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@section QMC Formulation
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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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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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@tex
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$$
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$$
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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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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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(\vec{r})}
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$$
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$$
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@end tex
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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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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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@tex
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$$
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$$
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d_{\sigma \sigma}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r_N}) \left[
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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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\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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$$
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$$
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d_{\sigma {\bar \sigma}}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N)
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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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\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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$$
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$$
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@end tex
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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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The electron pair localization function is bounded in the @math{[-1,1]} interval,
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electrons is smaller than the shortest distance separating electrons of same
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and is defined as
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spin, the EPLF takes positive values and indicates pairing of anti-parallel
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@tex
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electrons. In contrast, if the shortest distance separating anti-parallel
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$$
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electrons is larger than the shortest distance separating electrons of same
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{\rm EPLF}(\vec{r}) =
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spin, the EPLF takes negative values and indicates pairing of parallel
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{ d_{\sigma \sigma} (\vec{r})
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electrons (which in practice never happens). If the shortest distance
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- d_{\sigma {\bar \sigma}} (\vec{r}) \over
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separating anti-parallel electrons is equivalent to the shortest distance
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d_{\sigma \sigma} (\vec{r})
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separating electrons of same spin, the EPLF takes values close to zero and
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+ d_{\sigma {\bar \sigma}} (\vec{r}) }
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indicates no electron pairing.
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$$
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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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The original formulation of EPLF is extremely easy to compute in the quantum
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This localization function does not depend on the type of wave function, as
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Monte Carlo framework. However, it is not possible to compute it analytically
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opposed to the Electron Localization Function@footnote{A. D. Becke and K. E. Edgecombe
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due to the presence of the min function in the definitions of @math{d_{\sigma
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@i{J. Chem. Phys.} @b{92}, 5397 (1990).} (ELF) where the
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\sigma}} and @math{d_{\sigma \bar \sigma}} .
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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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@section Analytical Formulation
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@subsection The EPLF operator
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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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To evaluate exactly the average of the min function is not possible. We
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@i{Theor. Chem. Acc.} @b{126}, 243 (2010).}
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proposed@footnote{``Structural and optical properties of a neutral Nickel bisdithiolene
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The @math{\min} function is approximated
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complex. Density Functional versus ab-initio methods.'', F. Alary, J.-L.
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using:
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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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@tex
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$$
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$$
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\min_{j\neq i} |\vec{r}_i - \vec{r}_j| = \lim_{\gamma \rightarrow \infty}
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\min_{j\neq i} r_{ij} = \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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\sqrt{ -{1\over\gamma} \ln f(\gamma;r_{ij})}
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\vec{r}_j|^2 } ) }
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$$
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$$
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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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$$
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$$
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d_{\sigma \sigma}(\vec{r}) \sim \lim_{\gamma \rightarrow \infty}
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f(\gamma;r_{ij}) =
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\sqrt{-{1 \over \gamma} \ln \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N)
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\sum_{j \neq i} e^{ -\gamma r_{ij}^2 }
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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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$$
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$$
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@end tex
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@end tex
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@subsection Single determinant wave functions
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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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@tex
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$$
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$$
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\Psi(r_1,\dots,r_N) = D_0 = |\phi_1,\dots,\phi_N|
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\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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$$
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$$
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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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$$
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$$
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@end tex
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@end tex
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can be approximated as
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@section Multi-determinant wave functions
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@tex
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@tex
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$$
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$$
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\Psi = \sum_k c_k D_k
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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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$$
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$$
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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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@end tex
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The expectation values of the minimum distances are now given by:
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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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@tex
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$$
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$$
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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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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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$$
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$$
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@end tex
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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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with the two-electron integrals:
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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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@tex
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$$
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$$
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d_{\rm max} = \sqrt{-{1 \over \gamma} \ln \epsilon}
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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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$$
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$$
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@end tex
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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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If @math{\gamma} is not large enough in regions where
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@math{N} of electrons contained in the sphere is
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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
|
||||||
|
@math{f_{\sigma \bar{\sigma}}} have almost always extremely small values. In this
|
||||||
|
case, the values of @math{f_{\sigma \sigma}} and @math{f_{\sigma \bar{\sigma}}} are in practice not
|
||||||
|
accurately representable with 64 bit floating point numbers, as they may be
|
||||||
|
rounded to zero. We propose @math{\gamma} to depend on the density, such that the
|
||||||
|
largest representable value of @math{d_{\sigma \sigma}} is equal to the radius of a
|
||||||
|
sphere that contains almost 0.1 electron:
|
||||||
@tex
|
@tex
|
||||||
$$
|
$$
|
||||||
N = {4\over 3} \pi d_{\rm max}^3 \rho(r)
|
\gamma(\vec{r}) = \left({4\pi \over 3 n} \rho(\vec{r}) \right)^{2/3} (-\ln (\epsilon))
|
||||||
$$
|
|
||||||
@end tex
|
|
||||||
The maximum possible distance can be parametrized by a target constant number @math{N}
|
|
||||||
of electrons in the sphere
|
|
||||||
@tex
|
|
||||||
$$
|
|
||||||
d_{\rm max}(r) = \left( {1\over N} {4\over 3} \pi \rho(r) \right)^{-1/3}
|
|
||||||
$$
|
$$
|
||||||
@end tex
|
@end tex
|
||||||
|
where @math{n=0.1} and @math{\epsilon} is the smallest floating point number representable with
|
||||||
|
64 bits (@math{\sim 2.10^{-308}}).
|
||||||
|
|
||||||
|
|
||||||
If @math{N} is chosen small enough (0.01 electron), the value of
|
|
||||||
@math{\langle \Psi | f(r) | \Psi \rangle} is very unlikely to exceed 1, and the
|
|
||||||
value of @math{\gamma} will also be sufficiently large in the valence regions.
|
|
||||||
|
|
||||||
|
@ -95,15 +95,13 @@ def write_ezfioFile(res,filename):
|
|||||||
ezfio.ao_basis_ao_expo = expo
|
ezfio.ao_basis_ao_expo = expo
|
||||||
|
|
||||||
# MOs
|
# MOs
|
||||||
NumOrbSym = [ s[1] for s in res.symmetries ]
|
|
||||||
mo_tot_num = sum(NumOrbSym)
|
|
||||||
ezfio.mo_basis_mo_tot_num = mo_tot_num
|
|
||||||
|
|
||||||
MoTag = res.mo_types[-1]
|
MoTag = res.mo_types[-1]
|
||||||
if MoTag == 'Natural':
|
if MoTag == 'Natural':
|
||||||
res.mo_types.pop()
|
res.mo_types.pop()
|
||||||
MoTag = res.mo_types[-1]
|
MoTag = res.mo_types[-1]
|
||||||
print MoTag
|
mo_tot_num = len(res.mo_sets[MoTag])
|
||||||
|
ezfio.mo_basis_mo_tot_num = mo_tot_num
|
||||||
|
|
||||||
try:
|
try:
|
||||||
newocc = [0. for i in range(mo_tot_num)]
|
newocc = [0. for i in range(mo_tot_num)]
|
||||||
while len(res.occ_num[MoTag]) < mo_tot_num:
|
while len(res.occ_num[MoTag]) < mo_tot_num:
|
||||||
|
Loading…
Reference in New Issue
Block a user