"""Calculator for the Embedded Atom Method Potential""" # eam.py # Embedded Atom Method Potential # These routines integrate with the ASE simulation environment # Paul White (Oct 2012) # UNCLASSIFIED # License: See accompanying license files for details import os import numpy as np from ase.neighborlist import NeighborList from ase.calculators.calculator import Calculator, all_changes from scipy.interpolate import InterpolatedUnivariateSpline as spline from ase.units import Bohr, Hartree class EAM(Calculator): r""" EAM Interface Documentation Introduction ============ The Embedded Atom Method (EAM) [1]_ is a classical potential which is good for modelling metals, particularly fcc materials. Because it is an equiaxial potential the EAM does not model directional bonds well. However, the Angular Dependent Potential (ADP) [2]_ which is an extended version of EAM is able to model directional bonds and is also included in the EAM calculator. Generally all that is required to use this calculator is to supply a potential file or as a set of functions that describe the potential. The files containing the potentials for this calculator are not included but many suitable potentials can be downloaded from The Interatomic Potentials Repository Project at https://www.ctcms.nist.gov/potentials/ Theory ====== A single element EAM potential is defined by three functions: the embedded energy, electron density and the pair potential. A two element alloy contains the individual three functions for each element plus cross pair interactions. The ADP potential has two additional sets of data to define the dipole and quadrupole directional terms for each alloy and their cross interactions. The total energy `E_{\rm tot}` of an arbitrary arrangement of atoms is given by the EAM potential as .. math:: E_\text{tot} = \sum_i F(\bar\rho_i) + \frac{1}{2}\sum_{i\ne j} \phi(r_{ij}) and .. math:: \bar\rho_i = \sum_j \rho(r_{ij}) where `F` is an embedding function, namely the energy to embed an atom `i` in the combined electron density `\bar\rho_i` which is contributed from each of its neighbouring atoms `j` by an amount `\rho(r_{ij})`, `\phi(r_{ij})` is the pair potential function representing the energy in bond `ij` which is due to the short-range electro-static interaction between atoms, and `r_{ij}` is the distance between an atom and its neighbour for that bond. The ADP potential is defined as .. math:: E_\text{tot} = \sum_i F(\bar\rho_i) + \frac{1}{2}\sum_{i\ne j} \phi(r_{ij}) + \frac{1}{2} \sum_{i,\alpha} (\mu_i^\alpha)^2 + \frac{1}{2} \sum_{i,\alpha,\beta} (\lambda_i^{\alpha\beta})^2 - \frac{1}{6} \sum_i \nu_i^2 where `\mu_i^\alpha` is the dipole vector, `\lambda_i^{\alpha\beta}` is the quadrupole tensor and `\nu_i` is the trace of `\lambda_i^{\alpha\beta}`. The fs potential is defined as .. math:: E_i = F_\alpha (\sum_{j\neq i} \rho_{\alpha \beta}(r_{ij})) + \frac{1}{2}\sum_{j\neq i}\phi_{\alpha \beta}(r_{ij}) where `\alpha` and `\beta` are element types of atoms. This form is similar to original EAM formula above, except that `\rho` and `\phi` are determined by element types. Running the Calculator ====================== EAM calculates the cohesive atom energy and forces. Internally the potential functions are defined by splines which may be directly supplied or created by reading the spline points from a data file from which a spline function is created. The LAMMPS compatible ``.alloy``, ``.fs`` and ``.adp`` formats are supported. The LAMMPS ``.eam`` format is slightly different from the ``.alloy`` format and is currently not supported. For example:: from ase.calculators.eam import EAM mishin = EAM(potential='Al99.eam.alloy') mishin.write_potential('new.eam.alloy') mishin.plot() slab.calc = mishin slab.get_potential_energy() slab.get_forces() The breakdown of energy contribution from the indvidual components are stored in the calculator instance ``.results['energy_components']`` Arguments ========= ========================= ==================================================== Keyword