Vibrational properties of metastable polymorph structures by machine learning
Fleur Legrain Ambroise van Roekeghem Stefano Curtarolo Jesus Carrete Georg K. H. Madsen Natalio Mingo




  Despite vibrational properties being critical for the ab initio prediction of the finite temperature stability and transport properties of solids,
    their inclusion in  ab initio materials repositories has been hindered by expensive computational requirements.
    Here we tackle the challenge, by showing that a good estimation of force constants and vibrational properties 
    can be quickly achieved from the knowledge of atomic equilibrium positions using machine learning.
    A random-forest algorithm trained on only 121 metastable structures of KZnF_3 reaches a maximum absolute error of 0.17 eV/Å^2 for the  interatomic force constants, 
    and it is much less expensive than training the complete force field for such compound.
    The predicted force constants are then used to estimate phonon spectral features, heat capacities, vibrational entropies, and vibrational free energies, 
    which compare well with the ab initio ones.
    The approach can be used for the rapid estimation of stability at finite temperatures.









§ INTRODUCTION

  Large databases of calculated material properties, such as AFLOW.org<cit.>, 
  the Materials Project<cit.>, and
  OQMD<cit.>,  
  have become powerful tools for accelerated materials design<cit.>. 
  Ab initio relaxed crystal structures and ground state energies are routinely provided in these repositories, and often used to evaluate phase diagrams starting from zero temperature
  or with simple approximations <cit.>. With this approach roughly 50% of the experimentally known compounds are found above the convex hull.<cit.> This can be due to the experimental structure being truly metastable. Another possible explanation could be the lack of accuracy of standard density functional approximations.<cit.> However, an important factor will undoubtedly be that phonon-related contributions are highly important at the temperatures of interest<cit.>. These contributions are often neglected, principally due to the high computational cost posed by the interatomic force constants () matrix, i.e.the Hessian, or second derivatives of the energy with respect to the atomic displacements. Similarly, structural global energy minimization methods, such as USPEX<cit.>, generate hundreds of relaxed candidate structures. However, both for large databases and global energy methods, the vibrational energy contributions are typically too expensive to be calculated with brute force. 
  A considerable advantage would 
  come from
  an on-the-fly estimation of vibrational free energies during the search.
  
Neglecting phonon contributions to the free energy is obviously wrong and this practice is mainly due to computational necessities. Obtaining the Hessian typically requires one or two orders of magnitude more computer time than the corresponding structural relaxation. However, neglecting phonons can have dramatic consequences. For example, 
  vibrational contributions have been shown to modify the sequence of reactions occurring as a function of temperature or pressure<cit.>, 
  to explain the precipitation sequence of metallurgical phases<cit.>, 
  or to alter the stability ordering of novel 2D material phases<cit.>. 
  Phonons also have been shown to be as important as configurational disorder for the prediction of alloy phase diagrams and thereby essential to obtain experimental agreement<cit.>. 
  Particularly relevant is the problem of polymorphs, i.e. materials sharing the same chemical composition but having different crystal structures.
  Calculations on organic molecules have shown that ∼69% of polymorph pairs reversed their relative stability when increasing the temperature, due to the vibrational contribution to the free energy<cit.>. 
Also, roughly 50% of the compounds in the Materials Project database are metastable with a median energy above the convex hull of 15 meV/atom<cit.>
  and similar values apply to the ICSD <cit.> repository within AFLOW.org.
  This energy is comparable to typical phonon free energy differences between polymorphs<cit.>, 
  highlighting the importance of including the phonon vibrational energy when determining the finite temperature ground states.
  The high-throughput prediction of phase diagrams at finite temperatures is still a major challenge for computational materials design, 
  mostly because of the difficulty to quickly compute Hessians <cit.>. 
  Clearly, there is an urgent need for a rapid and reliable approach to predict the .

  Machine learning () algorithms can be used to avoid costly calculations.
   has been successfully used to predict  for compounds from the same crystal structure but different chemical composition <cit.>,
  which was subsequently shown to be a major factor determining the vibrational free energy of compounds<cit.>. 
However, the more complex problem of predicting  of competing structures of the same composition has not been addressed. 
  Often the relaxed structures are already known and the challenge is to predict only the computationally expensive Hessians. 
  This is the case with large ab initio databases, which contain many metastable structures or artificial configurations for sampling the phase space <cit.>.
  Contrary to force-field fitting where a continuum of “deformations→forces” states has to be sampled,
  here accurate representations of the potential energy surface around diverse and potentially uncorrelated metastable states is needed. Is this doable?
  In the present work we tackle the challenge by finding
  an efficient solution with the help of random forests, trained with only one hundred metastable structures, but still capable of predicting accurate , spectral properties and thermodynamic quantities.



