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<HTML>
<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
</CENTER>
<HR>
<H3>pair_style bop command
</H3>
<P><B>Syntax:</B>
</P>
<PRE>pair_style bop keyword ...
</PRE>
<UL><LI>zero or more keywords may be appended
<LI>keyword = <I>table</I> or <I>save</I> or <I>sigmaoff</I>
<PRE> <I>table</I> = BOP potential file has tabulated form
<I>save</I> = pre-compute and save some values
<I>sigmaoff</I> = assume a_sigma = 0
</PRE>
</UL>
<P><B>Examples:</B>
</P>
<PRE>pair_style bop
pair_coeff * * ../potentials/CdTe_bop Cd Te
pair_style bop table save
pair_coeff * * ../potentials/CdTe.bop.table Cd Te Te
communicate single cutoff 14.70
</PRE>
<P><B>Description:</B>
</P>
<P>The <I>bop</I> pair style computes Bond-Order Potentials (BOP) based on
quantum mechanical theory incorporating both sigma and pi bondings.
By analytically deriving the BOP from quantum mechanical theory its
transferability to different phases can approach that of quantum
mechanical methods. This particlular BOP is extremely effective at
modeling III-V and II-VI compounds such as GaAs and CdTe. This
potential is similar to the original BOP developed by Pettifor
(<A HREF = "#Pettifor_1">Pettifor_1</A>, <A HREF = "#Pettifor_2">Pettifor_2</A>,
<A HREF = "#Pettifor_3">Pettifor_3</A>) and later updated by Murdick, Zhou, and Ward
(<A HREF = "#Murdick">Murdick</A>, <A HREF = "#Ward">Ward</A>).
</P>
<P>The BOP potential consists of three terms:
</P>
<CENTER><IMG SRC = "Eqs/pair_bop.jpg">
</CENTER>
<P>where phi_ij(r_ij) is a short-range two-body function representing the
repulsion between a pair of ion cores, beta_(sigma,ij)(r_ij) and
beta_(sigma,ij)(r_ij) are respectively sigma and pi bond ingtegrals,
THETA_(sigma,ij) and THETA_(pi,ij) are sigma and pi bond-orders, and
U_prom is the promotion energy for sp-valent systems.
</P>
<P>The detailed formulas for this potential are given in Ward
(<A HREF = "#Ward">Ward</A>); here we provide only a brief description.
</P>
<P>The repulsive energy phi_ij(r_ij) and the bond integrals
beta_(sigma,ij)(r_ij) and beta_(phi,ij)(r_ij) are functions of the
interatomic distance r_ij between atom i and j. Each of these
potentials has a smooth cutoff at a radius of r_(cut,ij). These
smooth cutoffs ensure stable behavior at situations with high sampling
near the cutoff such as melts and surfaces.
</P>
<P>The bond-orders can be viewed as environment-dependent local variables
that are ij bond specific. The maximum value of the sigma bond-order
(THETA_sigma) is 1, while that of the pi bond-order (THETA_pi) is 2,
attributing to a maximum value of the total bond-order
(THETA_sigma+THETA_pi) of 3. The sigma and pi bond-orders reflect the
ubiquitous single-, double-, and triple- bond behavior of
chemistry. Their analytical expressions can be derived from tight-
binding theory by recursively expanding an inter-site Green's function
as a continued fraction. To accurately represent the bonding with a
computationally efficient potential formulation suitable for MD
simulations, the derived BOP only takes (and retains) the first two
levels of the recursive representations for both the sigma and the pi
bond-orders. Bond-order terms can be understood in terms of molecular
orbital hopping paths based upon the Cyrot-Lackmann theorem
(<A HREF = "#Pettifor_1">Pettifor_1</A>). The sigma bond-order with a half-full
valence band filling. This pi bond-order expression also contains
also contains a three-member ring term that allows implementation of
an asymmetric density of states, which helps to either stabilize or
destabilize close-packed structures. The pi bond-order includes
hopping paths of length 4. This enables the incorporation of dihedral
angles effects.
</P>
<P>The cutoffs for the various interactions are defined in the BOP
potential file.
