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150 lines
5.9 KiB
Plaintext
150 lines
5.9 KiB
Plaintext
@section QMC Formulation
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The Electron Pair Localization Function (EPLF) is a three-dimensional function
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measuring locally the electron pairing in a molecular
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system.@footnote{A. Scemama, P. Chaquin and M. Caffarel, @i{J. Chem. Phys.}
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@b{121}, 1725-1735 (2004).}
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An electron @math{i} located at @math{\vec{r}_i} is said to be paired to an electron @math{j}
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located at @math{\vec{r}_j} if electron @math{j} is the closest electron to @math{i}. The
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amount of electron pairing at point @math{{\vec r}} is therefore proportional to the
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inverse of
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@tex
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$$
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d({\vec r}) = \sum_{i=1,N} \langle \Psi | \delta(\vec{r}-\vec{r}_i) \min_{j\ne i} r_{ij} | \Psi \rangle
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$$
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@end tex
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where
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@math{d({\vec r})} is the shortest electron-electron distance at @math{{\vec r}},
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@math{\Psi({\vec r}_1,\dots,{\vec r}_N)} is the @math{N}-electron wave function
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and
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@math{r_{ij} = |{\vec r}_j - {\vec r}_i|}.
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There are two different types of electron pairs: pairs of electrons with
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the same spin @math{\sigma}, and pairs of electrons with opposite spins @math{\sigma}
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and @math{\bar \sigma}.
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Hence, two quantities are introduced:
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@tex
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$$
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d_{\sigma \sigma}({\vec r}) = \sum_{i=1,N} \langle \Psi | \delta(\vec{r}-\vec{r}_i) \min_{j\ne i ; \sigma_i = \sigma_j} r_{ij} | \Psi \rangle
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$$
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$$
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d_{\sigma {\bar \sigma}}({\vec r}) = \sum_{i=1,N} \langle \Psi | \delta(\vec{r}-\vec{r}_i) \min_{j ; \sigma_i \ne \sigma_j} r_{ij} | \Psi \rangle
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$$
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@end tex
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The electron pair localization function is bounded in the @math{[-1,1]} interval,
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and is defined as
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@tex
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$$
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{\rm EPLF}(\vec{r}) =
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{ d_{\sigma \sigma} (\vec{r})
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- d_{\sigma {\bar \sigma}} (\vec{r}) \over
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d_{\sigma \sigma} (\vec{r})
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+ d_{\sigma {\bar \sigma}} (\vec{r}) }
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$$
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@end tex
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When the pairing of spin-unlike electrons is predominant, @math{d_{\sigma \sigma}
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> d_{\sigma {\bar \sigma}}} and @math{{\rm EPLF}({\vec r}) > 0}.
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When the pairing of spin-like electrons is predominant, @math{d_{\sigma \sigma}
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< d_{\sigma {\bar \sigma}}} and @math{{\rm EPLF}({\vec r}) < 0}.
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When the electron pairing of spin-like and spin-unlike electrons is equivalent,
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@math{{\rm EPLF}({\vec r}) \sim 0}.
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This localization function does not depend on the type of wave function, as
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opposed to the Electron Localization Function@footnote{A. D. Becke and K. E. Edgecombe
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@i{J. Chem. Phys.} @b{92}, 5397 (1990).} (ELF) where the
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wave function has to be expressed on a single-electron basis set.
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The EPLF can therefore measure electron pairing using any kind of wave
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function: Hartree-Fock (HF), configuration interaction (CI),
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multi-configurational self consistent field (MCSCF) as well as Slater-Jastrow,
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Diffusion Monte Carlo (DMC), Hylleraas wave functions, @i{etc}.
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Due to the presence of the @math{\min} function in the definition of @math{d_{\sigma
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\sigma}} and @math{d_{\sigma {\bar \sigma}}} these quantities cannot be evaluated
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analytically, so quantum Monte Carlo (QMC) methods have been
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used to compute three-dimensional EPLF grids via a statistical sampling of
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@math{|\Psi({\vec r}_1,\dots,{\vec r}_N)|^2}.
