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LAMMPS WWW Site - LAMMPS Documentation - LAMMPS Commands +
+ + + + + + +
+ +

fix qeq/comb command +

+

Syntax: +

+
fix ID group-ID qeq/comb Nevery precision keyword value ... 
+
+
  • ID, group-ID are documented in fix command + +
  • qeq/comb = style name of this fix command + +
  • Nevery = perform charge equilibration every this many steps + +
  • precision = convergence criterion for charge equilibration + +
  • zero or more keyword/value pairs may be appended + +
  • keyword = file + +
      file value = filename
+    filename = name of file to write QEQ equilibration info to 
+
    + +
+

Examples: +

+
fix 1 surface qeq/comb 10 0.0001 
+
+

Description: +

+

This fix is designed for use with the pair_style comb command which +implements the COMB COMB (Charge-Optimized Many-Body) potential as +described in (COMB_1) and (COMB_2). It performs +the charge equilibration portion of the calculation using the +so-called QEq method, whereby the charge on each atom is adjusted to +minimize the energy of the system. +

+

Only charges on the atoms in the specified group are equilibrated. +The fix relies on the pair style (COMB in this case) to calculate the +per-atom electronegativity (effective force on the charges). An +electronegativity equalization calculation (or QEq) is performed in an +interative fashion, which in parallel requires communication at each +iteration for processors to exchange charge information about nearby +atoms with each other. See Rappe_and_Goddard and +Rick_and_Stuart for details. +

+

During a run, charge equilibration is peformed every Nevery time +steps. Charge equilibration is also always enforced on the first step +of each run. The precision argument controls the tolerance for the +difference in electronegativity for all atoms during charge +equilibration. Precision is a trade-off between the cost of +performing charge equilibration (more iterations) and accuracy. +

+

If the file keyword is used, then information about each +equilibration calculation is written to the specifed file. +

+

Restart, fix_modify, output, run start/stop, minimize info: +

+

No information about this fix is written to binary restart +files. None of the fix_modify options +are relevant to this fix. +

+

This fix produces a per-atom vector which can be accessed by various +output commands. The vector stores the +gradient of the charge on each atom. The per-atom values be accessed +on any timestep. +

+

No parameter of this fix can be used with the start/stop keywords of +the run command. This fix is not invoked during energy +minimization. +

+

Restrictions: +

+

This fix command currently only supports pair style comb. +

+

Related commands: +

+

pair_style comb +

+

Default: +

+

No file output is performed. +

+
+ + + +

(COMB_1) J. Yu, S. B. Sinnott, S. R. Phillpot, Phys Rev B, 75, 085311 (2007), +

+ + +

(COMB_2) T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott, S. R. +Phillpot, Phys Rev B, 81, 125328 (2010). +

+ + +

(Rappe_and_Goddard) A. K. Rappe, W. A. Goddard, J Phys Chem 95, 3358 +(1991). +

+ + +

(Rick_and_Stuart) S. W. Rick, S. J. Stuart, B. J. Berne, J Chem Phys +101, 16141 (1994). +