§ APPROACH

The interatomic force constants between atom i and j constitute a second-order tensor defined by the second derivatives of the PES with respect to atomic displacements

    Φ_ij=(∇_r_i⊗∇_r_j) E

For  to predict the Φ_ij's we need to construct atom-centered descriptors based on an internal coordinate representation that is invariant with respect to the symmetries of the systems, as well as permutations among atoms of the same species. A similar challenge is faced in force-field fitting<cit.> but here we face the additional problem of generalizing the concept to tensors.

Scalar quantities of the physical system, like the energy, are expressed in this representation as functions of a set of scalar descriptors, {g_i,j^α}, based on these internal coordinates. 
  Vector quantities associated to the i^th-atom can similarly be expressed by descriptors 
g_ij^αr_ij, that transform contravariantly.
  More generally, however, one can produce quantities that transform as tensors by taking gradients of the scalar descriptors.

 We choose a series of Gaussians, similar to those used in force-field fitting<cit.>, to represent the pair part
  
    g_ij^α=e^-(r_ij a_α)^2

  where {a_α} are a set of radii spanning a few interatomic distances encompassing atoms i and j. Taking the gradients of these scalar descriptors leads to 3×3 matrices defined for each atomic pair (i,j) as:
  
    M^η,η'≡∂^2 ∂x_i^η∂x_j^η'g_ij^α
        = 2 a_α^2 g_ij^α[ δ^η,η'-2r^η_ij r^η'_jia_α^2],
 
  where η and η' run over the three Cartesian coordinates.
  While the δ term transforms as a scalar, the r_ij^η r_ji^η' term corresponds to the outer product of the gradients of scalar function g and transforms as a rank-2 tensor. 
  Therefore, descriptors of the type 
  
    𝐃_ij^(2)α=(∇_r_i⊗∇_r_j)g_ij^α∝ g_ij^αr_ij⊗r_ji,
 
  transform as rank-2 tensors and can be used for the regression of Hessians.
   Periodic boundary conditions within the supercell spanning the force cut-off range (here  5 × 5 × 5) require an extra modification of the descriptor as
    
    𝐃_i,j^(2)α = ∑_me^-|r_ij+R_m a_α|^2(r_ij + R_m) ⊗ (r_ji - R_m),
 
where R_m are the translation vectors connecting identical atoms in the supercell.

The set of descriptors above can be extended to higher orders, at an increased computational expense. For instance, the following set of rank-2 tensor descriptors would capture further 3-body interactions.
  
    𝐃_ij^(3)α,β,γ   =   ∑_k { g_ik^α g_kj^β∇_r_iθ_ikj⊗∇_r_jθ_ikj
                                                      + g_ij^γ g_jk^β∇_r_iθ_ijk⊗∇_r_jθ_ijk + g_ki^α g_ij^γ∇_r_iθ_kij⊗∇_r_jθ_kij}
       ≡   ∑_k 𝐃^α,β,γ_i,k,j.

  where θ_ijk is the angle formed by atoms i,j and k.
  The gradients of θ_ijk can be expressed in terms of cross products of pairs in {i,j,k} and 𝐃_ij^(3) transform as a tensor.
    There are other ways to define descriptors involving two and three-atom terms<cit.>. 
  However, to the best of our knowledge, direct regression of Hessians using invariant tensorial-form descriptors has not been attempted before.

  Descriptors D_i,j^(2) α are used to predict Φ_ij, i.e. the 3 × 3 matrices of  between different i- and j-atoms.
  Atomic force constants of the form, Φ_ii, which simply describes the forces on atom i due to its own displacement, cannot come from D_i,i^(2) α.
  Thus, the whole environment is included through the sum:
  
    𝐃_i,i^(diag)(2)α = ∑_j𝐃_i,j^(2)α,

  where j indices through all the atoms of the supercell , including i itself.
  To get the whole , two different  models are then trained:  D_i,j^(2) α MLΦ_i≠ j and D_i,i^(diag)(2)α MLΦ_ii.

  For clarity of the presentation, the dependence on chemical species, required for multi-component systems, has not been included in previous the formulas.
  Given a set of species {s}, the descriptors can be written as 
  𝐃_s,s';i,j^(2) α≡ (δ_s_i,sδ_s_j,s'+δ_s_i,s'δ_s_j,s)𝐃_i,j^(2)α, and 𝐃^(3)α,β,γ_s,s',s”,i,j=∑_k δ_s_k,s”(δ_s_i,sδ_s_j,s'+δ_s_i,s'δ_s_j,s) 𝐃^(3)α,β,γ_i,j,k,
  with s and s' species indices.



§ RESULTS AND DISCUSSION




 §.§ Data set.
 
