forked from lijiext/lammps
313 lines
13 KiB
Plaintext
313 lines
13 KiB
Plaintext
"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
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:link(lws,http://lammps.sandia.gov)
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:link(ld,Manual.html)
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:link(lc,Commands_all.html)
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:line
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compute sna/atom command :h3
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compute snad/atom command :h3
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compute snav/atom command :h3
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compute snap command :h3
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[Syntax:]
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compute ID group-ID sna/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
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compute ID group-ID snad/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
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compute ID group-ID snav/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
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compute ID group-ID snap rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ... :pre
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ID, group-ID are documented in "compute"_compute.html command :ulb,l
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sna/atom = style name of this compute command :l
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rcutfac = scale factor applied to all cutoff radii (positive real) :l
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rfac0 = parameter in distance to angle conversion (0 < rcutfac < 1) :l
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twojmax = band limit for bispectrum components (non-negative integer) :l
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R_1, R_2,... = list of cutoff radii, one for each type (distance units) :l
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w_1, w_2,... = list of neighbor weights, one for each type :l
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zero or more keyword/value pairs may be appended :l
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keyword = {rmin0} or {switchflag} or {bzeroflag} or {quadraticflag} :l
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{rmin0} value = parameter in distance to angle conversion (distance units)
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{switchflag} value = {0} or {1}
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{0} = do not use switching function
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{1} = use switching function
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{bzeroflag} value = {0} or {1}
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{0} = do not subtract B0
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{1} = subtract B0
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{quadraticflag} value = {0} or {1}
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{0} = do not generate quadratic terms
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{1} = generate quadratic terms :pre
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:ule
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[Examples:]
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compute b all sna/atom 1.4 0.99363 6 2.0 2.4 0.75 1.0 rmin0 0.0
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compute db all sna/atom 1.4 0.95 6 2.0 1.0
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compute vb all sna/atom 1.4 0.95 6 2.0 1.0
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compute snap all snap 1.4 0.95 6 2.0 1.0 :pre
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[Description:]
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Define a computation that calculates a set of quantities related to the
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bispectrum components of the atoms in a group. These computes are
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used primarily for calculating the dependence of energy, force, and
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stress components on the linear coefficients in the
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"snap pair_style"_pair_snap.html, which is useful when training a
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SNAP potential to match target data.
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Bispectrum components of an atom are order parameters characterizing
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the radial and angular distribution of neighbor atoms. The detailed
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mathematical definition is given in the paper by Thompson et
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al. "(Thompson)"_#Thompson20141
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The position of a neighbor atom {i'} relative to a central atom {i} is
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a point within the 3D ball of radius {R_ii' = rcutfac*(R_i + R_i')}
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Bartok et al. "(Bartok)"_#Bartok20101, proposed mapping this 3D ball
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onto the 3-sphere, the surface of the unit ball in a four-dimensional
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space. The radial distance {r} within {R_ii'} is mapped on to a third
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polar angle {theta0} defined by,
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:c,image(Eqs/compute_sna_atom1.jpg)
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In this way, all possible neighbor positions are mapped on to a subset
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of the 3-sphere. Points south of the latitude {theta0max=rfac0*Pi}
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are excluded.
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The natural basis for functions on the 3-sphere is formed by the 4D
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hyperspherical harmonics {U^j_m,m'(theta, phi, theta0).} These
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functions are better known as {D^j_m,m',} the elements of the Wigner
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{D}-matrices "(Meremianin"_#Meremianin2006,
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"Varshalovich)"_#Varshalovich1987.
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The density of neighbors on the 3-sphere can be written as a sum of
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Dirac-delta functions, one for each neighbor, weighted by species and
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radial distance. Expanding this density function as a generalized
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Fourier series in the basis functions, we can write each Fourier
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coefficient as
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:c,image(Eqs/compute_sna_atom2.jpg)
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The {w_i'} neighbor weights are dimensionless numbers that are chosen
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to distinguish atoms of different types, while the central atom is
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arbitrarily assigned a unit weight. The function {fc(r)} ensures that
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the contribution of each neighbor atom goes smoothly to zero at
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{R_ii'}:
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:c,image(Eqs/compute_sna_atom4.jpg)
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The expansion coefficients {u^j_m,m'} are complex-valued and they are
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not directly useful as descriptors, because they are not invariant
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under rotation of the polar coordinate frame. However, the following
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scalar triple products of expansion coefficients can be shown to be
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real-valued and invariant under rotation "(Bartok)"_#Bartok20101.