+ diff --git a/doc/fix_qeq.txt b/doc/fix_qeq.txt new file mode 100644 index 0000000000..ee787f0774 --- /dev/null +++ b/doc/fix_qeq.txt @@ -0,0 +1,99 @@ +"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c + +:link(lws,http://lammps.sandia.gov) +:link(ld,Manual.html) +:link(lc,Section_commands.html#comm) + +:line + +fix qeq/comb command :h3 + +[Syntax:] + +fix ID group-ID qeq/comb Nevery precision keyword value ... :pre + +ID, group-ID are documented in "fix"_fix.html command :ulb,l +qeq/comb = style name of this fix command :l +Nevery = perform charge equilibration every this many steps :l +precision = convergence criterion for charge equilibration :l +zero or more keyword/value pairs may be appended :l +keyword = {file} :l + {file} value = filename + filename = name of file to write QEQ equilibration info to :pre +:ule + +[Examples:] + +fix 1 surface qeq/comb 10 0.0001 :pre + +[Description:] + +This fix is designed for use with the pair_style comb command which +implements the COMB COMB (Charge-Optimized Many-Body) potential as +described in "(COMB_1)"_#COMB_1 and "(COMB_2)"_#COMB_2. It performs +the charge equilibration portion of the calculation using the +so-called QEq method, whereby the charge on each atom is adjusted to +minimize the energy of the system. + +Only charges on the atoms in the specified group are equilibrated. +The fix relies on the pair style (COMB in this case) to calculate the +per-atom electronegativity (effective force on the charges). An +electronegativity equalization calculation (or QEq) is performed in an +interative fashion, which in parallel requires communication at each +iteration for processors to exchange charge information about nearby +atoms with each other. See "Rappe_and_Goddard"_#Rappe_and_Goddard and +"Rick_and_Stuart"_#Rick_and_Stuart for details. + +During a run, charge equilibration is peformed every {Nevery} time +steps. Charge equilibration is also always enforced on the first step +of each run. The {precision} argument controls the tolerance for the +difference in electronegativity for all atoms during charge +equilibration. {Precision} is a trade-off between the cost of +performing charge equilibration (more iterations) and accuracy. + +If the {file} keyword is used, then information about each +equilibration calculation is written to the specifed file. + +[Restart, fix_modify, output, run start/stop, minimize info:] + +No information about this fix is written to "binary restart +files"_restart.html. None of the "fix_modify"_fix_modify.html options +are relevant to this fix. + +This fix produces a per-atom vector which can be accessed by various +"output commands"_Section_howto.html#4_15. The vector stores the +gradient of the charge on each atom. The per-atom values be accessed +on any timestep. + +No parameter of this fix can be used with the {start/stop} keywords of +the "run"_run.html command. This fix is not invoked during "energy +minimization"_minimize.html. + +[Restrictions:] + +This fix command currently only supports "pair style {comb}"_pair_comb.html. + +[Related commands:] + +"pair_style comb"_pair_comb.html + +[Default:] + +No file output is performed. + +:line + +:link(COMB_1) +[(COMB_1)] J. Yu, S. B. Sinnott, S. R. Phillpot, Phys Rev B, 75, 085311 (2007), + +:link(COMB_2) +[(COMB_2)] T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott, S. R. +Phillpot, Phys Rev B, 81, 125328 (2010). + +:link(Rappe_and_Goddard) +[(Rappe_and_Goddard)] A. K. Rappe, W. A. Goddard, J Phys Chem 95, 3358 +(1991). + +:link(Rick_and_Stuart) +[(Rick_and_Stuart)] S. W. Rick, S. J. Stuart, B. J. Berne, J Chem Phys +101, 16141 (1994). diff --git a/doc/pair_comb.html b/doc/pair_comb.html new file mode 100644 index 0000000000..aed88db02a --- /dev/null +++ b/doc/pair_comb.html @@ -0,0 +1,231 @@ + +
LAMMPS WWW Site - LAMMPS Documentation - LAMMPS Commands +
+ + + + + + +
+ +

pair_style comb command +

+

Syntax: +

+
pair_style comb 
+
+

Examples: +

+
pair_style comb
+pair_coeff * * ../potentials/ffield.comb Si
+pair_coeff * * ../potentials/ffield.comb Hf Si O 
+
+

Description: +

+

Style comb computes a variable charge COMB (Charge-Optimized +Many-Body) potential as described in (COMB_1) and +(COMB_2). The energy E of a system of atoms +is given by +

+
+
+

where ET is the total potential energy of the system, +ESi is the self-energy term of atom i, +Vij is the interatomic potential between the ith and +jth atoms, rij is the distance of the atoms i and +j, and qi and qj are charges of the atoms, +and EBBi is the bond-bending term of atom i. +

+

The interatomic potential energy Vij consists of four +components: two-body short-range repulsion, +URij, many-body short-range attraction, +UAij, long-range Coulombic electrostatic +interaction, UIij, and van der Waals energy, +UVij, which are defined as: +

+
+
+

The short-range repulsion and attraction are based on the +Tersoff potential (see the pair_style +tersoff command); thus for a zero-charge pure +element system with no van der Waals interaction, COMB potential +reduces to Tersoff potential. +

+

The COMB potential is a variable charge potential. The equilibrium +charge on each atom is calculated by the electronegativity +equalization (QEq) method. See Rick_and_Stuart for +furthur details). This is implemented by the fix +qeq/comb command, which should normally be +specified in the input script when running a model with the COMB +potential. The fix qeq/comb command has options +that determine how often charge equilibration is performed, its +convergence criterion, and which atoms are included in the +calculation. +