  The  approach is developed for a test chemical system: the metastable structures of KZnF_3 (a cubic perovskite at 0K, chosen for simplicity).
  The initial data set consists of 267 KZnF_3 structures with 10 atoms per unit cell, randomly generated by the first-generation run of the  code<cit.>, 
  and optimized using density functional theory ()<cit.> as implemented in  <cit.>.
  The projector augmented wave () method is employed to deal with the core and valence electrons<cit.>. 
  The data sets preparation follows AFLOW.org high-throughput recommendations <cit.>, 
  and the kinetic energy cutoffs are set to 450 eV for the plane wave basis.
  The force constant matrices and the phonon frequencies are computed at Γ using density functional perturbation theory<cit.>.


  The identification and reduction of symmetrically equivalent cells is performed through the following structural fingerprint.
  For every structure C, descriptors are computed for each i-, j-atom pair:
  D_s,s';i,j(C)^α≡ (δ_s_i,sδ_s_j,s'+δ_s_i,s'δ_s_j,s) g_ij^α(C), where {s,s'} are  the species indices
  (these same descriptors are also employed to predict the  matrix invariants, as detailed later).
  The sum over the pairs leades to a fingerprint K for a given structure C:  
  
    K^α_s,s'(C) ≡∑_i,j D_s,s';i,j^α(C) = ∑_i,j (δ_s_i,sδ_s_j,s'+δ_s_i,s'δ_s_j,s) g_ij^α(C)

  The distance d(C_1,C_2) between structures C_1 and C_2 is then defined as: 
  
    d(C_1,C_2) ≡∑_(s,s';α)| K^α_s,s'(C_1)- K^α_s,s'(C_2)|/| K^α_s,s'(C_1)|+| K^α_s,s'(C_2)|.

  After combinatorial analysis between the structures, an optimum distance's threshold of 0.35 is found by inspection.
  Only 121 inequivalent configurations are found, the remaining ones being discarded as duplicates (Supplementary Materials).
  The 121 cells are then used to build the  model and assess its performance. 




 §.§ Predicting force constants.

  Amongst the available regression algorithms, random forests () are chosen because they are non-parametric, require virtually no data pre-conditioning, and usually yield robust and reliable results.
  The scikit-learn implementation<cit.> is used to assess the performance of the model via 10-fold cross validation. 
  The forest contain 100 trees: better performance was not noticed with larger forests. 
  The three independent scalar invariants <cit.>
  { tr(Φ_ij), √(tr(Φ_ij^2)), √(tr(Φ_ij^3))}
  — derived from calculated or predicted Hessians and defined for each  
  between different atoms — are used to assess the quality of the model.
  The performance of the random forests is listed in Table <ref> and depicted in Figure <ref>.


 

  Upon trying with various different choices of radii a_α, the best results are achieved with a_α={1,2,3,...,30} Å. 
The outcome of the descriptor with periodicity (Eq. (<ref>)) is satisfactory.
  On the contrary, the performance is poor when periodicity is neglected (mean absolute errors are larger than 1eV/Å^2, see Supplementary Materials).
  Errors are larger on small supercells, and decrease if training is performed on larger systems.

 

  The full Hessian is tackled with the descriptors from Eq. (<ref>) 
and Eq. (<ref>). 
The individual force constants predicted versus calculated — are compared in Figure <ref> and listed in Table <ref>.


  The  results indicate that very good predictions can be obtained using only simple two-atom descriptors.
  The extension to the 3-atom environments,  Eq. <ref>,  
  does not noticeably improve the outcome,
  implying that the key factors determining the  are the species and relative positions between atoms' pairs, without much contribution from other environmental atoms. 
  3-atom descriptors take much longer to calculate, are much more numerous than the pair descriptors, and therefore impose constraints onto the number of accessible radii {α}, potentially leading to sub optimal results.
  Thus, it is possible that other descriptor algebraic formalisms and/or broader training sets – more systems and larger structures — could improve the outcome when
  3-atom environments are accessed. This is beyond the scope of the current work and it will be tackled in the future.

  How does the promising accuracy of predicted  translate into
  vibrational properties? 
  Errors do accumulate and even an apparently good
  prediction of forces could still violate conservation rules of the system leading to unphysical results.
  The phonon frequencies of the different KZnF_3 cells are computed
  from the -predicted . 
  Results are extremely sensitive to small inaccuracies in
  predicted forces and imaginary phonon frequencies appear.

  The  imaginary modes are removed by correcting each Hessian H to the “closest” semipositive definite matrix H'.
  A diagonal matrix D is obtained through the basis transformation H=PDP^T. A corrected D → D_ c is produced by 
  replacing the negative terms with zeroes. The object is rotated back to the original basis, PD_ cP^T.
  The term is further corrected by enforcing the acoustic sum rule
  leading to H_ c=(I-Q) (PD_ cP^T) (I-Q), with Q ≡∑_t=1,2,3v_t^T v_t, v_t a vector of size 3 N_ defined as 
  v_t,i=δ_i,k/√(N_), and k ≡[(t-1)3].