Description ========================= ==================================================== ``potential`` file of potential in ``.eam``, ``.alloy``, ``.adp`` or ``.fs`` format or file object (This is generally all you need to supply). In case of file object the ``form`` argument is required ``elements[N]`` array of N element abbreviations ``embedded_energy[N]`` arrays of embedded energy functions ``electron_density[N]`` arrays of electron density functions ``phi[N,N]`` arrays of pair potential functions ``d_embedded_energy[N]`` arrays of derivative embedded energy functions ``d_electron_density[N]`` arrays of derivative electron density functions ``d_phi[N,N]`` arrays of derivative pair potentials functions ``d[N,N], q[N,N]`` ADP dipole and quadrupole function ``d_d[N,N], d_q[N,N]`` ADP dipole and quadrupole derivative functions ``skin`` skin distance passed to NeighborList(). If no atom has moved more than the skin-distance since the last call to the ``update()`` method then the neighbor list can be reused. Defaults to 1.0. ``form`` the form of the potential ``eam``, ``alloy``, ``adp`` or ``fs``. This will be determined from the file suffix or must be set if using equations or file object ========================= ==================================================== Additional parameters for writing potential files ================================================= The following parameters are only required for writing a potential in ``.alloy``, ``.adp`` or ``fs`` format file. ========================= ==================================================== Keyword Description ========================= ==================================================== ``header`` Three line text header. Default is standard message. ``Z[N]`` Array of atomic number of each element ``mass[N]`` Atomic mass of each element ``a[N]`` Array of lattice parameters for each element ``lattice[N]`` Lattice type ``nrho`` No. of rho samples along embedded energy curve ``drho`` Increment for sampling density ``nr`` No. of radial points along density and pair potential curves ``dr`` Increment for sampling radius ========================= ==================================================== Special features ================ ``.plot()`` Plots the individual functions. This may be called from multiple EAM potentials to compare the shape of the individual curves. This function requires the installation of the Matplotlib libraries. Notes/Issues ============= * Although currently not fast, this calculator can be good for trying small calculations or for creating new potentials by matching baseline data such as from DFT results. The format for these potentials is compatible with LAMMPS_ and so can be used either directly by LAMMPS or with the ASE LAMMPS calculator interface. * Supported formats are the LAMMPS_ ``.alloy`` and ``.adp``. The ``.eam`` format is currently not supported. The form of the potential will be determined from the file suffix. * Any supplied values will override values read from the file. * The derivative functions, if supplied, are only used to calculate forces. * There is a bug in early versions of scipy that will cause eam.py to crash when trying to evaluate splines of a potential with one neighbor such as caused by evaluating a dimer. .. _LAMMPS: http://lammps.sandia.gov/ .. [1] M.S. Daw and M.I. Baskes, Phys. Rev. Letters 50 (1983) 1285. .. [2] Y. Mishin, M.J. Mehl, and D.A. Papaconstantopoulos, Acta Materialia 53 2005 4029--4041. End EAM Interface Documentation """ implemented_properties = ['energy', 'forces'] default_parameters = dict( skin=1.0, potential=None, header=[b'EAM/ADP potential file\n', b'Generated from eam.py\n', b'blank\n']) def __init__(self, restart=None, ignore_bad_restart_file=Calculator._deprecated, label=os.curdir, atoms=None, form=None, **kwargs): self.form = form if 'potential' in kwargs: self.read_potential(kwargs['potential']) Calculator.