l_j(\vec{r'}) d\vec{r'}
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$$
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$$
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- \int \phi^k_i(\vec{r})
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\phi^k_j(\vec{r'}) e^{-\gamma |\vec{r'}-\vec{r}|^2} \phi^l_j(\vec{r})
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\phi^l_i(\vec{r'}) d\vec{r'} $$
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$$
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\langle D_k | f_{\sigma\bar \sigma}^\gamma(\vec{r}) | D_l \rangle =
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\sum_{\sigma=\alpha,\beta} \sum_{i=1}^{N_\sigma} \sum_{j=1}^{N_{\bar \sigma}}
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\int \phi^k_i(\vec{r}) \phi^k_j(\vec{r'}) e^{-\gamma |\vec{r'}-\vec{r}|^2}
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\phi^l_i(\vec{r}) \phi^l_j(\vec{r'}) d\vec{r'} $$
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@end tex
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@subsection Expression of the gamma parameter
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In our definition of the modified EPLF the parameter @math{\gamma} is supposed to be (very)
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large. In practice, if it is chosen too small, the integrals @math{\langle \Psi |
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f_{\sigma{\bar \sigma}}^\gamma(\vec{r}) | \Psi \rangle} can take values larger
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than 1 in regions of high density, giving a negative value for the shortest
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average distance. If it is chosen too large numerical instabilities appear
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since the values of the integrals become extremely small. We propose to use a
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value of @math{\gamma} which depends on the electron density @math{\rho(\vec{r})} as:
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@tex
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$$
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\gamma(\vec{r}) = (-\ln \epsilon) \left( {1\over N}{4\over 3}\pi \rho(\vec{r}) \right)^{2/3}
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$$
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@end tex
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where @math{\epsilon} is the smallest possible floating point number that can
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be represented with 64 bits (@math{\sim 2.10^{-308}}). The reason for this
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choice is the following.
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The largest possible value of the approximate shortest distance can be computed as
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@tex
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$$
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d_{\rm max} = \sqrt{-{1 \over \gamma} \ln \epsilon}
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$$
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@end tex
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If the density is conidered constant in a sphere of radius @math{d_{\rm max}}, the number
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@math{N} of electrons contained in the sphere is
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@tex
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$$
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N = {4\over 3} \pi d_{\rm max}^3 \rho(r)
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$$
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@end tex
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The maximum possible distance can be parametrized by a target constant number @math{N}
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of electrons in the sphere
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@tex
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$$
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d_{\rm max}(r) = \left( {1\over N} {4\over 3} \pi \rho(r) \right)^{-1/3}
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$$
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@end tex
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If @math{N} is chosen small enough (0.01 electron), the value of
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@math{\langle \Psi | f(r) | \Psi \rangle} is very unlikely to exceed 1, and the
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value of @math{\gamma} will also be sufficiently large in the valence regions.
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@ -0,0 +1,64 @@
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@section Wave function preparation
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@macro gamess
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{@acronym{GAMESS} }
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@end macro
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@macro mcscf
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{@acronym{MCSCF} }
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@end macro
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@macro scf
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{@acronym{SCF} }
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@end macro
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@macro cas
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{@acronym{CAS-SCF} }
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@end macro
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@macro rhf
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{@acronym{RHF} }
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@end macro
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@macro ci
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{@acronym{CI} }
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@end macro
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Output files of Gaussian, Molpro and @gamess can be read to build the wave function files.
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A major constraint is to realize @emph{single point} a calculation.
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@subsection Using Gaussian
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In the Gaussian input file, use the keywords @code{GFPRINT} and @code{pop=Full}.
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In the case of @cas wave functions, use the @code{#p} keyword and the @code{SlaterDet}
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attribute of the @code{CAS} keyword.
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@subsection Using Molpro
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Use the following options in the Molpro input file:
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@itemize @bullet
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@item @code{print,basis;}
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@item @code{gprint,civector;}
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@item @code{gprint,orbital;}
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@item @code{gthresh,printci=0.;} for @mcscf calculations
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@end itemize
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An @rhf calculation is mandatory before any @mcscf calculation. Be sure to
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print @emph{all} molecular orbitals using the @code{orbprint} keyword.
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@subsection Using @gamess
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For @mcscf calculations, first compute the @mcscf single-point wave function
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with the @acronym{GUGA} algorithm. Then, put the the @mcscf orbitals in the
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@gamess input file, and run a single-point @acronym{GUGA} @ci calculation with
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the following keywords:
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@itemize @bullet
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@item
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@code{PRTTOL=0.0} in the @code{$GUGDIA} group
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@item
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@code{NPRT=2} in the @code{$CIDRT} group
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@item
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@code{PRTMO=.T.} in the @code{$GUESS} group
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@end itemize
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@ -4,9 +4,10 @@ BEGIN_PROVIDER [ real, eplf_gamma ]
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! Value of the gaussian for the EPLF
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END_DOC
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include 'constants.F'
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real :: eps
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real :: eps, N
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N = 1.
