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\title{Dynamical simulation of brittle fracture \\
using order-N electronic-structure method}

\author{Takeo Hoshi and Takeo Fujiwara}
\address{Dept. of Applied Physics, Univ. of Tokyo}
\email{Email:hoshi@coral.t.u-tokyo.ac.jp}


\noindent %%  YOUR SUMMARY BEGINS HERE

We have developed a variational and/or perturbative order-N 
algorithm
for large-scale electronic-structure calculations, 
based on the localized-state theory \cite{Ref}.
This algorithm was tested as molecular-dynamics simulations
of upto about 34000 atoms, 
using a single-CPU workstation with 1GB RAMs.
Such localized states are formally equivalent to 
a unitary transformation of eigen states and satisfy
the Kohn-Sham equation with off-diagonal matrix elements
($H_{\rm eff} \psi_j = \sum_{i}^{\rm occ.} \varepsilon_{ij} \psi_i $).

As an application of our algorithm,
we perform dynamical fracture simulations of silicon
with external loads, where 
we use a sp tight-binding Hamiltonian and isolated clusters with 
orientation-fixed sp$^3$ states at their surfaces.
In smaller systems,
we observe a satisfactory agreement of the stress values
and the crack-propagating velocity
among the order-N method, the exact diagonalization method,
and experimental results.
We also observe that the fracture occurs,
when the band-gap goes to be zero.
This is microscopically 
interpreted as bond-breaking processes or 
as vanishing the gap
between local bonding- and antibonding- levels.
Such microscopic pictures can be directly monitored 
in the present order-N simulation.

For investigation of the microscopic mechanism,
we monitor the occupation ratio of s-orbital 
for each localized state:
$f_{\rm s}^{(j)} \equiv | \langle {\rm s} | \psi_j \rangle |^2$.
This quantity describes the extent of the sp hybridization and,
for instance, is 1/4 for an ideal sp$^3$ state.
In general,
the extent of the sp hybridization 
is a quantum-mechanical freedom and 
plays a crucial role for atomic structures.
For detailed investigation of the electronic states,
we also monitor the projected density of states 
of the localized states in the bond-breaking processes.
We observe that after bond-breaking processes,
the localized states on the broken bonds
are stabilized 
with a high value of $f_{\rm s}^{(j)}$
($0.5 \le f_{\rm s}^{(j)} \le 0.8  $).
This means that 
such states are similar to an atomic s-state,
rather than a sp$^3$ dangling-bond state.
These \lq s-rich' states seem to be transitional 
and tend to re-bond 
somewhere later in the simulation.

These observations show a general theory 
of the brittle fracture.
The atomic s- and p- levels are $\varepsilon_{\rm s}=-5.45$eV 
and $\varepsilon_{\rm p}=1.2$eV, respectively.
Since the atomic configuration is s$^2$p$^2$,
the averaged occupation ratio of s-orbital is the half 
($f_{\rm s}=1/2$) and the electronic energy 
is given by  
$\varepsilon_{\rm atom} \equiv 
( \varepsilon_{\rm s} + \varepsilon_{\rm p})/2 = -2.13$eV per electron.
In perfect crystals,
all the localized states form the sp$^3$ bonds and 
the corresponding energy level is 
$\varepsilon_{\rm bulk}=-5.2$eV,
located at the weighted band center of the valence band \cite{Ref}.
So the bulk system is stabilized 
with the cohesive electronic-energy of 
$\Delta \varepsilon \equiv \varepsilon_{\rm atom}
- \varepsilon_{\rm bulk} (>0)$ per electron.
Here it is important that the atomic s-level is 
{\it lower} than the bulk level
$(\varepsilon_{\rm s} <\varepsilon_{\rm bulk})$.
Therefore the occupation ratio ($f_{\rm s}$)
is crucial to the cohesion of the perfect crystal.
In crack tips, on the other hand,
the sp$^3$-bonded network is {\it locally} broken and 
the existence of the low s level
stabilizes the transitional \lq s-rich' states.
Therefore the bonds near at crack tips 
tend to be broken much easier than those at the bulk regions.
This can be interpreted as 
the microscopic origin of the brittle fracture.


\begin{thebibliography}{9}
   \setlength{\itemsep}{-0.06in}
\bibitem{Ref} 
Takeo Hoshi and Takeo Fujiwara,
J. Phys. Soc. Jpn, vol. 69, No.12, pp3773-3776 (2000). 
\end{thebibliography}

\end{document}
