forked from lijiext/lammps
496 lines
21 KiB
Plaintext
496 lines
21 KiB
Plaintext
"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
|
|
|
|
:link(lws,http://lammps.sandia.gov)
|
|
:link(ld,Manual.html)
|
|
:link(lc,Commands_all.html)
|
|
|
|
:line
|
|
|
|
kspace_style command :h3
|
|
|
|
[Syntax:]
|
|
|
|
kspace_style style value :pre
|
|
|
|
style = {none} or {ewald} or {ewald/disp} or {ewald/omp} or {pppm} or {pppm/cg} or {pppm/disp} or {pppm/tip4p} or {pppm/stagger} or {pppm/disp/tip4p} or {pppm/gpu} or {pppm/kk} or {pppm/omp} or {pppm/cg/omp} or {pppm/tip4p/omp} or {msm} or {msm/cg} or {msm/omp} or {msm/cg/omp} or {scafacos} :ulb,l
|
|
{none} value = none
|
|
{ewald} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{ewald/disp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{ewald/omp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{ewald/dipole} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{ewald/dipole/spin} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/cg} values = accuracy (smallq)
|
|
accuracy = desired relative error in forces
|
|
smallq = cutoff for charges to be considered (optional) (charge units)
|
|
{pppm/disp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/tip4p} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/disp/tip4p} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/gpu} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/intel} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/kk} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/omp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/cg/omp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/disp/intel} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/tip4p/omp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/stagger} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/dipole} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{pppm/dipole/spin} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{msm} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{msm/cg} value = accuracy (smallq)
|
|
accuracy = desired relative error in forces
|
|
smallq = cutoff for charges to be considered (optional) (charge units)
|
|
{msm/omp} value = accuracy
|
|
accuracy = desired relative error in forces
|
|
{msm/cg/omp} value = accuracy (smallq)
|
|
accuracy = desired relative error in forces
|
|
smallq = cutoff for charges to be considered (optional) (charge units)
|
|
{scafacos} values = method accuracy
|
|
method = fmm or p2nfft or p3m or ewald or direct
|
|
accuracy = desired relative error in forces :pre
|
|
:ule
|
|
|
|
[Examples:]
|
|
|
|
kspace_style pppm 1.0e-4
|
|
kspace_style pppm/cg 1.0e-5 1.0e-6
|
|
kspace style msm 1.0e-4
|
|
kspace style scafacos fmm 1.0e-4
|
|
kspace_style none :pre
|
|
|
|
[Description:]
|
|
|
|
Define a long-range solver for LAMMPS to use each timestep to compute
|
|
long-range Coulombic interactions or long-range 1/r^6 interactions.
|
|
Most of the long-range solvers perform their computation in K-space,
|
|
hence the name of this command.
|
|
|
|
When such a solver is used in conjunction with an appropriate pair
|
|
style, the cutoff for Coulombic or 1/r^N interactions is effectively
|
|
infinite. If the Coulombic case, this means each charge in the system
|
|
interacts with charges in an infinite array of periodic images of the
|
|
simulation domain.
|
|
|
|
Note that using a long-range solver requires use of a matching "pair
|
|
style"_pair_style.html to perform consistent short-range pairwise
|
|
calculations. This means that the name of the pair style contains a
|
|
matching keyword to the name of the KSpace style, as in this table:
|
|
|
|
Pair style : KSpace style
|
|
coul/long : ewald or pppm
|
|
coul/msm : msm
|
|
lj/long or buck/long : disp (for dispersion)
|
|
tip4p/long : tip4p :tb(s=:,ea=c)
|
|
|
|
:line
|
|
|
|
The {ewald} style performs a standard Ewald summation as described in
|
|
any solid-state physics text.
|
|
|
|
The {ewald/disp} style adds a long-range dispersion sum option for
|
|
1/r^6 potentials and is useful for simulation of interfaces
|
|
"(Veld)"_#Veld. It also performs standard Coulombic Ewald summations,
|
|
but in a more efficient manner than the {ewald} style. The 1/r^6
|
|
capability means that Lennard-Jones or Buckingham potentials can be
|
|
used without a cutoff, i.e. they become full long-range potentials.
|
|
The {ewald/disp} style can also be used with point-dipoles, see
|
|
"(Toukmaji)"_#Toukmaji.
|
|
|
|
The {ewald/dipole} style adds long-range standard Ewald summations
|
|
for dipole-dipole interactions, see "(Toukmaji)"_#Toukmaji.
|
|
|
|
The {ewald/dipole/spin} style adds long-range standard Ewald
|
|
summations for magnetic dipole-dipole interactions between
|
|
magnetic spins.
|
|
|
|
:line
|
|
|
|
The {pppm} style invokes a particle-particle particle-mesh solver
|
|
"(Hockney)"_#Hockney which maps atom charge to a 3d mesh, uses 3d FFTs
|
|
to solve Poisson's equation on the mesh, then interpolates electric
|
|
fields on the mesh points back to the atoms. It is closely related to
|
|
the particle-mesh Ewald technique (PME) "(Darden)"_#Darden used in
|
|
AMBER and CHARMM. The cost of traditional Ewald summation scales as
|
|
N^(3/2) where N is the number of atoms in the system. The PPPM solver
|
|
scales as Nlog(N) due to the FFTs, so it is almost always a faster
|
|
choice "(Pollock)"_#Pollock.
|
|
|
|
The {pppm/cg} style is identical to the {pppm} style except that it
|
|
has an optimization for systems where most particles are uncharged.
|
|
Similarly the {msm/cg} style implements the same optimization for {msm}.
|
|
The optional {smallq} argument defines the cutoff for the absolute
|
|
charge value which determines whether a particle is considered charged
|
|
or not. Its default value is 1.0e-5.
|
|
|
|
The {pppm/dipole} style invokes a particle-particle particle-mesh solver
|
|
for dipole-dipole interactions, following the method of "(Cerda)"_#Cerda2008.
|
|
|
|
The {pppm/dipole/spin} style invokes a particle-particle particle-mesh solver
|
|
for magnetic dipole-dipole interactions between magnetic spins.
|
|
|
|
The {pppm/tip4p} style is identical to the {pppm} style except that it
|
|
adds a charge at the massless 4th site in each TIP4P water molecule.
|
|
It should be used with "pair styles"_pair_style.html with a
|
|
{tip4p/long} in their style name.
|
|
|
|
The {pppm/stagger} style performs calculations using two different
|
|
meshes, one shifted slightly with respect to the other. This can
|
|
reduce force aliasing errors and increase the accuracy of the method
|
|
for a given mesh size. Or a coarser mesh can be used for the same
|
|
target accuracy, which saves CPU time. However, there is a trade-off
|
|
since FFTs on two meshes are now performed which increases the
|
|
computation required. See "(Cerutti)"_#Cerutti, "(Neelov)"_#Neelov,
|
|
and "(Hockney)"_#Hockney for details of the method.
|
|
|
|
For high relative accuracy, using staggered PPPM allows the mesh size
|
|
to be reduced by a factor of 2 in each dimension as compared to
|
|
regular PPPM (for the same target accuracy). This can give up to a 4x
|
|
speedup in the KSpace time (8x less mesh points, 2x more expensive).
|
|
However, for low relative accuracy, the staggered PPPM mesh size may
|
|
be essentially the same as for regular PPPM, which means the method
|
|
will be up to 2x slower in the KSpace time (simply 2x more expensive).
|
|
For more details and timings, see the "Speed tips"_Speed_tips.html doc
|
|
page.
|
|
|
|
NOTE: Using {pppm/stagger} may not give the same increase in the
|
|
accuracy of energy and pressure as it does in forces, so some caution
|
|
must be used if energy and/or pressure are quantities of interest,
|
|
such as when using a barostat.
|
|
|
|
:line
|
|
|
|
The {pppm/disp} and {pppm/disp/tip4p} styles add a mesh-based long-range
|
|
dispersion sum option for 1/r^6 potentials "(Isele-Holder)"_#Isele-Holder2012,
|
|
similar to the {ewald/disp} style. The 1/r^6 capability means
|
|
that Lennard-Jones or Buckingham potentials can be used without a cutoff,
|
|
i.e. they become full long-range potentials.