</P>
<P>IMPORTANT NOTE: You must use the <A HREF = "communicate.html">communicate cutoff</A>
command to insure ghost atoms are acquired at a distance 3x further
than the largest BOP cutoff (for a particular pair of elements).
E.g. if the BOP cutoff is 4.9 Angstroms, then the ghost atom
communication needs to be 14.7 Angstroms or greater as in the example
above. This is because the BOP formulation uses neighbors of
neighbors of neighbors to enumerate all the required many-body
interactions. LAMMPS will generate an error if you do not use an
appropriate setting for the <A HREF = "communicate.html">communicate cutoff</A>
command.
</P>
<P>Several options can be specified as keywords with the pair_style
command.
</P>
<P>The <I>table</I> keyword tells LAMMPS what format the BOP potential file is
in. The default is a non-tabulated form. If the <I>table</I> keyword is
used, the file is in a tabulated form containing pre-tabulated pair
functions for phi_ij(r_ij), beta_(sigma,ij)(r_ij), and
beta_(pi,ij)(r_ij). This allows you to use your own functional
form for various interactions.
</P>
<P>The <I>save</I> keyword gives you the option to calculate and store in
advance a set of distances, angles, and derivatives of angles. The
default is to not do this, but to calculate the various quantities
on-the-fly each time they are needed. The former may be faster, but
takes more memory. The latter requires less memory, but may be
slower. It is best to test this option to see if it makes a
difference on your machine for the specific problem you are modeling.
</P>
<P>The <I>sigmaoff</I> keyword optimizes the BOP equations for the case of
a_sigma = 0. For some published BOP potentials, a_sigma = 0 and
several terms in the BOP equationas drop out. If this is the case,
specifying <I>sigmaoff</I> will typically speed up the BOP pair style.
</P>
<HR>
<P>Only a single pair_coeff command is used with the <I>bop</I> style which
specifies a BOP potential file, with parameters for all needed
elements. These are mapped to LAMMPS atom types by specifying
N additional arguments after the filename in the pair_coeff command,
where N is the number of LAMMPS atom types:
</P>
<UL><LI>filename
<LI>N element names = mapping of BOP elements to atom types
</UL>
<P>See the <A HREF = "pair_coeff.html">pair_coeff</A> doc page for alternate ways
to specify the path for the potential file.
</P>
<P>As an example, imagine the CdTe.bop file has BOP values for Cd
and Te. If your LAMMPS simulation has 4 atoms types and you want the
1st 3 to be Cd, and the 4th to be Te, you would use the following
pair_coeff command:
</P>
<PRE>pair_coeff * * CdTe Cd Cd Cd Te
</PRE>
<P>The 1st 2 arguments must be * * so as to span all LAMMPS atom types.
The first three Cd arguments map LAMMPS atom types 1,2,3 to the Cd
element in the BOP file. The final Te argument maps LAMMPS atom type
4 to the Te element in the BOP file. If a mapping value is specified
as NULL, the mapping is not performed. This can be used when a
<I>bop</I> potential is used as part of the <I>hybrid</I> pair style. The
NULL values are placeholders for atom types that will be used with
other potentials.
</P>
<P>BOP files in the <I>potentials</I> directory of the LAMMPS distribution
have a ".bop" or ".bop.table" suffix, depending on whether they are of
the non-tabulated or tabulated form, as described above.
</P>
<P>The parameters/coefficients format for the both kinds of BOP files are
given below with variables matching the formulation of Ward
(<A HREF = "#Ward">Ward</A>). Each header line containing a ":" is preceded by a
blank line.
</P>
<UL><LI>Line 1: elements: (header)
<LI>Line 2: #elements <I>N</I>
</UL>
<P>The first two lines are followed by N lines containing the atomic
number and mass of each element.
</P>
<HR>
<P><B>Non-tabulated potential file format</B>:
</P>
<P>Following the definition of the elements is the block of global
variables for spline and quadratic fits of THETA_(S,ij) and its
components THETA_0, THETA_1, and S.
