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ArticleJune 29, 2018

Machine Learning Adaptive Basis Sets for Efficient Large Scale Density Functional Theory Simulation
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Ole Schütt
Ole Schütt
Department of Materials, ETH Zürich, 8093 Zürich, Switzerland
More by Ole Schütt
Joost VandeVondele*
Joost VandeVondele
Department of Materials, ETH Zürich, 8093 Zürich, Switzerland
Swiss National Supercomputing Centre (CSCS), 6900 Lugano, Switzerland
*E-mail: [email protected]
More by Joost VandeVondele
http://orcid.org/0000-0002-0902-5111

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Journal of Chemical Theory and Computation

Cite this: J. Chem. Theory Comput. 2018, 14, 8, 4168–4175

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https://doi.org/10.1021/acs.jctc.8b00378

Published June 29, 2018

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Abstract

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It is chemically intuitive that an optimal atom centered basis set must adapt to its atomic environment, for example by polarizing toward nearby atoms. Adaptive basis sets of small size can be significantly more accurate than traditional atom centered basis sets of the same size. The small size and well conditioned nature of these basis sets leads to large saving in computational cost, in particular in a linear scaling framework. Here, it is shown that machine learning can be used to predict such adaptive basis sets using local geometrical information only. As a result, various properties of standard DFT calculations can be easily obtained at much lower costs, including nuclear gradients. In our approach, a rotationally invariant parametrization of the basis is obtained by employing a potential anchored on neighboring atoms to ultimately construct a rotation matrix that turns a traditional atom centered basis set into a suitable adaptive basis set. The method is demonstrated using MD simulations of liquid water, where it is shown that minimal basis sets yield structural properties in fair agreement with basis set converged results, while reducing the computational cost in the best case by a factor of 200 and the required flops by 4 orders of magnitude. Already a very small training set yields satisfactory results as the variational nature of the method provides robustness.

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1. Introduction

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The rapid increase in computational power and the development of linear scaling methods (1,2) now allow for easy single-point density functional theory (DFT) energy calculations of systems with 10,000–1,000,000 atoms. (3,4) However, the approach is computationally demanding for routine application, especially if first-principles molecular dynamics or relaxation is required. The computational cost of a DFT calculation depends critically on the size and condition number of the employed basis set. Traditional linear scaling DFT implementations employ basis sets which are atom centered, static, and isotropic. Since molecular systems are never isotropic, it is apparent that isotropic basis sets are suboptimal. Therefore, in this work a scheme is presented to define small adaptive basis sets as a function of the local chemical environment. These chemical environments are subject to change, e.g., during the aforementioned relaxations or sampling. In order to map chemical environments to basis functions in a predictable fashion, a machine learning (ML) approach is used. The analytic nature of a ML framework allows for the calculation of exact analytic forces, as required for dynamic simulations.

The idea of representing the electronic structure with adapted atomic or quasi-atomic basis functions dates back several decades. It underlays, e.g., many early tools used for the investigation of bonding order. (5−10) Also more recent methods for extracting atomic orbitals from molecular orbitals build on this idea. (11−16) Besides using adaptive basis sets for analytic tasks, they can also be used to speed up SCF algorithms, which was pioneered by Adams. (17−19) The approach was later refined by Lee and Head-Gordon (20,21) and subsequently applied to linear scaling DFT by Berghold et al. (22) Many linear scaling DFT packages have also developed their own adaptive basis set scheme: The CONQUEST program (4) uses local support functions, derived either from plane waves (23) or pseudoatomic orbitals. (24) The ONETEP program (25) uses nonorthogonal generalized Wannier functions (NGWFs). (26) The BigDFT program (27) uses a minimal set of on-the-fly optimized contracted basis functions. (28) Other related methods include numeric atomic orbital (29−32) and localized filter diagonalization. (33−37) Recently Mao et al. used perturbation theory to correct for the error introduced into a DFT calculation by a minimal adaptive basis. (38)

Here, we focus on polarized atomic orbitals (PAOs) and build on the work of Berghold et al. (22) PAOs are linear combinations of atomic orbitals (AOs) on a single atomic center, called primary basis in the following, that minimize the total energy when used as a basis. As a result, small PAO basis sets are usually of good quality and their variational aspect is advantageous when computing properties, such as, e.g., nuclear gradients. While there is no fundamental restriction on the PAO basis size, minimal PAO basis sets have been studied in the most detail and are also the focus of this work. Despite their qualities, the use of PAOs in simulation has been very limited, which we attribute to the difficulty of optimizing these PAOs for each molecular geometry in addition to the implied approximation. Our aim is to exploit the adaptivity of the PAO basis but to avoid this tedious optimization step by a machine learning approach.

The application of machine learning techniques to quantum chemistry is a rather young and very active field. For a recent review see Ramakrishnan and von Lilienfeld. (39) Its aim is to mitigate the high computational cost associated with quantum calculations. Initially, the research focused mostly on predicting observable properties directly from atomic positions. (40) For example, very successful recent applications include the derivation of force fields using neural network descriptions. (41−43) However, such end-to-end predictions pose a very challenging learning problem. As a consequence they require large amounts of training data with increasing system size, and the learning must be repeated for each property. Fortunately, the past decades of research have provided a wealth of quantum chemical insights. One can therefore build onto established approximations, such as DFT, and apply machine learning only to small, but expensive, subparts of the algorithms. Examples are schemes for learning the kinetic energy functional to perform orbital free DFT (44) or learning the electronic density of states at the Fermi energy. (45) Alternatively, machine learning can be used to improve the accuracy of semiempirical methods by making their parameters configuration-dependent. (46,47) In this work, machine learning is used to predict suitable PAO basis sets for a given chemical environment. The present method is thus essentially a standard DFT calculation in a geometry-dependent, optimized basis. Contrary to methods learning specific properties, including the total energy, the present method thus provides access to all properties in DFT calculations.

2. Methods

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2.1. Polarized Atomic Orbitals

The polarized atomic orbital basis is derived from a larger primary basis through linear combinations among functions centered on the same atom. In the following, the notation from Berghold et al. (22) has been adopted. Variables with a tilde denote objects in the smaller PAO basis, while undecorated variables refer to objects in the primary basis. Formally, a PAO basis function φ̃_μ can be written as a weighted sum of primary basis functions φ_ν, where μ and ν belong to the same atom:

(1)

As a consequence of the atom-wise contractions, the transformation matrix B assumes a rectangular block-diagonal structure. Since the primary basis is nonorthogonal, the tensor property of the involved matrices has to be taken into account. (48) Covariant matrices such as the Kohn–Sham matrix H and the overlap matrix S transform differently than the contravariant density matrix P. Hence, two transformation matrices A and B are introduced.

(2)

Notice that A^TB = B^TA =

gives the identity matrix in the PAO basis, while AB^T = BA^T is the projector onto the subspace spanned by the PAO basis within the primary basis. In order to treat the matrices A and B in a simple and unified fashion, they are rewritten as a product of three matrices:

(3)

Due to the atom-wise contractions, the matrices N, U, and Y are block-diagonal as well. The matrices N^±1 transform into the orthonormal basis, in which co- and contravariance coincide and the distinction can be dropped. The unitary matrix U rotates the orthonormalized primary basis functions of each atom such that the desired PAO basis functions become the first m_I components. The selector matrix Y is a rectangular matrix, which selects for each atom the first m_I components. Each atomic block Y_I of the selector matrix is a rectangular identity matrix of dimension n_I·m_I, where n_I denotes the size of the primary basis and m_I the PAO basis size for the given atom I:

(4)

In the formulation from eq 3 the PAO basis is now solely determined by the unitary diagonal blocks of matrix U, without any loss of generality. None of the matrix multiplications required in the transformation is expensive to compute, because the matrices either are block-diagonal or expressed in the small PAO basis.

2.2. Potential Parametrization

The PAO basis is determined by the unitary matrix U. In order to ensure the unitariness of U, it is constructed from the eigenvectors of an auxiliary atomic Hamiltonian H_aux:

(5)

(6)

Effectivly, the lowest m states of the auxiliary Hamiltonian are taken as PAO basis functions. Here, the atomic Hamiltonian H₀ describes the isolated spherical atom, and V is the polarization potential that models the influence of neighboring atoms. In the absence of V the PAO basis will reproduce the isolated atom exactly.

In the context of machine learning a parametrization should also be rotationally invariant. A parametrization without rotational invariance, on the contrary, would require training data for all possible orientations and still bear the risk of introducing artificial torque forces. In this work rotational invariant parameters X are obtained by expanding the potential V into terms V_i that are anchored on the neighboring atoms:

(7)

When the system is rotated, the potential terms V_i change accordingly, while the X_i remain invariant. As a consquence, the optimal X⃗ is independent of the system’s orientation.

Explicit Form of the Potential Terms

The explicit form of the potential terms V_i must be sufficiently flexible to span the relevant subspace. Yet, they must also depend smoothly on atomic positions, be independent of the atom ordering, and be sufficiently local in nature. While we expect that more advanced forms can be found, the following scheme has been employed:

(8)

where

(9)

is a potential that results from spherical Gaussian potentials centered on all neighboring atoms.

(10)

is a projector on shells of basis functions that share a common radial part, and the same angular momentum number l, but have different m quantum numbers. Specializing the terms by different l quantum number and radial part introduces the needed flexibility, while retaining the rotational invariance. Nonlocal pseudopotentials have some resemblance to this scheme. Finally, additional terms are added that just result from the central atom, these are give by

(11)

Trivially degenerated terms with l_u = l_v = 0 are included only once. The weights w_J and exponents β_J could be used for fine-tunning the potential terms. However, througout this work simply w_J = 1, β_J = 2, and k ≤ 2 are used.

2.3. Machine Learning

Machine learning essentially means to approximate a complex, usually unknown, function from a given set of training points. The amount of required training data grows with the difficulty of the learning problem. Therefore, the learning problem should be kept as small as possible by exploiting a priori knowledge about the function’s domain and codomain.

For the co-domain side this simplification is achieved through the previously described potential parametrization. It takes as input a PAO parameter vector and returns the unitary matrix that eventually determines the PAO basis: X⃗ → U.

For the domain side a so-called descriptor is used. It takes as input all atom positions and returns a low-dimensional feature vector that characterizes the chemical environment: {R⃗_I} → q⃗. The search for a good general-purpose descriptor is an ongoing research effort. (49−51) For this work a variant of the descriptor proposed by Sadeghi et al. (52) and inspired by Rupp et al. (53) was chosen. For each atom I an overlap matrix of its surrouding atoms is constructed:

(12)

The eigenvalues of this overlap matrix are then used as descriptor. They are invariant under rotation of the system and permutation of equivalent atoms. The exponent σ_I acts as a screening parameter, while β_J allows the descriptor to distinguish between different atomic species. With these two simplifications in place, the learning machinery only has to perform the remaining mapping of feature vectors onto PAO parameter vectors: q⃗ → X⃗. A number of different learning methods have been proposed, including neural networks (54) and regression. (40) For this work a Gaussian process (GP) (55) was chosen as a relatively simple, but well characterized, ML procedure. As kernel served the popular squared exponential covariance function:

(13)

However, the PAO-ML scheme makes no assumptions about the employed ML algorithm and can be used in combination with any other machine learning method. Finally, a small number of hyper-parameters had to be optimized to achieve good results. While fixing the descriptor screening to σ_I = 1 and the GP noise level to ϵ = 10^–4, the descriptor’s β_J and the GP length scale σ were determined with a derivative-free optimizer as β_O = 0.09, β_H = 0.23, and σ = 0.46 au. For an overview of the entire PAO-ML scheme see Figure 1.

Figure 1. Overview of the PAO-ML scheme for using the potential parametrization and machine learning to calculate the PAO basis from given atomic positions.

2.4. Analytic Forces

In order to run molecular dynamics simulations, accurate forces are essential. Forces are the derivative of the total energy with respect to atom positions. While a variationally optimized PAO basis does not contribute any additional force terms, the same does not hold for approximately optimized PAO basis sets. The advantage of using a pretrained machine learning scheme is the possibility to calculate accurate forces nevertheless.

The PAO-ML scheme contributes two force terms that have to be added to the common DFT forces F⃗_DFT. One term originates from the potential terms V_i from eq 8, which are anchored on neighboring atoms. The other force term arises from the descriptor, which takes atom positions as input. Both additional terms can be calculated analytically:

(14)

2.5. Training Data Acquisition

Training data are obtained by explicitly optimizing the PAO basis for a set of training structures. This poses an intricate minimization problem because the total energy must be minimal with respect to the rotation matrix U and the density matrix P̃. Additionally, the solution has to be self-consistent because the Kohn–Sham matrix H depends on the density matrix. Significant speedups can be obtained from temporarily relaxing the self-consistency by fixing the Kohn–Sham matrix H during an optimization cycle of P̃ and U.

Regularization

For high-quality training data the optimal parameters X⃗ should be unique and vary smoothly with atomic positions. To this end, two carefully designed regularization terms were introduced. The first term is inspired by Tikhonov regularization (56) and penalizes expansion on linearly dependent potential terms in eq 7. The second term is a L₂ regularization for the excess degrees of freedom in the potential V. Together both regularizations can be expressed via the overlap matrix of the potential terms:

(15)

(16)

Througout this work the values α = 10^–6 and β = c = 1 mHa are used.

3. Results

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In this section, the performance of the method for bulk liquid water is explored. This system has a long tradition within the first-principles MD community, as it is both important and difficult to describe. (57) From an energetic point of view, the challenge arises from the delicate balance between directional hydrogen bonding and nondirectional interactions such as van der Waals interactions. (58) The relatively weak interaction can furthermore be influenced by technical aspects, such as basis set quality. Additionally, the liquid is a disordered state, which requires sampling of configurations for a proper description. The disorder makes it also an interesting test case for the ML approach, as the variability of the environment of each molecule can be large.

3.1. Learning Curve

In order to validate the PAO-ML method a learning curve is recorded. To do this, 71 frames containing 64 water molecules, spaced 100 fs apart, are taken from an earlier MP2 MD simulation at ambient temperature and pressure. (59) The first 30 frames are used as training data while the last 30 frames serve as a test set. For each training frame the optimal PAO basis is determined via explicit optimization using DZVP-MOLOPT-GTH as the primary basis. The PAO-ML method is then used to predict basis sets for all test frames based on an increasing number of training frames. The learning curve in Figure 2 shows the standard deviation of the energy difference with respect to the primary basis taken across all 64 water molecules in all 30 test frames. It shows that already a single frame, i.e., 64 molecular geometries, is sufficient training data to yield an error below 0.1 mHa per water molecule. The curve furthermore shows good resilience against overfitting as the error continues to decrease even for large training sets, eventually reaching 0.083 mHa per molecule. In comparison, a traditional minimal (SZV-MOLOPT-GTH) basis set exhibits an error of 0.360 mHa. The learning can at best reach the accuracy of the underlying PAO approximation (0.074 mHa). It is unlikely that the current descriptor would be sufficient to attain that bound.

Figure 2. Learning curve showing the decreasing error of PAO-ML (blue) with increased training set size. For comparison the error of a variationally optimized PAO basis (green) and a traditional minimal SZV-MOLOPT-GTH (red) basis set are shown. With very little training data, the variational limit is approached by the ML method.

3.2. Consistency of Energy and Forces

In order to validate that the forces provided by the PAO-ML implementation are consistent with its energies, a series of short molecular dynamics simulations with different time steps was performed on a water dimer. For the integration of Newton’s law of motion the velocity–Verlet algorithm (60) has been employed, which has an integration error that is of second order in the time step. Figure 3 shows the fluctuations obtained with time steps of 0.4, 0.2, and 0.1 fs. The standard variations extracted from these fluctuation curves are 5.00, 1.23, and 0.31 μHa. This matches nicely the 4-fold decrease expected for a time step halving and confirms the consistency of the PAO-ML implementation.

Figure 3. Energy fluctuation during a series of MD simulation of a water dimer using the PAO-ML scheme. The simulations were conducted in the *NVE* ensemble using different time steps Δt to demonstrate the consistency of the forces and thus the controllability of the integration error.

3.3. PAO-ML Molecular Dynamics of Liquid Water

So far, we have tested the performance of the method based on frames sampled with a traditional approach. More challenging for a ML method is sampling configurations based on predicted energies, in particular, to verify that instabilities and unphysical behavior are absent when the method is given the freedom to explore phase space. To test and verify the performance, molecular dynamics simulations have been performed for 64 molecules of water at experimental density and 300 K, producing trajectories between 20 and 40 ps depending on the method. Besides PAO-ML, a traditional minimal (SZV-MOLOPT-GTH) basis set, a standard basis sets of triple-ζ quality (TZV2P-MOLOPT-GTH), and density functional tight binding (DFTB) (61,62) were used. TZV2P serves as a reference converged result, while SZV and DFTB provide insight in the performance of methods with a basis set size identical to PAO-ML. The oxygen–oxygen pair correlation functions of liquid water are shown in Figure 4. First, these results show that the PAO-ML simulation is similar to the reference TZV2P-MOLOPT-GTH. The position of the first peak in the O–O pair correlation function matches well the one of the experimental reference, which is a significant improvement over the result obtained with a SZV-MOLOPT-GTH basis. Compared to the experiment, overstructuring of the first peak can be mostly attributed to the employed PBE exchange and correlation functional, as it also shows up with the triple-ζ basis set. Comparing to the DFTB results, the difference is most significant near the second solvation shell, which is mostly absent or strongly shifted to larger distances with DFTB, whereas the PAO-ML reproduces the reference results rather accurately.

Figure 4. Shown are oxygen–oxygen pair correlation functions for liquid water at 300 K. As reference the experimental (green, ref (63)) and TZV2P-MOLOPT-GTH basis sets (blue) results are shown. The SZV-MOLOPT-GTH curve (red) and DFTB (orange) are examples of results typically obtained from a minimal basis sets. The adaptive basis set PAO-ML (black) reproduces the reference (TZV2P) better than any of the alternative minimal basis set methods.

3.4. Check for Unphysical Minima

We checked that the PAO-ML potential energy surface is free from unphysical minima. To this end, the 30 test frames of bulk liquid water employed in section 3.1 were geometry optimized using PAO-ML. During this optimization the energy dropped on average by 3.14 mHa per water molecule and each atom moved on average 0.212 Å. Afterward, starting from the PAO-ML minima configuration, a second geometry optimization was performed using the DZVP-MOLOPT-GTH basis. Confirming the physical nature of the PAO-ML minima, the average energy difference between the configurations optimized with PAO-ML and DZVP is a neglibile 0.028 mHa per molecule and the positions changed on average by only 0.014 Å per atom. This confirms the quality of the PAO-ML basis.

3.5. PAO-ML Speedup

The speedup obtained with PAO-ML in the context of linear scaling calculations will be quantified. As a test system, a cubic unit cell containing 6912 water molecules (∼20000 atoms) is employed. The simulations were run on a Cray XC40 using between 64 and 400 nodes each with two CPUs. Table 1 shows the timings for both the full energy calculation and the sparse matrix multiplication part alone. Linear scaling calculations are typically dominated by matrix multiplication, which made it the target of the PAO-ML method. The largest speedup for this part is observable on a few nodes, in which case the PAO-ML scheme yields a 200× wall time reduction. The number of flops actually executed decreases by 4 orders of magnitude from 61.63 × 10¹⁵ flops for DZVP-MOLOPT-GTH to only 4.07 × 10¹² flops for PAO-ML. This speedup can only be partially attributed to the smaller basis set, as the reduction in flops in the dense case would be only 56× (6 vs 23 basis functions per water molecule). This demonstrates the importance of the condition number of the overlap matrix in sparse linear algebra, because the PAO basis exhibits a condition number around 6, which is more than 2 orders of magnitude lower than for the primary DZVP basis set. Due to the large speedup of the matrix multiplication, the Kohn–Sham matrix construction becomes a major contribution to the timings. Nevertheless, on 64 nodes the PAO-ML method speedup the full calculation by 60×. Running on 400 nodes allows one to perform an SCF step in just 3.3 s.

Table 1. Timings (seconds) for the Complete CP2K Energy Calculation (Full) and the Matrix Multiplication Part (mult) on a System Consisting of ∼20000 Atoms, As Described in the Text^a

nodes	64	100	169	256	400
PAO-ML
full	87	58	41	33	24
mult	23	17	13	11	8
DZVP-MOLOPT-GTH
full	5215	2765	1996	1840	1201
mult	5036	2655	1922	1779	1165

^{^a}

The PAO-ML method outperforms a standard DFT run with a DZVP-MOLOPT-GTH basis by a factor of at least 50×.

3.6. Computational Setup

All the calculations were performed using the CP2K software. (64−66) CP2K combines a primary contracted Gaussian basis with an auxiliary plane-wave (PW) basis. This Gaussian and plane-wave (GPW) (67) scheme allows for an efficient linear-scaling calculation of the Kohn–Sham matrix. The auxiliary PW basis is used to calculate the Hartree (Coulomb) energy in linear-scaling time using fast Fourier transforms. The transformation between the Gaussian and PW basis can be computed rapidly. The cutoff for the PW basis set was chosen to be at least 400 Ry in all simulations. While the PW basis is efficient for the Hartree energy, the primary Gaussian basis set is local in nature and allows for a sparse representation of the Kohn–Sham matrix. For the simulations, the Perdew–Burke–Ernzerhof (68) (PBE) exchange and correlation (XC) functional and Goedecker–Teter–Hutter (GTH) pseudopotentials (69) were used. The linear-scaling calculations were performed with the implementation as described in ref (3), which in particular allows for variable sparsity patterns of the matrices. All SCF optimization used the TRS4 (70) algorithm. The SCF optimization was converged to a threshold (EPS_SCF) of 10^–8 or tighter; the filtering threshold EPS_FILTER was to 10^–7 or tighter. The default accuracy (EPS_DEFAULT) was set to 10^–10 or tighter. All simulations were run in double precision. CP2K input files are available in the Supporting Information.

4. Discussion and Conclusions

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In this work, the PAO-ML scheme has been presented and tested. PAO-ML employs machine learning techniques to infer geometry adapted atom centered basis sets from training data in a general way. The scheme can serve as an almost drop-in replacement for conventional basis sets to speedup otherwise standard DFT calculations. The method is similar to semiempirical models based on minimal basis sets but offers improved accuracy and quasi-automatic parametrization.

The PAO-ML approach has the interesting property that the optimal prediction of the parameters makes the energy minimal with respect to these parameters. During the actual simulation, this implies a certain stability of the simulation, as regions with poorly predicted parameters will be avoided due to their higher energy. Ultimately, the whole PAO-ML method provides basis sets that depend in an analytical way on the atomic coordinates. As such, analytic nuclear forces are available, making the method suitable for geometry optimization and energy conserving molecular dynamics simulations.

The performance of the method was demonstrated using MD simulations of liquid water, where it was shown that small basis sets yield structural properties that outperform those of other minimal basis set approaches. Interestingly, very small samples of training data yielded satisfactory results. Compared to the standard approach, the number of flops needed in matrix–matrix multiplications decreased by over 4 orders of magnitude, leading to an effective 60-fold run-time speedup.

Finally, it is clear that the approach presented in this work can be further refined and extended. Some early results have been published in a Ph.D. thesis. (71) Possible directions for improvements include the following: (a) systematic storage and extension of reference data to yield a general purpose machine learned framework for large scale simulation, including a more rigorous quantification of the expected error, which will improve usability; (b) refined parametrization of the PAO basis sets, reducing the number of parameters needed and the enhancing the robustness of the method; (c) nonminimal PAO basis sets; (d) extensions of the method to yield basis sets for fragments or molecules rather than atoms, which will increase accuracy and efficiency; (e) more advanced machine learning techniques and alternative descriptors, which will allow for larger training sets and improved transferability of reference results. These directions should be explored in future work.

Supporting Information

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.8b00378.

Representative input files for most simulations (ZIP)

ct8b00378_si_001.zip (1.84 MB)

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Author Information

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Corresponding Author
- Joost VandeVondele - Department of Materials, ETH Zürich, 8093 Zürich, Switzerland; Swiss National Supercomputing Centre (CSCS), 6900 Lugano, Switzerland; http://orcid.org/0000-0002-0902-5111; Email: [email protected]
Author
- Ole Schütt - Department of Materials, ETH Zürich, 8093 Zürich, Switzerland
Funding
This work was supported by the European Union FP7 with an ERC Starting Grant under Contract No. 277910 and by a grant from the Swiss National Supercomputing Centre (CSCS) under Project ID ch5.
Notes
The authors declare no competing financial interest.

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Criteria for the optimal choice of finite linear combinations of Slater-type orbitals (S.T.O.) adequate to approx. closely S.C.F. M.O. were examd. in the light of computations on HF and other mols. Some aspects of the A.O. (generalized Heitler-London) method were discussed. The inherent limitations on the meaning of charges on atoms in a mol., or of degree of ionic character, were discussed. Unacceptable at. charges are found from an L.C.A.O. mol. orbital population analysis made on S.C.F. M.O. wave functions, approximated by using unbalanced S.T.O. sets, while acceptable results are obtained with judiciously balanced and sufficiently complete S.T.O. sets. The effects of insufficiently complete and unbalanced S.T.O. sets on the computed dipole moments of S.C.F. M.O. wave functions were discussed, with examples.
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Davidson, E. R. Electronic Population Analysis of Molecular Wavefunctions. J. Chem. Phys. 1967, 46, 3320– 3324, DOI: 10.1063/1.1841219
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Electronic population analysis of molecular wavefunctions
Davidson, Ernest Roy
Journal of Chemical Physics (1967), 46 (9), 3320-4CODEN: JCPSA6; ISSN:0021-9606.
A general method for carrying out population analysis of wavefunctions calcd. with arbitrary basis sets is presented. The difficulties in defining orbital populations is discussed. Results are presented for the sequence BF, CO, and N2 which indicate that back-transfer of charge in the π bond cancels the normal transfer of the charge in the σ bond. Contrary to popular belief, the more electroneg. element has the larger degree of hybridization in each case. The amt. of promotion of 2s electrons is greater on the less-electroneg. element, as expected.
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Roby, K. R. Quantum theory of chemical valence concepts. Mol. Phys. 1974, 27, 81– 104, DOI: 10.1080/00268977400100071
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Heinzmann, R.; Ahlrichs, R. Population analysis based on occupation numbers of modified atomic orbitals (MAOs). Theor. Chim. Acta. 1976, 42, 33– 45, DOI: 10.1007/BF00548289
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Population analysis based on occupation numbers of modified atomic orbitals (MAOs)
Heinzmann, Rolf; Ahlrichs, Reinhart
Theoretica Chimica Acta (1976), 42 (1), 33-45CODEN: TCHAAM; ISSN:0040-5744.
A new interpretational scheme is proposed for the anal. of mol. wavefunctions. Starting from the mol. d. operator a minimal set of MAOs is constructed from the requirement that the MOs can be represented as closely as possible by the MAOs. The MAOs are used to compute at. occupation nos. N and shared electron nos. σ. The mol. d. is then discussed in terms of N and σ. This approach has the following advantages: (1) it is generally applicable, (2) the quantities N and σ are virtually basis set independent, (3) the quantities N and σ fulfil the intuitively expected boundary conditions, (4) the simultaneous consideration of N and σ allows for a more reliable description of chem. bonding than consideration of at. charges only.
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Ehrhardt, C.; Ahlrichs, R. Population analysis based on occupation numbers II. Relationship between shared electron numbers and bond energies and characterization of hypervalent contributions. Theor. Chim. Acta. 1985, 68, 231– 245, DOI: 10.1007/BF00526774
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Population analysis based on occupation numbers. II. Relationship between shared electron numbers and bond energies and characterization of hypervalent contributions
Ehrhardt, Claus; Ahlrichs, Reinhart
Theoretica Chimica Acta (1985), 68 (3), 231-45CODEN: TCHAAM; ISSN:0040-5744.
The population anal. based on occupation nos. is briefly reviewed. A new way is proposed to det. modified AOs and to characterize hypervalent contributions. This is discussed in application to the mols. NSF, NSF3, SF6, OPCl, OPCl2, O2PCl, SO2, and ClO4-. The connection was studied between shared electron nos. σ - considered as a measure of covalent bond strength - and bond energies. The σ is a reliable measure of bond energies.
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Reed, A. E.; Curtiss, L. A.; Weinhold, F. Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chem. Rev. 1988, 88, 899– 926, DOI: 10.1021/cr00088a005
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Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint
Reed, Alan E.; Curtiss, Larry A.; Weinhold, Frank
Chemical Reviews (Washington, DC, United States) (1988), 88 (6), 899-926CODEN: CHREAY; ISSN:0009-2665.
A review with >135 refs. is given with discussion of natural bond orbital (NBO) anal., intermol. interaction models based on NBO anal., relation of donor-acceptor and electrostatic models.
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Lee, M. S.; Head-Gordon, M. Extracting polarized atomic orbitals from molecular orbital calculations. Int. J. Quantum Chem. 2000, 76, 169– 184, DOI: 10.1002/(SICI)1097-461X(2000)76:2<169::AID-QUA7>3.0.CO;2-G
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Extracting polarized atomic orbitals from molecular orbital calculations
Lee, Michael S.; Head-Gordon, Martin
International Journal of Quantum Chemistry (2000), 76 (2), 169-184CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
We present a class of methods for extg. a polarized AO (EPAO) minimal basis set from a converged MO calcn. Unlike minimal basis sets obtained from previous approaches, EPAOs rigorously contain the occupied MO space. EPAOs achieve this exactness because their spatial extent is not restricted. Nonetheless, EPAOs are optimally localized with respect to a localization criterion and are essentially single-centered. EPAOs provide an alternative scheme for partitioning the electron d. into at. subspaces. Therefore, they can be used to det. at. and chem. group properties such as charge populations. Since EPAOs provide a compact description to the occupied space, they may have other computational applications such as in local correlation methods. Addnl., the EPAOs give a description of valence antibonding orbitals that may be appropriate for nondynamical electron correlation.
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Mayer, I. Orthogonal effective atomic orbitals in the topological theory of atoms. Can. J. Chem. 1996, 74, 939– 942, DOI: 10.1139/v96-103
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Cioslowski, J.; Liashenko, A. Atomic orbitals in molecules. J. Chem. Phys. 1998, 108, 4405– 4412, DOI: 10.1063/1.475853
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Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg, K. Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals. J. Chem. Phys. 2004, 120, 2629– 2637, DOI: 10.1063/1.1638731
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Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals
Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg, K.
Journal of Chemical Physics (2004), 120 (6), 2629-2637CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A method is presented for expressing the occupied SCF orbitals of a mol. exactly in terms of chem. deformed at. minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The MOs referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact at. SCF orbitals, and the formulation of the deformed "quasiat. minimal-basis-sets" is independent of the calculational AO basis used. The resulting resoln. of MOs in terms of quasiat. minimal basis set orbitals is therefore intrinsic to the exact mol. wave functions. The deformations are analyzed in terms of interat. contributions. The Mulliken population anal. is formulated in terms of the quasiat. minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quant. ab initio formulation of the qual. model of virtual valence orbitals, which are useful for calcg. electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham d. functional theory orbitals and is easily generalized to valence MCSCF orbitals.
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Laikov, D. N. Intrinsic minimal atomic basis representation of molecular electronic wavefunctions. Int. J. Quantum Chem. 2011, 111, 2851– 2867, DOI: 10.1002/qua.22767
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Intrinsic minimal atomic basis representation of molecular electronic wavefunctions
Laikov, Dimitri N.
International Journal of Quantum Chemistry (2011), 111 (12), 2851-2867CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
The problem of finding an effective minimal at. basis that spans the exact occupied wavefunctions of a mean-field theory at a given mol. geometry, which has a no. of special properties, is studied and a new general procedure is developed that (1) solves for a raw minimal set of strongly atom-centered functions-products of spherical harmonics and mol.-optimized radial parts-that approx. span the occupied mol. wavefunctions and minimize the sum of their energies, (2) uses projection operators to get a new set of deformed atom-centered functions that exactly span the occupied space and fall into core and valence subsets, (3) applies a new zero-bond-dipole orthogonalization scheme to the core-orthogonalized valence subset so that for each two-center product of these functions the projection of its dipole moment along the line going through the two centers is zero. The resulting effective minimal at. basis is intrinsic to the mol. problem and does not need a free-atoms input. Some interesting features of the zero-bond-dipole orthogonalization are showing up in the at. population anal. of a diverse set of mols. The new procedure may be useful for the interpretation of electronic structure, for the construction of model Hamiltonians in terms of transferable mol. integrals, and for the definition of active valence space in the treatment of electron correlation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011.
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Knizia, G. Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts. J. Chem. Theory Comput. 2013, 9, 4834– 4843, DOI: 10.1021/ct400687b
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Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts
Knizia, Gerald
Journal of Chemical Theory and Computation (2013), 9 (11), 4834-4843CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
Modern quantum chem. can make quant. predictions on an immense array of chem. systems. However, the interpretation of those predictions is often complicated by the complex wave function expansions used. Here we show that an exceptionally simple algebraic construction allows for defining at. core and valence orbitals, polarized by the mol. environment, which can exactly represent SCF wave functions. This construction provides an unbiased and direct connection between quantum chem. and empirical chem. concepts, and can be used, for example, to calc. the nature of bonding in mols., in chem. terms, from first principles. In particular, we find consistency with electronegativities (χ), C 1s core-level shifts, resonance substituent parameters (σR), Lewis structures, and oxidn. states of transition-metal complexes.
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Adams, W. H. On the Solution of the Hartree-Fock Equation in Terms of Localized Orbitals. J. Chem. Phys. 1961, 34, 89– 102, DOI: 10.1063/1.1731622
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The solution of the Hartree-Fock equation in terms of localized orbitals
Adams, William H.
Journal of Chemical Physics (1961), 34 (), 89-102CODEN: JCPSA6; ISSN:0021-9606.
The Hartree-Fock method was discussed, with emphasis on the transformation properties of the Hartree-Fock equation. This equation can be solved in terms of nonorthogonal one-electron functions. In some cases it may be more convenient to choose such solns. Equations were developed which define the localized one-electron functions. For a system of closedshell atoms or ions, it was suggested that the localized orbitals of each atom or ion can be expanded in terms of functions centered on its nucleus. This was based on the success of the ionic theory of crystals. Because of the symmetry of a crystal, use of the localized orbitals could lead to expressions for the 1st-order, Hartree-Fock d. matrix and the Hartree-Fock energy of a crystal.
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Adams, W. H. Orbital Theories of Electronic Structure. J. Chem. Phys. 1962, 37, 2009– 2018, DOI: 10.1063/1.1733420
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Adams, W. Distortion of interacting atoms and ions. Chem. Phys. Lett. 1971, 12, 295– 298, DOI: 10.1016/0009-2614(71)85068-6
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Lee, M. S.; Head-Gordon, M. Polarized atomic orbitals for self-consistent field electronic structure calculations. J. Chem. Phys. 1997, 107, 9085– 9095, DOI: 10.1063/1.475199
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Polarized atomic orbitals for self-consistent field electronic structure calculations
Lee, Michael S.; Head-Gordon, Martin
Journal of Chemical Physics (1997), 107 (21), 9085-9095CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present a new SCF approach which, given a large "secondary" basis set of AOs, variationally optimizes MOs in terms of a small "primary" basis set of distorted AOs, which are simultaneously optimized. If the primary basis is taken as a minimal basis, the resulting functions are termed polarized AOs (PAO's) because they are valence (or core) AOs which have distorted or polarized in an optimal way for their mol. environment. The PAO's derive their flexibility from the fact that they are formed from atom-centered linear-combinations of the larger set of secondary AOs. The variational conditions satisfied by PAO's are defined, and an iterative method for performing a PAO-SCF calcn. is introduced. We compare the PAO-SCF approach against full SCF calcns. for the energies, dipoles, and mol. geometries of various mols. The PAO's are potentially useful for studying large systems that are currently intractable with larger than minimal basis sets, as well as offering potential interpretative benefits relative to calcns. in extended basis sets.
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Lee, M. S.; Head-Gordon, M. Absolute and relative energies from polarized atomic orbital self-consistent field calculations and a second order correction.: Convergence with size and composition of the secondary basis. Comput. Chem. 2000, 24, 295– 301, DOI: 10.1016/S0097-8485(99)00086-8
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Absolute and relative energies from polarized atomic orbital self-consistent field calculations and a second order correction. Convergence with size and composition of the secondary basis
Lee, Michael S.; Head-Gordon, Martin
Computers & Chemistry (Oxford) (2000), 24 (3,4), 295-301CODEN: COCHDK; ISSN:0097-8485. (Elsevier Science Ltd.)
Polarized AOs (PAO's) are mol.-adapted minimal basis functions that are variationally obtained as an atom-blocked transformation from a conventional extended basis set, as a Hartree-Fock calcn. is performed in the PAO basis. This approxn. yields a higher energy than a HF calcn. performed in the extended basis, although the two results converge to the same limit as the extended basis approaches completeness on each atom. To test the rate of convergence, PAO-HF calcns. were performed using cc-pVXZ and aug-cc-pVXZ basis sets for the water monomer and dimer, and six substituted ethylenes. The results show that the quality of PAO calcns. converges smoothly with X. The use of augmented functions is recommended. To correct a PAO-HF calcn. for residual deficiencies, a noniterative second order correction is introduced. This correction corresponds to an energy-weighted steepest descent step, and substantially improves the quality of PAO energies.
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Berghold, G.; Parrinello, M.; Hutter, J. Polarized atomic orbitals for linear scaling methods. J. Chem. Phys. 2002, 116, 1800– 1810, DOI: 10.1063/1.1431270
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Bowler, D. R.; Miyazaki, T.; Gillan, M. J. Recent progress in linear scaling ab initio electronic structure techniques. J. Phys.: Condens. Matter 2002, 14, 2781, DOI: 10.1088/0953-8984/14/11/303
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Recent progress in linear scaling ab initio electronic structure techniques
Bowler, D. R.; Miyazaki, T.; Gillan, M. J.
Journal of Physics: Condensed Matter (2002), 14 (11), 2781-2798CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
A review. We describe recent progress in developing linear scaling ab initio electronic structure methods, referring in particular to our highly parallel code CONQUEST. After reviewing the state of the field, we present the basic ideas underlying almost all linear scaling methods, and discuss specific practical details of the implementation. We also note the connection between linear scaling methods and embedding techniques.
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Torralba, A. S.; Todorović, M.; Brázdová, V.; Choudhury, R.; Miyazaki, T.; Gillan, M. J.; Bowler, D. R. Pseudo-atomic orbitals as basis sets for the DFT code CONQUEST. J. Phys.: Condens. Matter 2008, 20 (29), 294206, DOI: 10.1088/0953-8984/20/29/294206
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Pseudo-atomic orbitals as basis sets for the O(N) DFT code CONQUEST
Torralba, A. S.; Todorovic, M.; Brazdova, V.; Choudhury, R.; Miyazaki, T.; Gillan, M. J.; Bowler, D. R.
Journal of Physics: Condensed Matter (2008), 20 (29), 294206/1-294206/8CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
Various aspects of the implementation of pseudo-AOs (PAOs) as basis functions for the linear scaling CONQUEST code are presented. Preliminary results for the assignment of a large set of PAOs to a smaller space of support functions are encouraging, and an important related proof on the necessary symmetry of the support functions is shown. Details of the generation and integration schemes for the PAOs are also given.
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Skylaris, C.-K.; Haynes, P. D.; Mostofi, A. A.; Payne, M. C. Introducing ONETEP: Linear-scaling density functional simulations on parallel computers. J. Chem. Phys. 2005, 122, 084119, DOI: 10.1063/1.1839852
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Introducing ONETEP: Linear-scaling density functional simulations on parallel computers
Skylaris, Chris-Kriton; Haynes, Peter D.; Mostofi, Arash A.; Payne, Mike C.
Journal of Chemical Physics (2005), 122 (8), 084119/1-084119/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present ONETEP (order-N electronic total energy package), a d. functional program for parallel computers whose computational cost scales linearly with the no. of atoms and the no. of processors. ONETEP is based on our reformulation of the plane wave pseudopotential method which exploits the electronic localization that is inherent in systems with a nonvanishing band gap. We summarize the theor. developments that enable the direct optimization of strictly localized quantities expressed in terms of a delocalized plane wave basis. These same localized quantities lead us to a phys. way of dividing the computational effort among many processors to allow calcns. to be performed efficiently on parallel supercomputers. We show with examples that ONETEP achieves excellent speedups with increasing nos. of processors and confirm that the time taken by ONETEP as a function of increasing no. of atoms for a given no. of processors is indeed linear. What distinguishes our approach is that the localization is achieved in a controlled and math. consistent manner so that ONETEP obtains the same accuracy as conventional cubic-scaling plane wave approaches and offers fast and stable convergence. We expect that calcns. with ONETEP have the potential to provide quant. theor. predictions for problems involving thousands of atoms such as those often encountered in nanoscience and biophysics.
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Skylaris, C.-K.; Mostofi, A. A.; Haynes, P. D.; Diéguez, O.; Payne, M. C. Nonorthogonal generalized Wannier function pseudopotential plane-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 035119, DOI: 10.1103/PhysRevB.66.035119
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Nonorthogonal generalized Wannier function pseudopotential plane-wave method
Skylaris, Chris-Kriton; Mostofi, Arash A.; Haynes, Peter D.; Dieguez, Oswaldo; Payne, Mike C.
Physical Review B: Condensed Matter and Materials Physics (2002), 66 (3), 035119/1-035119/12CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
We present a reformulation of the plane-wave pseudopotential method for insulators. This new approach allows us to perform d.-functional calcns. by solving directly for "nonorthogonal generalized Wannier functions" rather than extended Bloch states. We outline the theory on which our method is based and present test calcns. on a variety of systems. Comparison of our results with a std. plane-wave code shows that they are equiv. Apart from the usual advantages of the plane-wave approach such as the applicability to any lattice symmetry and the high accuracy, our method also benefits from the localization properties of our functions in real space. The localization of all our functions greatly facilitates the future extension of our method to linear-scaling schemes or calcns. of the elec. polarization of cryst. insulators.
