forked from lijiext/lammps
263 lines
11 KiB
HTML
263 lines
11 KiB
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<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>
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<HR>
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<H3>compute sna/atom command
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</H3>
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<H3>compute snad/atom command
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</H3>
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<H3>compute snav/atom command
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</H3>
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<P><B>Syntax:</B>
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</P>
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<PRE>compute ID group-ID sna/atom ntypes rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
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compute ID group-ID snad/atom ntypes rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
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compute ID group-ID snav/atom ntypes rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
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</PRE>
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<UL><LI>ID, group-ID are documented in <A HREF = "compute.html">compute</A> command
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<LI>sna/atom = style name of this compute command
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<LI>rcutfac = scale factor applied to all cutoff radii (positive real)
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<LI>rfac0 = parameter in distance to angle conversion (0 < rcutfac < 1)
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<LI>twojmax = band limit for bispectrum components (non-negative integer)
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<LI>R_1, R_2,... = list of cutoff radii, one for each type (distance units)
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<LI>w_1, w_2,... = list of neighbor weights, one for each type
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<LI>zero or more keyword/value pairs may be appended
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<LI>keyword = <I>diagonal</I> or <I>rmin0</I> or <I>switchflag</I>
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<PRE> <I>diagonal</I> value = <I>0</I> or <I>1</I> or <I>2</I> or <I>3</I>
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<I>0</I> = all j1, j2, j <= twojmax, j2 <= j1
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<I>1</I> = subset satisfying j1 == j2
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<I>2</I> = subset satisfying j1 == j2 == j3
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<I>3</I> = subset satisfying j2 <= j1 <= j
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<I>rmin0</I> value = parameter in distance to angle conversion (distance units)
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<I>switchflag</I> value = <I>0</I> or <I>1</I>
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<I>0</I> = do not use switching function
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<I>1</I> = use switching function
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</PRE>
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</UL>
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<P><B>Examples:</B>
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</P>
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<PRE>compute b all sna/atom 1.4 0.99363 6 2.0 2.4 0.75 1.0 diagonal 3 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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</PRE>
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<P><B>Description:</B>
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</P>
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<P>Define a computation that calculates a set of bispectrum components
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for each atom in a group.
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</P>
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<P>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. <A HREF = "#Thompson2014">(Thompson)</A>
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</P>
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<P>The position of a neighbor atom <I>i'</I> relative to a central atom <I>i</I> is
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a point within the 3D ball of radius <I>R_ii' = rcutfac*(R_i + R_i')</I>
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</P>
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<P>Bartok et al. <A HREF = "#Bartok2010">(Bartok)</A>, 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 <I>r</I> within <I>R_ii'</I> is mapped on to a third
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polar angle <I>theta0</I> defined by,
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</P>
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<CENTER><IMG SRC = "Eqs/compute_sna_atom1.jpg">
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</CENTER>
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<P>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 <I>theta0max=rfac0*Pi</I>
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are excluded.
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</P>
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<P>The natural basis for functions on the 3-sphere is formed by the 4D
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hyperspherical harmonics <I>U^j_m,m'(theta, phi, theta0).</I> These
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functions are better known as <I>D^j_m,m',</I> the elements of the Wigner
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<I>D</I>-matrices <A HREF = "#Meremianin2006">(Meremianin</A>,
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<A HREF = "#Varshalovich1987">Varshalovich)</A>.
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</P>
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<P>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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</P>
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<CENTER><IMG SRC = "Eqs/compute_sna_atom2.jpg">
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</CENTER>
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<P>The <I>w_i'</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 <I>fc(r)</I> ensures that
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the contribution of each neighbor atom goes smoothly to zero at
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<I>R_ii'</I>:
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</P>
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<CENTER><IMG SRC = "Eqs/compute_sna_atom4.jpg">
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</CENTER>
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<P>The expansion coefficients <I>u^j_m,m'</I> 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 <A HREF = "#Bartok2010">(Bartok)</A>.