</P>
<UL><LI>Line 1: global: (header)
<LI>Line 2: delta_1-delta_7 (if all are not used in the particular
formulation, set unused values to 0.0)
<LI>Line 3: ncutoff, r_big, r_small (r_big and r_small are parameters for
pairwise paramters gamma typically set to 0.99 and 0.01, respectively)
<LI>Line 4: which, alpha, nfunc (these are options for the spline
which=1.0 (means using a smooth function); which=2.0 (spline), alpha is
a parameter in the spline, nfunc is the type of GSP function (f_ij)
(nfunc=1 is the published equation from Ward (<A HREF = "#Ward">Ward</A>); nfunc=2
f_ij(r_ij)=exp(n_ij*r_ij); nfunc=3 f_ij(r_ij)=1/(r_ij)^(n_ij)).
<LI>Line 5: alpha_1, beta_1, gamma_1 (alpha_1=first coefficient for
THETA_0; beta_1=first exponent for THETA_0; gamma_1=second exponent for
THETA_0)
<LI>Line 6: alpha_2, beta_2 (alpha_2=second coefficient for S; beta_2=first
exponent for S)
<LI>Line 7: alpha_3, beta_3 (alpha_3=first coefficient for THETA_1;
beta_3=second coefficient for THETA_1)
</UL>
<P>The next block contains constants for the environment depend
promotional energy for sp-valent systems, each of which are species
dependent. Refer to Pettifor (<A HREF = "#Pettifor_3">Pettifor_3</A>) for constant
definitions. As well as one species dependent parameter p_pi.
</P>
<UL><LI>Line 1: ptrs: (header)
</UL>
<P>Following the ptrs header there are N lines for e_1-e_N containing
(A_ij)^(mu*nu), delta^mu, p_pi
</P>
<UL><LI>Line 2: (A_ij)^(mu*nu), delta^mu, p_pi (for e_1)
<LI>Line 3: (A_ij)^(mu*nu), delta^mu, p_pi (for e_2 and continues to e_N)
</UL>
<P>The next block contains constants for the pair interactions.
</P>
<UL><LI>Line 1: pairs: (header)
</UL>
<P>Following the header the block contains a series of constants for the
number of pair interaction types, the block will be broken up into
parameters for e_i-e_j with i=0->N, j=i->N. Each single interaction
section for this block will contain (see <A HREF = "#Ward">Ward</A> for parameter
definitions):
</P>
<UL><LI>Line 2: r_0, r_c, r_1, r_cut (for e_1-e_2 interactions)
<LI>Line 3: m, n, n_c
<LI>Line 4: phi_0, beta_(sigma,0), beta_(pi,0)
<LI>Line 5: a_sigma, c_sigma, delta_sigma (From complete formulation of 1/2
full valance shell for this particular formulation delta_sigma=0)
<LI>Line 6: a_pi, c_pi, delta_pi
<LI>Line 7: f_sigma, k_sigma, delta_3 (This delta_3 is similar to that of
the previous section but is interaction type dependent)
<LI>Line 8: r_0, r_c, r_1, r_cut (for e_1-e_2 interactions and repeats as
above)
</UL>
<P>The next block contains tris.
</P>
<UL><LI>Line 1: tris: (header)
</UL>
<P>Following the header there is a line for each three body interaction
types as e_j-e_i-e_k with i->N, j->N, k=j->N
</P>
<UL><LI>Line 2: g_sigma0, g_sigma1, g_sigma2 (these are coefficients for
g_(sigma,ijk)(theta_ijk) for e_1-e_1-e_1 interaction. <A HREF = "#Ward">Ward</A>
contains the full expressions for the constants as functions of
b_(sigma,ijk), p_(sigma,ijk), u(sigma,ijk)
<LI>Line 3: g_sigma0, g_sigma1, g_sigma2 (for e_1-e_1-e_2)
</UL>
<P>This would be the end of the potential parameter file without pre-
tabulated data.
</P>
<HR>
<P><B>Tabulated potential file format</B>:
</P>
<P>The parameters/coefficients format for the BOP potentials input file
containing pre-tabulated functions of is given below with variables
matching the formulation of Ward (<A HREF = "#Ward">Ward</A>).