+

Only a single pair_coeff command is used with the comb style which +specifies the COMB potential file with parameters for all needed +elements. These are mapped to LAMMPS atom types by specifying N +additional arguments after the potential file in the pair_coeff +command, where N is the number of LAMMPS atom types. The provided +potential file ffield.comb contains all currently-available COMB +parameterizations: for Si, Cu, Hf, Ti, O, their oxides and Zr, Zn and +U metals. +

+

For example, if your LAMMPS simulation of a Si/SiO2/ +HfO2 interface has 4 atom types, and you want the 1st and +last to be Si, the 2nd to be Hf, and the 3rd to be O, and you would +use the following pair_coeff command: +

+
pair_coeff * * ../potentials/ffield.comb Si Hf O Si 
+
+

The first two arguments must be * * so as to span all LAMMPS atom +types. The first and last Si arguments map LAMMPS atom types 1 and 4 +to the Si element in the ffield.comb file. The second Hf argument +maps LAMMPS atom type 2 to the Hf element, and the third O argument +maps LAMMPS atom type 3 to the O element in the potential file. If a +mapping value is specified as NULL, the mapping is not performed. +This can be used when a tersoff potential is used as part of the +hybrid pair style. The NULL values are placeholders for atom types +that will be used with other potentials. +

+

The ffield.comb potential file is in the potentials directory of +the LAMMPS distribution. Lines that are not blank or comments +(starting with #) define parameters for a triplet of elements. The 49 +parameters in a single entry correspond to coefficients in the formula +above: +

+
  • element 1 (the center atom in a 3-body interaction) +
  • element 2 (the atom bonded to the center atom) +
  • element 3 (the atom influencing the 1-2 bond in a bond-order sense) +
  • m +
  • c +
  • d +
  • h (cos_theta0 (can be a value -1 or 1)) +
  • n +
  • beta +
  • lambda21, lambda2 of element 1 (1/distance units) +
  • lambda22, lambda2 of element 2 (1/distance units) +
  • B of element 1 (energy units) +
  • B of element 2 (energy units) +
  • R (cutoff, distance units, 0.5*(r_outer + r_inner)) +
  • D (cutoff, distance units, R - r_inner) +
  • lambda11, lambda1 of element 1 (1/distance units) +
  • lambda12, lambda1 of element 2 (1/distance units) +
  • A of element 1 (energy units) +
  • A of element 2 (energy units) +
  • K_LP_1 (energy units, 1st order Legendre polynomial coefficient) +
  • K_LP_3 (energy units, 3rd order Legendre polynomial coefficient) +
  • K_LP_6 (energy units, 6th order Legendre polynomial coefficient) +
  • A123 (cos_theta, theta = equilibrium MOM or OMO bond angles) +
  • Aconf (cos_theta, theta = equilibrium MOM or OMO bond-bending coefficient) +
  • addrep (energy units, additional repulsion) +
  • R_omiga_a (unit-less scaler for A) +
  • R_omiga_b (unit-less scaler for B) +
  • R_omiga_c (unit-less scaler for 0.5*(lambda21+lambda22)) +
  • R_omiga_d (unit-less scaler for 0.5*(lambda11+lambda12)) +
  • QL1 (charge units, lower charge limit for element 1) +
  • QU1 (charge units, upper charge limit for element 1) +
  • DL1 (distance units, ion radius of element 1 with charge QL1) +
  • DU1 (distance units, ion radius of element 1 with charge QU1) +
  • QL2 (charge units, lower charge limit for element 2) +
  • QU2 (charge units, upper charge limit for element 2) +
  • DL2 (distance units, ion radius of element 2 with charge QL2) +
  • DU2 (distance units, ion radius of element 2 with charge QU2) +
  • chi (energy units, self energy 1st power term) +
  • dJ (energy units, self energy 2nd power term) +
  • dK (energy units, self energy 3rd power term) +
  • dL (energy units, self energy 4th power term) +
  • dM (energy units, self energy 6th power term) +
  • esm (distance units, orbital exponent) +
  • cmn1 (self energy penalty, rho 1 of element 1) +
  • cml1 (self energy penalty, rho 1 of element 2) +
  • cmn2 (self energy penalty, rho 2 of element 1) +
  • cmn2 (self energy penalty, rho 2 of element 2) +
  • coulcut (long range Coulombic cutoff, distance units) +
  • hfocor (coordination term) +
+

The parameterization of COMB potentials start with a pure element +(e.g. Si, Cu) then extend to its oxide and polymorphs +(e.g. SiO2, Cu2O). For interactions not +involving oxygen (e.g. Si-Cu or Hf-Zr), the COMB potential uses a +mixing rule to generate these parameters. For furthur details on the +parameterization and parameters, see the Tersoff +doc page and the COMB publications (COMB_1) and +(COMB_2). For more details on 3-body interaction types +(e.g. SiSiO vs SiOSi), the mixing rule, and how to generate the +potential file, please see the Tersoff doc page. +

+

In the potentials directory, the file ffield.comb provides the +LAMMPS parameters for COMB's Si, Cu, Ti, Hf and their oxides, as well +as pure U, Zn and Zr metals. This file can be used for pure elements +(e.g. Si, Zr), binary oxides, binary alloys (e.g. SiCu, TiZr), and +complex systems. Note that alloys and complex systems require all +3-body entries be pre-defined in the potential file. +

+
+ +

Mixing, shift, table, tail correction, restart, rRESPA info: +

+

For atom type pairs I,J and I != J, where types I and J correspond to +two different element types, mixing is performed by LAMMPS as +described above from values in the potential file. +

+

This pair style does not support the pair_modify +shift, table, and tail options. +

+

This pair style does not write its information to binary restart +files, since it is stored in potential files. Thus, you +need to re-specify the pair_style, pair_coeff and fix_qeq commands in +an input script that reads a restart file. +

+

This pair style can only be used via the pair keyword of the +run_style respa command. It does not support the +inner, middle, outer keywords. +

+
+ +

Restrictions: +

+

This pair style is part of the "manybody" package. It is only enabled +if LAMMPS was built with that package (which it is by default). See +the Making LAMMPS section for more info. +

+

This pair style requires the newton setting to be "on" +for pair interactions. +

+

The COMB potentials in the ffield.comb file provided with LAMMPS +(see the potentials directory) are parameterized for metal +units. You can use the COMB potential with any LAMMPS +units, but you would need to create your own COMB potential file with +coefficients listed in the appropriate units if your simulation +doesn't use "metal" units. +

+

Related commands: +

+

pair_style, pair_coeff, +fix_qeq +

+

Default: none +

+
+ + + +

(COMB_1) J. Yu, S. B. Sinnott, S. R. Phillpot, Phys Rev B, 75, 085311 (2007), +

+ + +

(COMB_2) T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott, S. R. +Phillpot, Phys Rev B, 81, 125328 (2010). +

+ + +

(Tersoff) J. Tersoff, Phys Rev B, 37, 6991 (1988). +

+ + +

(Rick_and_Stuart) S. W. Rick, S. J. Stuart, B. J. Berne, J Chem Phys +101, 16141 (1994). +