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eps = -real(dlog(tiny(1.d0)))
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eplf_gamma = (4./3.*pi*density_value_p)**(2./3.) * eps
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eplf_gamma = (4./(3.*N)*pi*density_value_p)**(2./3.) * eps
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END_PROVIDER
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BEGIN_PROVIDER [ double precision, ao_eplf_integral_matrix, (ao_num,ao_num) ]
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@ -40,13 +41,12 @@ BEGIN_PROVIDER [ double precision, mo_eplf_integral_matrix, (mo_num,mo_num) ]
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do k=1,ao_num
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if (abs(mo_coef(k,i)) /= 0.) then
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if (mo_coef(k,i) /= 0.) then
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do l=1,ao_num
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t = mo_coef(k,i)*ao_eplf_integral_matrix(l,k)
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if (abs(ao_eplf_integral_matrix(l,k))>1.d-16) then
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do j=i,mo_num
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mo_eplf_integral_matrix(j,i) = mo_eplf_integral_matrix(j,i) + &
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t*mo_coef_transp(j,l)
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mo_eplf_integral_matrix(j,i) += t*mo_coef_transp(j,l)
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enddo
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endif
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enddo
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@ -70,43 +70,53 @@ END_PROVIDER
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END_DOC
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integer :: i, j
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double precision :: thr
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thr = 1.d-12 / eplf_gamma
|
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eplf_up_up = 0.d0
|
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eplf_up_dn = 0.d0
|
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|
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PROVIDE mo_coef_transp
|
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|
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do j=1,elec_beta_num
|
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do i=1,elec_beta_num
|
||||
do j=1,mo_closed_num
|
||||
do i=1,mo_closed_num
|
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eplf_up_up += 2.d0*mo_value_p(i)* ( &
|
||||
mo_value_p(i)*mo_eplf_integral_matrix(j,j) - &
|
||||
mo_value_p(j)*mo_eplf_integral_matrix(i,j) )
|
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enddo
|
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|
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do i=elec_beta_num+1,elec_alpha_num
|
||||
eplf_up_up += mo_value_p(i)* ( &
|
||||
mo_value_p(i)*mo_eplf_integral_matrix(j,j) - &
|
||||
mo_value_p(j)*mo_eplf_integral_matrix(i,j) )
|
||||
eplf_up_dn += 2.d0*mo_value_p(j)**2 * &
|
||||
mo_eplf_integral_matrix(i,i)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
do j=elec_beta_num+1,elec_alpha_num
|
||||
do i=1,elec_alpha_num
|
||||
eplf_up_up += mo_value_p(i)* ( &
|
||||
mo_value_p(i)*mo_eplf_integral_matrix(j,j) - &
|
||||
mo_value_p(j)*mo_eplf_integral_matrix(i,j) )
|
||||
enddo
|
||||
enddo
|
||||
integer :: k,l,m,n,p
|
||||
double precision :: ckl
|
||||
do k=1,det_num
|
||||
do l=1,det_num
|
||||
|
||||
ckl = det_coef(k)*det_coef(l)
|
||||
|
||||
do p=1,2
|
||||
do m=1,elec_num_2(p)-mo_closed_num
|
||||
j = det(m,k,p)
|
||||
do n=1,elec_num_2(p)-mo_closed_num
|
||||
i = det(n,l,p)
|
||||
eplf_up_up += ckl*mo_value_p(i)* ( &
|
||||
mo_value_p(i)*mo_eplf_integral_matrix(j,j) - &
|
||||
mo_value_p(j)*mo_eplf_integral_matrix(i,j) )
|
||||
enddo
|
||||
enddo
|
||||
enddo
|
||||
|
||||
do m=1,elec_beta_num-mo_closed_num
|
||||
j = det(m,k,2)
|
||||
do n=1,elec_alpha_num-mo_closed_num
|
||||
i = det(n,k,1)
|
||||
eplf_up_dn += ckl * ( mo_value_p(i)**2 * mo_eplf_integral_matrix(j,j) &
|
||||
+ mo_value_p(j)**2 * mo_eplf_integral_matrix(i,i) )
|
||||
enddo
|
||||
enddo
|
||||
|
||||
do j=1,elec_beta_num
|
||||
do i=1,elec_alpha_num
|
||||
eplf_up_dn += mo_value_p(i)**2 * &
|
||||
mo_eplf_integral_matrix(j,j)
|
||||
enddo
|
||||
enddo
|
||||
eplf_up_dn = 2.d0*eplf_up_dn
|
||||
|
||||
|
||||
END_PROVIDER
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue