Atomic-Scale Representation and Statistical Learning of Tensorial Properties
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Atomic-Scale Representation and Statistical Learning of Tensorial Properties

  • Andrea Grisafi
    Andrea Grisafi
    Laboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
    More by Andrea Grisafi
  • , 
  • David M. Wilkins
    David M. Wilkins
    Laboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
    More by David M. Wilkins
  • , 
  • Michael J. Willatt
    Michael J. Willatt
    Laboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
    More by Michael J. Willatt
  • , and 
  • Michele Ceriotti *
    Michele Ceriotti
    Laboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
    *E-mail: [email protected]
    More by Michele Ceriotti
DOI: 10.1021/bk-2019-1326.ch001
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Publication Date (Web):November 20, 2019
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Copyright © 2019 American Chemical Society. This publication is available under these Terms of Use.
Machine Learning in Chemistry: Data-Driven Algorithms, Learning Systems, and Predictions
Chapter 1pp 1-21
ACS Symposium SeriesVol. 1326
ISBN13: 9780841235052eISBN: 9780841235045

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Abstract

This chapter discusses the importance of incorporating three-dimensional symmetries in the context of statistical learning models geared towards the interpolation of the tensorial properties of atomic-scale structures. We focus on Gaussian process regression, and in particular on the construction of structural representations, and the associated kernel functions, that are endowed with the geometric covariance properties compatible with those of the learning targets. We summarize the general formulation of such a symmetry-adapted Gaussian process regression model, and how it can be implemented based on a scheme that generalizes the popular smooth overlap of atomic positions representation. We give examples of the performance of this framework when learning the polarizability, the hyperpolarizability, and the ground-state electron density of a molecule.

This publication is licensed for personal use by The American Chemical Society.

Introduction

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The purpose of a statistical learning model is the prediction of regression targets by means of simple and easily accessible input parameters ( 1). In chemistry, physics and materials science, regression targets are usually scalars or tensors, including electronic energies ( 2, 3, 4, 5), quantum-mechanical forces ( 6, 7, 8), electronic multipoles ( 9, 10, 11), response functions and scalar fields like the electron density ( 12, 13, 14, 15, 16, 17, 18). For ground-state properties, the regression input usually consists of all the information connected with the atomic structure at a given point of the Born-Oppenheimer surface (e.g., nuclear charges and atomic positions). A more or less complex manipulation of these primitive inputs leads to what is usually called a structural descriptor, or representation ((Figure 1)).

Figure 1

Figure 1. Structural descriptors should identify unequivocally and concisely the geometry and composition of a molecule or condensed phase.
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It is widely recognized that an essential ingredient for maximizing the efficiency of machine learning models is to use representations that mirror the properties one wants to predict. Here we discuss an effective approach to build linear regression models for tensors. The notion that the representation should mirror the property means when a symmetry operation is applied to an atomic structure, the associated representation should transform in a way that mimics the transformation of the properties of the structure. It should be stressed that it is completely possible to build a ML model that does not incorporate such transformation properties. The universal symmetries of the property must then be learned by the model through exposure to data in the training set, making the training process less efficient. A crucial focus of this chapter is the creation of symmetry-adapted representations. Once one has a symmetry-adapted representation at hand, the linear regression model is bound to fulfill the symmetry requirements imposed by the property ( 19, 20, 21, 22, 23). There is, however, another important consideration when building a model for tensors, expressed in terms of a Cartesian reference system. It is well known that any tensor can be decomposed into a set of spherical components that transform independently under rotations ( 24, 25). Particularly for high-order tensors, the irreducible spherical decomposition of a tensor simplifies greatly the learning task, compared to the Cartesian representation, as we will discuss later on.
The process of symmetry-adapting a representation is general but rather abstract, and for it to be practical one must choose the initial representation with care. For this purpose we use the smooth overlap of atomic positions (SOAP) framework, which is based on the representation of atom-centered environments constructed from a smooth atom density built up using Gaussians centered on each neighbor of the central atom. This density-based representation can be adapted to incorporate correlations between atoms to any order. It has been applied successfully to a vast number of ML investigations for physical properties of atomic structures ( 26, 27, 28). After summarizing the derivation and efficient implementation of an extension to SOAP, called λ-SOAP, which is particularly well-suited to the learning of tensorial properties, we present a few examples to demonstrate its effectiveness for this task.

Linear Regression

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Suppose one wanted to build a linear regression model to predict a scalar property y(X) for an input X,
In this equation |w〉 represents the weight vector we wish to learn and |X〉 is a representation of the input. The usual approach for learning the weight vector is to suppose the properties are independently and normally distributed, that is,
One then maximizes the log likelihood of a set of N observations {yn} with respect to the weight vector. The log likelihood (loss or cost function) is given by
where the regularizer α2 〈w|w〉 appears if one introduces a Gaussian prior on w with variance α−2. L(w) attains its maximum at
where the covariance Ĉ is
and η = α/σ.
The preceding linear regression scheme in which one handles the representation |X〉 explicitly is often called the primal formulation. There is in fact another, complementary formulation called the dual (kernel ridge regression [KRR] or Gaussian process regression [GPR]) in which the equations take a slightly different form. In the dual, one does not handle the representation explicitly but rather introduces a kernel function which, roughly speaking, measures the similarity between two inputs. The link between the primal and dual lies in the observation that a positive-definite kernel k(X,X′) can always be written as an inner product ( 1),
This means that given a kernel one can always construct a representation and vice versa. From the perspective of GPR, the kernel is interpreted as the covariance between the properties of its two arguments,
The properties are assumed to be normally distributed, which means one can straightforwardly find the conditional distribution of the property y(X) given a set of observations in a training set {yn}. The mean of this distribution is given by
where the jth component of k(X) is k(X, Xj), Kjk = k(Xj, Xk) and y is a vector formed from {yn}.
When the feature space associated with a kernel is known explicitly, and finite-dimensional, the primal and dual formulations are formally equivalent, and the choice of which to use is an important but purely practical question. Constructing a primal model requires inversion of the covariance matrix, while the dual requires inversion of the kernel matrix K. If the feature space (i.e., the space occupied by the representation) is larger than the training set then the GPR approach is more convenient. Of course, the real utility of the kernel trick becomes apparent when the kernel is a complex, non-linear function for which the feature space is unknown and/or infinite-dimensional. In these circumstances, working in the dual makes it possible to formulate regression as a linear problem, where reference configurations (or a sparse set of representative states) is used to define a basis for the target, as in the right hand side of eq (8). As such, all the complexity of the input space representation is contained in the definition of the kernel function.

Tensors, Symmetries and Correlations

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The previous discussion defines the general architecture of regression models which can be used to predict any scalar quantity associated with the molecular geometry. We now discuss the implications of learning tensors, or, similarly, any quantity that is not invariant under a rigid rotation or reflection of the atomic structure. In so doing, we will introduce a formalism which is general enough to encompass both proper Cartesian tensors, such as molecular polarizabilities, and three-dimensional scalar fields that can be conveniently decomposed in atom-centered contributions, such as the ground-state charge density of a molecule.
Let us start by considering the prototypical case of a Cartesian tensor y ≡ yαβ... of rank r, with the combination of indices {αβ...} running over a number of Cartesian components equal to 3r. Given any arbitrary distorted atomic structure with no particular internal symmetry, we are interested in characterizing the transformations of the tensor under only three families of symmetry operations (viz., translations, rotations and reflections). Since these symmetry operations do not affect the internal geometry of an atomic structure, we can think equivalently in terms of active transformations, in which the system undergoes the symmetry operation and the reference frame remains fixed, or in terms of passive transformations, in which the reference frame undergoes the symmetry operation and the system remains fixed. In the following, we summarize the symmetry operations by adopting an active picture and assume the system is not subjected to an external field.

Translations

Any physical property of an atomic structure X remains unchanged under a rigid translation t̂ of atomic positions, that is,

Rotations

Under the application of a rigid rotation R̂ to an atomic structure X, we assume that each Cartesian component of the tensor undergoes a covariant linear transformation. Using Einstein notation for convenience, and representing by R the rotation matrix corresponding to R̂, the rotated tensor is

Reflections

Applying a reflection operator Q̂ to an atomic structure X through any mirror plane leads to the following reflected tensor,

Covariant Descriptors

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In general terms, a primitive representation that mirrors a tensor of a given rank r could formally be built by considering
where |X〉 is an arbitrary description of the system, while |α〉 represents a set of Cartesian axes which is rigidly attached to the system. When using this primitive representation in a linear regression model, the tensor component corresponding to αβ... would be
or
After maximizing the log likelihood, the former possibility leads to a model that predicts every component to be the same, while the latter ignores the known correlations between the components and is therefore likely to overfit. For example, consider a training set in which only one of the tensor components is non-zero. All but one of the regression weights {|wαβ...〉} would be driven towards zero to maximize the log likelihood, so the trained model would only predict a finite value for the component it had been explicitly exposed to in the training set. The model would therefore incorrectly predict the tensor components for a structure differing only by a rigid rotation from one in the training set.
To address these problems, one should adapt the primitive descriptor so that it fulfills each of the symmetries detailed in eqs ((9)-(11)). Since the Cartesian basis vectors are invariant under translations, eq (9) implies the core representation should itself be invariant under translations. Using Haar integration one can construct a core representation that is invariant under translations by integrating an arbitrary representation over the translation operator t̂ ( 29). One can then proceed to consider covariance under SO(3) group operations. Eq (10) implies that a covariant representation for |Xαβ...〉 should satisfy the invariance relationship
for any rotation R̂. Starting from the primitive definition of eq (12), there are a variety of ways to enforce this invariance relationship. One possibility is to use
where the operator R̂X→ is defined to rotate X into a specified orientation which is common to all the molecules of the dataset ((Figure 2)).

Figure 2

Figure 2. Provided that one can define a local reference system, it is possible to learn tensorial properties by aligning each molecule (or environment) into a fixed reference frame.
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This works under the assumption that it is always possible to define a unique (and therefore unambiguous) internal reference frame to rotate X into a specified orientation, which might be possible when the system involved has a particularly rigid internal structure. A more general strategy, which does not require any assumption on the molecular geometry to be made, consists in considering the covariant integration over the operator R̂ (Haar integration),
On the top of this definition, the requirement that a representation be covariant in O(3), including the reflection symmetry of the tensor as in eq (11), means that improper rotations must be included (i.e., O(3) = SO(3) × {Î, Q̂}) with Q̂ representing a reflection operator. This this is done by a simple linear combination of the SO(3) descriptor with its reflected counterpart with respect to any arbitrary mirror plane of the system; that is,
Any other reflection operation can be automatically included by having made the descriptor covariant under rotations.

Covariant Regression

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Having shown how to build a symmetry-adapted representation of the system, let us see the implications of this procedure for linear regression. Using a symmetry-adapted representation in a linear regression model leads to the following solution for the regression weight,
where the covariance is
Note that the solution for the linear regression weight does not change when the training structures and corresponding tensors simultaneously undergo a symmetry operation that the representation has been adapted to. In other words, the same model results regardless of the arbitrary orientation of structures in the training set.
When moving to the dual, we find the kernel to be
This result corresponds to
As stressed earlier, performing the linear regression in the dual using this kernel leads to a formally-equivalent model to that resulting from the primal formulation described above, yet this kernel appears to be more complicated than a symmetry-adapted descriptor since it involves two integrations over rotations. If, however, we assume the core representation |X〉 undergoes a unitary transformation when the system is rotated,
the kernel reduces to
where k(X, X′) = (X|X′) is the kernel corresponding to the core representation. The requirement that the core representation should undergo a unitary transformation when the system is rotated is reasonable since, if it were not true, the autocorrelation k(X, X) would depend on the absolute orientation of X, which is unphysical given our assumption of the absence of external fields. Note that upon defining a collective tensorial index {αβ...}, a kernel matrix of size 3rN × 3rN can be constructed by stacking together each of the 3r × 3r vector-valued correlation functions. Then, a covariant tensorial prediction of the property of interest can eventually be carried out according to the GPR prescription of eq (8). It should be noted that the symmetry-adapted kernel of eq (24) is just a generalization of the covariant kernels that have been introduced in Glielmo et al. ( 7) to learn forces. Taking scalar products of symmetry-adapted representations provide a route to design easy-to-compute covariant kernels for tensors of arbitrary order.
It is instructive to compare the symmetry-adapted kernel definition of eq (24) to the kernel that one gets from the aligned descriptors of eq (16). In this case, building a kernel function on the top of this descriptor effectively means carrying out the structural comparison in a common reference frame where the two molecules are mutually aligned. One can then conveniently learn the tensor of interest component-by-component through a much simpler scalar regression framework. For the simple case of rank-1 tensors, for instance, we would get,
where we have defined the best alignment operator as . This strategy has been successfully used in the learning of electronic multipoles of organic molecules as well as for predicting optical response functions of water molecules in their liquid environments ( 10, 12). For the latter example, a representation of the best-alignment structural comparison is reported in (Figure 3).
This method for tensor learning has the clear drawback of relying on the definition of a rigid molecular geometry, for which an internal reference frame can be effectively used to perform the procedure of best alignment. Following this line of thought, the availability of a covariant kernel function allows us to implicitly carry out both the structural comparison and the geometric alignment of two molecules simultaneously, neglecting any prior consideration about the internal structure of the molecule at hand.

Figure 3

Figure 3. Representation of the reciprocal alignment between water environments.
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Spherical Representation

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The family of Cartesian symmetry-adapted descriptors previously introduced can be effectively used, in principle, to predict any Cartesian tensor of arbitrary order. However, we should notice that having a tensor product for each additional Cartesian axis makes the cost of the regression scale unfavorably with the tensor order, producing a global kernel matrix of dimension (3r)2. In fact, it is well established that a more natural representation of Cartesian tensors is given by their irreducible spherical components (ISC) ( 25). As described in Stone ( 25), the transformation matrix from Cartesian to spherical tensors can be found recursively, starting from the known transformation for rank-2 tensors.
Upon trivial manipulations, that might account for the non-symmetric nature of the tensor, each ISC transforms separately as spherical harmonics . Spherical harmonics form a complete basis set of the SO(3) group. In particular, each λ-component of the tensor spans an orthogonal subspace of dimension 2λ + 1. For instance, the 9 components of a rank-2 tensor separate out into a term (proportional to the trace) that transforms like a scalar, three terms that transform like , and five terms that transform like . When using a spherical representation, the kernel matrix is block diagonal, which greatly reduces the number of non-zero entries, and makes it possible to learn separately the different components. An additional advantage is that the possible symmetry of the tensor can be naturally incorporated by retaining only the spherical components λ that have the same parity as the tensor rank r. For instance, the λ = 1 component of a symmetric rank-2 tensor vanishes identically, meaning that only the 6 surviving elements of the tensor need to be considered when doing the regression. Especially for high rank tensors, this property means that the number of components can be cut down significantly.
In light of the discussion carried out for Cartesian tensors, it is straightforward to realize how a symmetry-adapted descriptor that transforms covariantly with spherical harmonics of order λ should look. Since each ISC is effectively a vector of dimension 2λ + 1, we can first write a primitive spherical harmonic representation as
where |λµ〉 is an angular momentum state of order λ, such that . Its symmetry-adapted counterpart, which is covariant in SO(3), is
Finally, since the parity of |λµ〉 with respect to the inversion operator î is determined by λ, a spherical tensor descriptor that is covariant in O(3) can be obtained by considering
Note that a tensorial kernel function built on the top of this descriptor would transform under rotations as the Wigner-D matrix of order λ, :
In addition to being the most natural strategy to perform the regression of Cartesian tensors, using a representation like that of eq (28) comes in handy when building regression models for the many physical properties that can be decomposed in a basis of atom-centered spherical harmonics. In the following sections, we will give an example of this kind by predicting the ground-state electronic charge density of molecular systems.

SOAP Representation

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We now proceed to characterize the exact functional form of a symmetry-adapted representation of order λ which can be used to carry out a covariant prediction of any property that transforms as a spherical harmonic. In the section above, it was pointed out that, within a framework of linear regression, both the primal and the dual formulation can be adopted to actually implement the interpolation of a given tensorial property. In what follows, however, we will focus our attention on the dual formulation, discussing in parallel the feature vector associated with the λ-SOAP representation and the corresponding kernel function. This choice is justified by the greater flexibility of the kernel formulation, allowing a non-linear extension of the framework as discussed below.
An atom-centered environment Xj describes the set of atoms that are included within a spherical cutoff rcut around the central atom j. We will label as |Xj〉 the abstract vector which describes the local structure. A convenient definition of |Xj〉 in real space can be obtained by writing a smooth probability amplitude, for each atomic species α, as a superposition of Gaussians with spread σ that are centered on the positions {ri} of the atoms that surround the central atom j:
This definition descends naturally from the requirement of translational invariance of a representation of the entire structure and corresponds to the construction that is used in Bartok et al. ( 21) to define the SOAP kernel ( 29). Formally, one can then write
with the ket |α〉 tagging the identity of each species. Even though it might be convenient to use a lower-dimensional chemical space ( 30), particularly when building models for dataset containing many elements, in what follows we will assume that each element is associated with an orthogonal subspace (i.e., 〈α|β〉 = δαβ). This implies that, when using this representation to define a scalar-product kernel, only the density distributions of the same atomic type are overlapped,
With this choice, the two adjustable parameters rcut and σ determine respectively the range and the resolution of the representation. To simplify the notation, we will omit the α labels, assuming that a single element is present. The extension to the case with multiple chemical species follows straightforwardly.

λ-SOAP(1) Representation

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To the first order in structural correlations, including the environmental state |Xj〉 in the definition of a local symmetry-adapted descriptor of order λ reads
The real space representation of can be understood as a rotational average of the environmental density which is rigidly attached to a spherical harmonic of order λ,
A more concise, and easily-computed version of this representation results from projecting on a basis of spherical harmonics, in which the integral over rotations can be performed analytically,
It is clear that many of the indices in this representation are redundant, and would have no effect when taking an inner product between two such representations. The most concise form that produces the same scalar product kernel as eq (34) is
where we introduced the spherical density component
This ket corresponds to the kernel
which is straightforward to calculate using a quadrature in r or an expansion on a radial basis.
It is insightful to consider the explicit expression for eq (36) in terms of the atom density. Taking for instance the case of λ = 1, µ = 0, for which :
One sees that the 2-body λ-SOAP representation corresponds to moments of the smooth atom density, resolved over different shells around the central atom. A linear model built on these features can respond to changes in the atomic density at different distances, simultaneously adapting the magnitude and geometric orientation of the target property.

λ-SOAP(2) Representation

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Describing an atomic environment in a way that goes beyond the two-body structural correlations (ν > 1) is of fundamental importance, because information on distances alone is not sufficient to uniquely determine an atomic structure. Building on the definition of eq (33), and on the symmetrized-atom-density framework of Willatt et al. ( 29), this can be achieved by introducing an additional tensor product in the environmental state |Xj〉 within the rotational average,
By projecting on a real-space basis, the representation becomes
Similarly to the ν = 1 case, one can compute the ket without an explicit rotational average by projecting on a basis of spherical harmonics,
where the parentheses denote a Wigner 3j symbol. Just as for the λ-SOAP(1) case considered earlier, it is clear that many of the indices in this expression are redundant. When taking an inner product between two such representations, one can use orthogonality of Wigner 3j symbols to simplify to an inner product between two objects with the following form,
The Clebsch-Gordan coefficient 〈lk, lk′|λµ〉 has the role of combining two angular momentum components of the atomic environment Xj to be compatible with the spherical tensor order λ. This contains all the essential information of the abstract representation that is needed for λ-SOAP(2) linear regression. Note that 〈lk, lk'|λµ〉 is zero unless k + k′ = µ, that the indices l, l' and λ must satisfy the inequality |l − l'| ≤ λ ≤ l + l' and that the representation is invariant under transposition of r and r'.
Let us see how the representation changes under inversion. Given the parity of the spherical harmonics,
it follows that
This condition implies that a representation that is covariant in O(3), eq (28), can be easily obtained by retaining only the components of the feature vectors for which l + l′ +λ is even.
In practice it is often more convenient to use real spherical harmonics instead of |λν〉 in the representation. Using real spherical harmonics ensures the kernel is purely real, but the components of the representation need not be because the phases are unimportant. In fact, what one finds upon replacing |λµ〉 with a real spherical harmonic is that the components are either purely real or purely imaginary, depending on whether l + l′ + λ is even or odd. For example, the representation for µ > 0 becomes
which satisfies
and the same relation also holds for the other real spherical harmonics for ν < 0 and |λ0〉 for ν = 0). One can therefore discard all imaginary components of the representation to enforce inversion invariance.
Generalization of this procedure to higher orders of λ-SOAP is tedious but straightforward using well-known formulae for integrals of products of Wigner-D matrices over rotations.

Non-linearity

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As already mentioned in the introduction, a crucial aspect to improve regression performance is to incorporate non-linearities in the construction of the representation. For instance, tensor products of the scalar representation introduce higher body order correlations, in a way that can be easily implemented in a kernel framework by raising the kernel to an integer power ( 29). When working with tensorial representations, however, one has to be careful to avoid breaking the covariant transformation properties of the feature vector. Taking products of kets would require re-projecting the product onto the irreducible representations of the group, which would be as cumbersome as increasing the body order exponent ν. One obvious solution to this problem is to multiply the spherical kernel of order λ by its scalar and rotationally invariant counterpart, which can then be raised to an integer power ζ without breaking the tensorial nature of the kernel. For any generic order ν and ν′ in structural correlations, this procedure consists in considering the tensor product
which leads to the kernel definition
For ζ = 1, one recovers the original tensorial kernel, while a non-linear behavior is introduced for ζ > 1. A considerable improvement of the learning power is usually obtained when using ζ = 2, while negligible further improvement is observed for ζ > 2.
These considerations also apply to the use of fully non-linear ML models like a neural network. To guarantee that the prediction of the model is consistent with the group covariances, the tensorial λ-SOAP features must enter the network at the last layer, and all the previous non-linear layers can only contribute to different linear combinations of the tensorial features, for example,
where each of the frr′ll′ can be an arbitrary non-linear combination of the scalar SOAP features Similar ideas have already been implemented in the context of generalizing the construction of spherical convolutional neural networks ( 31).

Implementation

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In the previous discussion it was pointed out that beyond the formal definition of the structural descriptor in real space, the kernel evaluation eventually requires the computation of the SOAP density power spectrum . In turn, computing this quantity requires the evaluation of the density expansion coefficients 〈rlm|Xj〉. In practice, the continuous variable r can be replaced by an expansion over a discrete set of orthogonal radial functions Rn(r) that are defined within the spherical cutoff rcut. For this reason, we will refer, from now on, to the density expansion coefficients as 〈nlm|Xj〉.
Having represented the environmental density distribution as a superposition of Gaussian functions centered on each atom, the spherical harmonics projection can be carried out analytically ( 32), leading to:
where the sum over i runs over the neighboring atoms of a given chemical element, and ιl represents a modified spherical Bessel function of the first kind. Under suitable choices of the functions Rn(r), the radial integration can also be carried out analytically, too.
One possibility is to start with non-orthogonal Gaussian type functions, R̃k(r), reminiscent of Gaussian-type orbitals commonly used in quantum chemistry:
where Nk is a normalization factor, such that . The set of Gaussian widths {σk} can be chosen to effectively span the radial interval involved in the environment definition. For instance, one can take , obtaining functions that have equally-spaced peaks between 0 and rcut. The explicit functional form of the primitive radial integrals is
where Γ is the Gamma function, while 1F1 is the confluent hypergeometric function of the first kind. These primitive integrals can be finally orthogonalized by applying the orthogonalization matrix S−1/2, with S representing the overlap matrix between primitive functions,
for which well-known analytical expressions exist ( 33).

Examples

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In this section, the effectiveness of a KRR model that is adapted to the fundamental physical symmetries of the target is demonstrated, considering two very different quantities as examples. As a first example, we consider the prediction of the dielectric response series of the Zundel cation, when training a λ-SOAP(2) regression model on the ISC of the tensors at hand. In the second example, we show how to predict the charge density ρ(r) of a small, yet flexible, hydrocarbon molecule like butane, by decomposing ρ(r) into atom-centered spherical harmonics. In both cases, a comparison of the prediction performance is carried out between λ-SOAP(2) descriptors that are covariant in SO(3), which were used in previous work, and those that have been made fully O(3) compliant by symmetrization over î.

Dielectric Response Series

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Consider the dielectric response series of a molecule including the dipole µ, the polarizability α and the hyperpolarizability β. The latter, for instance, is a rank-3 tensor describing the third-order response of the molecular energy U with respect to an applied electric field E, with components. By construction this tensor is symmetric, meaning that it can be decomposed into two spherical components, 3 of λ = 1 symmetry and 7 of λ = 3 symmetry. The total number of components to be learned is thus 10, consistently with the number of non-equivalent components of the Cartesian tensor. The dataset is made of 1000 configurations, of which 800 are randomly selected to train the regression model, while the remaining 200 are using to test the prediction performances. λ-SOAP(2) kernels that are adapted to SO(3) and O(3) group symmetry were constructed using a Gaussian smearing of σ = 0.3 Å and an environment cutoff of rcut = 4.0 Å.
The performances of each independent learning exercises (λ = 1, 2, 3) are reported in (Figure 4). For all the spherical components we observe a systematic improvement of the regression when endowing the kernel with the inversion symmetry about the atomic centers.

Figure 4

Figure 4. Learning curves of the Zundel cation dielectric response series µ,α and β as decomposed in their anisotropic (λ > 0) spherical tensor components. Full and dashed lines refer to predictions that are carried out with λ-SOAP kernel functions that are covariant in SO(3) and O(3) respectively.
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This improvement is particularly pronounced for few training points, while it becomes less relevant for larger training set sizes, where the symmetry under inversion is eventually learned by the SO(3)-kernel as well.

Electronic Charge Densities

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Another learning task that can benefit from a symmetry-adapted regression scheme involves the learning of scalar fields such as the electron charge density. ML models for the charge density have been proposed based on the coefficients in a plane wave basis–this is convenient due to orthogonality, but leads to poor transferability when considering flexible molecules, or learning across different molecular species–or based on direct prediction of the density on a real-space grid ( 16, 17, 34). By expanding the density on an atom-centred basis set, composed of radial functions multiplied by spherical harmonics,
one obtains a model that is localized and transferable, concise, and easily integrated with the many electronic structure codes that are based on atom-centered basis functions. The coefficients in the expansion transform under rotations like spherical harmonics, and can therefore be learned efficiently using a symmetry adapted GPR model,
where the sum runs over a set of reference environments Zi centered around atoms of the same kind as i, and the weights are computed by a regression procedure that is complicated by the fact that the basis set is not orthogonal ( 18).

Figure 5

Figure 5. Learning curves of the predicted charge density of 200 randomly selected butane molecules, when considering up to 800 reference molecules to train the model. The molecular geometries and computational details are the same as in Grisafi et al. ( 18) The black full line refers to the prediction error as reported in Grisafi et al. ( 18) Blue lines refer to the result obtained with the RI-cc-pV5Z basis, both with a λ-SOAP(2) descriptor covariant in SO (3) (full) and O(3) (dashed). Dotted lines refer to the basis set error. In both cases, 100 reference atomic environments have been used to define the problem dimensionality.
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In (Figure 5) we report the result obtained for a dataset of butane molecules (C4H10), for which 1000 reference pseudo-valence densities have been computed at the DFT/PBE level. The dimensionality of the regression problem is defined by considering the 100 most diverse atomic environments, out of a total of 14,000, selected by farthest point sampling through the 0-SOAP(2) distance metric ( 35). Given that in our previous work the learning performance was essentially limited by the basis set expansion error for the density, we decided to compare the optimized basis set used in Grisafi et al. ( 18) with a resolution of the identity (RI) basis set, usually adopted in the context of avoiding the computation of the four-center Hartree integral in electronic structure theory ( 36). When considering in particular the RI-cc-pV5Z basis, which accounts for basis functions up to l = 4 of angular momentum, we find that the basis set decomposition error is almost halved (~0.6%) with respect to Grisafi et al. ( 18), as shown by the asymptotic convergence in (Figure 5). The figure also compares, in the case of the RI basis, the learning performances associated with λ-SOAP(2) descriptors that have been made covariant in SO(3) and O(3) respectively. As seen for the case of polarizability, the O(3) features improve, although only slightly, the prediction accuracy. The improvement is more substantial at the smallest training set size, where the incorporation of prior knowledge on the symmetries of the system can make up for the scarcity of data.

Conclusions

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The previous examples show how statistical learning of a tensorial quantity across the configurational space of atomic coordinates and composition represents a challenging methodological task which requires considerable modifications to the architecture of more familiar scalar learning models. The efficiency of a regression model benefits greatly from the incorporation of symmetry, as it effectively reduces the dimensionality of the space in which the algorithm is asked to interpolate the values of the target property. Symmetry of tensorial quantities should be included in two distinct ways. First, one should decompose the tensor into ISC, so as to minimize the amount of information that is needed to account for geometric covariance. Particularly for high-rank Cartesian tensors, the matrix of correlations between tensor elements can be made block diagonal, which reflects on the size and complexity of the associated kernel matrices. Second, by constructing representations of the molecular structure that are made isomorphic with the tensor of interest, one can obtain a linear basis that satisfies the expected covariant transformations. An important aspect to consider is that, in order to preserve the properties of the symmetry-matched basis, non-linearities have to be treated with care. We discuss how it is possible to do so in the context of KRR models, and how one should proceed to design a covariant neural network that can be used to efficiently accomplish a symmetry-adapted regression task.
We discuss a practical implementation of these ideas within the framework of the SOAP representations, that uses a spherical-harmonics representation of the atom density and is therefore particularly well-suited to incorporate SO(3) covariance. We discuss an extension, that we refer to as λ-SOAP, that provides a natural linear basis to regress quantities that transform like spherical harmonics, and can be made to represent arbitrarily high body-order correlations between atomic coordinates. As an original result of this work, we also discuss how to satisfy the inversion symmetry of the tensor, showing that representations that incorporate the full O(3) covariances improve the performance of the ML model, particularly in the limit of a small training set. We also show an example of the use of λ-SOAP representations to learn a scalar field in three-dimension as a sum of atom-centered contributions, choosing the electron density as a physically relevant example. We believe that this strategy–although more complex than alternatives that use orthogonal basis functions or a real-space grid–has the best promise to be transferable across different systems, and to be combined with standard electronic structure packages.

Acknowledgments

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The Authors acknowledge support by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 677013-HBMAP).

References

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This chapter references 36 other publications.

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    We review some recently published methods to represent at. neighborhood environments, and analyze their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave nos. are used to expand the at. neighborhood d. function. Using the example system of small clusters, we quant. show that this expansion needs to be carried to higher and higher wave nos. as the no. of neighbors increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether different approach, called Smooth Overlap of Atomic Positions, that sidesteps these difficulties by directly defining the similarity between any two neighborhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.
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    We introduce a scheme for mol. simulations, the deep potential mol. dynamics (DPMD) method, based on a many-body potential and interat. forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is first-principles based in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and mols. In all these cases, DPMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size.
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    Evaluating the (dis)similarity of cryst., disordered and mol. compds. is a crit. step in the development of algorithms to navigate automatically the configuration space of complex materials. For instance, a structural similarity metric is crucial for classifying structures, searching chem. space for better compds. and materials, and driving the next generation of machine-learning techniques for predicting the stability and properties of mols. and materials. In the last few years several strategies have been designed to compare at. coordination environments. In particular, the smooth overlap of at. positions (SOAPs) has emerged as an elegant framework to obtain translation, rotation and permutation-invariant descriptors of groups of atoms, underlying the development of various classes of machine-learned inter-at. potentials. Here we discuss how one can combine such local descriptors using a regularized entropy match (REMatch) approach to describe the similarity of both whole mol. and bulk periodic structures, introducing powerful metrics that enable the navigation of alchem. and structural complexities within a unified framework. Furthermore, using this kernel and a ridge regression method we can predict atomization energies for a database of small org. mols. with a mean abs. error below 1 kcal mol-1, reaching an important milestone in the application of machine-learning techniques for the evaluation of mol. properties.
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    Mol. crystals play an important role in several fields of science and technol. They frequently crystallize in different polymorphs with substantially different phys. properties. To help guide the synthesis of candidate materials, at.-scale modeling can be used to enumerate the stable polymorphs and to predict their properties, as well as to propose heuristic rules to rationalize the correlations between crystal structure and materials properties. Here we show how a recently-developed machine-learning (ML) framework can be used to achieve inexpensive and accurate predictions of the stability and properties of polymorphs, and a data-driven classification that is less biased and more flexible than typical heuristic rules. We discuss, as examples, the lattice energy and property landscapes of pentacene and two azapentacene isomers that are of interest as org. semiconductor materials. We show that we can est. force field or DFT lattice energies with sub-kJ mol-1 accuracy, using only a few hundred ref. configurations, and reduce by a factor of ten the computational effort needed to predict charge mobility in the crystal structures. The automatic structural classification of the polymorphs reveals a more detailed picture of mol. packing than that provided by conventional heuristics, and helps disentangle the role of hydrogen bonded and π-stacking interactions in detg. mol. self-assembly. This observation demonstrates that ML is not just a black-box scheme to interpolate between ref. calcns., but can also be used as a tool to gain intuitive insights into structure-property relations in mol. crystal engineering.
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    The applications of machine learning techniques to chem. and materials science become more numerous by the day. The main challenge is to devise representations of at. systems that are at the same time complete and concise, so as to reduce the no. of ref. calcns. that are needed to predict the properties of different types of materials reliably. This has led to a proliferation of alternative ways to convert an at. structure into an input for a machine-learning model. We introduce an abstr. definition of chem. environments that is based on a smoothed at. d., using a bra-ket notation to emphasize basis set independence and to highlight the connections with some popular choices of representations for describing at. systems. The correlations between the spatial distribution of atoms and their chem. identities are computed as inner products between these feature kets, which can be given an explicit representation in terms of the expansion of the atom d. on orthogonal basis functions, that is equiv. to the smooth overlap of at. positions power spectrum, but also in real space, corresponding to n-body correlations of the atom d. This formalism lays the foundations for a more systematic tuning of the behavior of the representations, by introducing operators that represent the correlations between structure, compn., and the target properties. It provides a unifying picture of recent developments in the field and indicates a way forward toward more effective and computationally affordable machine-learning schemes for mols. and materials. (c) 2019 American Institute of Physics.
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    Machine-learning of at.-scale properties amts. to extg. correlations between structure, compn. and the quantity that one wants to predict. Representing the input structure in a way that best reflects such correlations makes it possible to improve the accuracy of the model for a given amt. of ref. data. When using a description of the structures that is transparent and well-principled, optimizing the representation might reveal insights into the chem. of the data set. Here, we show how one can generalize the SOAP kernel to introduce a distance-dependent wt. that accounts for the multi-scale nature of the interactions, and a description of correlations between chem. species. We show that this improves substantially the performance of ML models of mol. and materials stability, while making it easier to work with complex, multi-component systems and to extend SOAP to coarse-grained intermol. potentials. The element correlations that give the best performing model show striking similarities with the conventional periodic table of the elements, providing an inspiring example of how machine learning can rediscover, and generalize, intuitive concepts that constitute the foundations of chem.
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    Increasingly, it is recognized that new automated forms of anal. are required to understand the high-dimensional output obtained from atomistic simulations. Recently, we introduced a new dimensionality redn. algorithm, sketch-map, that was designed specifically to work with data from mol. dynamics trajectories. In what follows, we provide more details on how this algorithm works and on how to set its parameters. We also test it on two well-studied Lennard-Jones clusters and show that the coordinates we ext. using this algorithm are extremely robust. In particular, we demonstrate that the coordinates constructed for one particular Lennard-Jones cluster can be used to describe the configurations adopted by a second, different cluster and even to tell apart different phases of bulk Lennard-Jonesium.
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  • Figure 1

    Figure 1. Structural descriptors should identify unequivocally and concisely the geometry and composition of a molecule or condensed phase.
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    Figure 2

    Figure 2. Provided that one can define a local reference system, it is possible to learn tensorial properties by aligning each molecule (or environment) into a fixed reference frame.
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    Figure 3

    Figure 3. Representation of the reciprocal alignment between water environments.
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    Figure 4

    Figure 4. Learning curves of the Zundel cation dielectric response series µ,α and β as decomposed in their anisotropic (λ > 0) spherical tensor components. Full and dashed lines refer to predictions that are carried out with λ-SOAP kernel functions that are covariant in SO(3) and O(3) respectively.
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    Figure 5

    Figure 5. Learning curves of the predicted charge density of 200 randomly selected butane molecules, when considering up to 800 reference molecules to train the model. The molecular geometries and computational details are the same as in Grisafi et al. ( 18) The black full line refers to the prediction error as reported in Grisafi et al. ( 18) Blue lines refer to the result obtained with the RI-cc-pV5Z basis, both with a λ-SOAP(2) descriptor covariant in SO (3) (full) and O(3) (dashed). Dotted lines refer to the basis set error. In both cases, 100 reference atomic environments have been used to define the problem dimensionality.
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  • References

    CHAPTER SECTIONS
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    This chapter references 36 other publications.

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      Bartok, Albert P.; Payne, Mike C.; Kondor, Risi; Csanyi, Gabor
      Physical Review Letters (2010), 104 (13), 136403/1-136403/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We introduce a class of interat. potential models that can be automatically generated from data consisting of the energies and forces experienced by atoms, as derived from quantum mech. calcns. The models do not have a fixed functional form and hence are capable of modeling complex potential energy landscapes. They are systematically improvable with more data. We apply the method to bulk crystals, and test it by calcg. properties at high temps. Using the interat. potential to generate the long mol. dynamics trajectories required for such calcns. saves orders of magnitude in computational cost.
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      Jain A. Ong S. P. Hautier G. Chen W. Richards W. D. Dacek S. Cholia S. Gunter D. Skinner D. Ceder G. Persson K. A. Commentary: The Materials Project: A materials genome approach to accelerating materials innovation APL Mater. 2013 1 011002
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      Commentary: The Materials Project: A materials genome approach to accelerating materials innovation
      Jain, Anubhav; Ong, Shyue Ping; Hautier, Geoffroy; Chen, Wei; Richards, William Davidson; Dacek, Stephen; Cholia, Shreyas; Gunter, Dan; Skinner, David; Ceder, Gerbrand; Persson, Kristin A.
      APL Materials (2013), 1 (1), 011002/1-011002/11CODEN: AMPADS; ISSN:2166-532X. (American Institute of Physics)
      Accelerating the discovery of advanced materials is essential for human welfare and sustainable, clean energy. In this paper, we introduce the Materials Project (www.materialsproject.org), a core program of the Materials Genome Initiative that uses high-throughput computing to uncover the properties of all known inorg. materials. This open dataset can be accessed through multiple channels for both interactive exploration and data mining. The Materials Project also seeks to create open-source platforms for developing robust, sophisticated materials analyses. Future efforts will enable users to perform rapid-prototyping'' of new materials in silico, and provide researchers with new avenues for cost-effective, data-driven materials design. (c) 2013 American Institute of Physics.
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      Calderon C. E. Plata J. J. Toher C. Oses C. Levy O. Fornari M. Natan A. Mehl M. J. Hart G. Nardelli M. B. Curtarolo S. The AFLOW standard for high-throughput materials science calculations Comput. Mater. Sci. 2015 108 233 238
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      The AFLOW standard for high-throughput materials science calculations
      Calderon, Camilo E.; Plata, Jose J.; Toher, Cormac; Oses, Corey; Levy, Ohad; Fornari, Marco; Natan, Amir; Mehl, Michael J.; Hart, Gus; Buongiorno Nardelli, Marco; Curtarolo, Stefano
      Computational Materials Science (2015), 108 (Part_A), 233-238CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
      The Automatic-Flow (AFLOW) std. for the high-throughput construction of materials science electronic structure databases is described. Electronic structure calcns. of solid state materials depend on a large no. of parameters which must be understood by researchers, and must be reported by originators to ensure reproducibility and enable collaborative database expansion. We therefore describe std. parameter values for k-point grid d., basis set plane wave kinetic energy cut-off, exchange-correlation functionals, pseudopotentials, DFT+U parameters, and convergence criteria used in AFLOW calcns.
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      Ward L. Wolverton C. Atomistic calculations and materials informatics: A review Curr. Opin. Solid State Mater. Sci. 2017 21 167 176
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      Atomistic calculations and materials informatics: A review
      Ward, Logan; Wolverton, Chris
      Current Opinion in Solid State & Materials Science (2017), 21 (3), 167-176CODEN: COSSFX; ISSN:1359-0286. (Elsevier Ltd.)
      A review. In recent years, there has been a large effort in the materials science community to employ materials informatics to accelerate materials discovery or to develop new understanding of materials behavior. Materials informatics methods utilize machine learning techniques to ext. new knowledge or predictive models out of existing materials data. In this review, we discuss major advances in the intersection between data science and atom-scale calcns. with a particular focus on studies of solid-state, inorg. materials. The examples discussed in this review cover methods for accelerating the calcn. of computationally-expensive properties, identifying promising regions for materials discovery based on existing data, and extg. chem. intuition automatically from datasets. We also identify key issues in this field, such as limited distribution of software necessary to utilize these techniques, and opportunities for areas of research that would help lead to the wider adoption of materials informatics in the atomistic calcns. community.
    6. 6
      Li Z. Kermode J. R. De Vita A. Molecular Dynamics with On-the-Fly Machine Learning of Quantum-Mechanical Forces Phys. Rev. Lett. 2015 114 096405
      6
      Molecular dynamics with on-the-fly machine learning of quantum-mechanical forces
      Li, Zhenwei; Kermode, James R.; De Vita, Alessandro
      Physical Review Letters (2015), 114 (9), 096405/1-096405/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We present a mol. dynamics scheme which combines first-principles and machine-learning (ML) techniques in a single information-efficient approach. Forces on atoms are either predicted by Bayesian inference or, if necessary, computed by on-the-fly quantum-mech. (QM) calcns. and added to a growing ML database, whose completeness is, thus, never required. As a result, the scheme is accurate and general, while progressively fewer QM calls are needed when a new chem. process is encountered for the second and subsequent times, as demonstrated by tests on cryst. and molten silicon.
    7. 7
      Glielmo A. Sollich P. De Vita A. Accurate interatomic force fields via machine learning with covariant kernels Phys. Rev. B 2017 95 214302
      7
      Accurate interatomic force fields via machine learning with covariant kernels
      Glielmo, Aldo; Sollich, Peter; De Vita, Alessandro
      Physical Review B (2017), 95 (21), 214302/1-214302/10CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      A review. We present a novel scheme to accurately predict at. forces as vector quantities, rather than sets of scalar components, by Gaussian process (GP) regression. This is based on matrix-valued kernel functions, on which we impose the requirements that the predicted force rotates with the target configuration and is independent of any rotations applied to the configuration database entries. We show that such covariant GP kernels can be obtained by integration over the elements of the rotation group SO(d) for the relevant dimensionality d. Remarkably, in specific cases the integration can be carried out anal. and yields a conservative force field that can be recast into a pair interaction form. Finally, we show that restricting the integration to a summation over the elements of a finite point group relevant to the target system is sufficient to recover an accurate GP. The accuracy of our kernels in predicting quantum-mech. forces in real materials is investigated by tests on pure and defective Ni, Fe, and Si cryst. systems.
    8. 8
      Glielmo A. Zeni C. De Vita A. Efficient nonparametric n-body force fields from machine learning Phys. Rev. B 2018 97 184307
      8
      Efficient nonparametric n-body force fields from machine learning
      Glielmo, Aldo; Zeni, Claudio; De Vita, Alessandro
      Physical Review B (2018), 97 (18), 184307CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      A review. We provide a definition and explicit expressions for n-body Gaussian process (GP) kernels, which can learn any interat. interaction occurring in a phys. system, up to n-body contributions, for any value of n. The series is complete, as it can be shown that the "universal approximator" squared exponential kernel can be written as a sum of n-body kernels. These recipes enable the choice of optimally efficient force models for each target system, as confirmed by extensive testing on various materials. We furthermore describe how the n-body kernels can be "mapped" on equiv. representations that provide database-size-independent predictions and are thus crucially more efficient. We explicitly carry out this mapping procedure for the first nontrivial (three-body) kernel of the series, and we show that this reproduces the GP-predicted forces with meV/Å accuracy while being orders of magnitude faster. These results pave the way to using novel force models (here named "M-FFs") that are computationally as fast as their corresponding std. parametrized n-body force fields, while retaining the nonparametric character, the ease of training and validation, and the accuracy of the best recently proposed machine-learning potentials.
    9. 9
      Yuan Y. Mills M. J. Popelier P. L. Multipolar electrostatics based on the Kriging machine learning method: an application to serine J. Mol. Model. 2014 20 2172
      9
      Multipolar electrostatics based on the Kriging machine learning method: an application to serine
      Yuan Yongna; Mills Matthew J L; Popelier Paul L A
      Journal of molecular modeling (2014), 20 (4), 2172 ISSN:.
      A multipolar, polarizable electrostatic method for future use in a novel force field is described. Quantum Chemical Topology (QCT) is used to partition the electron density of a chemical system into atoms, then the machine learning method Kriging is used to build models that relate the multipole moments of the atoms to the positions of their surrounding nuclei. The pilot system serine is used to study both the influence of the level of theory and the set of data generator methods used. The latter consists of: (i) sampling of protein structures deposited in the Protein Data Bank (PDB), or (ii) normal mode distortion along either (a) Cartesian coordinates, or (b) redundant internal coordinates. Wavefunctions for the sampled geometries were obtained at the HF/6-31G(d,p), B3LYP/apc-1, and MP2/cc-pVDZ levels of theory, prior to calculation of the atomic multipole moments by volume integration. The average absolute error (over an independent test set of conformations) in the total atom-atom electrostatic interaction energy of serine, using Kriging models built with the three data generator methods is 11.3 kJ mol-1 (PDB), 8.2 kJ mol-1 (Cartesian distortion), and 10.1 kJ mol-1 (redundant internal distortion) at the HF/6-31G(d,p) level. At the B3LYP/apc-1 level, the respective errors are 7.7 kJ mol-1, 6.7 kJ mol-1, and 4.9 kJ mol-1, while at the MP2/cc-pVDZ level they are 6.5 kJ mol-1, 5.3 kJ mol-1, and 4.0 kJ mol-1. The ranges of geometries generated by the redundant internal coordinate distortion and by extraction from the PDB are much wider than the range generated by Cartesian distortion. The atomic multipole moment and electrostatic interaction energy predictions for the B3LYP/apc-1 and MP2/cc-pVDZ levels are similar, and both are better than the corresponding predictions at the HF/6-31G(d,p) level.
    10. 10
      Bereau T. Andrienko D. von Lilienfeld O. A. Transferable Atomic Multipole Machine Learning Models for Small Organic Molecules J. Chem. Theory Comput. 2015 11 3225 3233
      10
      Transferable Atomic Multipole Machine Learning Models for Small Organic Molecules
      Bereau, Tristan; Andrienko, Denis; von Lilienfeld, O. Anatole
      Journal of Chemical Theory and Computation (2015), 11 (7), 3225-3233CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Accurate representation of the mol. electrostatic potential, which is often expanded in distributed multipole moments, is crucial for an efficient evaluation of intermol. interactions. Here we introduce a machine learning model for multipole coeffs. of atom types H, C, O, N, S, F, and Cl in any mol. conformation. The model is trained on quantum-chem. results for atoms in varying chem. environments drawn from thousands of org. mols. Multipoles in systems with neutral, cationic, and anionic mol. charge states are treated with individual models. The models' predictive accuracy and applicability are illustrated by evaluating intermol. interaction energies of nearly 1,000 dimers and the cohesive energy of the benzene crystal.
    11. 11
      Bereau T. DiStasio R. A. Tkatchenko A. von Lilienfeld O. A. Non-covalent interactions across organic and biological subsets of chemical space: Physics-based potentials parametrized from machine learning J. Chem. Phys. 2018 148 241706
      11
      Non-covalent interactions across organic and biological subsets of chemical space: Physics-based potentials parametrized from machine learning
      Bereau, Tristan; Di Stasio, Robert A.; Tkatchenko, Alexandre; von Lilienfeld, O. Anatole
      Journal of Chemical Physics (2018), 148 (24), 241706/1-241706/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Classical intermol. potentials typically require an extensive parametrization procedure for any new compd. considered. To do away with prior parametrization, we propose a combination of physics-based potentials with machine learning (ML), coined IPML, which is transferable across small neutral org. and biol. relevant mols. ML models provide on-the-fly predictions for environment-dependent local at. properties: electrostatic multipole coeffs. (significant error redn. compared to previously reported), the population and decay rate of valence at. densities, and polarizabilities across conformations and chem. compns. of H, C, N, and O atoms. These parameters enable accurate calcns. of intermol. contributions-electrostatics, charge penetration, repulsion, induction/polarization, and many-body dispersion. Unlike other potentials, this model is transferable in its ability to handle new mols. and conformations without explicit prior parametrization: All local at. properties are predicted from ML, leaving only eight global parameters-optimized once and for all across compds. We validate IPML on various gas-phase dimers at and away from equil. sepn., where we obtain mean abs. errors between 0.4 and 0.7 kcal/mol for several chem. and conformationally diverse datasets representative of non-covalent interactions in biol. relevant mols. We further focus on hydrogen-bonded complexes - essential but challenging due to their directional nature - where datasets of DNA base pairs and amino acids yield an extremely encouraging 1.4 kcal/mol error. Finally, and as a first look, we consider IPML for denser systems: water clusters, supramol. host-guest complexes, and the benzene crystal. (c) 2018 American Institute of Physics.
    12. 12
      Liang C. Tocci G. Wilkins D. M. Grisafi A. Roke S. Ceriotti M. Solvent fluctuations and nuclear quantum effects modulate the molecular hyperpolarizability of water Phys. Rev. B 2017 96 041407
      12
      Solvent fluctuations and nuclear quantum effects modulate the molecular hyperpolarizability of water
      Liang, Chungwen; Tocci, Gabriele; Wilkins, David M.; Grisafi, Andrea; Roke, Sylvie; Ceriotti, Michele
      Physical Review B (2017), 96 (4), 041407/1-041407/6CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      Second-harmonic scattering (SHS) expts. provide a unique approach to probe noncentrosym. environments in aq. media, from bulk solns. to interfaces, living cells, and tissue. A central assumption made in analyzing SHS expts. is that each mol. scatters light according to a const. mol. hyperpolarizability tensor β(2). Here, we investigate the dependence of the mol. hyperpolarizability of water on its environment and internal geometric distortions, in order to test the hypothesis of const. β(2). We use quantum chem. calcns. of the hyperpolarizability of a mol. embedded in point-charge environments obtained from simulations of bulk water. We demonstrate that both the heterogeneity of the solvent configurations and the quantum mech. fluctuations of the mol. geometry introduce large variations in the nonlinear optical response of water. This finding has the potential to change the way SHS expts. are interpreted: In particular, isotopic differences between H2O and D2O could explain recent SHS observations. Finally, we show that a machine-learning framework can predict accurately the fluctuations of the mol. hyperpolarizability. This model accounts for the microscopic inhomogeneity of the solvent and represents a step towards quant. modeling of SHS expts.
    13. 13
      Grisafi A. Wilkins D. M. Csányi G. Ceriotti M. Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems Phys. Rev. Lett. 2018 120 036002
      13
      Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems
      Grisafi, Andrea; Wilkins, David M.; Csanyi, Gabor; Ceriotti, Michele
      Physical Review Letters (2018), 120 (3), 036002CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      A review. Statistical learning methods show great promise in providing an accurate prediction of materials and mol. properties, while minimizing the need for computationally demanding electronic structure calcns. The accuracy and transferability of these models are increased significantly by encoding into the learning procedure the fundamental symmetries of rotational and permutational invariance of scalar properties. However, the prediction of tensorial properties requires that the model respects the appropriate geometric transformations, rather than invariance, when the ref. frame is rotated. We introduce a formalism that extends existing schemes and makes it possible to perform machine learning of tensorial properties of arbitrary rank, and for general mol. geometries. To demonstrate it, we derive a tensor kernel adapted to rotational symmetry, which is the natural generalization of the smooth overlap of at. positions kernel commonly used for the prediction of scalar properties at the at. scale. The performance and generality of the approach is demonstrated by learning the instantaneous response to an external elec. field of water oligomers of increasing complexity, from the isolated mol. to the condensed phase.
    14. 14
      Wilkins D. M. Grisafi A. Yang Y. Lao K. U. DiStasio R. A. Ceriotti M. Accurate molecular polarizabilities with coupled cluster theory and machine learning Proc. Natl. Acad. Sci. 2019 116 3401 3406
      14
      Accurate molecular polarizabilities with coupled cluster theory and machine learning
      Wilkins, David M.; Grisafi, Andrea; Yang, Yang; Lao, Ka Un; Di Stasio, Robert A., Jr.; Ceriotti, Michele
      Proceedings of the National Academy of Sciences of the United States of America (2019), 116 (9), 3401-3406CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      The mol. dipole polarizability describes the tendency of a mol. to change its dipole moment in response to an applied elec. field. This quantity governs key intra- and intermol. interactions, such as induction and dispersion; plays a vital role in detg. the spectroscopic signatures of mols.; and is an essential ingredient in polarizable force fields. Compared with other ground-state properties, an accurate prediction of the mol. polarizability is considerably more difficult, as this response quantity is quite sensitive to the underlying electronic structure description. In this work, we present highly accurate quantum mech. calcns. of the static dipole polarizability tensors of 7,211 small org. mols. computed using linear response coupled cluster singles and doubles theory (LR-CCSD). Using a symmetry-adapted machine-learning approach, we demonstrate that it is possible to predict the LR-CCSD mol. polarizabilities of these small mols. with an error that is an order of magnitude smaller than that of hybrid d. functional theory (DFT) at a negligible computational cost. The resultant model is robust and transferable, yielding mol. polarizabilities for a diverse set of 52 larger mols. (including challenging conjugated systems, carbohydrates, small drugs, amino acids, nucleobases, and hydrocarbon isomers) at an accuracy that exceeds that of hybrid DFT. The atom-centered decompn. implicit in our machine-learning approach offers some insight into the shortcomings of DFT in the prediction of this fundamental quantity of interest.
    15. 15
      Christensen A. S. Faber F. A. von Lilienfeld O. A. Operators in quantum machine learning: Response properties in chemical space J. Chem. Phys. 2019 150 064105
      15
      Operators in quantum machine learning: Response properties in chemical space
      Christensen, Anders S.; Faber, Felix A.; von Lilienfeld, O. Anatole
      Journal of Chemical Physics (2019), 150 (6), 064105/1-064105/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The role of response operators is well established in quantum mechanics. We investigate their use for universal quantum machine learning models of response properties in mols. After introducing a theor. basis, we present and discuss numerical evidence based on measuring the potential energy's response with respect to at. displacement and to elec. fields. Prediction errors for corresponding properties, at. forces, and dipole moments improve in a systematic fashion with training set size and reach high accuracy for small training sets. Prediction of normal modes and IR-spectra of some small mols. demonstrates the usefulness of this approach for chem. (c) 2019 American Institute of Physics.
    16. 16
      Brockherde F. Vogt L. Li L. Tuckerman M. E. Burke K. Mu¨ller K.-R. Bypassing the Kohn-Sham equations with machine learning Nat. Commun. 2017 8 872
      16
      Bypassing the Kohn-Sham equations with machine learning
      Brockherde Felix; Muller Klaus-Robert; Brockherde Felix; Vogt Leslie; Tuckerman Mark E; Li Li; Burke Kieron; Tuckerman Mark E; Tuckerman Mark E; Burke Kieron; Muller Klaus-Robert; Muller Klaus-Robert
      Nature communications (2017), 8 (1), 872 ISSN:.
      Last year, at least 30,000 scientific papers used the Kohn-Sham scheme of density functional theory to solve electronic structure problems in a wide variety of scientific fields. Machine learning holds the promise of learning the energy functional via examples, bypassing the need to solve the Kohn-Sham equations. This should yield substantial savings in computer time, allowing larger systems and/or longer time-scales to be tackled, but attempts to machine-learn this functional have been limited by the need to find its derivative. The present work overcomes this difficulty by directly learning the density-potential and energy-density maps for test systems and various molecules. We perform the first molecular dynamics simulation with a machine-learned density functional on malonaldehyde and are able to capture the intramolecular proton transfer process. Learning density models now allows the construction of accurate density functionals for realistic molecular systems.Machine learning allows electronic structure calculations to access larger system sizes and, in dynamical simulations, longer time scales. Here, the authors perform such a simulation using a machine-learned density functional that avoids direct solution of the Kohn-Sham equations.
    17. 17
      Alred J. M. Bets K. V. Xie Y. Yakobson B. I. Machine learning electron density in sulfur crosslinked carbon nanotubes Compos. Sci. Technol. 2018 166 3 9
      17
      Machine learning electron density in sulfur crosslinked carbon nanotubes
      Alred, John M.; Bets, Ksenia V.; Xie, Yu; Yakobson, Boris I.
      Composites Science and Technology (2018), 166 (), 3-9CODEN: CSTCEH; ISSN:0266-3538. (Elsevier Ltd.)
      Mech. strengthening of composite materials that include carbon nanotubes (CNT) requires strong inter-bonding to achieve significant CNT-CNT or CNT-matrix load transfer. The same principle is applicable to the improvement of CNT bundles and calls for covalent crosslinks between individual tubes. In this work, sulfur crosslinks are studied using a combination of d. functional theory (DFT) and classical mol. dynamics (MD). Atomic chains of at least two sulfur atoms or more are shown to be stable between both zigzag and armchair CNTs. All types of crosslinked CNTs exhibit significantly improved load transfer. Moreover, sulfur crosslinks show evidence of a cooperative self-healing mechanism allowing for links to rebond once broken leading to sustained load transfer under shear loading. Addnl., a general approach for utilizing machine learning for assessing the ground state electron d. is developed and applied to these sulfur crosslinked CNTs.
    18. 18
      Grisafi A. Fabrizio A. Meyer B. Wilkins D. M. Corminboeuf C. Ceriotti M. Transferable Machine-Learning Model of the Electron Density ACS Cent. Sci. 2019 5 57 64
      18
      Transferable Machine-Learning Model of the Electron Density
      Grisafi, Andrea; Fabrizio, Alberto; Meyer, Benjamin; Wilkins, David M.; Corminboeuf, Clemence; Ceriotti, Michele
      ACS Central Science (2019), 5 (1), 57-64CODEN: ACSCII; ISSN:2374-7951. (American Chemical Society)
      The electronic charge d. plays a central role in detg. the behavior of matter at the at. scale, but its computational evaluation requires demanding electronic-structure calcns. We introduce an atom-centered, symmetry-adapted framework to machine-learn the valence charge d. based on a small no. of ref. calcns. The model is highly transferable, meaning it can be trained on electronic-structure data of small mols. and used to predict the charge d. of larger compds. with low, linear-scaling cost. Applications are shown for various hydrocarbon mols. of increasing complexity and flexibility, and demonstrate the accuracy of the model when predicting the d. on octane and octatetraene after training exclusively on butane and butadiene. This transferable, data-driven model can be used to interpret expts., accelerate electronic structure calcns., and compute electrostatic interactions in mols. and condensed-phase systems.
    19. 19
      Braams B. J. Bowman J. M. Permutationally invariant potential energy surfaces in high dimensionality Int. Rev. Phys. Chem. 2009 28 577 606
      19
      Permutationally invariant potential energy surfaces in high dimensionality
      Braams, Bastiaan J.; Bowman, Joel M.
      International Reviews in Physical Chemistry (2009), 28 (4), 577-606CODEN: IRPCDL; ISSN:0144-235X. (Taylor & Francis Ltd.)
      We review recent progress in developing potential energy and dipole moment surfaces for polyat. systems with up to 10 atoms. The emphasis is on global linear least squares fitting of tens of thousands of scattered ab initio energies using a special, compact fitting basis of permutationally invariant polynomials in Morse-type variables of all the internuclear distances. The computational mathematics underlying this approach is reviewed first, followed by a review of the practical approaches used to obtain the data for the fits. A straightforward symmetrization approach is also given, mainly for pedagogical purposes. The methods are illustrated for potential energy surfaces for CH+5, (H2O)2 and CH3CHO. The relationship of this approach to other approaches is also briefly reviewed.
    20. 20
      Behler J. Parrinello M. Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces Phys. Rev. Lett. 2007 98 146401
      20
      Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces
      Behler, Jorg; Parrinello, Michele
      Physical Review Letters (2007), 98 (14), 146401/1-146401/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The accurate description of chem. processes often requires the use of computationally demanding methods like d.-functional theory (DFT), making long simulations of large systems unfeasible. In this Letter we introduce a new kind of neural-network representation of DFT potential-energy surfaces, which provides the energy and forces as a function of all at. positions in systems of arbitrary size and is several orders of magnitude faster than DFT. The high accuracy of the method is demonstrated for bulk silicon and compared with empirical potentials and DFT. The method is general and can be applied to all types of periodic and nonperiodic systems.
    21. 21
      Bartók A. P. Kondor R. Csányi G. On representing chemical environments Phys. Rev. B 2013 87 184115
      21
      On representing chemical environments
      Bartok, Albert P.; Kondor, Risi; Csanyi, Gabor
      Physical Review B: Condensed Matter and Materials Physics (2013), 87 (18), 184115/1-184115/16CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We review some recently published methods to represent at. neighborhood environments, and analyze their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave nos. are used to expand the at. neighborhood d. function. Using the example system of small clusters, we quant. show that this expansion needs to be carried to higher and higher wave nos. as the no. of neighbors increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether different approach, called Smooth Overlap of Atomic Positions, that sidesteps these difficulties by directly defining the similarity between any two neighborhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.
    22. 22
      Shapeev A. Moment Tensor Potentials: A Class of Systematically Improvable Interatomic Potentials Multiscale Model. Sim. 2016 14 1153 1173
      There is no corresponding record for this reference.
    23. 23
      Zhang L. Han J. Wang H. Car R. E W. Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics Phys. Rev. Lett. 2018 120 143001
      23
      Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics
      Zhang, Linfeng; Han, Jiequn; Wang, Han; Car, Roberto; E, Weinan
      Physical Review Letters (2018), 120 (14), 143001CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      We introduce a scheme for mol. simulations, the deep potential mol. dynamics (DPMD) method, based on a many-body potential and interat. forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is first-principles based in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and mols. In all these cases, DPMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size.
    24. 24
      Weinert U. Spherical tensor representation Arch. Ration. Mech. Anal. 1980 74 165 196
      There is no corresponding record for this reference.
    25. 25
      Stone A. J. Transformation between cartesian and spherical tensors Mol. Phys. 1975 29 1461 1471
      25
      Transformation between cartesian and spherical tensors
      Stone, A. J.
      Molecular Physics (1975), 29 (5), 1461-71CODEN: MOPHAM; ISSN:0026-8976.
      A std. unitary transformation is detd. for interconversion between cartesian and spherical tensors and between equations including such tensors. The effects of symmetry with respect to permutation of cartesian tensor suffixes were examd. The angle dependence of the circular intensity differential of Rayleigh scattering from a dimer was derived.
    26. 26
      De S. Bartók A. P. Csányi G. Ceriotti M. Comparing molecules and solids across structural and alchemical space Phys. Chem. Chem. Phys. 2016 18 13754 13769
      26
      Comparing molecules and solids across structural and alchemical space
      De, Sandip; Bartok, Albert P.; Csanyi, Gabor; Ceriotti, Michele
      Physical Chemistry Chemical Physics (2016), 18 (20), 13754-13769CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      Evaluating the (dis)similarity of cryst., disordered and mol. compds. is a crit. step in the development of algorithms to navigate automatically the configuration space of complex materials. For instance, a structural similarity metric is crucial for classifying structures, searching chem. space for better compds. and materials, and driving the next generation of machine-learning techniques for predicting the stability and properties of mols. and materials. In the last few years several strategies have been designed to compare at. coordination environments. In particular, the smooth overlap of at. positions (SOAPs) has emerged as an elegant framework to obtain translation, rotation and permutation-invariant descriptors of groups of atoms, underlying the development of various classes of machine-learned inter-at. potentials. Here we discuss how one can combine such local descriptors using a regularized entropy match (REMatch) approach to describe the similarity of both whole mol. and bulk periodic structures, introducing powerful metrics that enable the navigation of alchem. and structural complexities within a unified framework. Furthermore, using this kernel and a ridge regression method we can predict atomization energies for a database of small org. mols. with a mean abs. error below 1 kcal mol-1, reaching an important milestone in the application of machine-learning techniques for the evaluation of mol. properties.
    27. 27
      Musil F. De S. Yang J. Campbell J. E. J. Day G. G. M. Ceriotti M. Machine learning for the structure-energy-property landscapes of molecular crystals Chem. Sci. 2018 9 1289 1300
      27
      Machine learning for the structure-energy-property landscapes of molecular crystals
      Musil, Felix; De, Sandip; Yang, Jack; Campbell, Joshua E.; Day, Graeme M.; Ceriotti, Michele
      Chemical Science (2018), 9 (5), 1289-1300CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
      Mol. crystals play an important role in several fields of science and technol. They frequently crystallize in different polymorphs with substantially different phys. properties. To help guide the synthesis of candidate materials, at.-scale modeling can be used to enumerate the stable polymorphs and to predict their properties, as well as to propose heuristic rules to rationalize the correlations between crystal structure and materials properties. Here we show how a recently-developed machine-learning (ML) framework can be used to achieve inexpensive and accurate predictions of the stability and properties of polymorphs, and a data-driven classification that is less biased and more flexible than typical heuristic rules. We discuss, as examples, the lattice energy and property landscapes of pentacene and two azapentacene isomers that are of interest as org. semiconductor materials. We show that we can est. force field or DFT lattice energies with sub-kJ mol-1 accuracy, using only a few hundred ref. configurations, and reduce by a factor of ten the computational effort needed to predict charge mobility in the crystal structures. The automatic structural classification of the polymorphs reveals a more detailed picture of mol. packing than that provided by conventional heuristics, and helps disentangle the role of hydrogen bonded and π-stacking interactions in detg. mol. self-assembly. This observation demonstrates that ML is not just a black-box scheme to interpolate between ref. calcns., but can also be used as a tool to gain intuitive insights into structure-property relations in mol. crystal engineering.
    28. 28
      Bartók A. P. De S. Poelking C. Bernstein N. Kermode J. R. Csányi G. Ceriotti M. Machine learning unifies the modeling of materials and molecules Sci. Adv. 2017 3
      There is no corresponding record for this reference.
    29. 29
      Willatt M. J. Musil F. Ceriotti M. Atom-density representations for machine learning J. Chem. Phys. 2019 150 154110
      29
      Atom-density representations for machine learning
      Willatt, Michael J.; Musil, Felix; Ceriotti, Michele
      Journal of Chemical Physics (2019), 150 (15), 154110/1-154110/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The applications of machine learning techniques to chem. and materials science become more numerous by the day. The main challenge is to devise representations of at. systems that are at the same time complete and concise, so as to reduce the no. of ref. calcns. that are needed to predict the properties of different types of materials reliably. This has led to a proliferation of alternative ways to convert an at. structure into an input for a machine-learning model. We introduce an abstr. definition of chem. environments that is based on a smoothed at. d., using a bra-ket notation to emphasize basis set independence and to highlight the connections with some popular choices of representations for describing at. systems. The correlations between the spatial distribution of atoms and their chem. identities are computed as inner products between these feature kets, which can be given an explicit representation in terms of the expansion of the atom d. on orthogonal basis functions, that is equiv. to the smooth overlap of at. positions power spectrum, but also in real space, corresponding to n-body correlations of the atom d. This formalism lays the foundations for a more systematic tuning of the behavior of the representations, by introducing operators that represent the correlations between structure, compn., and the target properties. It provides a unifying picture of recent developments in the field and indicates a way forward toward more effective and computationally affordable machine-learning schemes for mols. and materials. (c) 2019 American Institute of Physics.
    30. 30
      Willatt M. J. Musil F. Ceriotti M. Feature Optimization for Atomistic Machine Learning Yields a Data-Driven Construction of the Periodic Table of the Elements Phys. Chem. Chem. Phys. 2018 20 29661 29668
      30
      Feature optimization for atomistic machine learning yields a data-driven construction of the periodic table of the elements
      Willatt, Michael J.; Musil, Felix; Ceriotti, Michele
      Physical Chemistry Chemical Physics (2018), 20 (47), 29661-29668CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      Machine-learning of at.-scale properties amts. to extg. correlations between structure, compn. and the quantity that one wants to predict. Representing the input structure in a way that best reflects such correlations makes it possible to improve the accuracy of the model for a given amt. of ref. data. When using a description of the structures that is transparent and well-principled, optimizing the representation might reveal insights into the chem. of the data set. Here, we show how one can generalize the SOAP kernel to introduce a distance-dependent wt. that accounts for the multi-scale nature of the interactions, and a description of correlations between chem. species. We show that this improves substantially the performance of ML models of mol. and materials stability, while making it easier to work with complex, multi-component systems and to extend SOAP to coarse-grained intermol. potentials. The element correlations that give the best performing model show striking similarities with the conventional periodic table of the elements, providing an inspiring example of how machine learning can rediscover, and generalize, intuitive concepts that constitute the foundations of chem.
    31. 31
      Kondor R. Zhen L. Trivedi S. Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network arXiv:1806.09231 2018
      There is no corresponding record for this reference.
    32. 32
      Kaufmann K. Baumeister W. Single-centre expansion of Gaussian basis functions and the angular decomposition of their overlap integrals J. Phys. B: At. Mol. Opt. 1989 22 1
      32
      Single-center expansion of Gaussian basis functions and the angular decomposition of their overlap integrals
      Kaufmann, Karl; Baumeister, Werner
      Journal of Physics B: Atomic, Molecular and Optical Physics (1989), 22 (1), 1-12CODEN: JPAPEH; ISSN:0953-4075.
      Single-center partial-wave expansions are derived for several Gaussian-type functions: simple, solid harmonic, and spheric Gaussians. Single-center expansions for the most commonly used Cartesian Gaussians are obtained by expanding these functions in spherical Gaussians. Transformation matrixes for expanding Cartsian in spherical Gaussians are given for s-, p-, d-, and f-type functions. The single-center expansions are used to calc. the partial-wave decompn. of overlap integrals for all Gaussian-type functions specified. The formulas given are suitable for fast numerical computation, and were tested with programs developed for this purpose.
    33. 33
      Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products, 7th ed.; Elsevier/Academic Press, Amsterdam, 2007; pp xlviii+1171, Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
      There is no corresponding record for this reference.
    34. 34
      Chandrasekaran A. Kamal D. Batra R. Kim C. Chen L. Ramprasad R. Solving the electronic structure problem with machine learning Npj Comput. Mater. 2019 5 22
      There is no corresponding record for this reference.
    35. 35
      Ceriotti M. Tribello G. A. Parrinello M. Demonstrating the Transferability and the Descriptive Power of Sketch-Map J. Chem. Theory Comput. 2013 9 1521 1532
      35
      Demonstrating the Transferability and the Descriptive Power of Sketch-Map
      Ceriotti, Michele; Tribello, Gareth A.; Parrinello, Michele
      Journal of Chemical Theory and Computation (2013), 9 (3), 1521-1532CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Increasingly, it is recognized that new automated forms of anal. are required to understand the high-dimensional output obtained from atomistic simulations. Recently, we introduced a new dimensionality redn. algorithm, sketch-map, that was designed specifically to work with data from mol. dynamics trajectories. In what follows, we provide more details on how this algorithm works and on how to set its parameters. We also test it on two well-studied Lennard-Jones clusters and show that the coordinates we ext. using this algorithm are extremely robust. In particular, we demonstrate that the coordinates constructed for one particular Lennard-Jones cluster can be used to describe the configurations adopted by a second, different cluster and even to tell apart different phases of bulk Lennard-Jonesium.
    36. 36
      Hättig C. Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Corevalence and quintuple- basis sets for H to Ar and QZVPP basis sets for Li to Kr Phys. Chem. Chem. Phys. 2005 7 59 66
      There is no corresponding record for this reference.