Source code for pymatgen.analysis.path_finder

"""
This module finds diffusion paths through a structure based on a given
potential field.

If you use PathFinder algorithm for your research, please consider citing the
following work::

    Ziqin Rong, Daniil Kitchaev, Pieremanuele Canepa, Wenxuan Huang, Gerbrand
    Ceder, The Journal of Chemical Physics 145 (7), 074112
"""

from abc import ABCMeta
import math
import logging
from scipy.interpolate import interp1d
import scipy.signal
import scipy.stats
import numpy as np
import numpy.linalg as la


from pymatgen.core.structure import Structure
from pymatgen.core.sites import PeriodicSite
from pymatgen.io.vasp.inputs import Poscar

__author__ = "Daniil Kitchaev"
__version__ = "1.0"
__maintainer__ = "Daniil Kitchaev, Ziqin Rong"
__email__ = "dkitch@mit.edu, rongzq08@mit.edu"
__status__ = "Development"
__date__ = "March 17, 2015"

logger = logging.getLogger(__name__)


[docs]class NEBPathfinder: """ General pathfinder for interpolating between two structures, where the interpolating path is calculated with the elastic band method with respect to the given static potential for sites whose indices are given in relax_sites, and is linear otherwise. """ def __init__(self, start_struct, end_struct, relax_sites, v, n_images=20, mid_struct=None): """ Args: start_struct, end_struct: Endpoint structures to interpolate relax_sites: List of site indices whose interpolation paths should be relaxed v: Static potential field to use for the elastic band relaxation n_images: Number of interpolation images to generate mid_struct: (optional) additional structure between the start and end structures to help """ self.__s1 = start_struct self.__s2 = end_struct self.__mid = mid_struct self.__relax_sites = relax_sites self.__v = v self.__n_images = n_images self.__images = None self.interpolate()
[docs] def interpolate(self): """ Finds a set of n_images from self.s1 to self.s2, where all sites except for the ones given in relax_sites, the interpolation is linear (as in pymatgen.core.structure.interpolate), and for the site indices given in relax_sites, the path is relaxed by the elastic band method within the static potential V. If a mid point is defined we will interpolate from s1--> mid -->s2 The final number of images will still be n_images. """ if self.__mid is not None: # to make arithmatic easier we will do the interpolation in two parts with n images each # then just take every other image at the end, this results in exactly n images images_0 = self.__s1.interpolate(self.__mid, nimages=self.__n_images, interpolate_lattices=False)[:-1] images_1 = self.__mid.interpolate(self.__s2, nimages=self.__n_images, interpolate_lattices=False) images = images_0 + images_1 images = images[::2] else: images = self.__s1.interpolate(self.__s2, nimages=self.__n_images, interpolate_lattices=False) for site_i in self.__relax_sites: start_f = images[0].sites[site_i].frac_coords end_f = images[-1].sites[site_i].frac_coords path = NEBPathfinder.string_relax( NEBPathfinder.__f2d(start_f, self.__v), NEBPathfinder.__f2d(end_f, self.__v), self.__v, n_images=(self.__n_images + 1), dr=[ self.__s1.lattice.a / self.__v.shape[0], self.__s1.lattice.b / self.__v.shape[1], self.__s1.lattice.c / self.__v.shape[2] ]) for image_i, image in enumerate(images): image.translate_sites( site_i, NEBPathfinder.__d2f(path[image_i], self.__v) - image.sites[site_i].frac_coords, frac_coords=True, to_unit_cell=True) self.__images = images
@property def images(self): """ Returns a list of structures interpolating between the start and endpoint structures. """ return self.__images