+ diff --git a/doc/pair_comb.txt b/doc/pair_comb.txt new file mode 100644 index 0000000000..f5e81db659 --- /dev/null +++ b/doc/pair_comb.txt @@ -0,0 +1,222 @@ +"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c + +:link(lws,http://lammps.sandia.gov) +:link(ld,Manual.html) +:link(lc,Section_commands.html#comm) + +:line + +pair_style comb command :h3 + +[Syntax:] + +pair_style comb :pre + +[Examples:] + +pair_style comb +pair_coeff * * ../potentials/ffield.comb Si +pair_coeff * * ../potentials/ffield.comb Hf Si O :pre + +[Description:] + +Style {comb} computes a variable charge COMB (Charge-Optimized +Many-Body) potential as described in "(COMB_1)"_#COMB_1 and +"(COMB_2)"_#COMB_2. The energy E of a system of atoms +is given by + +:c,image(Eqs/pair_comb_1.jpg) + +where {ET} is the total potential energy of the system, +{ESi} is the self-energy term of atom {i}, +{Vij} is the interatomic potential between the {i}th and +{j}th atoms, {rij} is the distance of the atoms {i} and +{j}, and {qi} and {qj} are charges of the atoms, +and {EBBi} is the bond-bending term of atom {i}. + +The interatomic potential energy {Vij} consists of four +components: two-body short-range repulsion, +{URij}, many-body short-range attraction, +{UAij}, long-range Coulombic electrostatic +interaction, {UIij}, and van der Waals energy, +{UVij}, which are defined as: + +:c,image(Eqs/pair_comb_2.jpg) + +The short-range repulsion and attraction are based on the +"Tersoff"_#Tersoff potential (see the "pair_style +tersoff"_pair_tersoff.html command); thus for a zero-charge pure +element system with no van der Waals interaction, COMB potential +reduces to Tersoff potential. + +The COMB potential is a variable charge potential. The equilibrium +charge on each atom is calculated by the electronegativity +equalization (QEq) method. See "Rick_and_Stuart"_#Rick_and_Stuart for +furthur details). This is implemented by the "fix +qeq/comb"_fix_qeq_comb.html command, which should normally be +specified in the input script when running a model with the COMB +potential. The "fix qeq/comb"_fix_qeq_comb.html command has options +that determine how often charge equilibration is performed, its +convergence criterion, and which atoms are included in the +calculation. + +Only a single pair_coeff command is used with the {comb} style which +specifies the COMB potential file with parameters for all needed +elements. These are mapped to LAMMPS atom types by specifying N +additional arguments after the potential file in the pair_coeff +command, where N is the number of LAMMPS atom types. The provided +potential file {ffield.comb} contains all currently-available COMB +parameterizations: for Si, Cu, Hf, Ti, O, their oxides and Zr, Zn and +U metals. + +For example, if your LAMMPS simulation of a Si/SiO2/ +HfO2 interface has 4 atom types, and you want the 1st and +last to be Si, the 2nd to be Hf, and the 3rd to be O, and you would +use the following pair_coeff command: + +pair_coeff * * ../potentials/ffield.comb Si Hf O Si :pre + +The first two arguments must be * * so as to span all LAMMPS atom +types. The first and last Si arguments map LAMMPS atom types 1 and 4 +to the Si element in the {ffield.comb} file. The second Hf argument +maps LAMMPS atom type 2 to the Hf element, and the third O argument +maps LAMMPS atom type 3 to the O element in the potential file. If a +mapping value is specified as NULL, the mapping is not performed. +This can be used when a {tersoff} potential is used as part of the +{hybrid} pair style. The NULL values are placeholders for atom types +that will be used with other potentials. + +The {ffield.comb} potential file is in the {potentials} directory of +the LAMMPS distribution. Lines that are not blank or comments +(starting with #) define parameters for a triplet of elements. The 49 +parameters in a single entry correspond to coefficients in the formula +above: + +element 1 (the center atom in a 3-body interaction) +element 2 (the atom bonded to the center atom) +element 3 (the atom influencing the 1-2 bond in a bond-order sense) +m +c +d +h (cos_theta0 (can be a value -1 or 1)) +n +beta +lambda21, lambda2 of element 1 (1/distance units) +lambda22, lambda2 of element 2 (1/distance units) +B of element 1 (energy units) +B of element 2 (energy units) +R (cutoff, distance units, 0.5*(r_outer + r_inner)) +D (cutoff, distance units, R - r_inner) +lambda11, lambda1 of element 1 (1/distance units) +lambda12, lambda1 of element 2 (1/distance units) +A of element 1 (energy units) +A of element 2 (energy units) +K_LP_1 (energy units, 1st order Legendre polynomial