</P>
<UL><LI>Line 1: # elements N
</UL>
<P>The first two lines are followed by N lines containing the atomic
number and mass of each element THETA_0 and THETA_1 (see
<A HREF = "#Ward">Ward</A>).
</P>
<P>Following the definition of the elements several global variables for
the tabulated functions are given.
</P>
<UL><LI>Line 1: nr, nBOt (nr is the number of divisions the radius is broken
into for function tables and MUST be a factor of 5; nBOt is the number
of divisions for the tabulated values of THETA_(S,ij)
<LI>Line 2: delta_1-delta_7 (if all are not used in the particular
<LI>formulation, set unused values to 0.0)
</UL>
<P>Following this N lines for e_1-e_N containing p_pi.
</P>
<UL><LI>Line 3: p_pi (for e_1)
<LI>Line 4: p_pi (for e_2 and continues to e_N)
</UL>
<P>The next section contains several pair constants for the number of
interaction types e_i-e_j, with i=1->N, j=i->N
</P>
<UL><LI>Line 1: r_cut (for e_1-e_1 interactions)
<LI>Line 2: c_sigma, a_sigma, c_pi, a_pi
<LI>Line 3: delta_sigma, delta_pi
<LI>Line 4: f_sigma, k_sigma, delta_3 (This delta_3 is similar to that of
the previous section but is interaction type dependent)
</UL>
<P>The next section contains a line for each three body interaction type
e_j-e_i-e_k with i=0->N, j=0->N, k=j->N
</P>
<UL><LI>Line 1: g_(sigma0), g_(sigma1), g_(sigma2) (These are coefficients for
g_(sigma,jik)(THETA_ijk) for e_1-e_1-e_1 interaction. <A HREF = "#Ward">Ward</A>
contains the full expressions for the constants as functions of
b_(sigma,ijk), p_(sigma,ijk), u_(sigma,ijk))
<LI>Line 2: g_(sigma0), g_(sigma1), g_(sigma2) (for e_1-e_1-e_2)
</UL>
<P>The next section contains a block for each interaction type for the
phi_ij(r_ij). Each block has nr entries with 5 entries per line.
</P>
<UL><LI>Line 1: phi(r1), phi(r2), phi(r3), phi(r4), phi(r5) (for the e_1-e_1
interaction type)
<LI>Line 2: phi(r6), phi(r7), phi(r8), phi(r9), phi(r10) (this continues
until nr)
<LI>...
<LI>Line nr/5_1: phi(r1), phi(r2), phi(r3), phi(r4), phi(r5), (for the
e_1-e_1 interaction type)
</UL>
<P>The next section contains a block for each interaction type for the
beta_(sigma,ij)(r_ij). Each block has nr entries with 5 entries per
line.
</P>
<UL><LI>Line 1: beta_sigma(r1), beta_sigma(r2), beta_sigma(r3), beta_sigma(r4),
beta_sigma(r5) (for the e_1-e_1 interaction type)
<LI>Line 2: beta_sigma(r6), beta_sigma(r7), beta_sigma(r8), beta_sigma(r9),
beta_sigma(r10) (this continues until nr)
<LI>...
<LI>Line nr/5+1: beta_sigma(r1), beta_sigma(r2), beta_sigma(r3),
beta_sigma(r4), beta_sigma(r5) (for the e_1-e_2 interaction type)
</UL>
<P>The next section contains a block for each interaction type for
beta_(pi,ij)(r_ij). Each block has nr entries with 5 entries per line.
</P>
<UL><LI>Line 1: beta_pi(r1), beta_pi(r2), beta_pi(r3), beta_pi(r4), beta_pi(r5)
(for the e_1-e_1 interaction type)
<LI>Line 2: beta_pi(r6), beta_pi(r7), beta_pi(r8), beta_pi(r9),
beta_pi(r10) (this continues until nr)
<LI>...
<LI>Line nr/5+1: beta_pi(r1), beta_pi(r2), beta_pi(r3), beta_pi(r4),
beta_pi(r5) (for the e_1-e_2 interaction type)
</UL>
<P>The next section contains a block for each interaction type for the
THETA_(S,ij)((THETA_(sigma,ij))^(1/2), f_(sigma,ij)). Each block has
nBOt entries with 5 entries per line.