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Mohr, S.; Ratcliff, L. E.; Genovese, L.; Caliste, D.; Boulanger, P.; Goedecker, S.; Deutsch, T. Accurate and efficient linear scaling DFT calculations with universal applicability. Phys. Chem. Chem. Phys. 2015, 17, 31360– 31370, DOI: 10.1039/C5CP00437C
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Accurate and efficient linear scaling DFT calculations with universal applicability
Mohr, Stephan; Ratcliff, Laura E.; Genovese, Luigi; Caliste, Damien; Boulanger, Paul; Goedecker, Stefan; Deutsch, Thierry
Physical Chemistry Chemical Physics (2015), 17 (47), 31360-31370CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
D. functional theory calcns. are computationally extremely expensive for systems contg. many atoms due to their intrinsic cubic scaling. This fact has led to the development of so-called linear scaling algorithms during the last few decades. In this way it becomes possible to perform ab initio calcns. for several tens of thousands of atoms within reasonable walltimes. However, even though the use of linear scaling algorithms is phys. well justified, their implementation often introduces some small errors. Consequently most implementations offering such a linear complexity either yield only a limited accuracy or, if one wants to go beyond this restriction, require a tedious fine tuning of many parameters. In our linear scaling approach within the BigDFT package, we were able to overcome this restriction. Using an ansatz based on localized support functions expressed in an underlying Daubechies wavelet basis - which offers ideal properties for accurate linear scaling calcns. - we obtain an amazingly high accuracy and a universal applicability while still keeping the possibility of simulating large system with linear scaling walltimes requiring only a moderate demand of computing resources. We prove the effectiveness of our method on a wide variety of systems with different boundary conditions, for single-point calcns. as well as for geometry optimizations and mol. dynamics.
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Mohr, S.; Ratcliff, L. E.; Boulanger, P.; Genovese, L.; Caliste, D.; Deutsch, T.; Goedecker, S. Daubechies wavelets for linear scaling density functional theory. J. Chem. Phys. 2014, 140, 204110, DOI: 10.1063/1.4871876
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Daubechies wavelets for linear scaling density functional theory
Mohr, Stephan; Ratcliff, Laura E.; Boulanger, Paul; Genovese, Luigi; Caliste, Damien; Deutsch, Thierry; Goedecker, Stefan
Journal of Chemical Physics (2014), 140 (20), 204110/1-204110/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calcd. in this basis with the same accuracy as if they were calcd. directly in a Daubechies wavelets basis, provided that the amplitude of these adaptively contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of d. functional theory calcns., and can be combined with sparse matrix algebra to obtain linear scaling with respect to the no. of electrons in the system. Calcns. on systems of 10 000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calcns. of neutral and charged systems. (c) 2014 American Institute of Physics.
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Ozaki, T. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 155108, DOI: 10.1103/PhysRevB.67.155108
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Variationally optimized atomic orbitals for large-scale electronic structures
Ozaki, T.
Physical Review B: Condensed Matter and Materials Physics (2003), 67 (15), 155108/1-155108/5CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
A simple and practical method for variationally optimizing numerical AOs used in d. functional calcns. is presented based on the force theorem. The derived equation provides the same procedure for the optimization of AOs as that for the geometry optimization. The optimized orbitals well reproduce convergent results calcd. by a larger no. of unoptimized orbitals. In addn., we demonstrate that the optimized orbitals significantly reduce the computational effort in the geometry optimization, while keeping a high degree of accuracy.
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Ozaki, T.; Kino, H. Numerical atomic basis orbitals from H to Kr. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 195113, DOI: 10.1103/PhysRevB.69.195113
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Numerical atomic basis orbitals from H to Kr
Ozaki, T.; Kino, H.
Physical Review B: Condensed Matter and Materials Physics (2004), 69 (19), 195113/1-195113/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
We present a systematic study for numerical at. basis orbitals ranging from H to Kr, which could be used in large scale O(N) electronic structure calcns. based on d.-functional theories (DFT). The comprehensive investigation of convergence properties with respect to our primitive basis orbitals provides a practical guideline in an optimum choice of basis sets for each element, which well balances the computational efficiency and accuracy. Moreover, starting from the primitive basis orbitals, a simple and practical method for variationally optimizing basis orbitals is presented based on the force theorem, which enables us to maximize both the computational efficiency and accuracy. The optimized orbitals well reproduce convergent results calcd. by a larger no. of primitive orbitals. As illustrations of the orbital optimization, we demonstrate two examples: the geometry optimization coupled with the orbital optimization of a C60 mol. and the preorbital optimization for a specific group such as proteins. They clearly show that the optimized orbitals significantly reduce the computational efforts, while keeping a high degree of accuracy, thus indicating that the optimized orbitals are quite suitable for large scale DFT calcns.
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Junquera, J.; Paz, O.; Sánchez-Portal, D.; Artacho, E. Numerical atomic orbitals for linear-scaling calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 64, 235111, DOI: 10.1103/PhysRevB.64.235111
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Numerical atomic orbitals for linear-scaling calculations
Junquera, Javier; Paz, Oscar; Sanchez-Portal, Daniel; Artacho, Emilio
Physical Review B: Condensed Matter and Materials Physics (2001), 64 (23), 235111/1-235111/9CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
The performance of basis sets made of numerical AOs is explored in d.-functional calcns. of solids and mols. With the aim of optimizing basis quality while maintaining strict localization of the orbitals, as needed for linear-scaling calcns., several schemes have been tried. The best performance is obtained for the basis sets generated according to a new scheme presented here, a flexibilization of previous proposals. Strict localization is maintained while ensuring the continuity of the basis-function deriv. at the cutoff radius. The basis sets are tested vs. converged plane-wave calcns. on a significant variety of systems, including covalent, ionic, and metallic. Satisfactory convergence is obtained for reasonably small basis sizes, with a clear improvement over previous schemes. The transferability of the obtained basis sets is tested in several cases and it is found to be satisfactory as well.
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Basanta, M.; Dappe, Y.; Jelínek, P.; Ortega, J. Optimized atomic-like orbitals for first-principles tight-binding molecular dynamics. Comput. Mater. Sci. 2007, 39, 759– 766, DOI: 10.1016/j.commatsci.2006.09.003
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Optimized atomic-like orbitals for first-principles tight-binding molecular dynamics
Basanta, M. A.; Dappe, Y. J.; Jelinek, P.; Ortega, J.
Computational Materials Science (2007), 39 (4), 759-766CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
We analyze the optimization of at.-like minimal basis sets for the hydrocarbons and for materials made up only of C atoms, e.g. C-nanotubes. In our approach the optimized numerical AOs (NAOs) are obtained as a linear combination of only two primitive NAOs. We find that the optimized basis sets yield an important lowering of the total energy, and bond lengths in very good agreement with the exptl. evidence. Also, we find that a good "universal" minimal basis set for the hydrocarbons and C-only materials can be obtained using this simple method. The approach discussed is a promising tool for the simulation of complex org. materials, beyond the hydrocarbons, using optimized minimal basis sets.
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Rayson, M. J.; Briddon, P. R. Highly efficient method for Kohn-Sham density functional calculations of 500–10000 atom systems. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 205104, DOI: 10.1103/PhysRevB.80.205104
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Highly efficient method for Kohn-Sham density functional calculations of 500-10 000 atom systems
Rayson, M. J.; Briddon, P. R.
Physical Review B: Condensed Matter and Materials Physics (2009), 80 (20), 205104/1-205104/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
A method for the soln. of the self-consistent Kohn-Sham equations using Gaussian-type orbitals is presented. Accurate relative energies and forces are demonstrated to be achievable at a fraction of the computational expense for large systems. With this approach calcns. involving around 1000 atoms can easily be performed with a serial desktop computer and ∼10 000 atom systems are within reach of relatively modest parallel computational resources. The method is applicable to arbitrary systems including metals. The approach generates a minimal basis on the fly while retaining the accuracy of the large underpinning basis set. Convergence of energies and forces are given for clusters as well as cubic cells of silicon and aluminum, for which the formation energies of defects are calcd. in systems up to 8000 and 4000 atoms, resp. For these systems the method exhibits linear scaling with the no. of atoms in the presently important size range of ∼500-3000 atoms.
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Rayson, M. Rapid filtration algorithm to construct a minimal basis on the fly from a primitive Gaussian basis. Comput. Phys. Commun. 2010, 181, 1051– 1056, DOI: 10.1016/j.cpc.2010.02.012
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Rapid filtration algorithm to construct a minimal basis on the fly from a primitive Gaussian basis
Rayson, M. J.
Computer Physics Communications (2010), 181 (6), 1051-1056CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
In a recent work an implementation of filter diagonalization with localization constraints was shown to provide an accurate and highly efficient method to solve the Kohn-Sham equations using a primitive Gaussian basis for systems contg. several thousand electrons. In this work, an alternative filtration algorithm is proposed, based on a rational approxn. to the filtration function, to replace the kernel of this algorithm. This approach is considerably faster than the diagonalization approach used previously and also its performance is largely independent of the filtration temp., aiding a more flexible approach to the construction of filtered basis sets.
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Nakata, A.; Bowler, D. R.; Miyazaki, T. Efficient Calculations with Multisite Local Orbitals in a Large-Scale DFT Code CONQUEST. J. Chem. Theory Comput. 2014, 10, 4813– 4822, DOI: 10.1021/ct5004934
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Efficient Calculations with Multisite Local Orbitals in a Large-Scale DFT Code CONQUEST
Nakata, Ayako; Bowler, David R.; Miyazaki, Tsuyoshi
Journal of Chemical Theory and Computation (2014), 10 (11), 4813-4822CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
Multisite local orbitals, which are formed from linear combinations of pseudoat. orbitals from a target atom and its neighbor atoms, have been introduced in the large-scale d. functional theory calcn. code CONQUEST. Multisite local orbitals correspond to local MOs so that the no. of required local orbitals can be minimal. The multisite support functions are detd. by using the localized filter diagonalization (LFD) method. Two new methods, the double cutoff method and the smoothing method, are introduced to the LFD method to improve efficiency and stability. The Hamiltonian and overlap matrixes with multisite local orbitals are constructed by efficient sparse-matrix multiplications in CONQUEST. The investigation of the calcd. energetic and geometrical properties and band structures of bulk Si, Al, and DNA systems demonstrate the accuracy and the computational efficiency of the present method. The representability of both occupied and unoccupied band structures with the present method has been also confirmed.
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Lin, L.; Lu, J.; Ying, L.; E, W. Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation. J. Comput. Phys. 2012, 231, 2140– 2154, DOI: 10.1016/j.jcp.2011.11.032
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Lin, L.; Lu, J.; Ying, L.; E, W. Optimized local basis set for Kohn-Sham density functional theory. J. Comput. Phys. 2012, 231, 4515– 4529, DOI: 10.1016/j.jcp.2012.03.009
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Mao, Y.; Horn, P. R.; Mardirossian, N.; Head-Gordon, T.; Skylaris, C.-K.; Head-Gordon, M. Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation. J. Chem. Phys. 2016, 145, 044109, DOI: 10.1063/1.4959125
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Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation
Mao, Yuezhi; Horn, Paul R.; Mardirossian, Narbe; Head-Gordon, Teresa; Skylaris, Chris-Kriton; Head-Gordon, Martin
Journal of Chemical Physics (2016), 145 (4), 044109/1-044109/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Recently developed d. functionals have good accuracy for both thermochem. (TC) and non-covalent interactions (NC) if very large AO basis sets are used. To approach the basis set limit with potentially lower computational cost, a new SCF scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each at. site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calcn., similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces <0.15 kcal/mol root-mean-square deviations for most of the tested TC datasets, and <0.1 kcal/mol for most of the NC datasets. The performance of d. functionals near the basis set limit can be even better reproduced. With further improvement to its implementation, MAB-SCF (PC) is a promising lower-cost substitute for conventional large-basis calcns. as a method to approach the basis set limit of modern d. functionals. (c) 2016 American Institute of Physics.
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Ramakrishnan, R.; von Lilienfeld, O. A. Rev. Comput. Chem.; John Wiley & Sons, Inc., 2017; pp 225– 256.
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Hansen, K.; Montavon, G.; Biegler, F.; Fazli, S.; Rupp, M.; Scheffler, M.; von Lilienfeld, O. A.; Tkatchenko, A.; Müller, K.-R. Assessment and Validation of Machine Learning Methods for Predicting Molecular Atomization Energies. J. Chem. Theory Comput. 2013, 9, 3404– 3419, DOI: 10.1021/ct400195d
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Assessment and Validation of Machine Learning Methods for Predicting Molecular Atomization Energies
Hansen, Katja; Montavon, Gregoire; Biegler, Franziska; Fazli, Siamac; Rupp, Matthias; Scheffler, Matthias; von Lilienfeld, O. Anatole; Tkatchenko, Alexandre; Mueller, Klaus-Robert
Journal of Chemical Theory and Computation (2013), 9 (8), 3404-3419CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
The accurate and reliable prediction of properties of mols. typically requires computationally intensive quantum-chem. calcns. Recently, machine learning techniques applied to ab initio calcns. have been proposed as an efficient approach for describing the energies of mols. in their given ground-state structure throughout chem. compd. space. In this paper we outline a no. of established machine learning techniques and investigate the influence of the mol. representation on the methods performance. The best methods achieve prediction errors of 3 kcal/mol for the atomization energies of a wide variety of mols. Rationales for this performance improvement are given together with pitfalls and challenges when applying machine learning approaches to the prediction of quantum-mech. observables.
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Handley, C. M.; Popelier, P. L. A. Potential Energy Surfaces Fitted by Artificial Neural Networks. J. Phys. Chem. A 2010, 114, 3371– 3383, DOI: 10.1021/jp9105585
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Potential Energy Surfaces Fitted by Artificial Neural Networks
Handley, Chris M.; Popelier, Paul L. A.
Journal of Physical Chemistry A (2010), 114 (10), 3371-3383CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
A review. Mol. mechanics is the tool of choice for the modeling of systems that are so large or complex that it is impractical or impossible to model them by ab initio methods. For this reason there is a need for accurate potentials that are able to quickly reproduce ab initio quality results at the fraction of the cost. The interactions within force fields are represented by a no. of functions. Some interactions are well understood and can be represented by simple math. functions while others are not so well understood and their functional form is represented in a simplistic manner or not even known. In the last 20 years there have been the first examples of a new design ethic, where novel and contemporary methods using machine learning, in particular, artificial neural networks, have been used to find the nature of the underlying functions of a force field. Here we appraise what has been achieved over this time and what requires further improvements, while offering some insight and guidance for the development of future force fields.
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Behler, J. Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations. Phys. Chem. Chem. Phys. 2011, 13, 17930– 17955, DOI: 10.1039/c1cp21668f
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Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations
Behler, Joerg
Physical Chemistry Chemical Physics (2011), 13 (40), 17930-17955CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
A review. The accuracy of the results obtained in mol. dynamics or Monte Carlo simulations crucially depends on a reliable description of the at. interactions. A large variety of efficient potentials has been proposed in the literature, but often the optimum functional form is difficult to find and strongly depends on the particular system. In recent years, artificial neural networks (NN) have become a promising new method to construct potentials for a wide range of systems. They offer a no. of advantages: they are very general and applicable to systems as different as small mols., semiconductors and metals; they are numerically very accurate and fast to evaluate; and they can be constructed using any electronic structure method. Significant progress has been made in recent years and a no. of successful applications demonstrate the capabilities of neural network potentials. In this Perspective, the current status of NN potentials is reviewed, and their advantages and limitations are discussed.
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Morawietz, T.; Singraber, A.; Dellago, C.; Behler, J. How van der Waals interactions determine the unique properties of water. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 8368– 8373, DOI: 10.1073/pnas.1602375113
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How van der Waals interactions determine the unique properties of water
Morawietz, Tobias; Singraber, Andreas; Dellago, Christoph; Behler, Joerg
Proceedings of the National Academy of Sciences of the United States of America (2016), 113 (30), 8368-8373CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
Whereas the interactions between water mols. are dominated by strongly directional hydrogen bonds (HBs), it was recently proposed that relatively weak, isotropic van der Waals (vdW) forces are essential for understanding the properties of liq. water and ice. This insight was derived from ab initio computer simulations, which provide an unbiased description of water at the at. level and yield information on the underlying mol. forces. However, the high computational cost of such simulations prevents the systematic investigation of the influence of vdW forces on the thermodn. anomalies of water. Here, we develop efficient ab initio-quality neural network potentials and use them to demonstrate that vdW interactions are crucial for the formation of water's d. max. and its neg. vol. of melting. Both phenomena can be explained by the flexibility of the HB network, which is the result of a delicate balance of weak vdW forces, causing, e.g., a pronounced expansion of the second solvation shell upon cooling that induces the d. max.
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Snyder, J. C.; Rupp, M.; Hansen, K.; Blooston, L.; Mueller, K.-R.; Burke, K. Orbital-free bond breaking via machine learning. J. Chem. Phys. 2013, 139, 224104, DOI: 10.1063/1.4834075
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Orbital-free bond breaking via machine learning
Snyder, John C.; Rupp, Matthias; Hansen, Katja; Blooston, Leo; Mueller, Klaus-Robert; Burke, Kieron