__init__(self, restart, ignore_bad_restart_file, label, atoms, **kwargs) valid_args = ('potential', 'elements', 'header', 'drho', 'dr', 'cutoff', 'atomic_number', 'mass', 'a', 'lattice', 'embedded_energy', 'electron_density', 'phi', # derivatives 'd_embedded_energy', 'd_electron_density', 'd_phi', 'd', 'q', 'd_d', 'd_q', # adp terms 'skin', 'Z', 'nr', 'nrho', 'mass') # set any additional keyword arguments for arg, val in self.parameters.items(): if arg in valid_args: setattr(self, arg, val) else: raise RuntimeError('unknown keyword arg "%s" : not in %s' % (arg, valid_args)) def set_form(self, name): """set the form variable based on the file name suffix""" extension = os.path.splitext(name)[1] if extension == '.eam': self.form = 'eam' elif extension == '.alloy': self.form = 'alloy' elif extension == '.adp': self.form = 'adp' elif extension == '.fs': self.form = 'fs' else: raise RuntimeError('unknown file extension type: %s' % extension) def read_potential(self, filename): """Reads a LAMMPS EAM file in alloy or adp format and creates the interpolation functions from the data """ if isinstance(filename, str): with open(filename) as fd: self._read_potential(fd) else: fd = filename self._read_potential(fd) def _read_potential(self, fd): if self.form is None: self.set_form(fd.name) lines = fd.readlines() def lines_to_list(lines): """Make the data one long line so as not to care how its formatted """ data = [] for line in lines: data.extend(line.split()) return data if self.form == 'eam': # single element eam file (aka funcfl) self.header = lines[:1] data = lines_to_list(lines[1:]) # eam form is just like an alloy form for one element self.Nelements = 1 self.Z = np.array([data[0]], dtype=int) self.mass = np.array([data[1]]) self.a = np.array([data[2]]) self.lattice = [data[3]] self.nrho = int(data[4]) self.drho = float(data[5]) self.nr = int(data[6]) self.dr = float(data[7]) self.cutoff = float(data[8]) n = 9 + self.nrho self.embedded_data = np.array([np.float_(data[9:n])]) self.rphi_data = np.zeros([self.Nelements, self.Nelements, self.nr]) effective_charge = np.float_(data[n:n + self.nr]) # convert effective charges to rphi according to # http://lammps.sandia.gov/doc/pair_eam.html self.rphi_data[0, 0] = Bohr * Hartree * (effective_charge**2) self.density_data = np.array( [np.float_(data[n + self.nr:n + 2 * self.nr])]) elif self.form in ['alloy', 'adq']: self.header = lines[:3] i = 3 data = lines_to_list(lines[i:]) self.Nelements = int(data[0]) d = 1 self.elements = data[d:d + self.Nelements] d += self.Nelements self.nrho = int(data[d]) self.drho = float(data[d + 1]) self.nr = int(data[d + 2]) self.dr = float(data[d + 3]) self.cutoff = float(data[d + 4]) self.embedded_data = np.zeros([self.Nelements, self.nrho]) self.density_data = np.zeros([self.Nelements, self.nr]) self.Z = np.zeros([self.Nelements], dtype=int) self.mass = np.zeros([self.Nelements]) self.a = np.zeros([self.Nelements]) self.lattice = [] d += 5 # reads in the part of the eam file for each element for elem in range(self.Nelements): self.Z[elem] = int(data[d]) self.mass[elem] = float(data[d + 1]) self.a[elem] = float(data[d + 2]) self.lattice.append(data[d + 3]) d += 4 self.embedded_data[elem] = np.float_( data[d:(d + self.nrho)]) d += self.nrho self.density_data[elem] = np.float_(data[d:(d + self.nr)]) d += self.nr # reads in the r*phi data for each interaction between elements self.rphi_data = np.zeros([self.Nelements, self.Nelements, self.nr]) for i in range(self.Nelements): for j in range(i + 1): self.rphi_data[j, i] = np.float_(data[d:(d + self.nr)]) d += self.nr elif self.form == 'fs': self.header = lines[:3] i = 3 data = lines_to_list(lines[i:]) self.Nelements = int(data[0]) d = 1 self.elements = data[d:d + self.Nelements] d += self.Nelements self.nrho = int(data[d]) self.drho = float(data[d + 1]) self.nr = int(data[d + 2]) self.dr = float(data[d + 3]) self.cutoff = float(data[d + 4]) self.embedded_data = np.zeros([self.Nelements, self.nrho]) self.density_data = np.zeros([self.Nelements, self.Nelements, self.nr]) self.Z = np.zeros([self.Nelements], dtype=int) self.mass = np.zeros([self.Nelements]) self.a = np.zeros([self.Nelements]) self.lattice = [] d += 5 # reads in the part of the eam file for each element for elem in range(self.Nelements): self.Z[elem] = int(data[d]) self.mass[elem] = float(data[d + 1]) self.a[elem] = float(data[d + 2]) self.lattice.append(data[d + 3]) d += 4 self.embedded_data[elem] = np.float_( data[d:(d + self.nrho)]) d += self.nrho self.density_data[elem, :, :] = np.float_( data[d:(d + self.nr*self.Nelements)]).reshape([self.Nelements, self.nr]) d += self.nr*self.Nelements # reads in the r*phi data for each interaction between elements self.rphi_data = np.zeros([self.Nelements, self.Nelements, self.nr]) for i in range(self.Nelements): for j in range(i + 1): self.rphi_data[j, i] = np.float_(data[d:(d + self.nr)]) d += self.nr self.r = np.arange(0, self.nr) * self.dr self.rho = np.arange(0, self.nrho) * self.drho # choose the set_splines method according to the type if self.form == 'fs': self.set_fs_splines() else: self.set_splines() if (self.form == 'adp'): self.read_adp_data(data, d) self.set_adp_splines() def set_splines(self): # this section turns the file data into three functions (and # derivative functions) that define the potential self.embedded_energy = np.empty(self.Nelements, object) self.electron_density = np.empty(self.Nelements, object) self.d_embedded_energy = np.empty(self.Nelements, object) self.d_electron_density = np.empty(self.Nelements, object) for i in range(self.Nelements): self.embedded_energy[i] = spline(self.rho, self.embedded_data[i], k=3) self.electron_density[i] = spline(self.r, self.density_data[i], k=3) self.d_embedded_energy[i] = self.deriv(self.embedded_energy[i]) self.d_electron_density[i] = self.deriv(self.electron_density[i]) self.phi = np.empty([self.Nelements, self.Nelements], object) self.d_phi = np.empty([self.Nelements, self.Nelements], object) # ignore the first point of the phi data because it is forced # to go through zero due to the r*phi format in alloy and adp for i in range(self.Nelements): for j in range(i, self.Nelements): self.phi[i, j] = spline( self.r[1:], self.rphi_data[i, j][1:] / self.r[1:], k=3) self.d_phi[i, j] = self.deriv(self.phi[i, j]) if j != i: self.phi[j, i] = self.phi[i, j] self.d_phi[j, i] = self.d_phi[i, j] def set_fs_splines(self): self.embedded_energy = np.empty(self.Nelements, object) self.electron_density = np.empty( [self.Nelements, self.Nelements], object) self.d_embedded_energy = np.empty(self.Nelements, object) self.d_electron_density = np.empty( [self.Nelements, self.Nelements], object) for i in range(self.Nelements): self.embedded_energy[i] = spline(self.rho, self.embedded_data[i], k=3) self.d_embedded_energy[i] = self.deriv(self.embedded_energy[i]) for j in range(self.Nelements): self.electron_density[i, j] = spline( self.r, self.density_data[i, j], k=3) self.d_electron_density[i, j] = self.deriv( self.electron_density[i, j]) self.phi = np.empty([self.Nelements, self.Nelements], object) self.d_phi = np.empty([self.Nelements, self.Nelements], object) for i in range(self.Nelements): for j in range(i, self.Nelements): self.phi[i, j] = spline( self.r[1:], self.rphi_data[i, j][1:] / self.r[1:], k=3) self.d_phi[i, j] = self.deriv(self.phi[i, j]) if j != i: self.phi[j, i] = self.phi[i, j] self.d_phi[j, i] = self.d_phi[i, j] def set_adp_splines(self): self.d = np.empty([self.Nelements, self.Nelements], object) self.d_d = np.empty([self.Nelements, self.Nelements], object) self.q = np.empty([self.Nelements, self.Nelements], object) self.d_q = np.empty([self.Nelements, self.Nelements], object) for i in range(self.Nelements): for j in range(i, self.Nelements): self.d[i, j] = spline(self.r[1:], self.d_data[i, j][1:], k=3) self.d_d[i, j] = self.deriv(self.d[i, j]) self.q[i, j] = spline(self.r[1:], self.q_data[i, j][1:], k=3) self.d_q[i, j] = self.deriv(self.q[i, j]) # make symmetrical if j != i: self.d[j, i] = self.d[i, j] self.d_d[j, i] = self.d_d[i, j] self.q[j, i] = self.q[i, j] self.d_q[j, i] = self.d_q[i, j] def