Many Body Tensor Representation

MBTR is a global descriptor for a molecule/unit cell. It eliminates rotational, translational, and permutation variances by gathering information about different configurations of $K$ atoms into tensors that are stratified by the involved chemical elements. All element combinations have an associated gaussian-smeared exponentially-weighted histogram.

The Tensor

The tensor comprises of combinations of elements in different numbers. So, K1 is 1 element, K2 is 2 elements, and so on. These K's represent different expression of the molecule/unit-cell.

K1

K1 represents the gaussian-smeared histogram for counts of each element type. So, in essense it is a matrix of size MxN, where M is the number of elements, and N is the number of bins.

K2

K2 represents the gaussian-smeared exponentially-weighted histogram inverse distances of pairs of elements. So, this becomes a tensor of size MxMxN, where M is the number of elements, and N is the number of bins.

K3

K3 represents the gaussian-smeared exponentially-weighted histogram angles between triplets of elements. So, this becomes a tensor of size MxMxMxN, where M is the number of elements, and N is the number of bins.

Note: the dscribe package has implementation of MBTR up to K3

K4

K4 represents the gaussian-smeared exponantially-weighted histogram di-hedral angles between quadruplets of elements. So, this becomes a tensor of size MxMxMxMxN, where M is the number of elements, and N is the number of bins.

Weighting

All the tensors, but K1, are weighted. This ensures that contributions from nearby atoms is higher, than from farther ones.

For more info about MBTR see: Huo, Haoyan, and Matthias Rupp. arXiv preprint arXiv:1704.06439 (2017)

For calculating MBTR, we use the DScribe package as developed by Surfaces and Interfaces at the Nanoscale, Aalto

Example

We are going to see MBTR in action for a simple NaCl system.



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# --- INITIAL DEFINITIONS ---
from dscribe.descriptors import MBTR
import numpy as np
from visualise import view
from ase import Atoms
import matplotlib.pyplot as mpl

Atom description

We'll make an ase.Atoms class for NaCl



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# Define the system under study: NaCl in a conventional cell.
NaCl_conv = Atoms(
    cell=[
        [5.6402, 0.0, 0.0],
        [0.0, 5.6402, 0.0],
        [0.0, 0.0, 5.6402]
    ],
    scaled_positions=[
        [0.0, 0.5, 0.0],
        [0.0, 0.5, 0.5],
        [0.0, 0.0, 0.5],
        [0.0, 0.0, 0.0],
        [0.5, 0.5, 0.5],
        [0.5, 0.5, 0.0],
        [0.5, 0.0, 0.0],
        [0.5, 0.0, 0.5]
    ],
    symbols=["Na", "Cl", "Na", "Cl", "Na", "Cl", "Na", "Cl"],
)
view(NaCl_conv)

Setting MBTR hyper-parameters

Next we set-up hyper-parameters:

species, the chemical elements to include in the MBTR, helps comparing two structures with missing elements
k, list/set of K's to be computed
grid: dictionary for K1, K2, K3 with min, max: are the min and max values for each distribution sigma, the exponent coefficient for smearing n, number of bins.
weights: dictionary of weighting functions to be used.

Note: The dscribe package has implementation of MBTR up to K3



In [ ]:

    
# Create the MBTR desciptor for the system
mbtr = MBTR(
    species=['Na', 'Cl'], # Na and Cl
    periodic=True,
    k1={
        "geometry": { "function": "atomic_number" },
        "grid": { "min": 10, "max": 18, "sigma": 0.1, "n": 200 },
    },
    k2={
        "geometry": {"function": "inverse_distance"},
        "grid": { "min": 0, "max": 0.7, "sigma": 0.01, "n": 200 },
        "weighting": {"function": "exp", "scale": 0.75, "cutoff": 1e-3}
    },
    k3={
        "geometry": {"function": "cosine"},
        "grid": { "min": -1.0, "max": 1.0, "sigma": 0.05, "n": 200 },
        "weighting": {"function": "exp", "scale": 0.5, "cutoff": 1e-3}
    },
    
    flatten=False,
    sparse=False)
print("Number of features: {}".format(mbtr.get_number_of_features()))

Calculate MBTR

We call the create functin of mbtr class over our System(ase.Atoms) object



In [ ]:

    
#Create Descriptor
desc = mbtr.create(NaCl_conv)

Plotting

We will now plot all the tensors, in the same plot, for K1, K2, and K3



In [ ]:

    
#plot K1
x1 = mbtr.get_k1_axis()
mpl.plot(x1, desc["k1"][0, :], label="Na", color="blue")
mpl.plot(x1, desc["k1"][1, :], label="Cl", color="orange")
mpl.ylabel("$\phi$ (arbitrary units)", size=14)
mpl.xlabel("Atomic number", size=14)
mpl.title("The element count", size=20)
mpl.legend()
mpl.show()



In [ ]:

    
# Plot K2
x2 = mbtr.get_k2_axis()
mpl.plot(x2, desc["k2"][0, 1, :], label="NaCl, ClNa", color="blue")
mpl.plot(x2, desc["k2"][1, 1, :], label="ClCl", color="orange")
mpl.plot(x2, desc["k2"][0, 0, :], label="NaNa", color="green")
mpl.ylabel("$\phi$ (arbitrary units)", size=14)
mpl.xlabel("Inverse distance (1/angstrom)", size=14)
mpl.title("Exponentially weighted inverse distance distribution", size=20)
mpl.legend()
mpl.show()



In [ ]:

    
# Plot K3
x3 = mbtr.get_k3_axis()
mpl.plot(x3, desc["k3"][0, 0, 0, :], label="NaNaNa, ClClCl", color="blue")
mpl.plot(x3, desc["k3"][0, 0, 1, :], label="NaNaCl, NaClCl", color="orange")
mpl.plot(x3, desc["k3"][1, 0, 1, :], label="NaClNa, ClNaCl", color="green")
mpl.ylabel("$\phi$ (arbitrary units)", size=14)
mpl.xlabel("cos(angle)", size=14)
mpl.title("Exponentially weighted angle distribution", size=20)
mpl.legend()
mpl.show()

Remark

The MBTR is a fingerprint of the entire system. Thus, it can be used to:

Compare the similarity of two chemical system by taking comparing the MBTR values.
Machine learn total properties, like energies, dipole moment, etc.

Exercise

Verify that the MBTR is translationally and rotationally invariant.



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