</P>
<UL><LI>Line 1: THETA_(S,ij)(r1), THETA_(S,ij)(r2), THETA_(S,ij)(r3),
THETA_(S,ij)(r4), THETA_(S,ij)(r5) (for the e_1-e_2 interaction type)
<LI>Line 2: THETA_(S,ij)(r6), THETA_(S,ij)(r7), THETA_(S,ij)(r8),
THETA_(S,ij)(r9), THETA_(S,ij)(r10) (this continues until nBOt)
<LI>...
<LI>Line nBOt/5+1: THETA_(S,ij)(r1), THETA_(S,ij)(r2), THETA_(S,ij)(r3),
THETA_(S,ij)(r4), THETA_(S,ij)(r5) (for the e_1-e_2 interaction type)
</UL>
<P>The next section contains a block of N lines for e_1-e_N
</P>
<UL><LI>Line 1: delta^mu (for e_1)
<LI>Line 2: delta^mu (for e_2 and repeats to e_N)
</UL>
<P>The last section contains more constants for e_i-e_j interactions with
i=0->N, j=i->N
</P>
<UL><LI>Line 1: (A_ij)^(mu*nu) (for e1-e1)
<LI>Line 2: (A_ij)^(mu*nu) (for e1-e2 and repeats as above)
</UL>
<HR>
<P><B>Mixing, shift, table tail correction, restart</B>:
</P>
<P>This pair style does not support the <A HREF = "pair_modify.html">pair_modify</A>
mix, shift, table, and tail options.
</P>
<P>This pair style does not write its information to <A HREF = "restart.html">binary restart
files</A>, since it is stored in potential files. Thus, you
need to re-specify the pair_style and pair_coeff commands in an input
script that reads a restart file.
</P>
<P>This pair style can only be used via the <I>pair</I> keyword of the
<A HREF = "run_style.html">run_style respa</A> command. It does not support the
<I>inner</I>, <I>middle</I>, <I>outer</I> keywords.
</P>
<HR>
<P><B>Restrictions:</B>
</P>
<P>These pair styles are part of the MANYBODY package. They are only
enabled if LAMMPS was built with that package (which it is by default).
See the <A HREF = "Section_start.html#start_3">Making LAMMPS</A> section for more
info.
</P>
<P>These pair potentials require the <A HREF = "newton.html">newtion</A> setting to be
"on" for pair interactions.
</P>
<P>The CdTe.bop and GaAs.bop potential files provided with LAMMPS (see the
potentials directory) are parameterized for metal <A HREF = "units.html">units</A>.
You can use the BOP potential with any LAMMPS units, but you would need
to create your own BOP potential file with coefficients listed in the
appropriate units if your simulation does not use "metal" units.
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "pair_coeff.html">pair_coeff</A>
</P>
<P><B>Default:</B>
</P>
<P>non-tabulated potential file, a_0 is non-zero.
</P>
<HR>
<A NAME = "Pettofor_1"></A>
<P><B>(Pettifor_1)</B> D.G. Pettifor and I.I. Oleinik, Phys. Rev. B, 59, 8487
(1999).
</P>
<A NAME = "Pettofor_2"></A>
<P><B>(Pettifor_2)</B> D.G. Pettifor and I.I. Oleinik, Phys. Rev. Lett., 84,
4124 (2000).
</P>
<A NAME = "Pettofor_3"></A>
<P><B>(Pettifor_3)</B> D.G. Pettifor and I.I. Oleinik, Phys. Rev. B, 65, 172103
(2002).
</P>
<A NAME = "Murdick"></A>
<P><B>(Murdick)</B> D.A. Murdick, X.W. Zhou, H.N.G. Wadley, D. Nguyen-Manh, R.
Drautz, and D.G. Pettifor, Phys. Rev. B, 73, 45206 (2006).
</P>
<A NAME = "Ward"></A>
<P><B>(Ward)</B> D.K. Ward, X.W. Zhou, B.M. Wong, F.P. Doty, and J.A.
Zimmerman, Phys. Rev. B, 85,115206 (2012).
</P>
</HTML>
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