  The phonon frequencies are computed at Γ from the corrected Hessian H_ c.
  Following Ref. doi:10.1021/acs.chemmater.7b00789, 
the zero frequencies are replaced by <ω_>/2  (ω_ represent the optical frequencies) in the calculation of vibrational properties. 
  From here, phonon spectral distribution (mean, max, and variance) and thermodynamic properties are then obtained.

  Figure <ref> displays the square root of the variance, the mean, and the maximum of the compound frequencies, computed with  and predicted with . 
  There is good correlation and the statistical analysis is summarized in Table <ref>.




  The specific heats at constant volume, vibrational entropies and free energies are computed at 300 K from the phonon frequencies ω following Ref. Landau
  (N_atoms is the number of atoms in the cell and n_ω is the Bose-Einstein distribution):
  
    C_   =   1/N_atoms∑_ωħ^2ω^2/k_BT^2exp(-ħω/k_BT)/(1-exp(-ħω/k_BT))^2
    
        S_   =   1/N_atoms∑_ω{ħω/Texp(-ħω/k_BT)/1-exp{-ħω/k_BT)-k_Bln[1-exp(-ħω/k_BT)]}
    
        F_   =   E_-TS_ =-1/N_atoms∑_ω(ħω/2+k_BTlnn_ω)


  The quantities are depicted in Figure <ref>: values computed with  are on the x-axis while values predicted with  are on the y-axis. 
  The plots show that the  approach gives a good approximation of the heat capacities, vibrational free energies and vibrational entropies of the different structures of KZnF_3. 
  The statistical analysis is summarized in Table <ref>.




  The results  illustrate that descriptors transforming as two-index tensors, in combination with  regression algorithms and relatively small training sets, can be used to predict  for generic metastable crystal structures. 
  Other approaches can be used to further improve the outcome:
  amongst them there are 
  support vector machines or neural networks as algorithms,
  different descriptor definitions, inhomogeneous or adjustable grids for the radii,
  replacing the Gaussian by different functions, considering the species as separate descriptors from the structural ones.
  In addition,  training on more and larger supercells, could improve the accuracy and enhance the relevance of 3-atom environmental descriptors, which could also take a different functional form from the presented one. 
  However, regardless of potential improvements, here it has been shown that it is possible to predict force constants by training exclusively on metastable structures — very abundant in online respositories — without the necessity to include unstable configurations.
  Atomic configurations can be viewed as points in a high-dimensional space of rototranslation-permutation invariant descriptors: close points → similar properties. 
  As such, to overcome similarity  force fields are typically trained on tens of thousands of atomic configurations not corresponding to local energy minima, in order to have a sufficiently dense coverage of the representative hyper-volume in the configuration space:
  any new configuration is close to some other points in the training set, allowing for a good prediction of the energy/forces. 
  In contrast, our case deals only with atomic configurations corresponding to local energy minima: metastable structures are intrinsically dissimilar, belonging to attraction basins separated by energy barriers in the configurational space.
  A priori there is no reason why the properties of such different structures should be related to each other. 
  A posteriori, the results show that the Hessians of different local minima are indeed inter-related and strongly determined by pair-wise interactions. 
  This unveils an underlying regularity in the character of the inter-atomic interactions that persists across the different metastable structures, enabling the prediction of force constants of an unknown metastable 
  structure by training only on the other metastable systems available, without having to include any unstable structures to populate the empty configurational space between energy minima. 
  The property can be leveraged for the quick estimation of vibrational contributions to phase stability and transport properties of materials, and to enable the high-throughput ab initio screening of these properties at finite temperatures.



§ CONCLUSIONS


  We have shown that Hessians and associated vibrational properties of multi-component metastable structures 
  can be efficiently predicted by machine learning regressions without the need of developing full force fields. 
  The key factors determining the interatomic force constants are captured by tensor descriptors depending only on the species and distance between atoms' pairs.
  The main features of the vibrational spectrum — maximum, mean and variance — are correctly reproduced.
   predictions of thermodynamic properties — specific heat, vibrational free energy and entropy — correlate well with the  calculations. 
  Once trained, the model allows for
  the rapid vibrational characterization of relaxed structures with arbitrary complexity at low computational cost
  and the efficient comparison of polymorphs competing for stability at finite temperature.
  It is envisioned that machine learning vibrational-approaches will enable
  the use of the abundant online repositories information for efficient 
  high-throughput screening of stability and transport at finite temperature.



§ ACKNOWLEDGEMENTS

The work is supported by M-era.net through the ICETS project (DFG: MA 5487/4-1 and ANR-14-MERA-0003-03) and ANR through the Carnot MAPPE project.
S.C. acknowledges DOD-ONR (N00014-15-1-2863), the Alexander von Humboldt Foundation and the Max Planck Society for financial support





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