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Bidimensionality of 8-atom clusters of Au: first principles study and comparison with Ag clusters


We have calculated the lowest energy structures of 8-atom neutral gold clusters using the density functional theory approach. In contrast with current literature that finds kinetic energy to be the determinant component, we have found that the 2D structure is energetically favored due to a higher electron delocalization that stems from the relativistic contraction of Au atom size which cause 3D clusters to deform. This higher delocalization lowers the total energy of the 2D structures against the 3D ones. Silver clusters do not suffer this size contraction, hence there is no higher delocalization in the 2D clusters, and their fundamental structure will be 3D.

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    It is to be noted that ELF value interpretation sometimes presents problems: a notable case is the H2 molecule, case in which we find ELF = 1 for all space.


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We wish to thank the help of the Centro de Supercomputación de Galicia (CESGA) for providing both computational resources and the technical support necessary for this research. Our work was supported by the Spanish Ministerio de Educación y Ciencia under the Project. No. MAT2006-10027. We also want to thank the program Isabel Barreto.

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Correspondence to Jorge Botana.



From the original Becke and Edgecombe’s definition of ELF (see footnote 1) we have that, for open-shell atomic systems:

\( {\text{ELF}} = \frac{1}{1 + \chi }\;{\text{with}}\;\chi = \frac{{\tau_{\sigma } - \frac{1}{4}\frac{{\left( {\nabla \rho_{\sigma } } \right)^{2} }}{{\rho_{\sigma } }}}}{{2^{{\frac{5}{3}}} C_{\text{F}} \rho_{\sigma }^{{\frac{5}{3}}} }} = \frac{{\sum\nolimits_{i}^{\sigma } {\left| {\nabla \psi_{i} } \right|^{2} - \frac{1}{4}\frac{{\left( {\nabla \rho_{\sigma } } \right)^{2} }}{{\rho_{\sigma } }}} }}{{2^{{\frac{5}{3}}} C_{\text{F}} \rho_{\sigma }^{{\frac{5}{3}}} }} \) where τ σ is the kinetic energy density for the σ-spin, ρ σ is the electron density for the σ-spin and C F is the Fermi constant, as in Ref. [33].

For closed-shell atoms: \( \tau_{\sigma } = \frac{1}{2}\tau = \frac{1}{2}\sum_{i}^{n} {\left| {\nabla \psi_{i} } \right|^{2} }. \)

Using the virial theorem for XC E XC[ρ] and kinetic energy T S[ρ] functionals from Levy and Perdew [34]:

$$ E_{\text{X}} \left[ \rho \right] = - \int {\rho \left( {\vec{r}} \right)\vec{r}\nabla \left( {\frac{{\partial E_{\text{X}} }}{\partial \rho }} \right){\text{d}}\vec{r}} $$
$$ T_{\text{S}} \left[ \rho \right] = - \frac{1}{2}\int {\rho \left( {\vec{r}} \right)\vec{r}\nabla \left( {\frac{{\partial T_{\text{S}} }}{\partial \rho }} \right){\text{d}}\vec{r}} $$

with Fuentealba’s approximation [34]: \( T_{\text{S}} \left[ \rho \right] = - \frac{1}{2}E_{\text{X}} \left[ \rho \right] - \frac{1}{2}\int {\rho \left( {\vec{r}} \right)\vec{r}\nabla \phi \left( {\vec{r}} \right){\text{d}}^{3} r} - \frac{1}{2}T_{\text{C}} \left[ \rho \right]\;{\text{where}}\;\phi \left( {\vec{r}} \right) = \frac{Z}{r} - \int {\frac{{\rho \left( {\vec{r}} \right)}}{{\left( {\vec{r} - \vec{r}^{\prime}} \right)}}{\text{d}}^{3} \vec{r}^{\prime}} \) and T C is the kinetic part of the correlation energy.

Since \( E_{\text{XC}} \left[ \rho \right] = E_{\text{X}} \left[ \rho \right] + E_{\text{C}} \left[ \rho \right] \) and having in account that \( E_{\text{X}} \left[ \rho \right] \gg E_{\text{C}} \left[ \rho \right], \) and from Levy and Perdew [34]: T C = −E C, T S can be approximated as:

$$ T_{\text{S}} \left[ \rho \right] = - \frac{1}{2}E_{\text{X}} \left[ \rho \right] - \frac{1}{2}\int {\rho \left( {\vec{r}} \right)\vec{r}\nabla \phi \left( {\vec{r}} \right){\text{d}}^{3} r} $$

By definition: \( T_{\text{S}} \left[ \rho \right] = \int {t_{\text{S}} \left( \rho \right){\text{d}}^{3} r} \;{\text{and}}\;T_{\text{S}} \left[ \rho \right] = - \frac{1}{2}\int {\left. {\nabla_{r}^{2} \gamma_{S} \left( {r,r^{\prime}} \right)} \right|_{{r^{\prime} = r}} {\text{d}}^{3} r} \;{\text{where}}\;\gamma_{\text{S}} \left( {r,r^{\prime}} \right) \) and where is the first-order reduced density matrix.

From this, as seen in Fuentealba’s work [36], we have: \( t_{\text{S}} \left( \rho \right) = \frac{1}{2}\sum_{i = 1}^{n} {\left| {\nabla \psi_{i} \left( {r^{\prime}} \right)} \right|^{2} - \frac{1}{4}\nabla^{2} \rho \left( r \right)} \;{\text{hence}} \) \( \tau_{\sigma } = \frac{1}{2}\sum_{i = 1}^{n} {\left| {\nabla \psi_{i} \left( {r^{\prime}} \right)} \right|^{2} = - \frac{1}{2}e_{x} - \frac{1}{2}\rho \left( r \right)\vec{r}} \nabla \phi \left( r \right) + \frac{1}{4}\nabla^{2} \rho \left( r \right). \)

So, ELF approximated form is:

$$ {\text{ELF}} = \left( {1 + \frac{{ - \frac{1}{2}e_{x} - \frac{1}{2}\rho \left( r \right)\vec{r}\nabla \phi \left( r \right) + \frac{1}{4}\nabla^{2} \rho \left( r \right) - \frac{1}{4}\frac{{\left( {\nabla \rho_{\sigma } } \right)^{2} }}{{\rho_{\sigma } }}}}{{2^{{\frac{5}{3}}} C_{\text{F}} \rho_{\sigma }^{{\frac{5}{3}}} }}} \right)^{ - 1} $$

expression from which we take the dependence between XC energy and localization.

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Botana, J., Pereiro, M., Baldomir, D. et al. Bidimensionality of 8-atom clusters of Au: first principles study and comparison with Ag clusters. Theor Chem Account 122, 297–304 (2009).

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  • Small clusters
  • Gold clusters
  • Ab initio
  • DFT