Journal of Chemical Physics (2013), 139 (22), 224104/1-224104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Using a one-dimensional model, we explore the ability of machine learning to approx. the non-interacting kinetic energy d. functional of diatomics. This nonlinear interpolation between Kohn-Sham ref. calcns. can (i) accurately dissoc. a diat., (ii) be systematically improved with increased ref. data and (iii) generate accurate self-consistent densities via a projection method that avoids directions with no data. With relatively few densities, the error due to the interpolation is smaller than typical errors in std. exchange-correlation functionals. (c) 2013 American Institute of Physics.
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Schütt, K. T.; Glawe, H.; Brockherde, F.; Sanna, A.; Müller, K. R.; Gross, E. K. U. How to represent crystal structures for machine learning: Towards fast prediction of electronic properties. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 205118, DOI: 10.1103/PhysRevB.89.205118
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How to represent crystal structures for machine learning: towards fast prediction of electronic properties
Schuett, K. T.; Glawe, H.; Brockherde, F.; Sanna, A.; Mueller, K. R.; Gross, E. K. U.
Physical Review B: Condensed Matter and Materials Physics (2014), 89 (20), 205118/1-205118/5, 5 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
High-throughput d. functional calcns. of solids are highly time-consuming. As an alternative, we propose a machine learning approach for the fast prediction of solid-state properties. To achieve this, local spin-d. approxn. calcns. are used as a training set. We focus on predicting the value of the d. of electronic states at the Fermi energy. We find that conventional representations of the input data, such as the Coulomb matrix, are not suitable for the training of learning machines in the case of periodic solids. We propose a novel crystal structure representation for which learning and competitive prediction accuracies become possible within an unrestricted class of spd systems of arbitrary unit-cell size.
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Dral, P. O.; von Lilienfeld, O. A.; Thiel, W. Machine Learning of Parameters for Accurate Semiempirical Quantum Chemical Calculations. J. Chem. Theory Comput. 2015, 11, 2120– 2125, DOI: 10.1021/acs.jctc.5b00141
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Machine Learning of Parameters for Accurate Semiempirical Quantum Chemical Calculations
Dral, Pavlo O.; von Lilienfeld, O. Anatole; Thiel, Walter
Journal of Chemical Theory and Computation (2015), 11 (5), 2120-2125CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We investigate possible improvements in the accuracy of semiempirical quantum chem. (SQC) methods through the use of machine learning (ML) models for the parameters. For a given class of compds., ML techniques require sufficiently large training sets to develop ML models that can be used for adapting SQC parameters to reflect changes in mol. compn. and geometry. The ML-SQC approach allows the automatic tuning of SQC parameters for individual mols., thereby improving the accuracy without deteriorating transferability to mols. with mol. descriptors very different from those in the training set. The performance of this approach is demonstrated for the semiempirical OM2 method using a set of 6095 constitutional isomers C7H10O2, for which accurate ab initio atomization enthalpies are available. The ML-OM2 results show improved av. accuracy and a much reduced error range compared with those of std. OM2 results, with mean abs. errors in atomization enthalpies dropping from 6.3 to 1.7 kcal/mol. They are also found to be superior to the results from specific OM2 reparameterizations (rOM2) for the same set of isomers. The ML-SQC approach thus holds promise for fast and reasonably accurate high-throughput screening of materials and mols.
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Kranz, J. J.; Kubillus, M.; Ramakrishnan, R.; von Lilienfeld, O. A.; Elstner, M. Generalized Density-Functional Tight-Binding Repulsive Potentials from Unsupervised Machine Learning. J. Chem. Theory Comput. 2018, 14, 2341– 2352, DOI: 10.1021/acs.jctc.7b00933
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Generalized Density-Functional Tight-Binding Repulsive Potentials from Unsupervised Machine Learning
Kranz, Julian J.; Kubillus, Maximilian; Ramakrishnan, Raghunathan; von Lilienfeld, O. Anatole; Elstner, Marcus
Journal of Chemical Theory and Computation (2018), 14 (5), 2341-2352CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We combine the approx. d.-functional tight-binding (DFTB) method with unsupervised machine learning. This allows us to improve transferability and accuracy, make use of large quantum chem. data sets for the parametrization, and efficiently automatize the parametrization process of DFTB. For this purpose, generalized pair-potentials are introduced, where the chem. environment is included during the learning process, leading to more specific effective two-body potentials. We train on energies and forces of equil. and nonequil. structures of 2100 mols., and test on ∼130,000 org. mols. contg. O, N, C, H, and F atoms. Atomization energies of the ref. method can be reproduced within an error of ∼2.6 kcal/mol, indicating drastic improvement over std. DFTB.
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White, C. A.; Maslen, P.; Lee, M. S.; Head-Gordon, M. The tensor properties of energy gradients within a non-orthogonal basis. Chem. Phys. Lett. 1997, 276, 133– 138, DOI: 10.1016/S0009-2614(97)88046-3
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Bartók, A. P.; Kondor, R.; Csányi, G. On representing chemical environments. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 184115, DOI: 10.1103/PhysRevB.87.184115
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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.
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De, S.; Bartok, A. P.; Csanyi, G.; Ceriotti, M. Comparing molecules and solids across structural and alchemical space. Phys. Chem. Chem. Phys. 2016, 18, 13754– 13769, DOI: 10.1039/C6CP00415F
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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.
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Zhu, L.; Amsler, M.; Fuhrer, T.; Schäfer, B.; Faraji, S.; Rostami, S.; Ghasemi, S. A.; Sadeghi, A.; Grauzinyte, M.; Wolverton, C.; Goedecker, S. A fingerprint based metric for measuring similarities of crystalline structures. J. Chem. Phys. 2016, 144, 034203, DOI: 10.1063/1.4940026
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A fingerprint based metric for measuring similarities of crystalline structures
Zhu, Li; Amsler, Maximilian; Fuhrer, Tobias; Schaefer, Bastian; Faraji, Somayeh; Rostami, Samare; Ghasemi, S. Alireza; Sadeghi, Ali; Grauzinyte, Migle; Wolverton, Chris; Goedecker, Stefan
Journal of Chemical Physics (2016), 144 (3), 034203/1-034203/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Measuring similarities/dissimilarities between at. structures is important for the exploration of potential energy landscapes. However, the cell vectors together with the coordinates of the atoms, which are generally used to describe periodic systems, are quantities not directly suitable as fingerprints to distinguish structures. Based on a characterization of the local environment of all atoms in a cell, we introduce crystal fingerprints that can be calcd. easily and define configurational distances between cryst. structures that satisfy the math. properties of a metric. This distance between two configurations is a measure of their similarity/dissimilarity and it allows in particular to distinguish structures. The new method can be a useful tool within various energy landscape exploration schemes, such as min. hopping, random search, swarm intelligence algorithms, and high-throughput screenings. (c) 2016 American Institute of Physics.
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Sadeghi, A.; Ghasemi, S. A.; Schäfer, B.; Mohr, S.; Lill, M. A.; Goedecker, S. Metrics for measuring distances in configuration spaces. J. Chem. Phys. 2013, 139, 184118, DOI: 10.1063/1.4828704
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Metrics for measuring distances in configuration spaces
Sadeghi, Ali; Ghasemi, S. Alireza; Schaefer, Bastian; Mohr, Stephan; Lill, Markus A.; Goedecker, Stefan
Journal of Chemical Physics (2013), 139 (18), 184118/1-184118/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
In order to characterize mol. structures we introduce configurational fingerprint vectors which are counterparts of quantities used exptl. to identify structures. The Euclidean distance between the configurational fingerprint vectors satisfies the properties of a metric and can therefore safely be used to measure dissimilarities between configurations in the high dimensional configuration space. In particular we show that these metrics are a perfect and computationally cheap replacement for the root-mean-square distance (RMSD) when one has to decide whether two noise contaminated configurations are identical or not. We introduce a Monte Carlo approach to obtain the global min. of the RMSD between configurations, which is obtained from a global minimization over all translations, rotations, and permutations of at. indexes. (c) 2013 American Institute of Physics.
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Rupp, M.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning. Phys. Rev. Lett. 2012, 108, 058301, DOI: 10.1103/PhysRevLett.108.058301
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Fast and accurate modeling of molecular atomization energies with machine learning
Rupp, Matthias; Tkatchenko, Alexandre; Mueller, Klaus-Robert; von Lilienfeld, O. Anatole
Physical Review Letters (2012), 108 (5), 058301/1-058301/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
The authors introduce a machine learning model to predict atomization energies of a diverse set of org. mols., based on nuclear charges and at. positions only. The problem of solving the mol. Schrodinger equation is mapped onto a nonlinear statistical regression problem of reduced complexity. Regression models are trained on and compared to atomization energies computed with hybrid d.-functional theory. Cross validation over more than seven thousand org. mols. yields a mean abs. error of ∼10 kcal/mol. Applicability is demonstrated for the prediction of mol. atomization potential energy curves.
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Behler, J. Constructing high-dimensional neural network potentials: A tutorial review. Int. J. Quantum Chem. 2015, 115, 1032– 1050, DOI: 10.1002/qua.24890
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Constructing high-dimensional neural network potentials: A tutorial review
Behler, Joerg
International Journal of Quantum Chemistry (2015), 115 (16), 1032-1050CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
A lot of progress has been made in recent years in the development of atomistic potentials using machine learning (ML) techniques. In contrast to most conventional potentials, which are based on phys. approxns. and simplifications to derive an analytic functional relation between the at. configuration and the potential-energy, ML potentials rely on simple but very flexible math. terms without a direct phys. meaning. Instead, in case of ML potentials the topol. of the potential-energy surface is "learned" by adjusting a no. of parameters with the aim to reproduce a set of ref. electronic structure data as accurately as possible. Due to this bias-free construction, they are applicable to a wide range of systems without changes in their functional form, and a very high accuracy close to the underlying first-principles data can be obtained. Neural network potentials (NNPs), which have first been proposed about two decades ago, are an important class of ML potentials. Although the first NNPs have been restricted to small mols. with only a few degrees of freedom, they are now applicable to high-dimensional systems contg. thousands of atoms, which enables addressing a variety of problems in chem., physics, and materials science. In this tutorial review, the basic ideas of NNPs are presented with a special focus on developing NNPs for high-dimensional condensed systems. A recipe for the construction of these potentials is given and remaining limitations of the method are discussed. © 2015 Wiley Periodicals, Inc.
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Rasmussen, C. E.; Williams, C. K. I. Gaussian Processes for Machine Learning; The MIT Press, 2006; http://www.gaussianprocess.org/gpml/ (accessed Apr. 20, 2018).
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Tikhonov, A. N.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G. Numerical Methods for the Solution of Ill-Posed Problems; Kluwer Academic, 1995.
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Gillan, M. J.; Alfé, D.; Michaelides, A. Perspective: How good is DFT for water. J. Chem. Phys. 2016, 144, 130901, DOI: 10.1063/1.4944633
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Perspective: How good is DFT for water?
Gillan, Michael J.; Alfe, Dario; Michaelides, Angelos
Journal of Chemical Physics (2016), 144 (13), 130901/1-130901/33CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A review. Kohn-Sham d. functional theory (DFT) has become established as an indispensable tool for investigating aq. systems of all kinds, including those important in chem., surface science, biol., and the earth sciences. Nevertheless, many widely used approxns. for the exchange-correlation (XC) functional describe the properties of pure water systems with an accuracy that is not fully satisfactory. The explicit inclusion of dispersion interactions generally improves the description, but there remain large disagreements between the predictions of different dispersion-inclusive methods. We present here a review of DFT work on water clusters, ice structures, and liq. water, with the aim of elucidating how the strengths and weaknesses of different XC approxns. manifest themselves across this variety of water systems. Our review highlights the crucial role of dispersion in describing the delicate balance between compact and extended structures of many different water systems, including the liq. By referring to a wide range of published work, we argue that the correct description of exchange-overlap interactions is also extremely important, so that the choice of semi-local or hybrid functional employed in dispersion-inclusive methods is crucial. The origins and consequences of beyond-2-body errors of approx. XC functionals are noted, and we also discuss the substantial differences between different representations of dispersion. We propose a simple numerical scoring system that rates the performance of different XC functionals in describing water systems, and we suggest possible future developments. (c) 2016 American Institute of Physics.
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Del Ben, M.; Schönherr, M.; Hutter, J.; VandeVondele, J. Bulk Liquid Water at Ambient Temperature and Pressure from MP2 Theory. J. Phys. Chem. Lett. 2013, 4, 3753– 3759, DOI: 10.1021/jz401931f
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Bulk Liquid Water at Ambient Temperature and Pressure from MP2 Theory
Del Ben, Mauro; Schonherr, Mandes; Hutter, Jurg; Vande Vondele, Joost
Journal of Physical Chemistry Letters (2013), 4 (21), 3753-3759CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
MP2 provides a good description of hydrogen bonding in water clusters and includes long-range dispersion interactions without the need to introduce empirical elements in the description of the interat. potential. To assess its performance for bulk liq. water under ambient conditions, an isobaric-isothermal (NpT) Monte Carlo simulation at the second-order Moller-Plesset perturbation theory level (MP2) has been performed. The obtained value of the water d. is excellent (1.02 g/mL), and the calcd. radial distribution functions are in fair agreement with exptl. data. The MP2 results are compared to a few d. functional approxns., including semilocal functionals, hybrid functionals, and functionals including empirical dispersion corrections. These results demonstrate the feasibility of directly sampling the potential energy surface of condensed-phase systems using correlated wave function theory, and their quality paves the way for further applications.
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Del Ben, M.; Hutter, J.; VandeVondele, J. Probing the structural and dynamical properties of liquid water with models including non-local electron correlation. J. Chem. Phys. 2015, 143, 054506, DOI: 10.1063/1.4927325
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Probing the structural and dynamical properties of liquid water with models including non-local electron correlation
Del Ben, Mauro; Hutter, Jurg; Vande Vondele, Joost
Journal of Chemical Physics (2015), 143 (5), 054506/1-054506/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Water is a ubiquitous liq. that displays a wide range of anomalous properties and has a delicate structure that challenges expt. and simulation alike. The various intermol. interactions that play an important role, such as repulsion, polarization, hydrogen bonding, and van der Waals interactions, are often difficult to reproduce faithfully in atomistic models. Here, electronic structure theories including all these interactions at equal footing, which requires the inclusion of non-local electron correlation, are used to describe structure and dynamics of bulk liq. water. Isobaric-isothermal (NpT) ensemble simulations based on the RPA yield excellent d. (0.994 g/mL) and fair radial distribution functions, while various other d. functional approxns. produce scattered results (0.8-1.2 g/mL). Mol. dynamics simulation in the microcanonical (NVE) ensemble based on Moller-Plesset perturbation theory (MP2) yields dynamical properties in the condensed phase, namely, the IR spectrum and diffusion const. At the MP2 and RPA levels of theory, ice is correctly predicted to float on water, resolving one of the anomalies as resulting from a delicate balance between van der Waals and hydrogen bonding interactions. For several properties, obtaining quant. agreement with expt. requires correction for nuclear quantum effects (NQEs), highlighting their importance, for structure, dynamics, and electronic properties. A computed NQE shift of 0.6 eV for the band gap and absorption spectrum illustrates the latter. Giving access to both structure and dynamics of condensed phase systems, non-local electron correlation will increasingly be used to study systems where weak interactions are of paramount importance. (c) 2015 American Institute of Physics.
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Benchmark oxygen-oxygen pair-distribution function of ambient water from x-ray diffraction measurements with a wide Q-range
Skinner, Lawrie B.; Huang, Congcong; Schlesinger, Daniel; Pettersson, Lars G. M.; Nilsson, Anders; Benmore, Chris J.
Journal of Chemical Physics (2013), 138 (7), 074506/1-074506/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Four recent x-ray diffraction measurements of ambient liq. water are reviewed here. Each of these measurements represents a significant development of the x-ray diffraction technique applied to the study of liq. water. Sources of uncertainty from statistical noise, Q-range, Compton scattering, and self-scattering are discussed. The oxygen-hydrogen contribution to the measured x-ray scattering pattern was subtracted using literature data to yield an exptl. detn., with error bars, of the oxygen-oxygen pair-distribution function, gOO(r), which essentially describes the distribution of mol. centers. The extended Q-range and low statistical noise of these measurements has significantly reduced truncation effects and related errors in the gOO(r) functions obtained. From these measurements and error anal., the position and height of the nearest neighbor max. in gOO(r) were found to be 2.80(1) Å and 2.57(5) resp. Numerical data for the coherent differential x-ray scattering cross-section IX(Q), the oxygen-oxygen structure factor SOO(Q), and the derived gOO(r) are provided as benchmarks for calibrating force-fields for water. (c) 2013 American Institute of Physics.
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Wiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (1), 15-25CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)
A review. Cp2k has become a versatile open-source tool for the simulation of complex systems on the nanometer scale. It allows for sampling and exploring potential energy surfaces that can be computed using a variety of empirical and first principles models. Excellent performance for electronic structure calcns. is achieved using novel algorithms implemented for modern and massively parallel hardware. This review briefly summarizes the main capabilities and illustrates with recent applications the science cp2k has enabled in the field of atomistic simulation. WIREs Comput Mol Sci 2014, 4:15-25. doi: 10.1002/wcms.1159 The authors have declared no conflicts of interest in relation to this article. For further resources related to this article, please visit the WIREs website.
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Abstract
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Figure 1
Figure 1. Overview of the PAO-ML scheme for using the potential parametrization and machine learning to calculate the PAO basis from given atomic positions.
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Figure 2
Figure 2. Learning curve showing the decreasing error of PAO-ML (blue) with increased training set size. For comparison the error of a variationally optimized PAO basis (green) and a traditional minimal SZV-MOLOPT-GTH (red) basis set are shown. With very little training data, the variational limit is approached by the ML method.