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:c,image(Eqs/compute_sna_atom3.jpg)
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The constants {H^jmm'_j1m1m1'_j2m2m2'} are coupling coefficients,
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analogous to Clebsch-Gordan coefficients for rotations on the
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2-sphere. These invariants are the components of the bispectrum and
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these are the quantities calculated by the compute {sna/atom}. They
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characterize the strength of density correlations at three points on
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the 3-sphere. The j2=0 subset form the power spectrum, which
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characterizes the correlations of two points. The lowest-order
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components describe the coarsest features of the density function,
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while higher-order components reflect finer detail. Note that the
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central atom is included in the expansion, so three point-correlations
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can be either due to three neighbors, or two neighbors and the central
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atom.
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Compute {snad/atom} calculates the derivative of the bispectrum components
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summed separately for each atom type:
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:c,image(Eqs/compute_sna_atom5.jpg)
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The sum is over all atoms {i'} of atom type {I}. For each atom {i},
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this compute evaluates the above expression for each direction, each
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atom type, and each bispectrum component. See section below on output
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for a detailed explanation.
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Compute {snav/atom} calculates the virial contribution due to the
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derivatives:
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:c,image(Eqs/compute_sna_atom6.jpg)
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Again, the sum is over all atoms {i'} of atom type {I}. For each atom
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{i}, this compute evaluates the above expression for each of the six
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virial components, each atom type, and each bispectrum component. See
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section below on output for a detailed explanation.
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Compute {snap} calculates a global array contains information related
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to all three of the above per-atom computes {sna/atom}, {snad/atom},
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and {snav/atom}. The first row of the array contains the summation of
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{sna/atom} over all atoms, but broken out by type. The last six rows
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of the array contain the summation of {snav/atom} over all atoms, broken
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out by type. In between these are 3*{N} rows containing the same values
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computed by {snad/atom} (these are already summed over all atoms and
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broken out by type). The element in the last column of each row contains
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the potential energy, force, or stress, according to the row.
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These quantities correspond to the user-specified reference potential
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that must be subtracted from the target data when fitting SNAP.
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The potential energy calculation uses the built in compute {thermo_pe}.
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The stress calculation uses a compute called {snap_press} that is
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automatically created behind the scenes, according to the following
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command:
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compute snap_press all pressure NULL virial :pre
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See section below on output for a detailed explanation of the data
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layout in the global array.
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The value of all bispectrum components will be zero for atoms not in
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the group. Neighbor atoms not in the group do not contribute to the
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bispectrum of atoms in the group.
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The neighbor list needed to compute this quantity is constructed each
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time the calculation is performed (i.e. each time a snapshot of atoms
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is dumped). Thus it can be inefficient to compute/dump this quantity
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too frequently.
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The argument {rcutfac} is a scale factor that controls the ratio of
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atomic radius to radial cutoff distance.
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The argument {rfac0} and the optional keyword {rmin0} define the
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linear mapping from radial distance to polar angle {theta0} on the
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3-sphere.
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The argument {twojmax} defines which
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bispectrum components are generated. See section below on output for a
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detailed explanation of the number of bispectrum components and the
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ordered in which they are listed.
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The keyword {switchflag} can be used to turn off the switching
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function.
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The keyword {bzeroflag} determines whether or not {B0}, the bispectrum
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components of an atom with no neighbors, are subtracted from
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the calculated bispectrum components. This optional keyword
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normally only affects compute {sna/atom}. However, when
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{quadraticflag} is on, it also affects {snad/atom} and {snav/atom}.
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The keyword {quadraticflag} determines whether or not the
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quadratic analogs to the bispectrum quantities are generated.
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These are formed by taking the outer product of the vector
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of bispectrum components with itself.
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See section below on output for a
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detailed explanation of the number of quadratic terms and the
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ordered in which they are listed.
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NOTE: If you have a bonded system, then the settings of
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"special_bonds"_special_bonds.html command can remove pairwise
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interactions between atoms in the same bond, angle, or dihedral. This
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is the default setting for the "special_bonds"_special_bonds.html
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command, and means those pairwise interactions do not appear in the
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neighbor list. Because this fix uses the neighbor list, it also means
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those pairs will not be included in the calculation. One way to get
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around this, is to write a dump file, and use the "rerun"_rerun.html
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command to compute the bispectrum components for snapshots in the dump
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file. The rerun script can use a "special_bonds"_special_bonds.html
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command that includes all pairs in the neighbor list.