[docs] def plot_images(self, outfile): """ Generates a POSCAR with the calculated diffusion path with respect to the first endpoint. :param outfile: Output file for the POSCAR """ sum_struct = self.__images[0].sites for image in self.__images: for site_i in self.__relax_sites: sum_struct.append( PeriodicSite(image.sites[site_i].specie, image.sites[site_i].frac_coords, self.__images[0].lattice, to_unit_cell=True, coords_are_cartesian=False)) sum_struct = Structure.from_sites(sum_struct, validate_proximity=False) p = Poscar(sum_struct) p.write_file(outfile)
[docs] @staticmethod def string_relax(start, end, V, n_images=25, dr=None, h=3.0, k=0.17, min_iter=100, max_iter=10000, max_tol=5e-6): """ Implements path relaxation via the elastic band method. In general, the method is to define a path by a set of points (images) connected with bands with some elasticity constant k. The images then relax along the forces found in the potential field V, counterbalanced by the elastic response of the elastic band. In general the endpoints of the band can be allowed to relax also to their local minima, but in this calculation they are kept fixed. Args: start, end: Endpoints of the path calculation given in discrete coordinates with respect to the grid in V V: potential field through which to calculate the path n_images: number of images used to define the path. In general anywhere from 20 to 40 seems to be good. dr: Conversion ratio from discrete coordinates to real coordinates for each of the three coordinate vectors h: Step size for the relaxation. h = 0.1 works reliably, but is slow. h=10 diverges with large gradients but for the types of gradients seen in CHGCARs, works pretty reliably k: Elastic constant for the band (in real units, not discrete) min_iter, max_iter: Number of optimization steps the string will take before exiting (even if unconverged) max_tol: Convergence threshold such that if the string moves by less than max_tol in a step, and at least min_iter steps have passed, the algorithm will terminate. Depends strongly on the size of the gradients in V, but 5e-6 works reasonably well for CHGCARs. """ # # This code is based on the MATLAB example provided by # Prof. Eric Vanden-Eijnden of NYU # (http://www.cims.nyu.edu/~eve2/main.htm) # # logger.debug("Getting path from {} to {} (coords wrt V grid)".format(start, end)) # Set parameters if not dr: dr = np.array( [1.0 / V.shape[0], 1.0 / V.shape[1], 1.0 / V.shape[2]]) else: dr = np.array(dr, dtype=float) keff = k * dr * n_images h0 = h # Initialize string g1 = np.linspace(0, 1, n_images) s0 = start s1 = end s = np.array([g * (s1 - s0) for g in g1]) + s0 ds = s - np.roll(s, 1, axis=0) ds[0] = (ds[0] - ds[0]) ls = np.cumsum(la.norm(ds, axis=1)) ls = ls / ls[-1] fi = interp1d(ls, s, axis=0) s = fi(g1) # Evaluate initial distances (for elastic equilibrium) ds0_plus = s - np.roll(s, 1, axis=0) ds0_minus = s - np.roll(s, -1, axis=0) ds0_plus[0] = (ds0_plus[0] - ds0_plus[0]) ds0_minus[-1] = (ds0_minus[-1] - ds0_minus[-1]) # Evaluate potential gradient outside the loop, as potential does not # change per step in this approximation. dV = np.gradient(V) # Evolve string for step in range(0, max_iter): if step > min_iter: # Gradually decay step size to prevent oscillations h = h0 * np.exp(-2.0 * (step - min_iter) / max_iter) else: h = h0 # Calculate forces acting on string d = V.shape s0 = s edV = np.array([[ dV[0][int(pt[0]) % d[0]][int(pt[1]) % d[1]][int(pt[2]) % d[2]] / dr[0], dV[1][int(pt[0]) % d[0]][int(pt[1]) % d[1]][int(pt[2]) % d[2]] / dr[0], dV[2][int(pt[0]) % d[0]][int(pt[1]) % d[1]][int(pt[2]) % d[2]] / dr[0] ] for pt in s]) # if(step % 100 == 0): # logger.debug(edV) # Update according to force due to potential and string elasticity ds_plus = s - np.roll(s, 1, axis=0) ds_minus = s - np.roll(s, -1, axis=0) ds_plus[0] = (ds_plus[0] - ds_plus[0]) ds_minus[-1] = (ds_minus[-1] - ds_minus[-1]) Fpot = edV Fel = keff * (la.norm(ds_plus) - la.norm(ds0_plus)) * (ds_plus / la.norm(ds_plus)) Fel += keff * (la.norm(ds_minus) - la.norm(ds0_minus)) * (ds_minus / la.norm(ds_minus)) s -= h * (Fpot + Fel) # Fix endpoints s[0] = s0[0] s[-1] = s0[-1] # Reparametrize string ds = s - np.roll(s, 1, axis=0) ds[0] = (ds[0] - ds[0]) ls = np.cumsum(la.norm(ds, axis=1)) ls = ls / ls[-1] fi = interp1d(ls, s, axis=0) s = fi(g1) tol = la.norm((s - s0) * dr) / n_images / h if tol > 1e10: raise ValueError( "Pathfinding failed, path diverged! Consider reducing h to " "avoid divergence.") if step > min_iter and tol < max_tol: logger.debug("Converged at step {}".format(step)) break if step % 100 == 0: logger.debug("Step {} - ds = {}".format(step, tol)) return s
@staticmethod def __f2d(frac_coords, v): """ Converts fractional coordinates to discrete coordinates with respect to the grid size of v """ # frac_coords = frac_coords % 1 return np.array([ int(frac_coords[0] * v.shape[0]), int(frac_coords[1] * v.shape[1]), int(frac_coords[2] * v.shape[2]) ]) @staticmethod def __d2f(disc_coords, v): """ Converts a point given in discrete coordinates withe respect to the grid in v to fractional coordinates. """ return np.array([ disc_coords[0] / v.shape[0], disc_coords[1] / v.shape[1], disc_coords[2] / v.shape[2] ])
[docs]class StaticPotential(metaclass=ABCMeta): """ Defines a general static potential for diffusion calculations. Implements grid-rescaling and smearing for the potential grid. Also provides a function to normalize the potential from 0 to 1 (recommended). """ def __init__(self, struct, pot): """ :param struct: atomic structure of the potential :param pot: volumentric data to be used as a potential """ self.__v = pot self.__s = struct
[docs] def get_v(self): """ Returns the potential """ return self.__v
[docs] def normalize(self): """ Sets the potential range 0 to 1. """ self.__v = self.__v - np.amin(self.__v) self.__v = self.__v / np.amax(self.__v)
[docs] def rescale_field(self, new_dim): """ Changes the discretization of the potential field by linear interpolation. This is necessary if the potential field obtained from DFT is strangely skewed, or is too fine or coarse. Obeys periodic boundary conditions at the edges of the cell. Alternatively useful for mixing potentials that originally are on different grids. :param new_dim: tuple giving the numpy shape of the new grid """ v_dim = self.__v.shape padded_v = np.lib.pad(self.__v, ((0, 1), (0, 1), (0, 1)), mode='wrap') ogrid_list = np.array([ list(c) for c in list(np.ndindex(v_dim[0] + 1, v_dim[1] + 1, v_dim[2] + 1)) ]) v_ogrid = padded_v.reshape( ((v_dim[0] + 1) * (v_dim[1] + 1) * (v_dim[2] + 1), -1)) ngrid_a, ngrid_b, ngrid_c = (np.mgrid[0:v_dim[0]:v_dim[0] / new_dim[0], 0:v_dim[1]:v_dim[1] / new_dim[1], 0:v_dim[2]:v_dim[2] / new_dim[2]]) v_ngrid = scipy.interpolate.griddata(ogrid_list, v_ogrid, (ngrid_a, ngrid_b, ngrid_c), method='linear').reshape( (new_dim[0], new_dim[1], new_dim[2])) self.__v = v_ngrid
[docs] def gaussian_smear(self, r): """ Applies an isotropic Gaussian smear of width (standard deviation) r to the potential field. This is necessary to avoid finding paths through narrow minima or nodes that may exist in the field (although any potential or charge distribution generated from GGA should be relatively smooth anyway). The smearing obeys periodic boundary conditions at the edges of the cell. :param r - Smearing width in cartesian coordinates, in the same units as the structure lattice vectors """ # Since scaling factor in fractional coords is not isotropic, have to # have different radii in 3 directions a_lat = self.