|
|
|
|
For these styles, you will possibly want to adjust the default choice
|
|
of parameters by using the "kspace_modify"_kspace_modify.html command.
|
|
This can be done by either choosing the Ewald and grid parameters, or
|
|
by specifying separate accuracies for the real and kspace
|
|
calculations. When not making any settings, the simulation will stop
|
|
with an error message. Further information on the influence of the
|
|
parameters and how to choose them is described in
|
|
"(Isele-Holder)"_#Isele-Holder2012,
|
|
"(Isele-Holder2)"_#Isele-Holder2013 and the "Howto
|
|
dispersion"_Howto_dispersion.html doc page.
|
|
|
|
:line
|
|
|
|
NOTE: All of the PPPM styles can be used with single-precision FFTs by
|
|
using the compiler switch -DFFT_SINGLE for the FFT_INC setting in your
|
|
lo-level Makefile. This setting also changes some of the PPPM
|
|
operations (e.g. mapping charge to mesh and interpolating electric
|
|
fields to particles) to be performed in single precision. This option
|
|
can speed-up long-range calculations, particularly in parallel or on
|
|
GPUs. The use of the -DFFT_SINGLE flag is discussed on the "Build
|
|
settings"_Build_settings.html doc page. MSM does not currently support
|
|
the -DFFT_SINGLE compiler switch.
|
|
|
|
:line
|
|
|
|
The {msm} style invokes a multi-level summation method MSM solver,
|
|
"(Hardy)"_#Hardy2006 or "(Hardy2)"_#Hardy2009, which maps atom charge
|
|
to a 3d mesh, and uses a multi-level hierarchy of coarser and coarser
|
|
meshes on which direct Coulomb solvers are done. This method does not
|
|
use FFTs and scales as N. It may therefore be faster than the other
|
|
K-space solvers for relatively large problems when running on large
|
|
core counts. MSM can also be used for non-periodic boundary conditions
|
|
and for mixed periodic and non-periodic boundaries.
|
|
|
|
MSM is most competitive versus Ewald and PPPM when only relatively
|
|
low accuracy forces, about 1e-4 relative error or less accurate,
|
|
are needed. Note that use of a larger Coulombic cutoff (i.e. 15
|
|
angstroms instead of 10 angstroms) provides better MSM accuracy for
|
|
both the real space and grid computed forces.
|
|
|
|
Currently calculation of the full pressure tensor in MSM is expensive.
|
|
Using the "kspace_modify"_kspace_modify.html {pressure/scalar yes}
|
|
command provides a less expensive way to compute the scalar pressure
|
|
(Pxx + Pyy + Pzz)/3.0. The scalar pressure can be used, for example,
|
|
to run an isotropic barostat. If the full pressure tensor is needed,
|
|
then calculating the pressure at every timestep or using a fixed
|
|
pressure simulation with MSM will cause the code to run slower.
|
|
|
|
:line
|
|
|
|
The {scafacos} style is a wrapper on the "ScaFaCoS Coulomb solver
|
|
library"_http://www.scafacos.de which provides a variety of solver
|
|
methods which can be used with LAMMPS. The paper by "(Who)"_#Who2012
|
|
gives an overview of ScaFaCoS.
|
|
|
|
ScaFaCoS was developed by a consortium of German research facilities
|
|
with a BMBF (German Ministry of Science and Education) funded project
|
|
in 2009-2012. Participants of the consortium were the Universities of
|
|
Bonn, Chemnitz, Stuttgart, and Wuppertal as well as the
|
|
Forschungszentrum Juelich.
|
|
|
|
The library is available for download at "http://scafacos.de" or can
|
|
be cloned from the git-repository
|
|
"git://github.com/scafacos/scafacos.git".
|
|
|
|
In order to use this KSpace style, you must download and build the
|
|
ScaFaCoS library, then build LAMMPS with the USER-SCAFACOS package
|
|
installed package which links LAMMPS to the ScaFaCoS library.
|
|
See details on "this page"_Section_packages.html#USER-SCAFACOS.