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;line
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[Output info:]
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Compute {sna/atom} calculates a per-atom array, each column
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corresponding to a particular bispectrum component. The total number
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of columns and the identity of the bispectrum component contained in
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each column depend of the value of {twojmax}, as
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described by the following piece of python code:
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for j1 in range(0,twojmax+1):
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for j2 in range(0,j1+1):
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for j in range(j1-j2,min(twojmax,j1+j2)+1,2):
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if (j>=j1): print j1/2.,j2/2.,j/2. :pre
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NOTE: the {diagonal} keyword allowing other possible choices
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for the number of bispectrum components was removed in 2019,
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since all potentials use the value of 3, corresponding to the
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above set of bispectrum components.
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Compute {snad/atom} evaluates a per-atom array. The columns are
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arranged into {ntypes} blocks, listed in order of atom type {I}. Each
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block contains three sub-blocks corresponding to the {x}, {y}, and {z}
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components of the atom position. Each of these sub-blocks contains
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one column for each bispectrum component, the same as for compute
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{sna/atom}
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Compute {snav/atom} evaluates a per-atom array. The columns are
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arranged into {ntypes} blocks, listed in order of atom type {I}. Each
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block contains six sub-blocks corresponding to the {xx}, {yy}, {zz},
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{yz}, {xz}, and {xy} components of the virial tensor in Voigt
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notation. Each of these sub-blocks contains one column for each
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bispectrum component, the same as for compute {sna/atom}
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Compute {snap} evaluates a global array.
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The columns are arranged into
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{ntypes} blocks, listed in order of atom type {I}. Each block
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contains one column for each bispectrum component, the same as for compute
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{sna/atom}. A final column contains the corresponding energy, force component
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on an atom, or virial stress component. The rows of the array appear
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in the following order:
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1 row: {sna/atom} quantities summed for all atoms of type {I}
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3*{N} rows: {snad/atom} quantities, with derivatives w.r.t. x, y, and z coordinate of atom {i} appearing in consecutive rows. The atoms are sorted based on atom ID.
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6 rows: {snav/atom} quantities summed for all atoms of type {I} :ul
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For example, if {K} =30 and ntypes=1, the number of columns in the per-atom
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arrays generated by {sna/atom}, {snad/atom}, and {snav/atom}
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are 30, 90, and 180, respectively. With {quadratic} value=1,
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the numbers of columns are 930, 2790, and 5580, respectively.
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The number of columns in the global array generated by {snap}
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are 31, and 931, respectively, while the number of rows is
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1+3*{N}+6, where {N} is the total number of atoms.
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If the {quadratic} keyword value is set to 1, then additional
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columns are generated, corresponding to
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the products of all distinct pairs of bispectrum components. If the
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number of bispectrum components is {K}, then the number of distinct pairs
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is {K}({K}+1)/2.
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For compute {sna/atom} these columns are appended to existing {K} columns.
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The ordering of quadratic terms is upper-triangular,
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(1,1),(1,2)...(1,{K}),(2,1)...({K}-1,{K}-1),({K}-1,{K}),({K},{K}).
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For computes {snad/atom} and {snav/atom} each set of {K}({K}+1)/2
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additional columns is inserted directly after each of sub-block
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of linear terms i.e. linear and quadratic terms are contiguous.
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So the nesting order from inside to outside is bispectrum component,
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linear then quadratic, vector/tensor component, type.
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These values can be accessed by any command that uses per-atom values
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from a compute as input. See the "Howto output"_Howto_output.html doc
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page for an overview of LAMMPS output options.
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[Restrictions:]
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These computes are part of the SNAP package. They are only enabled if
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LAMMPS was built with that package. See the "Build
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package"_Build_package.html doc page for more info.
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[Related commands:]
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"pair_style snap"_pair_snap.html
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[Default:]
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The optional keyword defaults are {rmin0} = 0,
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{switchflag} = 1, {bzeroflag} = 1, {quadraticflag} = 0,
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:line
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:link(Thompson20141)
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[(Thompson)] Thompson, Swiler, Trott, Foiles, Tucker, under review, preprint
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available at "arXiv:1409.3880"_http://arxiv.org/abs/1409.3880
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:link(Bartok20101)
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[(Bartok)] Bartok, Payne, Risi, Csanyi, Phys Rev Lett, 104, 136403 (2010).
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:link(Meremianin2006)
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[(Meremianin)] Meremianin, J. Phys. A, 39, 3099 (2006).
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:link(Varshalovich1987)
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[(Varshalovich)] Varshalovich, Moskalev, Khersonskii, Quantum Theory
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of Angular Momentum, World Scientific, Singapore (1987).
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