read_adp_data(self, data, d): """read in the extra adp data from the potential file""" self.d_data = np.zeros([self.Nelements, self.Nelements, self.nr]) # should be non symmetrical combinations of 2 for i in range(self.Nelements): for j in range(i + 1): self.d_data[j, i] = data[d:d + self.nr] d += self.nr self.q_data = np.zeros([self.Nelements, self.Nelements, self.nr]) # should be non symmetrical combinations of 2 for i in range(self.Nelements): for j in range(i + 1): self.q_data[j, i] = data[d:d + self.nr] d += self.nr def write_potential(self, filename, nc=1, numformat='%.8e'): """Writes out the potential in the format given by the form variable to 'filename' with a data format that is nc columns wide. Note: array lengths need to be an exact multiple of nc """ with open(filename, 'wb') as fd: self._write_potential(fd, nc=nc, numformat=numformat) def _write_potential(self, fd, nc, numformat): assert self.nr % nc == 0 assert self.nrho % nc == 0 for line in self.header: fd.write(line) fd.write('{0} '.format(self.Nelements).encode()) fd.write(' '.join(self.elements).encode() + b'\n') fd.write(('%d %f %d %f %f \n' % (self.nrho, self.drho, self.nr, self.dr, self.cutoff)).encode()) # start of each section for each element # rs = np.linspace(0, self.nr * self.dr, self.nr) # rhos = np.linspace(0, self.nrho * self.drho, self.nrho) rs = np.arange(0, self.nr) * self.dr rhos = np.arange(0, self.nrho) * self.drho for i in range(self.Nelements): fd.write(('%d %f %f %s\n' % (self.Z[i], self.mass[i], self.a[i], str(self.lattice[i]))).encode()) np.savetxt(fd, self.embedded_energy[i](rhos).reshape(self.nrho // nc, nc), fmt=nc * [numformat]) if self.form == 'fs': for j in range(self.Nelements): np.savetxt(fd, self.electron_density[i, j](rs).reshape( self.nr // nc, nc), fmt=nc * [numformat]) else: np.savetxt(fd, self.electron_density[i](rs).reshape( self.nr // nc, nc), fmt=nc * [numformat]) # write out the pair potentials in Lammps DYNAMO setfl format # as r*phi for alloy format for i in range(self.Nelements): for j in range(i, self.Nelements): np.savetxt(fd, (rs * self.phi[i, j](rs)).reshape(self.nr // nc, nc), fmt=nc * [numformat]) if self.form == 'adp': # these are the u(r) or dipole values for i in range(self.Nelements): for j in range(i + 1): np.savetxt(fd, self.d_data[i, j]) # these are the w(r) or quadrupole values for i in range(self.Nelements): for j in range(i + 1): np.savetxt(fd, self.q_data[i, j]) def update(self, atoms): # check all the elements are available in the potential self.Nelements = len(self.elements) elements = np.unique(atoms.get_chemical_symbols()) unavailable = np.logical_not( np.array([item in self.elements for item in elements])) if np.any(unavailable): raise RuntimeError('These elements are not in the potential: %s' % elements[unavailable]) # cutoffs need to be a vector for NeighborList cutoffs = self.cutoff * np.ones(len(atoms)) # convert the elements to an index of the position # in the eam format self.index = np.array([self.elements.index(el) for el in atoms.get_chemical_symbols()]) self.pbc = atoms.get_pbc() # since we need the contribution of all neighbors to the # local electron density we cannot just calculate and use # one way neighbors self.neighbors = NeighborList(cutoffs, skin=self.parameters.skin, self_interaction=False, bothways=True) self.neighbors.update(atoms) def calculate(self, atoms=None, properties=['energy'], system_changes=all_changes): """EAM Calculator atoms: Atoms object Contains positions, unit-cell, ... properties: list of str List of what needs to be calculated. Can be any combination of 'energy', 'forces' system_changes: list of str List of what has changed since last calculation. Can be any combination of these five: 'positions', 'numbers', 'cell', 'pbc', 'initial_charges' and 'initial_magmoms'. """ Calculator.calculate(self, atoms, properties, system_changes) # we shouldn't really recalc if charges or magmos change if len(system_changes) > 0: # something wrong with this way self.update(self.atoms) self.calculate_energy(self.atoms) if 'forces' in properties: self.calculate_forces(self.atoms) # check we have all the properties requested for property in properties: if property not in self.results: if property == 'energy': self.calculate_energy(self.atoms) if property == 'forces': self.calculate_forces(self.atoms) # we need to remember the previous state of parameters # if 'potential' in parameter_changes and potential != None: # self.read_potential(potential) def calculate_energy(self, atoms): """Calculate the energy the energy is made up of the ionic or pair interaction and the embedding energy of each atom into the electron cloud generated by its neighbors """ pair_energy = 0.0 embedding_energy = 0.0 mu_energy = 0.0 lam_energy = 0.0 trace_energy = 0.0 self.total_density = np.zeros(len(atoms)) if (self.form == 'adp'): self.mu = np.zeros([len(atoms), 3]) self.lam = np.zeros([len(atoms), 3, 3]) for i in range(len(atoms)): # this is the atom to be embedded neighbors, offsets = self.neighbors.get_neighbors(i) offset = np.dot(offsets, atoms.get_cell()) rvec = (atoms.positions[neighbors] + offset - atoms.positions[i]) # calculate the distance to the nearest neighbors r = np.sqrt(np.sum(np.square(rvec), axis=1)) # fast # r = np.apply_along_axis(np.linalg.norm, 1, rvec) # sloow nearest = np.arange(len(r))[r <= self.cutoff] for j_index in range(self.Nelements): use = self.index[neighbors[nearest]] == j_index if not use.any(): continue pair_energy += np.sum(self.phi[self.index[i], j_index]( r[nearest][use])) / 2. if self.form == 'fs': density = np.sum( self.electron_density[j_index, self.index[i]](r[nearest][use])) else: density = np.sum( self.electron_density[j_index](r[nearest][use])) self.total_density[i] += density if self.form == 'adp': self.mu[i] += self.adp_dipole( r[nearest][use], rvec[nearest][use], self.d[self.index[i], j_index]) self.lam[i] += self.adp_quadrupole( r[nearest][use], rvec[nearest][use], self.q[self.index[i], j_index]) # add in the electron embedding energy embedding_energy += self.embedded_energy[self.index[i]]( self.total_density[i]) components = dict(pair=pair_energy, embedding=embedding_energy) if self.form == 'adp': mu_energy += np.sum(self.mu ** 2) / 2. lam_energy += np.sum(self.lam ** 2) / 2. for i in range(len(atoms)): # this is the atom to be embedded trace_energy -= np.sum(self.lam[i].trace() ** 2) / 6. adp_result = dict(adp_mu=mu_energy, adp_lam=lam_energy, adp_trace=trace_energy) components.update(adp_result) self.positions = atoms.positions.copy() self.cell = atoms.get_cell().copy() energy = 0.0 for i in components.keys(): energy += components[i] self.energy_free = energy self.energy_zero = energy self.results['energy_components'] = components self.results['energy'] = energy def calculate_forces(self, atoms): # calculate the forces based on derivatives of the three EAM functions self.update(atoms) self.results['forces'] = np.zeros((len(atoms), 3)) for i in range(len(atoms)): # this is the atom to be embedded neighbors, offsets = self.neighbors.get_neighbors(i) offset = np.dot(offsets, atoms.get_cell()) # create a vector of relative positions of neighbors rvec = atoms.positions[neighbors] + offset - atoms.positions[i] r = np.sqrt(np.sum(np.square(rvec), axis=1)) nearest = np.arange(len(r))[r < self.cutoff] d_embedded_energy_i = self.d_embedded_energy[ self.index[i]](self.total_density[i]) urvec = rvec.copy() # unit directional vector for j in np.arange(len(neighbors)): urvec[j] = urvec[j] / r[j] for j_index in range(self.Nelements): use = self.index[neighbors[nearest]] == j_index if not use.any(): continue rnuse = r[nearest][use] density_j = self.total_density[neighbors[nearest][use]] if self.form == 'fs': scale = (self.d_phi[self.index[i], j_index](rnuse) + (d_embedded_energy_i * self.d_electron_density[j_index, self.index[i]](rnuse)) + (self.d_embedded_energy[j_index](density_j) * self.d_electron_density[self.index[i], j_index](rnuse))) else: scale = (self.d_phi[self.index[i], j_index](rnuse) + (d_embedded_energy_i * self.d_electron_density[j_index](rnuse)) + (self.d_embedded_energy[j_index](density_j) * self.d_electron_density[self.index[i]](rnuse))) self.results['forces'][i] += np.dot(scale, urvec[nearest][use]) if (self.form == 'adp'): adp_forces = self.angular_forces( self.mu[i], self.mu[neighbors[nearest][use]], self.lam[i], self.lam[neighbors[nearest][use]], rnuse, rvec[nearest][use], self.index[i], j_index) self.results['forces'][i] += adp_forces def angular_forces(self, mu_i, mu, lam_i, lam, r, rvec, form1, form2): # calculate the extra components for the adp forces # rvec are the relative positions to atom i psi = np.zeros(mu.shape) for gamma in range(3): term1 = (mu_i[gamma] - mu[:, gamma]) * self.d[form1][form2](r) term2 = np.sum((mu_i - mu) * self.d_d[form1][form2](r)[:, np.newaxis] * (rvec * rvec[:, gamma][:, np.newaxis] / r[:, np.newaxis]), axis=1) term3 = 2 * np.sum((lam_i[:, gamma] + lam[:, :, gamma]) * rvec * self.q[form1][form2](r)[:, np.newaxis], axis=1) term4 = 0.0 for alpha in range(3): for beta in range(3): rs = rvec[:, alpha] * rvec[:, beta] * rvec[:, gamma] term4 += ((lam_i[alpha, beta] + lam[:, alpha, beta]) * self.d_q[form1][form2](r) * rs) / r term5 = ((lam_i.trace() + lam.trace(axis1=1, axis2=2)) * (self.d_q[form1][form2](r) * r + 2 * self.q[form1][form2](r)) * rvec[:, gamma]) / 3. # the minus for term5 is a correction on the adp # formulation given in the 2005 Mishin Paper and is posted # on the NIST website with the AlH potential psi[:, gamma] = term1 + term2 + term3 + term4 - term5 return np.sum(psi, axis=0) def adp_dipole(self, r, rvec, d): # calculate the dipole contribution mu = np.sum((rvec * d(r)[:, np.newaxis]), axis=0) return mu # sign to agree with lammps def adp_quadrupole(self, r, rvec, q): # slow way of calculating the quadrupole contribution r = np.sqrt(np.sum(rvec ** 2, axis=1)) lam = np.zeros([rvec.shape[0], 3, 3]) qr = q(r) for alpha in range(3): for beta in range(3): lam[:, alpha, beta] += qr * rvec[:, alpha] * rvec[:, beta] return np.sum(lam, axis=0) def deriv(self, spline): """Wrapper for extracting the derivative from a spline""" def d_spline(aspline): return spline(aspline, 1) return d_spline def plot(self, name=''): """Plot the individual curves""" import matplotlib.pyplot as plt if self.form == 'eam' or self.form == 'alloy' or self.form == 'fs': nrow = 2 elif self.form == 'adp': nrow = 3 else: raise RuntimeError('Unknown form of potential: %s' % self.form) if hasattr(self, 'r'): r = self.r else: r = np.linspace(0, self.cutoff, 50) if hasattr(self, 'rho'): rho = self.rho else: rho = np.linspace(0, 10.0, 50) plt.subplot(nrow, 2, 1) self.elem_subplot(rho, self.embedded_energy, r'$\rho$', r'Embedding Energy $F(\bar\rho)$', name, plt) plt.subplot(nrow, 2, 2) if self.form == 'fs': self.multielem_subplot(r, self.electron_density, r'$r$', r'Electron Density $\rho(r)$', name, plt, half=False) else: self.elem_subplot(r, self.electron_density, r'$r$', r'Electron Density $\rho(r)$', name, plt) plt.subplot(nrow, 2, 3) self.multielem_subplot(r, self.phi, r'$r$', r'Pair Potential $\phi(r)$', name, plt) plt.ylim(-1.0, 1.0) # need reasonable values if self.form == 'adp': plt.subplot(nrow, 2, 5) self.multielem_subplot(r, self.d, r'$r$', r'Dipole Energy', name, plt) plt.subplot(nrow, 2, 6) self.multielem_subplot(r, self.q, r'$r$', r'Quadrupole Energy', name, plt) plt.plot() def elem_subplot(self, curvex, curvey, xlabel, ylabel, name, plt): plt.xlabel(xlabel) plt.ylabel(ylabel) for i in np.arange(self.Nelements): label = name + ' ' + self.elements[i] plt.plot(curvex, curvey[i](curvex), label=label) plt.legend() def multielem_subplot(self, curvex, curvey, xlabel, ylabel, name, plt, half=True): plt.xlabel(xlabel) plt.ylabel(ylabel) for i in np.arange(self.Nelements): for j in np.arange((i + 1) if half else self.Nelements): label = name + ' ' + self.elements[i] + '-' + self.elements[j] plt.plot(curvex, curvey[i, j](curvex), label=label) plt.legend()