Source code for

# flake8: noqa
import numpy as np

[docs]class DiffusionCoefficient: def __init__(self, traj, timestep, atom_indices=None, molecule=False): """ This class calculates the Diffusion Coefficient for the given Trajectory using the Einstein Equation: ..math:: \\left \\langle \\left | r(t) - r(0) \\right | ^{2} \\right \\rangle = 2nDt where r(t) is the position of atom at time t, n is the degrees of freedom and D is the Diffusion Coefficient Solved herein by fitting with :math:`y = mx + c`, i.e. :math:`\\frac{1}{2n} \\left \\langle \\left | r(t) - r(0) \\right | ^{2} \\right \\rangle = Dt`, with m = D and c = 0 wiki : Parameters: traj (Trajectory): Trajectory of atoms objects (images) timestep (Float): Timestep between *each image in the trajectory*, in ASE timestep units (For an MD simulation with timestep of N, and images written every M iterations, our timestep here is N * M) atom_indices (List of Int): The indices of atoms whose Diffusion Coefficient is to be calculated explicitly molecule (Boolean) Indicate if we are studying a molecule instead of atoms, therefore use centre of mass in calculations """ self.traj = traj self.timestep = timestep # Condition used if user wants to calculate diffusion coefficients for specific atoms or all atoms self.atom_indices = atom_indices if self.atom_indices == None: self.atom_indices = [i for i in range(len(traj[0]))] # Condition if we are working with the mobility of a molecule, need to manage arrays slightly differently self.is_molecule = molecule if self.is_molecule: self.types_of_atoms = ["molecule"] self.no_of_atoms = [1] else: self.types_of_atoms = sorted(set(traj[0].symbols[self.atom_indices])) self.no_of_atoms = [traj[0].get_chemical_symbols().count(symbol) for symbol in self.types_of_atoms] # Dummy initialisation for important results data object self._slopes = [] @property def no_of_types_of_atoms(self): """ Dynamically returns the number of different atoms in the system """ return len(self.types_of_atoms) @property def slopes(self): """ Method to return slopes fitted to datapoints. If undefined, calculate slopes """ if len(self._slopes) == 0: self.calculate() return self._slopes @slopes.setter def slopes(self, values): """ Method to set slopes as fitted to datapoints """ self._slopes = values def _initialise_arrays(self, ignore_n_images, number_of_segments): """ Private function to initialise data storage objects. This includes objects to store the total timesteps sampled, the average diffusivity for species in any given segment, and objects to store gradient and intercept from fitting. Parameters: ignore_n_images (Int): Number of images you want to ignore from the start of the trajectory, e.g. during equilibration number_of_segments (Int): Divides the given trajectory in to segments to allow statistical analysis """ total_images = len(self.traj) - ignore_n_images self.no_of_segments = number_of_segments self.len_segments = total_images // self.no_of_segments # These are the data objects we need when plotting information. First the x-axis, timesteps self.timesteps = np.linspace(0,total_images*self.timestep,total_images+1) # This holds all the data points for the diffusion coefficients, averaged over atoms self.xyz_segment_ensemble_average = np.zeros((self.no_of_segments,self.no_of_types_of_atoms,3,self.len_segments)) # This holds all the information on linear fits, from which we get the diffusion coefficients self.slopes = np.zeros((self.no_of_types_of_atoms,self.no_of_segments,3)) self.intercepts = np.zeros((self.no_of_types_of_atoms,self.no_of_segments,3)) self.cont_xyz_segment_ensemble_average = 0 def calculate(self, ignore_n_images=0, number_of_segments=1): """ Calculate the diffusion coefficients, using the previously supplied data. The user can break the data into segments and take the average over these trajectories, therefore allowing statistical analysis and derivation of standard deviations. Option