|
|
|
|
NOTE: Unlike other KSpace solvers in LAMMPS, ScaFaCoS computes all
|
|
Coulombic interactions, both short- and long-range. Thus you should
|
|
NOT use a Coulombic pair style when using kspace_style scafacos. This
|
|
also means the total Coulombic energy (short- and long-range) will be
|
|
tallied for "thermodynamic output"_thermo_style.html command as part
|
|
of the {elong} keyword; the {ecoul} keyword will be zero.
|
|
|
|
NOTE: See the current restriction below about use of ScaFaCoS in
|
|
LAMMPS with molecular charged systems or the TIP4P water model.
|
|
|
|
The specified {method} determines which ScaFaCoS algorithm is used.
|
|
These are the ScaFaCoS methods currently available from LAMMPS:
|
|
|
|
{fmm} = Fast Multi-Pole method
|
|
{p2nfft} = FFT-based Coulomb solver
|
|
{ewald} = Ewald summation
|
|
{direct} = direct O(N^2) summation
|
|
{p3m} = PPPM :ul
|
|
|
|
We plan to support additional ScaFaCoS solvers from LAMMPS in the
|
|
future. For an overview of the included solvers, refer to
|
|
"(Sutmann)"_#Sutmann2013
|
|
|
|
The specified {accuracy} is similar to the accuracy setting for other
|
|
LAMMPS KSpace styles, but is passed to ScaFaCoS, which can interpret
|
|
it in different ways for different methods it supports. Within the
|
|
ScaFaCoS library the {accuracy} is treated as a tolerance level
|
|
(either absolute or relative) for the chosen quantity, where the
|
|
quantity can be either the Columic field values, the per-atom Columic
|
|
energy or the total Columic energy. To select from these options, see
|
|
the "kspace_modify scafacos accuracy"_kspace_modify.html doc page.
|
|
|
|
The "kspace_modify scafacos"_kspace_modify.html command also explains
|
|
other ScaFaCoS options currently exposed to LAMMPS.
|
|
|
|
:line
|
|
|
|
The specified {accuracy} determines the relative RMS error in per-atom
|
|
forces calculated by the long-range solver. It is set as a
|
|
dimensionless number, relative to the force that two unit point
|
|
charges (e.g. 2 monovalent ions) exert on each other at a distance of
|
|
1 Angstrom. This reference value was chosen as representative of the
|
|
magnitude of electrostatic forces in atomic systems. Thus an accuracy
|
|
value of 1.0e-4 means that the RMS error will be a factor of 10000
|
|
smaller than the reference force.
|
|
|
|
The accuracy setting is used in conjunction with the pairwise cutoff
|
|
to determine the number of K-space vectors for style {ewald} or the
|
|
grid size for style {pppm} or {msm}.
|
|
|
|
Note that style {pppm} only computes the grid size at the beginning of
|
|
a simulation, so if the length or triclinic tilt of the simulation
|
|
cell increases dramatically during the course of the simulation, the
|
|
accuracy of the simulation may degrade. Likewise, if the
|
|
"kspace_modify slab"_kspace_modify.html option is used with
|
|
shrink-wrap boundaries in the z-dimension, and the box size changes
|
|
dramatically in z. For example, for a triclinic system with all three
|
|
tilt factors set to the maximum limit, the PPPM grid should be
|
|
increased roughly by a factor of 1.5 in the y direction and 2.0 in the
|
|
z direction as compared to the same system using a cubic orthogonal
|
|
simulation cell. One way to handle this issue if you have a long
|
|
simulation where the box size changes dramatically, is to break it
|
|
into shorter simulations (multiple "run"_run.html commands). This
|
|
works because the grid size is re-computed at the beginning of each
|
|
run. Another way to ensure the described accuracy requirement is met
|
|
is to run a short simulation at the maximum expected tilt or length,
|
|
note the required grid size, and then use the
|
|
"kspace_modify"_kspace_modify.html {mesh} command to manually set the
|
|
PPPM grid size to this value for the long run. The simulation then
|
|
will be "too accurate" for some portion of the run.