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</P>
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<CENTER><IMG SRC = "Eqs/compute_sna_atom3.jpg">
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</CENTER>
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<P>The constants <I>H^jmm'_j1m1m1'_j2m2m2'</I> 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 <I>sna/atom</I>. 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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</P>
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<P>Compute <I>snad/atom</I> calculates the derivative of the bispectrum components
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summed separately for each atom type:
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</P>
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<CENTER><IMG SRC = "Eqs/compute_sna_atom5.jpg">
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</CENTER>
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<P>The sum is over all atoms <I>i'</I> of atom type <I>I</I>. For each atom <I>i</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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</P>
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<P>Compute <I>snav/atom</I> calculates the virial contribution due to the
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derivatives:
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</P>
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<CENTER><IMG SRC = "Eqs/compute_sna_atom6.jpg">
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</CENTER>
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<P>Again, the sum is over all atoms <I>i'</I> of atom type <I>I</I>. For each atom
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<I>i</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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</P>
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<P>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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</P>
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<P>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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</P>
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<P>The argument <I>rcutfac</I> is a scale factor that controls the ratio of
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atomic radius to radial cutoff distance.
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</P>
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<P>The argument <I>rfac0</I> and the optional keyword <I>rmin0</I> define the
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linear mapping from radial distance to polar angle <I>theta0</I> on the
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3-sphere.
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</P>
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<P>The argument <I>twojmax</I> and the keyword <I>diagonal</I> define 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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</P>
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<P>The keyword <I>switchflag</I> can be used to turn off the switching
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function.
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</P>
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<P>IMPORTANT NOTE: If you have a bonded system, then the settings of
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<A HREF = "special_bonds.html">special_bonds</A> 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 <A HREF = "special_bonds.html">special_bonds</A>
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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 <A HREF = "rerun.html">rerun</A>
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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 <A HREF = "special_bonds.html">special_bonds</A>
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command that includes all pairs in the neighbor list.
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</P>
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<P>;line
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</P>
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<P><B>Output info:</B>
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</P>
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<P>Compute <I>sna/atom</I> 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 identities of the bispectrum component contained in
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each column depend on the values of <I>twojmax</I> and <I>diagonal</I>, as
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described by the following piece of python code:
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</P>
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<PRE>for j1 in range(0,twojmax+1):
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if(diagonal==2):
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print j1/2,j1/2,j1/2
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elif(diagonal==1):
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for j in range(0,min(twojmax,2*j1)+1,2):
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print j1/2,j1/2,j/2
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elif(diagonal==0):
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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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print j1/2,j2/2,j/2
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elif(diagonal==3):
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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
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</PRE>
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<P>Compute <I>snad/atom</I> evaluates a per-atom array. The columns are
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arranged into <I>ntypes</I> blocks, listed in order of atom type <I>I</I>. Each
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block contains three sub-blocks corresponding to the <I>x</I>, <I>y</I>, and <I>z</I>
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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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<I>sna/atom</I>
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</P>
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<P>Compute <I>snav/atom</I> evaluates a per-atom array. The columns are
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arranged into <I>ntypes</I> blocks, listed in order of atom type <I>I</I>. Each
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block contains six sub-blocks corresponding to the <I>xx</I>, <I>yy</I>, <I>zz</I>,
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<I>yz</I>, <I>xz</I>, and <I>xy</I> 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 <I>sna/atom</I>
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</P>
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<P>These values can be accessed by any command that uses per-atom values
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from a compute as input. See <A HREF = "Section_howto.html#howto_15">Section_howto
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15</A> for an overview of LAMMPS output
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options.
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</P>
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<P><B>Restrictions:</B>
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</P>
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<P>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 <A HREF = "Section_start.html#start_3">Making
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LAMMPS</A> section for more info.
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</P>
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<P><B>Related commands:</B>
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</P>
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<P><A HREF = "pair_snap.html">pair_style snap</A>
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</P>
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<P><B>Default:</B>
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</P>
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<P>The optional keyword defaults are <I>diagonal</I> = 0, <I>rmin0</I> = 0,
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<I>switchflag</I> = 1.
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</P>
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<HR>
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<A NAME = "Thompson2014"></A>
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<P><B>(Thompson)</B> Thompson, Swiler, Trott, Foiles, Tucker, under review, preprint
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available at <A HREF = "http://arxiv.org/abs/1409.3880">arXiv:1409.3880</A>
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</P>
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<A NAME = "Bartok2010"></A>
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<P><B>(Bartok)</B> Bartok, Payne, Risi, Csanyi, Phys Rev Lett, 104, 136403 (2010).
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</P>
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<A NAME = "Meremianin2006"></A>
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<P><B>(Meremianin)</B> Meremianin, J. Phys. A, 39, 3099 (2006).
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</P>
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<A NAME = "Varshalovich1987"></A>
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<P><B>(Varshalovich)</B> Varshalovich, Moskalev, Khersonskii, Quantum Theory
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of Angular Momentum, World Scientific, Singapore (1987).
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</P>
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</HTML>
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