is also provided to ignore initial images if data is perhaps unequilibrated initially. Parameters: ignore_n_images (Int): Number of images you want to ignore from the start of the trajectory, e.g. during equilibration number_of_segments (Int): Divides the given trajectory in to segments to allow statistical analysis """ # Setup all the arrays we need to store information self._initialise_arrays(ignore_n_images, number_of_segments) for segment_no in range(self.no_of_segments): start = segment_no*self.len_segments end = start + self.len_segments seg = self.traj[ignore_n_images+start:ignore_n_images+end] # If we are considering a molecular system, work out the COM for the starting structure if self.is_molecule: com_orig = np.zeros(3) for atom_no in self.atom_indices: com_orig += seg[0].positions[atom_no] / len(self.atom_indices) # For each image, calculate displacement. # I spent some time deciding if this should run from 0 or 1, as the displacement will be zero for # t = 0, but this is a data point that needs fitting too and so should be included for image_no in range(0,len(seg)): # This object collects the xyz displacements for all atom species in the image xyz_disp = np.zeros((self.no_of_types_of_atoms,3)) # Calculating for each atom individually, grouping by species type (e.g. solid state) if not self.is_molecule: # For each atom, work out displacement from start coordinate and collect information with like atoms for atom_no in self.atom_indices: sym_index = self.types_of_atoms.index(seg[image_no].symbols[atom_no]) xyz_disp[sym_index] += np.square(seg[image_no].positions[atom_no] - seg[0].positions[atom_no]) else: # Calculating for group of atoms (molecule) and work out squared displacement com_disp = np.zeros(3) for atom_no in self.atom_indices: com_disp += seg[image_no].positions[atom_no] / len(self.atom_indices) xyz_disp[0] += np.square(com_disp - com_orig) # For each atom species or molecule, use xyz_disp to calculate the average data for sym_index in range(self.no_of_types_of_atoms): # Normalise by degrees of freedom and average overall atoms for each axes over entire segment denominator = (2*self.no_of_atoms[sym_index]) for xyz in range(3): self.xyz_segment_ensemble_average[segment_no][sym_index][xyz][image_no] = (xyz_disp[sym_index][xyz]/denominator) # We've collected all the data for this entire segment, so now to fit the data. for sym_index in range(self.no_of_types_of_atoms): self.slopes[sym_index][segment_no], self.intercepts[sym_index][segment_no] = self._fit_data(self.timesteps[start:end], self.xyz_segment_ensemble_average[segment_no][sym_index]) def _fit_data(self, x, y): """ Private function that returns slope and intercept for linear fit to mean square diffusion data Parameters: x (Array of floats): Linear list of timesteps in the calculation y (Array of floats): Mean square displacement as a function of time. """ # Simpler implementation but disabled as fails Conda tests. # from scipy.stats import linregress # slope, intercept, r_value, p_value, std_err = linregress(x,y) # Initialise objects slopes = np.zeros(3) intercepts = np.zeros(3) # Convert into suitable format for lstsq x_edited = np.vstack([np.array(x), np.ones(len(x))]).T # Calculate slopes for x, y and z-axes for xyz in range(3): slopes[xyz], intercepts[xyz] = np.linalg.lstsq(x_edited, y[xyz], rcond=-1)[0] return slopes, intercepts def get_diffusion_coefficients(self): """ Returns diffusion coefficients for atoms (in alphabetical order) along with standard deviation. All data is currently passed out in units of Å^2/<ASE time units> To convert into Å^2/fs => multiply by ase.units.fs To convert from Å^2/fs to cm^2/s => multiply by (10^-8)^2 / 10^-15 = 10^-1 """ slopes = [np.mean(self.slopes[sym_index]) for sym_index in range(self.no_of_types_of_atoms)] std = [np.std(self.slopes[sym_index]) for sym_index in range(self.no_of_types_of_atoms)] return slopes, std def plot(self, ax=None, show=False): """ Auto-plot of Diffusion Coefficient data. Provides basic framework for visualising analysis. Parameters: ax (Matplotlib.axes.Axes) Axes object on to which plot can be created show (Boolean) Whether or not to show the created plot. Default: False """ # Necessary if user hasn't supplied an axis. import matplotlib.pyplot as plt # Convert from ASE time units to fs (aesthetic) from ase.units import fs as fs_conversion if ax is None: ax = plt.gca() # Define some aesthetic variables color_list =, 1, self.no_of_types_of_atoms)) xyz_labels=['X','Y','Z'] xyz_markers = ['o','s','^'] # Create an x-axis that is in a more intuitive format for the view graph_timesteps = self.timesteps / fs_conversion for segment_no in range(self.no_of_segments): start = segment_no*self.len_segments end = start + self.len_segments label = None for sym_index in range(self.no_of_types_of_atoms): for xyz in range(3): if segment_no == 0: label = 'Species: %s (%s)'%(self.types_of_atoms[sym_index], xyz_labels[xyz]) # Add scatter graph for the mean square displacement data in this segment ax.scatter(graph_timesteps[start:end], self.xyz_segment_ensemble_average[segment_no][sym_index][xyz], color=color_list[sym_index], marker=xyz_markers[xyz], label=label, linewidth=1, edgecolor='grey') # Print the line of best fit for segment line = np.mean(self.slopes[sym_index][segment_no])*fs_conversion*graph_timesteps[start:end]+np.mean(self.intercepts[sym_index][segment_no]) if segment_no == 0: label = 'Segment Mean : %s'%(self.types_of_atoms[sym_index]) ax.plot(graph_timesteps[start:end], line, color='C%d'%(sym_index), label=label, linestyle='--') # Plot separator at end of segment x_coord = graph_timesteps[end-1] ax.plot([x_coord, x_coord],[-0.001, 1.05*np.amax(self.xyz_segment_ensemble_average)], color='grey', linestyle=":") # Plot the overall mean (average of slopes) for each atom species # This only makes sense if the data is all plotted on the same x-axis timeframe, which currently we are not - everything is plotted sequentially #for sym_index in range(self.no_of_types_of_atoms): # line = np.mean(self.slopes[sym_index])*graph_timesteps+np.mean(self.intercepts[sym_index]) # label ='Mean, Total : %s'%(self.types_of_atoms[sym_index]) # ax.plot(graph_timesteps, line, color='C%d'%(sym_index), label=label, linestyle="-") # Aesthetic parts of the plot ax.set_ylim(-0.001, 1.05*np.amax(self.xyz_segment_ensemble_average)) ax.legend(loc='best') ax.set_xlabel('Time (fs)') ax.set_ylabel(r'Mean Square Displacement ($\AA^2$)') if show: def print_data(self): """ Output of statistical analysis for Diffusion Coefficient data. Provides basic framework for understanding calculation. """ from ase.units import fs as fs_conversion # Collect statistical data for diffusion coefficient over all segments slopes, std = self.get_diffusion_coefficients() # Useful notes for any consideration of conversion. # Converting gradient from Å^2/fs to more common units of cm^2/s => multiplying by (10^-8)^2 / 10^-15 = 10^-1 # Converting intercept from Å^2 to more common units of cm^2 => multiply by (10^-8)^2 = 10^-16 # # Note currently in ASE internal time units # Converting into fs => divide by 1/(fs_conversion) => multiply by (fs_conversion) # Print data for each atom, in each segment. for sym_index in range(self.no_of_types_of_atoms): print('---') print(r'Species: %4s' % self.types_of_atoms[sym_index]) print('---') for segment_no in range(self.no_of_segments): print(r'Segment %3d: Diffusion Coefficient = %.10f Å^2/fs; Intercept = %.10f Å^2;' % (segment_no, np.mean(self.slopes[sym_index][segment_no])*fs_conversion, np.mean(self.intercepts[sym_index][segment_no]))) # Print average overall data. print('---') for sym_index in range(self.no_of_types_of_atoms): print('Mean Diffusion Coefficient (X, Y and Z) : %s = %.10f Å^2/fs; Std. Dev. = %.10f Å^2/fs' % (self.types_of_atoms[sym_index], slopes[sym_index]*fs_conversion, std[sym_index]*fs_conversion)) print('---')