|
|
|
|
RMS force errors in real space for {ewald} and {pppm} are estimated
|
|
using equation 18 of "(Kolafa)"_#Kolafa, which is also referenced as
|
|
equation 9 of "(Petersen)"_#Petersen. RMS force errors in K-space for
|
|
{ewald} are estimated using equation 11 of "(Petersen)"_#Petersen,
|
|
which is similar to equation 32 of "(Kolafa)"_#Kolafa. RMS force
|
|
errors in K-space for {pppm} are estimated using equation 38 of
|
|
"(Deserno)"_#Deserno. RMS force errors for {msm} are estimated
|
|
using ideas from chapter 3 of "(Hardy)"_#Hardy2006, with equation 3.197
|
|
of particular note. When using {msm} with non-periodic boundary
|
|
conditions, it is expected that the error estimation will be too
|
|
pessimistic. RMS force errors for dipoles when using {ewald/disp}
|
|
or {ewald/dipole} are estimated using equations 33 and 46 of
|
|
"(Wang)"_#Wang. The RMS force errors for {pppm/dipole} are estimated
|
|
using the equations in "(Cerda)"_#Cerda2008.
|
|
|
|
|
|
See the "kspace_modify"_kspace_modify.html command for additional
|
|
options of the K-space solvers that can be set, including a {force}
|
|
option for setting an absolute RMS error in forces, as opposed to a
|
|
relative RMS error.
|
|
|
|
:line
|
|
|
|
Styles with a {gpu}, {intel}, {kk}, {omp}, or {opt} suffix are
|
|
functionally the same as the corresponding style without the suffix.
|
|
They have been optimized to run faster, depending on your available
|
|
hardware, as discussed on the "Speed packages"_Speed_packages.html doc
|
|
page. The accelerated styles take the same arguments and should
|
|
produce the same results, except for round-off and precision issues.
|
|
|
|
More specifically, the {pppm/gpu} style performs charge assignment and
|
|
force interpolation calculations on the GPU. These processes are
|
|
performed either in single or double precision, depending on whether
|
|
the -DFFT_SINGLE setting was specified in your lo-level Makefile, as
|
|
discussed above. The FFTs themselves are still calculated on the CPU.
|
|
If {pppm/gpu} is used with a GPU-enabled pair style, part of the PPPM
|
|
calculation can be performed concurrently on the GPU while other
|
|
calculations for non-bonded and bonded force calculation are performed
|
|
on the CPU.
|
|
|
|
The {pppm/kk} style also performs charge assignment and force
|
|
interpolation calculations on the GPU while the FFTs themselves are
|
|
calculated on the CPU in non-threaded mode.
|
|
|
|
These accelerated styles are part of the GPU, USER-INTEL, KOKKOS,
|
|
USER-OMP, and OPT packages respectively. They are only enabled if
|
|
LAMMPS was built with those packages. See the "Build
|
|
package"_Build_package.html doc page for more info.
|
|
|
|
See the "Speed packages"_Speed_packages.html doc page for more
|
|
instructions on how to use the accelerated styles effectively.
|
|
|
|
:line
|
|
|
|
[Restrictions:]
|
|
|
|
Note that the long-range electrostatic solvers in LAMMPS assume conducting
|
|
metal (tinfoil) boundary conditions for both charge and dipole
|
|
interactions. Vacuum boundary conditions are not currently supported.
|
|
|
|
The {ewald/disp}, {ewald}, {pppm}, and {msm} styles support
|
|
non-orthogonal (triclinic symmetry) simulation boxes. However,
|
|
triclinic simulation cells may not yet be supported by suffix versions
|
|
of these styles.
|
|
|
|
All of the kspace styles are part of the KSPACE package. They are
|
|
only enabled if LAMMPS was built with that package. See the "Build
|
|
package"_Build_package.html doc page for more info.
|
|
|
|
For MSM, a simulation must be 3d and one can use any combination of
|
|
periodic, non-periodic, or shrink-wrapped boundaries (specified using
|
|
the "boundary"_boundary.html command).
|
|
|
|
For Ewald and PPPM, a simulation must be 3d and periodic in all
|
|
dimensions. The only exception is if the slab option is set with
|
|
"kspace_modify"_kspace_modify.html, in which case the xy dimensions
|
|
must be periodic and the z dimension must be non-periodic.
|
|
|
|
The scafacos KSpace style will only be enabled if LAMMPS is built with
|
|
the USER-SCAFACOS package. See the "Build package"_Build_package.html
|
|
doc page for more info.