coefficient) +K_LP_3 (energy units, 3rd order Legendre polynomial coefficient) +K_LP_6 (energy units, 6th order Legendre polynomial coefficient) +A123 (cos_theta, theta = equilibrium MOM or OMO bond angles) +Aconf (cos_theta, theta = equilibrium MOM or OMO bond-bending coefficient) +addrep (energy units, additional repulsion) +R_omiga_a (unit-less scaler for A) +R_omiga_b (unit-less scaler for B) +R_omiga_c (unit-less scaler for 0.5*(lambda21+lambda22)) +R_omiga_d (unit-less scaler for 0.5*(lambda11+lambda12)) +QL1 (charge units, lower charge limit for element 1) +QU1 (charge units, upper charge limit for element 1) +DL1 (distance units, ion radius of element 1 with charge QL1) +DU1 (distance units, ion radius of element 1 with charge QU1) +QL2 (charge units, lower charge limit for element 2) +QU2 (charge units, upper charge limit for element 2) +DL2 (distance units, ion radius of element 2 with charge QL2) +DU2 (distance units, ion radius of element 2 with charge QU2) +chi (energy units, self energy 1st power term) +dJ (energy units, self energy 2nd power term) +dK (energy units, self energy 3rd power term) +dL (energy units, self energy 4th power term) +dM (energy units, self energy 6th power term) +esm (distance units, orbital exponent) +cmn1 (self energy penalty, rho 1 of element 1) +cml1 (self energy penalty, rho 1 of element 2) +cmn2 (self energy penalty, rho 2 of element 1) +cmn2 (self energy penalty, rho 2 of element 2) +coulcut (long range Coulombic cutoff, distance units) +hfocor (coordination term) :ul + +The parameterization of COMB potentials start with a pure element +(e.g. Si, Cu) then extend to its oxide and polymorphs +(e.g. SiO2, Cu2O). For interactions not +involving oxygen (e.g. Si-Cu or Hf-Zr), the COMB potential uses a +mixing rule to generate these parameters. For furthur details on the +parameterization and parameters, see the "Tersoff"_pair_tersoff.html +doc page and the COMB publications "(COMB_1)"_#COMB_1 and +"(COMB_2)"_#COMB_2. For more details on 3-body interaction types +(e.g. SiSiO vs SiOSi), the mixing rule, and how to generate the +potential file, please see the "Tersoff"_pair_tersoff.html doc page. + +In the potentials directory, the file {ffield.comb} provides the +LAMMPS parameters for COMB's Si, Cu, Ti, Hf and their oxides, as well +as pure U, Zn and Zr metals. This file can be used for pure elements +(e.g. Si, Zr), binary oxides, binary alloys (e.g. SiCu, TiZr), and +complex systems. Note that alloys and complex systems require all +3-body entries be pre-defined in the potential file. + +:line + +[Mixing, shift, table, tail correction, restart, rRESPA info]: + +For atom type pairs I,J and I != J, where types I and J correspond to +two different element types, mixing is performed by LAMMPS as +described above from values in the potential file. + +This pair style does not support the "pair_modify"_pair_modify.html +shift, table, and tail options. + +This pair style does not write its information to "binary restart +files"_restart.html, since it is stored in potential files. Thus, you +need to re-specify the pair_style, pair_coeff and fix_qeq commands in +an input script that reads a restart file. + +This pair style can only be used via the {pair} keyword of the +"run_style respa"_run_style.html command. It does not support the +{inner}, {middle}, {outer} keywords. + +:line + +[Restrictions:] + +This pair style is part of the "manybody" package. It is only enabled +if LAMMPS was built with that package (which it is by default). See +the "Making LAMMPS"_Section_start.html#2_3 section for more info. + +This pair style requires the "newton"_newton.html setting to be "on" +for pair interactions. + +The COMB potentials in the {ffield.comb} file provided with LAMMPS +(see the potentials directory) are parameterized for metal +"units"_units.html. You can use the COMB potential with any LAMMPS +units, but you would need to create your own COMB potential file with +coefficients listed in the appropriate units if your simulation +doesn't use "metal" units. + +[Related commands:] + +"pair_style"_pair_style.html, "pair_coeff"_pair_coeff.html, +"fix_qeq"_fix_qeq.html + +[Default:] none + +:line + +:link(COMB_1) +[(COMB_1)] J. Yu, S. B. Sinnott, S. R. Phillpot, Phys Rev B, 75, 085311 (2007), + +:link(COMB_2) +[(COMB_2)] T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott, S. R. +Phillpot, Phys Rev B, 81, 125328 (2010). + +:link(Tersoff) +[(Tersoff)] J. Tersoff, Phys Rev B, 37, 6991 (1988). + +:link(Rick_and_Stuart) +[(Rick_and_Stuart)] S. W. Rick, S. J. Stuart, B. J. Berne, J Chem Phys +101, 16141 (1994).