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Figure 3
Figure 3. Energy fluctuation during a series of MD simulation of a water dimer using the PAO-ML scheme. The simulations were conducted in the NVE ensemble using different time steps Δt to demonstrate the consistency of the forces and thus the controllability of the integration error.
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Figure 4
Figure 4. Shown are oxygen–oxygen pair correlation functions for liquid water at 300 K. As reference the experimental (green, ref (63)) and TZV2P-MOLOPT-GTH basis sets (blue) results are shown. The SZV-MOLOPT-GTH curve (red) and DFTB (orange) are examples of results typically obtained from a minimal basis sets. The adaptive basis set PAO-ML (black) reproduces the reference (TZV2P) better than any of the alternative minimal basis set methods.
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References
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  Bowler, D. R.; Miyazaki, T.
  Journal of Physics: Condensed Matter (2010), 22 (7), 074207/1-0074207/6CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
  An overview of the CONQUEST linear scaling d. functional theory (DFT) code is given, focusing particularly on the scaling behavior on modern high-performance computing platforms. We demonstrate that essentially perfect linear scaling and weak parallel scaling (with fixed no. of atoms per processor core) can be achieved, and that DFT calcns. on millions of atoms are now possible.
5. 5
  Mulliken, R. S. Criteria for the Construction of Good Self-Consistent-Field Molecular Orbital Wave Functions, and the Significance of LCAO-MO Population Analysis. J. Chem. Phys. 1962, 36, 3428– 3439, DOI: 10.1063/1.1732476
  5
  Criteria for the construction of good self-consistent-field molecular orbital wave functions, and the significance of L.C.A.O.M.O. population analysis
  Mulliken, R. S.
  Journal of Chemical Physics (1962), 36 (), 3428-39CODEN: JCPSA6; ISSN:0021-9606.
  Criteria for the optimal choice of finite linear combinations of Slater-type orbitals (S.T.O.) adequate to approx. closely S.C.F. M.O. were examd. in the light of computations on HF and other mols. Some aspects of the A.O. (generalized Heitler-London) method were discussed. The inherent limitations on the meaning of charges on atoms in a mol., or of degree of ionic character, were discussed. Unacceptable at. charges are found from an L.C.A.O. mol. orbital population analysis made on S.C.F. M.O. wave functions, approximated by using unbalanced S.T.O. sets, while acceptable results are obtained with judiciously balanced and sufficiently complete S.T.O. sets. The effects of insufficiently complete and unbalanced S.T.O. sets on the computed dipole moments of S.C.F. M.O. wave functions were discussed, with examples.
6. 6
  Davidson, E. R. Electronic Population Analysis of Molecular Wavefunctions. J. Chem. Phys. 1967, 46, 3320– 3324, DOI: 10.1063/1.1841219
  6
  Electronic population analysis of molecular wavefunctions
  Davidson, Ernest Roy
  Journal of Chemical Physics (1967), 46 (9), 3320-4CODEN: JCPSA6; ISSN:0021-9606.
  A general method for carrying out population analysis of wavefunctions calcd. with arbitrary basis sets is presented. The difficulties in defining orbital populations is discussed. Results are presented for the sequence BF, CO, and N2 which indicate that back-transfer of charge in the π bond cancels the normal transfer of the charge in the σ bond. Contrary to popular belief, the more electroneg. element has the larger degree of hybridization in each case. The amt. of promotion of 2s electrons is greater on the less-electroneg. element, as expected.
7. 7
  Roby, K. R. Quantum theory of chemical valence concepts. Mol. Phys. 1974, 27, 81– 104, DOI: 10.1080/00268977400100071
  There is no corresponding record for this reference.
8. 8
  Heinzmann, R.; Ahlrichs, R. Population analysis based on occupation numbers of modified atomic orbitals (MAOs). Theor. Chim. Acta. 1976, 42, 33– 45, DOI: 10.1007/BF00548289
  8
  Population analysis based on occupation numbers of modified atomic orbitals (MAOs)
  Heinzmann, Rolf; Ahlrichs, Reinhart
  Theoretica Chimica Acta (1976), 42 (1), 33-45CODEN: TCHAAM; ISSN:0040-5744.
  A new interpretational scheme is proposed for the anal. of mol. wavefunctions. Starting from the mol. d. operator a minimal set of MAOs is constructed from the requirement that the MOs can be represented as closely as possible by the MAOs. The MAOs are used to compute at. occupation nos. N and shared electron nos. σ. The mol. d. is then discussed in terms of N and σ. This approach has the following advantages: (1) it is generally applicable, (2) the quantities N and σ are virtually basis set independent, (3) the quantities N and σ fulfil the intuitively expected boundary conditions, (4) the simultaneous consideration of N and σ allows for a more reliable description of chem. bonding than consideration of at. charges only.
9. 9
  Ehrhardt, C.; Ahlrichs, R. Population analysis based on occupation numbers II. Relationship between shared electron numbers and bond energies and characterization of hypervalent contributions. Theor. Chim. Acta. 1985, 68, 231– 245, DOI: 10.1007/BF00526774
  9
  Population analysis based on occupation numbers. II. Relationship between shared electron numbers and bond energies and characterization of hypervalent contributions
  Ehrhardt, Claus; Ahlrichs, Reinhart
  Theoretica Chimica Acta (1985), 68 (3), 231-45CODEN: TCHAAM; ISSN:0040-5744.
  The population anal. based on occupation nos. is briefly reviewed. A new way is proposed to det. modified AOs and to characterize hypervalent contributions. This is discussed in application to the mols. NSF, NSF3, SF6, OPCl, OPCl2, O2PCl, SO2, and ClO4-. The connection was studied between shared electron nos. σ - considered as a measure of covalent bond strength - and bond energies. The σ is a reliable measure of bond energies.
10. 10
  Reed, A. E.; Curtiss, L. A.; Weinhold, F. Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chem. Rev. 1988, 88, 899– 926, DOI: 10.1021/cr00088a005
  10
  Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint
  Reed, Alan E.; Curtiss, Larry A.; Weinhold, Frank
  Chemical Reviews (Washington, DC, United States) (1988), 88 (6), 899-926CODEN: CHREAY; ISSN:0009-2665.
  A review with >135 refs. is given with discussion of natural bond orbital (NBO) anal., intermol. interaction models based on NBO anal., relation of donor-acceptor and electrostatic models.
11. 11
  Lee, M. S.; Head-Gordon, M. Extracting polarized atomic orbitals from molecular orbital calculations. Int. J. Quantum Chem. 2000, 76, 169– 184, DOI: 10.1002/(SICI)1097-461X(2000)76:2<169::AID-QUA7>3.0.CO;2-G
  11
  Extracting polarized atomic orbitals from molecular orbital calculations
  Lee, Michael S.; Head-Gordon, Martin
  International Journal of Quantum Chemistry (2000), 76 (2), 169-184CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
  We present a class of methods for extg. a polarized AO (EPAO) minimal basis set from a converged MO calcn. Unlike minimal basis sets obtained from previous approaches, EPAOs rigorously contain the occupied MO space. EPAOs achieve this exactness because their spatial extent is not restricted. Nonetheless, EPAOs are optimally localized with respect to a localization criterion and are essentially single-centered. EPAOs provide an alternative scheme for partitioning the electron d. into at. subspaces. Therefore, they can be used to det. at. and chem. group properties such as charge populations. Since EPAOs provide a compact description to the occupied space, they may have other computational applications such as in local correlation methods. Addnl., the EPAOs give a description of valence antibonding orbitals that may be appropriate for nondynamical electron correlation.
12. 12
  Mayer, I. Orthogonal effective atomic orbitals in the topological theory of atoms. Can. J. Chem. 1996, 74, 939– 942, DOI: 10.1139/v96-103
  There is no corresponding record for this reference.
13. 13
  Cioslowski, J.; Liashenko, A. Atomic orbitals in molecules. J. Chem. Phys. 1998, 108, 4405– 4412, DOI: 10.1063/1.475853
  There is no corresponding record for this reference.
14. 14
  Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg, K. Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals. J. Chem. Phys. 2004, 120, 2629– 2637, DOI: 10.1063/1.1638731
  14
  Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals
  Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg, K.
  Journal of Chemical Physics (2004), 120 (6), 2629-2637CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A method is presented for expressing the occupied SCF orbitals of a mol. exactly in terms of chem. deformed at. minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The MOs referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact at. SCF orbitals, and the formulation of the deformed "quasiat. minimal-basis-sets" is independent of the calculational AO basis used. The resulting resoln. of MOs in terms of quasiat. minimal basis set orbitals is therefore intrinsic to the exact mol. wave functions. The deformations are analyzed in terms of interat. contributions. The Mulliken population anal. is formulated in terms of the quasiat. minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quant. ab initio formulation of the qual. model of virtual valence orbitals, which are useful for calcg. electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham d. functional theory orbitals and is easily generalized to valence MCSCF orbitals.
15. 15
  Laikov, D. N. Intrinsic minimal atomic basis representation of molecular electronic wavefunctions. Int. J. Quantum Chem. 2011, 111, 2851– 2867, DOI: 10.1002/qua.22767
  15
  Intrinsic minimal atomic basis representation of molecular electronic wavefunctions
  Laikov, Dimitri N.
  International Journal of Quantum Chemistry (2011), 111 (12), 2851-2867CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
  The problem of finding an effective minimal at. basis that spans the exact occupied wavefunctions of a mean-field theory at a given mol. geometry, which has a no. of special properties, is studied and a new general procedure is developed that (1) solves for a raw minimal set of strongly atom-centered functions-products of spherical harmonics and mol.-optimized radial parts-that approx. span the occupied mol. wavefunctions and minimize the sum of their energies, (2) uses projection operators to get a new set of deformed atom-centered functions that exactly span the occupied space and fall into core and valence subsets, (3) applies a new zero-bond-dipole orthogonalization scheme to the core-orthogonalized valence subset so that for each two-center product of these functions the projection of its dipole moment along the line going through the two centers is zero. The resulting effective minimal at. basis is intrinsic to the mol. problem and does not need a free-atoms input. Some interesting features of the zero-bond-dipole orthogonalization are showing up in the at. population anal. of a diverse set of mols. The new procedure may be useful for the interpretation of electronic structure, for the construction of model Hamiltonians in terms of transferable mol. integrals, and for the definition of active valence space in the treatment of electron correlation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011.
16. 16
  Knizia, G. Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts. J. Chem. Theory Comput. 2013, 9, 4834– 4843, DOI: 10.1021/ct400687b
  16
  Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts
  Knizia, Gerald
  Journal of Chemical Theory and Computation (2013), 9 (11), 4834-4843CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  Modern quantum chem. can make quant. predictions on an immense array of chem. systems. However, the interpretation of those predictions is often complicated by the complex wave function expansions used. Here we show that an exceptionally simple algebraic construction allows for defining at. core and valence orbitals, polarized by the mol. environment, which can exactly represent SCF wave functions. This construction provides an unbiased and direct connection between quantum chem. and empirical chem. concepts, and can be used, for example, to calc. the nature of bonding in mols., in chem. terms, from first principles. In particular, we find consistency with electronegativities (χ), C 1s core-level shifts, resonance substituent parameters (σR), Lewis structures, and oxidn. states of transition-metal complexes.
17. 17
  Adams, W. H. On the Solution of the Hartree-Fock Equation in Terms of Localized Orbitals. J. Chem. Phys. 1961, 34, 89– 102, DOI: 10.1063/1.1731622
  17
  The solution of the Hartree-Fock equation in terms of localized orbitals
  Adams, William H.
  Journal of Chemical Physics (1961), 34 (), 89-102CODEN: JCPSA6; ISSN:0021-9606.
  The Hartree-Fock method was discussed, with emphasis on the transformation properties of the Hartree-Fock equation. This equation can be solved in terms of nonorthogonal one-electron functions. In some cases it may be more convenient to choose such solns. Equations were developed which define the localized one-electron functions. For a system of closedshell atoms or ions, it was suggested that the localized orbitals of each atom or ion can be expanded in terms of functions centered on its nucleus. This was based on the success of the ionic theory of crystals. Because of the symmetry of a crystal, use of the localized orbitals could lead to expressions for the 1st-order, Hartree-Fock d. matrix and the Hartree-Fock energy of a crystal.
18. 18
  Adams, W. H. Orbital Theories of Electronic Structure. J. Chem. Phys. 1962, 37, 2009– 2018, DOI: 10.1063/1.1733420
  There is no corresponding record for this reference.
19. 19
  Adams, W. Distortion of interacting atoms and ions. Chem. Phys. Lett. 1971, 12, 295– 298, DOI: 10.1016/0009-2614(71)85068-6
  There is no corresponding record for this reference.
20. 20
  Lee, M. S.; Head-Gordon, M. Polarized atomic orbitals for self-consistent field electronic structure calculations. J. Chem. Phys. 1997, 107, 9085– 9095, DOI: 10.1063/1.475199
  20
  Polarized atomic orbitals for self-consistent field electronic structure calculations
  Lee, Michael S.; Head-Gordon, Martin
  Journal of Chemical Physics (1997), 107 (21), 9085-9095CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present a new SCF approach which, given a large "secondary" basis set of AOs, variationally optimizes MOs in terms of a small "primary" basis set of distorted AOs, which are simultaneously optimized. If the primary basis is taken as a minimal basis, the resulting functions are termed polarized AOs (PAO's) because they are valence (or core) AOs which have distorted or polarized in an optimal way for their mol. environment. The PAO's derive their flexibility from the fact that they are formed from atom-centered linear-combinations of the larger set of secondary AOs. The variational conditions satisfied by PAO's are defined, and an iterative method for performing a PAO-SCF calcn. is introduced. We compare the PAO-SCF approach against full SCF calcns. for the energies, dipoles, and mol. geometries of various mols. The PAO's are potentially useful for studying large systems that are currently intractable with larger than minimal basis sets, as well as offering potential interpretative benefits relative to calcns. in extended basis sets.
21. 21
  Lee, M. S.; Head-Gordon, M. Absolute and relative energies from polarized atomic orbital self-consistent field calculations and a second order correction.: Convergence with size and composition of the secondary basis. Comput. Chem. 2000, 24, 295– 301, DOI: 10.1016/S0097-8485(99)00086-8
  21
  Absolute and relative energies from polarized atomic orbital self-consistent field calculations and a second order correction. Convergence with size and composition of the secondary basis
  Lee, Michael S.; Head-Gordon, Martin
  Computers & Chemistry (Oxford) (2000), 24 (3,4), 295-301CODEN: COCHDK; ISSN:0097-8485. (Elsevier Science Ltd.)
  Polarized AOs (PAO's) are mol.-adapted minimal basis functions that are variationally obtained as an atom-blocked transformation from a conventional extended basis set, as a Hartree-Fock calcn. is performed in the PAO basis. This approxn. yields a higher energy than a HF calcn. performed in the extended basis, although the two results converge to the same limit as the extended basis approaches completeness on each atom. To test the rate of convergence, PAO-HF calcns. were performed using cc-pVXZ and aug-cc-pVXZ basis sets for the water monomer and dimer, and six substituted ethylenes. The results show that the quality of PAO calcns. converges smoothly with X. The use of augmented functions is recommended. To correct a PAO-HF calcn. for residual deficiencies, a noniterative second order correction is introduced. This correction corresponds to an energy-weighted steepest descent step, and substantially improves the quality of PAO energies.
22. 22
  Berghold, G.; Parrinello, M.; Hutter, J. Polarized atomic orbitals for linear scaling methods. J. Chem. Phys. 2002, 116, 1800– 1810, DOI: 10.1063/1.1431270
  There is no corresponding record for this reference.
23. 23
  Bowler, D. R.; Miyazaki, T.; Gillan, M. J. Recent progress in linear scaling ab initio electronic structure techniques. J. Phys.: Condens. Matter 2002, 14, 2781, DOI: 10.1088/0953-8984/14/11/303
  23
  Recent progress in linear scaling ab initio electronic structure techniques
  Bowler, D. R.; Miyazaki, T.; Gillan, M. J.
  Journal of Physics: Condensed Matter (2002), 14 (11), 2781-2798CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
  A review. We describe recent progress in developing linear scaling ab initio electronic structure methods, referring in particular to our highly parallel code CONQUEST. After reviewing the state of the field, we present the basic ideas underlying almost all linear scaling methods, and discuss specific practical details of the implementation. We also note the connection between linear scaling methods and embedding techniques.
24. 24
  Torralba, A. S.; Todorović, M.; Brázdová, V.; Choudhury, R.; Miyazaki, T.; Gillan, M. J.; Bowler, D. R. Pseudo-atomic orbitals as basis sets for the DFT code CONQUEST. J. Phys.: Condens. Matter 2008, 20 (29), 294206, DOI: 10.1088/0953-8984/20/29/294206
  24
  Pseudo-atomic orbitals as basis sets for the O(N) DFT code CONQUEST
  Torralba, A. S.; Todorovic, M.; Brazdova, V.; Choudhury, R.; Miyazaki, T.; Gillan, M. J.; Bowler, D. R.
  Journal of Physics: Condensed Matter (2008), 20 (29), 294206/1-294206/8CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
  Various aspects of the implementation of pseudo-AOs (PAOs) as basis functions for the linear scaling CONQUEST code are presented. Preliminary results for the assignment of a large set of PAOs to a smaller space of support functions are encouraging, and an important related proof on the necessary symmetry of the support functions is shown. Details of the generation and integration schemes for the PAOs are also given.
25. 25
  Skylaris, C.-K.; Haynes, P. D.; Mostofi, A. A.; Payne, M. C. Introducing ONETEP: Linear-scaling density functional simulations on parallel computers. J. Chem. Phys. 2005, 122, 084119, DOI: 10.1063/1.1839852
  25
  Introducing ONETEP: Linear-scaling density functional simulations on parallel computers
  Skylaris, Chris-Kriton; Haynes, Peter D.; Mostofi, Arash A.; Payne, Mike C.
  Journal of Chemical Physics (2005), 122 (8), 084119/1-084119/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present ONETEP (order-N electronic total energy package), a d. functional program for parallel computers whose computational cost scales linearly with the no. of atoms and the no. of processors. ONETEP is based on our reformulation of the plane wave pseudopotential method which exploits the electronic localization that is inherent in systems with a nonvanishing band gap. We summarize the theor. developments that enable the direct optimization of strictly localized quantities expressed in terms of a delocalized plane wave basis. These same localized quantities lead us to a phys. way of dividing the computational effort among many processors to allow calcns. to be performed efficiently on parallel supercomputers. We show with examples that ONETEP achieves excellent speedups with increasing nos. of processors and confirm that the time taken by ONETEP as a function of increasing no. of atoms for a given no. of processors is indeed linear. What distinguishes our approach is that the localization is achieved in a controlled and math. consistent manner so that ONETEP obtains the same accuracy as conventional cubic-scaling plane wave approaches and offers fast and stable convergence. We expect that calcns. with ONETEP have the potential to provide quant. theor. predictions for problems involving thousands of atoms such as those often encountered in nanoscience and biophysics.
26. 26
  Skylaris, C.-K.; Mostofi, A. A.; Haynes, P. D.; Diéguez, O.; Payne, M. C. Nonorthogonal generalized Wannier function pseudopotential plane-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 035119, DOI: 10.1103/PhysRevB.66.035119
  26
  Nonorthogonal generalized Wannier function pseudopotential plane-wave method
  Skylaris, Chris-Kriton; Mostofi, Arash A.; Haynes, Peter D.; Dieguez, Oswaldo; Payne, Mike C.