|
|
|
|
The use of ScaFaCos in LAMMPS does not yet support molecular charged
|
|
systems where the short-range Coulombic interactions between atoms in
|
|
the same bond/angle/dihedral are weighted by the
|
|
"special_bonds"_special_bonds.html command. Likewise it does not
|
|
support the "TIP4P water style" where a fictitious charge site is
|
|
introduced in each water molecule.
|
|
Finally, the methods {p3m} and {ewald} do not support computing the
|
|
virial, so this contribution is not included.
|
|
|
|
[Related commands:]
|
|
|
|
"kspace_modify"_kspace_modify.html, "pair_style
|
|
lj/cut/coul/long"_pair_lj.html, "pair_style
|
|
lj/charmm/coul/long"_pair_charmm.html, "pair_style
|
|
lj/long/coul/long"_pair_lj_long.html, "pair_style
|
|
buck/coul/long"_pair_buck.html
|
|
|
|
[Default:]
|
|
|
|
kspace_style none :pre
|
|
|
|
:line
|
|
|
|
:link(Darden)
|
|
[(Darden)] Darden, York, Pedersen, J Chem Phys, 98, 10089 (1993).
|
|
|
|
:link(Deserno)
|
|
[(Deserno)] Deserno and Holm, J Chem Phys, 109, 7694 (1998).
|
|
|
|
:link(Hockney)
|
|
[(Hockney)] Hockney and Eastwood, Computer Simulation Using Particles,
|
|
Adam Hilger, NY (1989).
|
|
|
|
:link(Kolafa)
|
|
[(Kolafa)] Kolafa and Perram, Molecular Simulation, 9, 351 (1992).
|
|
|
|
:link(Petersen)
|
|
[(Petersen)] Petersen, J Chem Phys, 103, 3668 (1995).
|
|
|
|
:link(Wang)
|
|
[(Wang)] Wang and Holm, J Chem Phys, 115, 6277 (2001).
|
|
|
|
:link(Pollock)
|
|
[(Pollock)] Pollock and Glosli, Comp Phys Comm, 95, 93 (1996).
|
|
|
|
:link(Cerutti)
|
|
[(Cerutti)] Cerutti, Duke, Darden, Lybrand, Journal of Chemical Theory
|
|
and Computation 5, 2322 (2009)
|
|
|
|
:link(Neelov)
|
|
[(Neelov)] Neelov, Holm, J Chem Phys 132, 234103 (2010)
|
|
|
|
:link(Veld)
|
|
[(Veld)] In 't Veld, Ismail, Grest, J Chem Phys, 127, 144711 (2007).
|
|
|
|
:link(Toukmaji)
|
|
[(Toukmaji)] Toukmaji, Sagui, Board, and Darden, J Chem Phys, 113,
|
|
10913 (2000).
|
|
|
|
:link(Isele-Holder2012)
|
|
[(Isele-Holder)] Isele-Holder, Mitchell, Ismail, J Chem Phys, 137,
|
|
174107 (2012).
|
|
|
|
:link(Isele-Holder2013)
|
|
[(Isele-Holder2)] Isele-Holder, Mitchell, Hammond, Kohlmeyer, Ismail,
|
|
J Chem Theory Comput 9, 5412 (2013).
|
|
|
|
:link(Hardy2006)
|
|
[(Hardy)] David Hardy thesis: Multilevel Summation for the Fast
|
|
Evaluation of Forces for the Simulation of Biomolecules, University of
|
|
Illinois at Urbana-Champaign, (2006).
|
|
|
|
:link(Hardy2009)
|
|
[(Hardy2)] Hardy, Stone, Schulten, Parallel Computing, 35, 164-177
|
|
(2009).
|
|
|
|
:link(Sutmann2013)
|
|
[(Sutmann)] Sutmann, Arnold, Fahrenberger, et. al., Physical review / E 88(6), 063308 (2013)
|
|
|
|
:link(Cerda2008)
|
|
[(Cerda)] Cerda, Ballenegger, Lenz, Holm, J Chem Phys 129, 234104 (2008)
|
|
|
|
:link(Who2012)
|
|
[(Who)] Who, Author2, Author3, J of Long Range Solvers, 35, 164-177
|
|
(2012).
|