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@section Analytical Formulation
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Recently, we proposed an alternative definition of the EPLF which is
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analytically computable.@footnote{F. Alary, J.-L. Heully, A. Scemama, B. Garreau-de-Bonneval, K. I. Chane-Ching and M. Caffarel
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@i{Theor. Chem. Acc.} @b{126}, 243 (2010).}
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The @math{\min} function is approximated
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using:
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@tex
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$$
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\min_{j\neq i} r_{ij} = \lim_{\gamma \rightarrow \infty}
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\sqrt{ -{1\over\gamma} \ln f(\gamma;r_{ij})}
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$$
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$$
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f(\gamma;r_{ij}) =
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\sum_{j \neq i} e^{ -\gamma r_{ij}^2 }
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$$
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@end tex
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As @math{f(\gamma ; r_{ij})} has small
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fluctuations in the regions of interest (bonding regions, lone pairs,@dots),
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the integrals
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@tex
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$$
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\left\langle \Psi \left| \delta({\vec r}-{\vec r}_i) \left( \lim_{\gamma \rightarrow \infty} \sqrt{ -{1 \over \gamma} \ln f(\gamma;r_{ij})} \right) \right| \Psi \right\rangle
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$$
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@end tex
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can be approximated as
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@tex
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$$
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\lim_{\gamma \rightarrow \infty} \sqrt{ -{1 \over \gamma} \ln
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\langle \Psi | \delta({\vec r}-{\vec r}_i) f(\gamma;r_{ij}) | \Psi \rangle }
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$$
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@end tex
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The expectation values of the minimum distances are now given by:
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@tex
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$$
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d_{\sigma \sigma}(\vec{r}) \sim
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\lim_{\gamma \rightarrow \infty} \sqrt{-{1 \over \gamma} \ln
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f_{\sigma \sigma}(\gamma;\vec{r})}
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$$
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$$
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d_{\sigma \bar{\sigma}} (\vec{r}) \sim
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\lim_{\gamma \rightarrow \infty} \sqrt{-{1 \over \gamma} \ln
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f_{\sigma \bar{\sigma}} (\gamma;\vec{r})}
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$$
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@end tex
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with the two-electron integrals:
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@tex
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$$
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f_{\sigma \sigma}(\gamma;\vec{r}) =
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\left \langle \Psi \left| \sum_{i=1}^{N} \delta(\vec{r}-\vec{r}_i)
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\sum_{j \neq i ; \sigma_i = \sigma_j}^{N}
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e^{ -\gamma |\vec{r}_i - \vec{r}_j|^2 } \right| \Psi \right\rangle
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$$
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$$
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f_{\sigma \bar{\sigma}} (\gamma;\vec{r}) =
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\left \langle \Psi \left| \sum_{i=1}^{N} \delta(\vec{r}-\vec{r}_i)
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\sum_{j ; \sigma_i \neq \sigma_j}^{N}
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e^{ -\gamma |\vec{r}_i - \vec{r}_j|^2 } \right| \Psi \right \rangle
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$$
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@end tex
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If @math{\gamma} is not large enough in regions where
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electrons are close to each other, the value of @math{f_{\sigma \sigma}} or
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@math{f_{\sigma \bar{\sigma}}} can be greater than 1. In that case, the
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approximation of the min function is irrelevant since its value becomes negative.
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On the other hand, if @math{\gamma} is too large @math{f_{\sigma \sigma}} and
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@math{f_{\sigma \bar{\sigma}}} have almost always extremely small values. In this
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case, the values of @math{f_{\sigma \sigma}} and @math{f_{\sigma \bar{\sigma}}} are in practice not
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accurately representable with 64 bit floating point numbers, as they may be
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rounded to zero. We propose @math{\gamma} to depend on the density, such that the
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largest representable value of @math{d_{\sigma \sigma}} is equal to the radius of a
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sphere that contains almost 0.1 electron:
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@tex
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$$
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\gamma(\vec{r}) = \left({4\pi \over 3 n} \rho(\vec{r}) \right)^{2/3} (-\ln (\epsilon))
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$$
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@end tex
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where @math{n=0.1} and @math{\epsilon} is the smallest floating point number representable with
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64 bits (@math{\sim 2.10^{-308}}).
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