  Physical Review B: Condensed Matter and Materials Physics (2002), 66 (3), 035119/1-035119/12CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  We present a reformulation of the plane-wave pseudopotential method for insulators. This new approach allows us to perform d.-functional calcns. by solving directly for "nonorthogonal generalized Wannier functions" rather than extended Bloch states. We outline the theory on which our method is based and present test calcns. on a variety of systems. Comparison of our results with a std. plane-wave code shows that they are equiv. Apart from the usual advantages of the plane-wave approach such as the applicability to any lattice symmetry and the high accuracy, our method also benefits from the localization properties of our functions in real space. The localization of all our functions greatly facilitates the future extension of our method to linear-scaling schemes or calcns. of the elec. polarization of cryst. insulators.
27. 27
  Mohr, S.; Ratcliff, L. E.; Genovese, L.; Caliste, D.; Boulanger, P.; Goedecker, S.; Deutsch, T. Accurate and efficient linear scaling DFT calculations with universal applicability. Phys. Chem. Chem. Phys. 2015, 17, 31360– 31370, DOI: 10.1039/C5CP00437C
  27
  Accurate and efficient linear scaling DFT calculations with universal applicability
  Mohr, Stephan; Ratcliff, Laura E.; Genovese, Luigi; Caliste, Damien; Boulanger, Paul; Goedecker, Stefan; Deutsch, Thierry
  Physical Chemistry Chemical Physics (2015), 17 (47), 31360-31370CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  D. functional theory calcns. are computationally extremely expensive for systems contg. many atoms due to their intrinsic cubic scaling. This fact has led to the development of so-called linear scaling algorithms during the last few decades. In this way it becomes possible to perform ab initio calcns. for several tens of thousands of atoms within reasonable walltimes. However, even though the use of linear scaling algorithms is phys. well justified, their implementation often introduces some small errors. Consequently most implementations offering such a linear complexity either yield only a limited accuracy or, if one wants to go beyond this restriction, require a tedious fine tuning of many parameters. In our linear scaling approach within the BigDFT package, we were able to overcome this restriction. Using an ansatz based on localized support functions expressed in an underlying Daubechies wavelet basis - which offers ideal properties for accurate linear scaling calcns. - we obtain an amazingly high accuracy and a universal applicability while still keeping the possibility of simulating large system with linear scaling walltimes requiring only a moderate demand of computing resources. We prove the effectiveness of our method on a wide variety of systems with different boundary conditions, for single-point calcns. as well as for geometry optimizations and mol. dynamics.
28. 28
  Mohr, S.; Ratcliff, L. E.; Boulanger, P.; Genovese, L.; Caliste, D.; Deutsch, T.; Goedecker, S. Daubechies wavelets for linear scaling density functional theory. J. Chem. Phys. 2014, 140, 204110, DOI: 10.1063/1.4871876
  28
  Daubechies wavelets for linear scaling density functional theory
  Mohr, Stephan; Ratcliff, Laura E.; Boulanger, Paul; Genovese, Luigi; Caliste, Damien; Deutsch, Thierry; Goedecker, Stefan
  Journal of Chemical Physics (2014), 140 (20), 204110/1-204110/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calcd. in this basis with the same accuracy as if they were calcd. directly in a Daubechies wavelets basis, provided that the amplitude of these adaptively contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of d. functional theory calcns., and can be combined with sparse matrix algebra to obtain linear scaling with respect to the no. of electrons in the system. Calcns. on systems of 10 000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calcns. of neutral and charged systems. (c) 2014 American Institute of Physics.
29. 29
  Ozaki, T. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 155108, DOI: 10.1103/PhysRevB.67.155108
  29
  Variationally optimized atomic orbitals for large-scale electronic structures
  Ozaki, T.
  Physical Review B: Condensed Matter and Materials Physics (2003), 67 (15), 155108/1-155108/5CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  A simple and practical method for variationally optimizing numerical AOs used in d. functional calcns. is presented based on the force theorem. The derived equation provides the same procedure for the optimization of AOs as that for the geometry optimization. The optimized orbitals well reproduce convergent results calcd. by a larger no. of unoptimized orbitals. In addn., we demonstrate that the optimized orbitals significantly reduce the computational effort in the geometry optimization, while keeping a high degree of accuracy.
30. 30
  Ozaki, T.; Kino, H. Numerical atomic basis orbitals from H to Kr. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 195113, DOI: 10.1103/PhysRevB.69.195113
  30
  Numerical atomic basis orbitals from H to Kr
  Ozaki, T.; Kino, H.
  Physical Review B: Condensed Matter and Materials Physics (2004), 69 (19), 195113/1-195113/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  We present a systematic study for numerical at. basis orbitals ranging from H to Kr, which could be used in large scale O(N) electronic structure calcns. based on d.-functional theories (DFT). The comprehensive investigation of convergence properties with respect to our primitive basis orbitals provides a practical guideline in an optimum choice of basis sets for each element, which well balances the computational efficiency and accuracy. Moreover, starting from the primitive basis orbitals, a simple and practical method for variationally optimizing basis orbitals is presented based on the force theorem, which enables us to maximize both the computational efficiency and accuracy. The optimized orbitals well reproduce convergent results calcd. by a larger no. of primitive orbitals. As illustrations of the orbital optimization, we demonstrate two examples: the geometry optimization coupled with the orbital optimization of a C60 mol. and the preorbital optimization for a specific group such as proteins. They clearly show that the optimized orbitals significantly reduce the computational efforts, while keeping a high degree of accuracy, thus indicating that the optimized orbitals are quite suitable for large scale DFT calcns.
31. 31
  Junquera, J.; Paz, O.; Sánchez-Portal, D.; Artacho, E. Numerical atomic orbitals for linear-scaling calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 64, 235111, DOI: 10.1103/PhysRevB.64.235111
  31
  Numerical atomic orbitals for linear-scaling calculations
  Junquera, Javier; Paz, Oscar; Sanchez-Portal, Daniel; Artacho, Emilio
  Physical Review B: Condensed Matter and Materials Physics (2001), 64 (23), 235111/1-235111/9CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  The performance of basis sets made of numerical AOs is explored in d.-functional calcns. of solids and mols. With the aim of optimizing basis quality while maintaining strict localization of the orbitals, as needed for linear-scaling calcns., several schemes have been tried. The best performance is obtained for the basis sets generated according to a new scheme presented here, a flexibilization of previous proposals. Strict localization is maintained while ensuring the continuity of the basis-function deriv. at the cutoff radius. The basis sets are tested vs. converged plane-wave calcns. on a significant variety of systems, including covalent, ionic, and metallic. Satisfactory convergence is obtained for reasonably small basis sizes, with a clear improvement over previous schemes. The transferability of the obtained basis sets is tested in several cases and it is found to be satisfactory as well.
32. 32
  Basanta, M.; Dappe, Y.; Jelínek, P.; Ortega, J. Optimized atomic-like orbitals for first-principles tight-binding molecular dynamics. Comput. Mater. Sci. 2007, 39, 759– 766, DOI: 10.1016/j.commatsci.2006.09.003
  32
  Optimized atomic-like orbitals for first-principles tight-binding molecular dynamics
  Basanta, M. A.; Dappe, Y. J.; Jelinek, P.; Ortega, J.
  Computational Materials Science (2007), 39 (4), 759-766CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
  We analyze the optimization of at.-like minimal basis sets for the hydrocarbons and for materials made up only of C atoms, e.g. C-nanotubes. In our approach the optimized numerical AOs (NAOs) are obtained as a linear combination of only two primitive NAOs. We find that the optimized basis sets yield an important lowering of the total energy, and bond lengths in very good agreement with the exptl. evidence. Also, we find that a good "universal" minimal basis set for the hydrocarbons and C-only materials can be obtained using this simple method. The approach discussed is a promising tool for the simulation of complex org. materials, beyond the hydrocarbons, using optimized minimal basis sets.
33. 33
  Rayson, M. J.; Briddon, P. R. Highly efficient method for Kohn-Sham density functional calculations of 500–10000 atom systems. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 205104, DOI: 10.1103/PhysRevB.80.205104
  33
  Highly efficient method for Kohn-Sham density functional calculations of 500-10 000 atom systems
  Rayson, M. J.; Briddon, P. R.
  Physical Review B: Condensed Matter and Materials Physics (2009), 80 (20), 205104/1-205104/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
  A method for the soln. of the self-consistent Kohn-Sham equations using Gaussian-type orbitals is presented. Accurate relative energies and forces are demonstrated to be achievable at a fraction of the computational expense for large systems. With this approach calcns. involving around 1000 atoms can easily be performed with a serial desktop computer and ∼10 000 atom systems are within reach of relatively modest parallel computational resources. The method is applicable to arbitrary systems including metals. The approach generates a minimal basis on the fly while retaining the accuracy of the large underpinning basis set. Convergence of energies and forces are given for clusters as well as cubic cells of silicon and aluminum, for which the formation energies of defects are calcd. in systems up to 8000 and 4000 atoms, resp. For these systems the method exhibits linear scaling with the no. of atoms in the presently important size range of ∼500-3000 atoms.
34. 34
  Rayson, M. Rapid filtration algorithm to construct a minimal basis on the fly from a primitive Gaussian basis. Comput. Phys. Commun. 2010, 181, 1051– 1056, DOI: 10.1016/j.cpc.2010.02.012
  34
  Rapid filtration algorithm to construct a minimal basis on the fly from a primitive Gaussian basis
  Rayson, M. J.
  Computer Physics Communications (2010), 181 (6), 1051-1056CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
  In a recent work an implementation of filter diagonalization with localization constraints was shown to provide an accurate and highly efficient method to solve the Kohn-Sham equations using a primitive Gaussian basis for systems contg. several thousand electrons. In this work, an alternative filtration algorithm is proposed, based on a rational approxn. to the filtration function, to replace the kernel of this algorithm. This approach is considerably faster than the diagonalization approach used previously and also its performance is largely independent of the filtration temp., aiding a more flexible approach to the construction of filtered basis sets.
35. 35
  Nakata, A.; Bowler, D. R.; Miyazaki, T. Efficient Calculations with Multisite Local Orbitals in a Large-Scale DFT Code CONQUEST. J. Chem. Theory Comput. 2014, 10, 4813– 4822, DOI: 10.1021/ct5004934
  35
  Efficient Calculations with Multisite Local Orbitals in a Large-Scale DFT Code CONQUEST
  Nakata, Ayako; Bowler, David R.; Miyazaki, Tsuyoshi
  Journal of Chemical Theory and Computation (2014), 10 (11), 4813-4822CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  Multisite local orbitals, which are formed from linear combinations of pseudoat. orbitals from a target atom and its neighbor atoms, have been introduced in the large-scale d. functional theory calcn. code CONQUEST. Multisite local orbitals correspond to local MOs so that the no. of required local orbitals can be minimal. The multisite support functions are detd. by using the localized filter diagonalization (LFD) method. Two new methods, the double cutoff method and the smoothing method, are introduced to the LFD method to improve efficiency and stability. The Hamiltonian and overlap matrixes with multisite local orbitals are constructed by efficient sparse-matrix multiplications in CONQUEST. The investigation of the calcd. energetic and geometrical properties and band structures of bulk Si, Al, and DNA systems demonstrate the accuracy and the computational efficiency of the present method. The representability of both occupied and unoccupied band structures with the present method has been also confirmed.
36. 36
  Lin, L.; Lu, J.; Ying, L.; E, W. Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation. J. Comput. Phys. 2012, 231, 2140– 2154, DOI: 10.1016/j.jcp.2011.11.032
  There is no corresponding record for this reference.
37. 37
  Lin, L.; Lu, J.; Ying, L.; E, W. Optimized local basis set for Kohn-Sham density functional theory. J. Comput. Phys. 2012, 231, 4515– 4529, DOI: 10.1016/j.jcp.2012.03.009
  There is no corresponding record for this reference.
38. 38
  Mao, Y.; Horn, P. R.; Mardirossian, N.; Head-Gordon, T.; Skylaris, C.-K.; Head-Gordon, M. Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation. J. Chem. Phys. 2016, 145, 044109, DOI: 10.1063/1.4959125
  38
  Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation
  Mao, Yuezhi; Horn, Paul R.; Mardirossian, Narbe; Head-Gordon, Teresa; Skylaris, Chris-Kriton; Head-Gordon, Martin
  Journal of Chemical Physics (2016), 145 (4), 044109/1-044109/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Recently developed d. functionals have good accuracy for both thermochem. (TC) and non-covalent interactions (NC) if very large AO basis sets are used. To approach the basis set limit with potentially lower computational cost, a new SCF scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each at. site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calcn., similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces <0.15 kcal/mol root-mean-square deviations for most of the tested TC datasets, and <0.1 kcal/mol for most of the NC datasets. The performance of d. functionals near the basis set limit can be even better reproduced. With further improvement to its implementation, MAB-SCF (PC) is a promising lower-cost substitute for conventional large-basis calcns. as a method to approach the basis set limit of modern d. functionals. (c) 2016 American Institute of Physics.
39. 39
  Ramakrishnan, R.; von Lilienfeld, O. A. Rev. Comput. Chem.; John Wiley & Sons, Inc., 2017; pp 225– 256.
  There is no corresponding record for this reference.
40. 40
  Hansen, K.; Montavon, G.; Biegler, F.; Fazli, S.; Rupp, M.; Scheffler, M.; von Lilienfeld, O. A.; Tkatchenko, A.; Müller, K.-R. Assessment and Validation of Machine Learning Methods for Predicting Molecular Atomization Energies. J. Chem. Theory Comput. 2013, 9, 3404– 3419, DOI: 10.1021/ct400195d
  40
  Assessment and Validation of Machine Learning Methods for Predicting Molecular Atomization Energies
  Hansen, Katja; Montavon, Gregoire; Biegler, Franziska; Fazli, Siamac; Rupp, Matthias; Scheffler, Matthias; von Lilienfeld, O. Anatole; Tkatchenko, Alexandre; Mueller, Klaus-Robert
  Journal of Chemical Theory and Computation (2013), 9 (8), 3404-3419CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  The accurate and reliable prediction of properties of mols. typically requires computationally intensive quantum-chem. calcns. Recently, machine learning techniques applied to ab initio calcns. have been proposed as an efficient approach for describing the energies of mols. in their given ground-state structure throughout chem. compd. space. In this paper we outline a no. of established machine learning techniques and investigate the influence of the mol. representation on the methods performance. The best methods achieve prediction errors of 3 kcal/mol for the atomization energies of a wide variety of mols. Rationales for this performance improvement are given together with pitfalls and challenges when applying machine learning approaches to the prediction of quantum-mech. observables.
41. 41
  Handley, C. M.; Popelier, P. L. A. Potential Energy Surfaces Fitted by Artificial Neural Networks. J. Phys. Chem. A 2010, 114, 3371– 3383, DOI: 10.1021/jp9105585
  41
  Potential Energy Surfaces Fitted by Artificial Neural Networks
  Handley, Chris M.; Popelier, Paul L. A.
  Journal of Physical Chemistry A (2010), 114 (10), 3371-3383CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  A review. Mol. mechanics is the tool of choice for the modeling of systems that are so large or complex that it is impractical or impossible to model them by ab initio methods. For this reason there is a need for accurate potentials that are able to quickly reproduce ab initio quality results at the fraction of the cost. The interactions within force fields are represented by a no. of functions. Some interactions are well understood and can be represented by simple math. functions while others are not so well understood and their functional form is represented in a simplistic manner or not even known. In the last 20 years there have been the first examples of a new design ethic, where novel and contemporary methods using machine learning, in particular, artificial neural networks, have been used to find the nature of the underlying functions of a force field. Here we appraise what has been achieved over this time and what requires further improvements, while offering some insight and guidance for the development of future force fields.
42. 42
  Behler, J. Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations. Phys. Chem. Chem. Phys. 2011, 13, 17930– 17955, DOI: 10.1039/c1cp21668f
  42
  Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations
  Behler, Joerg
  Physical Chemistry Chemical Physics (2011), 13 (40), 17930-17955CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  A review. The accuracy of the results obtained in mol. dynamics or Monte Carlo simulations crucially depends on a reliable description of the at. interactions. A large variety of efficient potentials has been proposed in the literature, but often the optimum functional form is difficult to find and strongly depends on the particular system. In recent years, artificial neural networks (NN) have become a promising new method to construct potentials for a wide range of systems. They offer a no. of advantages: they are very general and applicable to systems as different as small mols., semiconductors and metals; they are numerically very accurate and fast to evaluate; and they can be constructed using any electronic structure method. Significant progress has been made in recent years and a no. of successful applications demonstrate the capabilities of neural network potentials. In this Perspective, the current status of NN potentials is reviewed, and their advantages and limitations are discussed.
43. 43
  Morawietz, T.; Singraber, A.; Dellago, C.; Behler, J. How van der Waals interactions determine the unique properties of water. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 8368– 8373, DOI: 10.1073/pnas.1602375113
  43
  How van der Waals interactions determine the unique properties of water
  Morawietz, Tobias; Singraber, Andreas; Dellago, Christoph; Behler, Joerg
  Proceedings of the National Academy of Sciences of the United States of America (2016), 113 (30), 8368-8373CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
  Whereas the interactions between water mols. are dominated by strongly directional hydrogen bonds (HBs), it was recently proposed that relatively weak, isotropic van der Waals (vdW) forces are essential for understanding the properties of liq. water and ice. This insight was derived from ab initio computer simulations, which provide an unbiased description of water at the at. level and yield information on the underlying mol. forces. However, the high computational cost of such simulations prevents the systematic investigation of the influence of vdW forces on the thermodn. anomalies of water. Here, we develop efficient ab initio-quality neural network potentials and use them to demonstrate that vdW interactions are crucial for the formation of water's d. max. and its neg. vol. of melting. Both phenomena can be explained by the flexibility of the HB network, which is the result of a delicate balance of weak vdW forces, causing, e.g., a pronounced expansion of the second solvation shell upon cooling that induces the d. max.
44. 44
  Snyder, J. C.; Rupp, M.; Hansen, K.; Blooston, L.; Mueller, K.-R.; Burke, K. Orbital-free bond breaking via machine learning. J. Chem. Phys. 2013, 139, 224104, DOI: 10.1063/1.4834075
  44
  Orbital-free bond breaking via machine learning
  Snyder, John C.; Rupp, Matthias; Hansen, Katja; Blooston, Leo; Mueller, Klaus-Robert; Burke, Kieron
  Journal of Chemical Physics (2013), 139 (22), 224104/1-224104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Using a one-dimensional model, we explore the ability of machine learning to approx. the non-interacting kinetic energy d. functional of diatomics. This nonlinear interpolation between Kohn-Sham ref. calcns. can (i) accurately dissoc. a diat., (ii) be systematically improved with increased ref. data and (iii) generate accurate self-consistent densities via a projection method that avoids directions with no data. With relatively few densities, the error due to the interpolation is smaller than typical errors in std. exchange-correlation functionals. (c) 2013 American Institute of Physics.