__s.lattice.a b_lat = self.__s.lattice.b c_lat = self.__s.lattice.c # Conversion factors for discretization of v v_dim = self.__v.shape r_frac = (r / a_lat, r / b_lat, r / c_lat) r_disc = (int(math.ceil(r_frac[0] * v_dim[0])), int(math.ceil(r_frac[1] * v_dim[1])), int(math.ceil(r_frac[2] * v_dim[2]))) # Apply smearing # Gaussian filter gauss_dist = np.zeros( (r_disc[0] * 4 + 1, r_disc[1] * 4 + 1, r_disc[2] * 4 + 1)) for g_a in np.arange(-2.0 * r_disc[0], 2.0 * r_disc[0] + 1, 1.0): for g_b in np.arange(-2.0 * r_disc[1], 2.0 * r_disc[1] + 1, 1.0): for g_c in np.arange(-2.0 * r_disc[2], 2.0 * r_disc[2] + 1, 1.0): g = np.array( [g_a / v_dim[0], g_b / v_dim[1], g_c / v_dim[2]]).T gauss_dist[int(g_a + r_disc[0])][int(g_b + r_disc[1])][int( g_c + r_disc[2])] = la.norm( np.dot(self.__s.lattice.matrix, g)) / r gauss = scipy.stats.norm.pdf(gauss_dist) gauss = gauss / np.sum(gauss, dtype=float) padded_v = np.pad(self.__v, ((r_disc[0], r_disc[0]), (r_disc[1], r_disc[1]), (r_disc[2], r_disc[2])), mode='wrap') smeared_v = scipy.signal.convolve(padded_v, gauss, mode='valid') self.__v = smeared_v
[docs]class ChgcarPotential(StaticPotential): """ Implements a potential field based on the charge density output from VASP. """ def __init__(self, chgcar, smear=False, normalize=True): """ :param chgcar: Chgcar object based on a VASP run of the structure of interest (Chgcar.from_file("CHGCAR")) :param smear: Whether or not to apply a Gaussian smearing to the potential :param normalize: Whether or not to normalize the potential to range from 0 to 1 """ v = chgcar.data['total'] v = v / (v.shape[0] * v.shape[1] * v.shape[2]) StaticPotential.__init__(self, chgcar.structure, v) if smear: self.gaussian_smear(2.0) if normalize: self.normalize()
[docs]class FreeVolumePotential(StaticPotential): """ Implements a potential field based on geometric distances from atoms in the structure - basically, the potential is lower at points farther away from any atoms in the structure. """ def __init__(self, struct, dim, smear=False, normalize=True): """ :param struct: Unit cell on which to base the potential :param dim: Grid size for the potential :param smear: Whether or not to apply a Gaussian smearing to the potential :param normalize: Whether or not to normalize the potential to range from 0 to 1 """ self.__s = struct v = FreeVolumePotential.__add_gaussians(struct, dim) StaticPotential.__init__(self, struct, v) if smear: self.gaussian_smear(2.0) if normalize: self.normalize() @staticmethod def __add_gaussians(s, dim, r=1.5): gauss_dist = np.zeros(dim) for a_d in np.arange(0.0, dim[0], 1.0): for b_d in np.arange(0.0, dim[1], 1.0): for c_d in np.arange(0.0, dim[2], 1.0): coords_f = np.array( [a_d / dim[0], b_d / dim[1], c_d / dim[2]]) d_f = sorted(s.get_sites_in_sphere(coords_f, s.lattice.a), key=lambda x: x[1])[0][1] # logger.debug(d_f) gauss_dist[int(a_d)][int(b_d)][int(c_d)] = d_f / r v = scipy.stats.norm.pdf(gauss_dist) return v
[docs]class MixedPotential(StaticPotential): """ Implements a potential that is a weighted sum of some other potentials """ def __init__(self, potentials, coefficients, smear=False, normalize=True): """ Args: potentials: List of objects extending the StaticPotential superclass coefficients: Mixing weights for the elements of the potentials list smear: Whether or not to apply a Gaussian smearing to the potential normalize: Whether or not to normalize the potential to range from 0 to 1 """ v = potentials[0].get_v() * coefficients[0] s = potentials[0].__s for i in range(1, len(potentials)): v += potentials[i].get_v() * coefficients[i] StaticPotential.__init__(self, s, v) if smear: self.gaussian_smear(2.0) if normalize: self.normalize()