45. 45
  Schütt, K. T.; Glawe, H.; Brockherde, F.; Sanna, A.; Müller, K. R.; Gross, E. K. U. How to represent crystal structures for machine learning: Towards fast prediction of electronic properties. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 205118, DOI: 10.1103/PhysRevB.89.205118
  45
  How to represent crystal structures for machine learning: towards fast prediction of electronic properties
  Schuett, K. T.; Glawe, H.; Brockherde, F.; Sanna, A.; Mueller, K. R.; Gross, E. K. U.
  Physical Review B: Condensed Matter and Materials Physics (2014), 89 (20), 205118/1-205118/5, 5 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
  High-throughput d. functional calcns. of solids are highly time-consuming. As an alternative, we propose a machine learning approach for the fast prediction of solid-state properties. To achieve this, local spin-d. approxn. calcns. are used as a training set. We focus on predicting the value of the d. of electronic states at the Fermi energy. We find that conventional representations of the input data, such as the Coulomb matrix, are not suitable for the training of learning machines in the case of periodic solids. We propose a novel crystal structure representation for which learning and competitive prediction accuracies become possible within an unrestricted class of spd systems of arbitrary unit-cell size.
46. 46
  Dral, P. O.; von Lilienfeld, O. A.; Thiel, W. Machine Learning of Parameters for Accurate Semiempirical Quantum Chemical Calculations. J. Chem. Theory Comput. 2015, 11, 2120– 2125, DOI: 10.1021/acs.jctc.5b00141
  46
  Machine Learning of Parameters for Accurate Semiempirical Quantum Chemical Calculations
  Dral, Pavlo O.; von Lilienfeld, O. Anatole; Thiel, Walter
  Journal of Chemical Theory and Computation (2015), 11 (5), 2120-2125CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We investigate possible improvements in the accuracy of semiempirical quantum chem. (SQC) methods through the use of machine learning (ML) models for the parameters. For a given class of compds., ML techniques require sufficiently large training sets to develop ML models that can be used for adapting SQC parameters to reflect changes in mol. compn. and geometry. The ML-SQC approach allows the automatic tuning of SQC parameters for individual mols., thereby improving the accuracy without deteriorating transferability to mols. with mol. descriptors very different from those in the training set. The performance of this approach is demonstrated for the semiempirical OM2 method using a set of 6095 constitutional isomers C7H10O2, for which accurate ab initio atomization enthalpies are available. The ML-OM2 results show improved av. accuracy and a much reduced error range compared with those of std. OM2 results, with mean abs. errors in atomization enthalpies dropping from 6.3 to 1.7 kcal/mol. They are also found to be superior to the results from specific OM2 reparameterizations (rOM2) for the same set of isomers. The ML-SQC approach thus holds promise for fast and reasonably accurate high-throughput screening of materials and mols.
47. 47
  Kranz, J. J.; Kubillus, M.; Ramakrishnan, R.; von Lilienfeld, O. A.; Elstner, M. Generalized Density-Functional Tight-Binding Repulsive Potentials from Unsupervised Machine Learning. J. Chem. Theory Comput. 2018, 14, 2341– 2352, DOI: 10.1021/acs.jctc.7b00933
  47
  Generalized Density-Functional Tight-Binding Repulsive Potentials from Unsupervised Machine Learning
  Kranz, Julian J.; Kubillus, Maximilian; Ramakrishnan, Raghunathan; von Lilienfeld, O. Anatole; Elstner, Marcus
  Journal of Chemical Theory and Computation (2018), 14 (5), 2341-2352CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We combine the approx. d.-functional tight-binding (DFTB) method with unsupervised machine learning. This allows us to improve transferability and accuracy, make use of large quantum chem. data sets for the parametrization, and efficiently automatize the parametrization process of DFTB. For this purpose, generalized pair-potentials are introduced, where the chem. environment is included during the learning process, leading to more specific effective two-body potentials. We train on energies and forces of equil. and nonequil. structures of 2100 mols., and test on ∼130,000 org. mols. contg. O, N, C, H, and F atoms. Atomization energies of the ref. method can be reproduced within an error of ∼2.6 kcal/mol, indicating drastic improvement over std. DFTB.
48. 48
  White, C. A.; Maslen, P.; Lee, M. S.; Head-Gordon, M. The tensor properties of energy gradients within a non-orthogonal basis. Chem. Phys. Lett. 1997, 276, 133– 138, DOI: 10.1016/S0009-2614(97)88046-3
  There is no corresponding record for this reference.
49. 49
  Bartók, A. P.; Kondor, R.; Csányi, G. On representing chemical environments. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 184115, DOI: 10.1103/PhysRevB.87.184115
  49
  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.
50. 50
  De, S.; Bartok, A. P.; Csanyi, G.; Ceriotti, M. Comparing molecules and solids across structural and alchemical space. Phys. Chem. Chem. Phys. 2016, 18, 13754– 13769, DOI: 10.1039/C6CP00415F
  50
  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.
51. 51
  Zhu, L.; Amsler, M.; Fuhrer, T.; Schäfer, B.; Faraji, S.; Rostami, S.; Ghasemi, S. A.; Sadeghi, A.; Grauzinyte, M.; Wolverton, C.; Goedecker, S. A fingerprint based metric for measuring similarities of crystalline structures. J. Chem. Phys. 2016, 144, 034203, DOI: 10.1063/1.4940026
  51
  A fingerprint based metric for measuring similarities of crystalline structures
  Zhu, Li; Amsler, Maximilian; Fuhrer, Tobias; Schaefer, Bastian; Faraji, Somayeh; Rostami, Samare; Ghasemi, S. Alireza; Sadeghi, Ali; Grauzinyte, Migle; Wolverton, Chris; Goedecker, Stefan
  Journal of Chemical Physics (2016), 144 (3), 034203/1-034203/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Measuring similarities/dissimilarities between at. structures is important for the exploration of potential energy landscapes. However, the cell vectors together with the coordinates of the atoms, which are generally used to describe periodic systems, are quantities not directly suitable as fingerprints to distinguish structures. Based on a characterization of the local environment of all atoms in a cell, we introduce crystal fingerprints that can be calcd. easily and define configurational distances between cryst. structures that satisfy the math. properties of a metric. This distance between two configurations is a measure of their similarity/dissimilarity and it allows in particular to distinguish structures. The new method can be a useful tool within various energy landscape exploration schemes, such as min. hopping, random search, swarm intelligence algorithms, and high-throughput screenings. (c) 2016 American Institute of Physics.
52. 52
  Sadeghi, A.; Ghasemi, S. A.; Schäfer, B.; Mohr, S.; Lill, M. A.; Goedecker, S. Metrics for measuring distances in configuration spaces. J. Chem. Phys. 2013, 139, 184118, DOI: 10.1063/1.4828704
  52
  Metrics for measuring distances in configuration spaces
  Sadeghi, Ali; Ghasemi, S. Alireza; Schaefer, Bastian; Mohr, Stephan; Lill, Markus A.; Goedecker, Stefan
  Journal of Chemical Physics (2013), 139 (18), 184118/1-184118/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  In order to characterize mol. structures we introduce configurational fingerprint vectors which are counterparts of quantities used exptl. to identify structures. The Euclidean distance between the configurational fingerprint vectors satisfies the properties of a metric and can therefore safely be used to measure dissimilarities between configurations in the high dimensional configuration space. In particular we show that these metrics are a perfect and computationally cheap replacement for the root-mean-square distance (RMSD) when one has to decide whether two noise contaminated configurations are identical or not. We introduce a Monte Carlo approach to obtain the global min. of the RMSD between configurations, which is obtained from a global minimization over all translations, rotations, and permutations of at. indexes. (c) 2013 American Institute of Physics.
53. 53
  Rupp, M.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning. Phys. Rev. Lett. 2012, 108, 058301, DOI: 10.1103/PhysRevLett.108.058301
  53
  Fast and accurate modeling of molecular atomization energies with machine learning
  Rupp, Matthias; Tkatchenko, Alexandre; Mueller, Klaus-Robert; von Lilienfeld, O. Anatole
  Physical Review Letters (2012), 108 (5), 058301/1-058301/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  The authors introduce a machine learning model to predict atomization energies of a diverse set of org. mols., based on nuclear charges and at. positions only. The problem of solving the mol. Schrodinger equation is mapped onto a nonlinear statistical regression problem of reduced complexity. Regression models are trained on and compared to atomization energies computed with hybrid d.-functional theory. Cross validation over more than seven thousand org. mols. yields a mean abs. error of ∼10 kcal/mol. Applicability is demonstrated for the prediction of mol. atomization potential energy curves.
54. 54
  Behler, J. Constructing high-dimensional neural network potentials: A tutorial review. Int. J. Quantum Chem. 2015, 115, 1032– 1050, DOI: 10.1002/qua.24890
  54
  Constructing high-dimensional neural network potentials: A tutorial review
  Behler, Joerg
  International Journal of Quantum Chemistry (2015), 115 (16), 1032-1050CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
  A lot of progress has been made in recent years in the development of atomistic potentials using machine learning (ML) techniques. In contrast to most conventional potentials, which are based on phys. approxns. and simplifications to derive an analytic functional relation between the at. configuration and the potential-energy, ML potentials rely on simple but very flexible math. terms without a direct phys. meaning. Instead, in case of ML potentials the topol. of the potential-energy surface is "learned" by adjusting a no. of parameters with the aim to reproduce a set of ref. electronic structure data as accurately as possible. Due to this bias-free construction, they are applicable to a wide range of systems without changes in their functional form, and a very high accuracy close to the underlying first-principles data can be obtained. Neural network potentials (NNPs), which have first been proposed about two decades ago, are an important class of ML potentials. Although the first NNPs have been restricted to small mols. with only a few degrees of freedom, they are now applicable to high-dimensional systems contg. thousands of atoms, which enables addressing a variety of problems in chem., physics, and materials science. In this tutorial review, the basic ideas of NNPs are presented with a special focus on developing NNPs for high-dimensional condensed systems. A recipe for the construction of these potentials is given and remaining limitations of the method are discussed. © 2015 Wiley Periodicals, Inc.
55. 55
  Rasmussen, C. E.; Williams, C. K. I. Gaussian Processes for Machine Learning; The MIT Press, 2006; http://www.gaussianprocess.org/gpml/ (accessed Apr. 20, 2018).
  There is no corresponding record for this reference.
56. 56
  Tikhonov, A. N.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G. Numerical Methods for the Solution of Ill-Posed Problems; Kluwer Academic, 1995.
  There is no corresponding record for this reference.
57. 57
  Gillan, M. J.; Alfé, D.; Michaelides, A. Perspective: How good is DFT for water. J. Chem. Phys. 2016, 144, 130901, DOI: 10.1063/1.4944633
  57
  Perspective: How good is DFT for water?
  Gillan, Michael J.; Alfe, Dario; Michaelides, Angelos
  Journal of Chemical Physics (2016), 144 (13), 130901/1-130901/33CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A review. Kohn-Sham d. functional theory (DFT) has become established as an indispensable tool for investigating aq. systems of all kinds, including those important in chem., surface science, biol., and the earth sciences. Nevertheless, many widely used approxns. for the exchange-correlation (XC) functional describe the properties of pure water systems with an accuracy that is not fully satisfactory. The explicit inclusion of dispersion interactions generally improves the description, but there remain large disagreements between the predictions of different dispersion-inclusive methods. We present here a review of DFT work on water clusters, ice structures, and liq. water, with the aim of elucidating how the strengths and weaknesses of different XC approxns. manifest themselves across this variety of water systems. Our review highlights the crucial role of dispersion in describing the delicate balance between compact and extended structures of many different water systems, including the liq. By referring to a wide range of published work, we argue that the correct description of exchange-overlap interactions is also extremely important, so that the choice of semi-local or hybrid functional employed in dispersion-inclusive methods is crucial. The origins and consequences of beyond-2-body errors of approx. XC functionals are noted, and we also discuss the substantial differences between different representations of dispersion. We propose a simple numerical scoring system that rates the performance of different XC functionals in describing water systems, and we suggest possible future developments. (c) 2016 American Institute of Physics.
58. 58
  Del Ben, M.; Schönherr, M.; Hutter, J.; VandeVondele, J. Bulk Liquid Water at Ambient Temperature and Pressure from MP2 Theory. J. Phys. Chem. Lett. 2013, 4, 3753– 3759, DOI: 10.1021/jz401931f
  58
  Bulk Liquid Water at Ambient Temperature and Pressure from MP2 Theory
  Del Ben, Mauro; Schonherr, Mandes; Hutter, Jurg; Vande Vondele, Joost
  Journal of Physical Chemistry Letters (2013), 4 (21), 3753-3759CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
  MP2 provides a good description of hydrogen bonding in water clusters and includes long-range dispersion interactions without the need to introduce empirical elements in the description of the interat. potential. To assess its performance for bulk liq. water under ambient conditions, an isobaric-isothermal (NpT) Monte Carlo simulation at the second-order Moller-Plesset perturbation theory level (MP2) has been performed. The obtained value of the water d. is excellent (1.02 g/mL), and the calcd. radial distribution functions are in fair agreement with exptl. data. The MP2 results are compared to a few d. functional approxns., including semilocal functionals, hybrid functionals, and functionals including empirical dispersion corrections. These results demonstrate the feasibility of directly sampling the potential energy surface of condensed-phase systems using correlated wave function theory, and their quality paves the way for further applications.
59. 59
  Del Ben, M.; Hutter, J.; VandeVondele, J. Probing the structural and dynamical properties of liquid water with models including non-local electron correlation. J. Chem. Phys. 2015, 143, 054506, DOI: 10.1063/1.4927325
  59
  Probing the structural and dynamical properties of liquid water with models including non-local electron correlation
  Del Ben, Mauro; Hutter, Jurg; Vande Vondele, Joost
  Journal of Chemical Physics (2015), 143 (5), 054506/1-054506/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Water is a ubiquitous liq. that displays a wide range of anomalous properties and has a delicate structure that challenges expt. and simulation alike. The various intermol. interactions that play an important role, such as repulsion, polarization, hydrogen bonding, and van der Waals interactions, are often difficult to reproduce faithfully in atomistic models. Here, electronic structure theories including all these interactions at equal footing, which requires the inclusion of non-local electron correlation, are used to describe structure and dynamics of bulk liq. water. Isobaric-isothermal (NpT) ensemble simulations based on the RPA yield excellent d. (0.994 g/mL) and fair radial distribution functions, while various other d. functional approxns. produce scattered results (0.8-1.2 g/mL). Mol. dynamics simulation in the microcanonical (NVE) ensemble based on Moller-Plesset perturbation theory (MP2) yields dynamical properties in the condensed phase, namely, the IR spectrum and diffusion const. At the MP2 and RPA levels of theory, ice is correctly predicted to float on water, resolving one of the anomalies as resulting from a delicate balance between van der Waals and hydrogen bonding interactions. For several properties, obtaining quant. agreement with expt. requires correction for nuclear quantum effects (NQEs), highlighting their importance, for structure, dynamics, and electronic properties. A computed NQE shift of 0.6 eV for the band gap and absorption spectrum illustrates the latter. Giving access to both structure and dynamics of condensed phase systems, non-local electron correlation will increasingly be used to study systems where weak interactions are of paramount importance. (c) 2015 American Institute of Physics.
60. 60
  Verlet, L. Computer ”Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Phys. Rev. 1967, 159, 98– 103, DOI: 10.1103/PhysRev.159.98
  60
  Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules
  Verlet, Loup
  Physical Review (1967), 159 (1), 98-103CODEN: PHRVAO; ISSN:0031-899X.
  The equation of motion of a system of 864 particles interacting through a Lennard-Jones potential was integrated for various values of the temp. and d., relative, generally, to a fluid state. The equil. properties agree with the corresponding properties of Ar. The equil. state of Ar can be described through a 2-body potential.
61. 61
  Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys. Rev. B: Condens. Matter Mater. Phys. 1998, 58, 7260– 7268, DOI: 10.1103/PhysRevB.58.7260
  61
  Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties
  Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, Th.; Suhai, S.; Seifert, G.
  Physical Review B: Condensed Matter and Materials Physics (1998), 58 (11), 7260-7268CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  We outline details about an extension of the tight-binding (TB) approach to improve total energies, forces, and transferability. The method is based on a second-order expansion of the Kohn-Sham total energy in d.-functional theory (DFT) with respect to charge-d. fluctuations. The zeroth-order approach is equiv. to a common std. non-self-consistent (TB) scheme, while at second-order a transparent, parameter-free, and readily calculable expression for generalized Hamiltonian matrix elements may be derived. These are modified by a self-consistent redistribution of Mulliken charges (SCC). Besides the usual "band structure" and short-range repulsive terms the final approx. Kohn-Sham energy addnl. includes a Coulomb interaction between charge fluctuations. At large distances this accounts for long-range electrostatic forces between two point charges and approx. includes self-interaction contributions of a given atom if the charges are located at one and the same atom. We apply the new SCC scheme to problems where deficiencies within the non-SCC std. TB approach become obvious. We thus considerably improve transferability.
62. 62
  Hu, H.; Lu, Z.; Elstner, M.; Hermans, J.; Yang, W. Simulating Water with the Self-Consistent-Charge Density Functional Tight Binding Method: From Molecular Clusters to the Liquid State. J. Phys. Chem. A 2007, 111, 5685– 5691, DOI: 10.1021/jp070308d
  62
  Simulating Water with the Self-Consistent-Charge Density Functional Tight Binding Method: From Molecular Clusters to the Liquid State
  Hu, Hao; Lu, Zhenyu; Elstner, Marcus; Hermans, Jan; Yang, Weitao
  Journal of Physical Chemistry A (2007), 111 (26), 5685-5691CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  The recently developed self-consistent-charge d. functional tight binding (SCCDFTB) method provides an accurate and inexpensive quantum mech. soln. to many mol. systems of interests. To examine the performance of the SCCDFTB method on (liq.) water, the most fundamental yet indispensable mol. in biol. systems, we report here the simulation results of water in sizes ranging from mol. clusters to the liq. state. The latter simulation was achieved through the use of the linear scaling divide-and-conquer approach. The results of liq. water simulation indicate that the SCCDFTB method can describe the structural and energetics of liq. water in qual. agreement with expts., and the results for water clusters suggest potential future improvements of the SCCDFTB method.
63. 63
  Skinner, L. B.; Huang, C.; Schlesinger, D.; Pettersson, L. G. M.; Nilsson, A.; Benmore, C. J. Benchmark oxygen-oxygen pair-distribution function of ambient water from x-ray diffraction measurements with a wide Q-range. J. Chem. Phys. 2013, 138, 074506, DOI: 10.1063/1.4790861
  63
  Benchmark oxygen-oxygen pair-distribution function of ambient water from x-ray diffraction measurements with a wide Q-range
  Skinner, Lawrie B.; Huang, Congcong; Schlesinger, Daniel; Pettersson, Lars G. M.; Nilsson, Anders; Benmore, Chris J.
  Journal of Chemical Physics (2013), 138 (7), 074506/1-074506/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Four recent x-ray diffraction measurements of ambient liq. water are reviewed here. Each of these measurements represents a significant development of the x-ray diffraction technique applied to the study of liq. water. Sources of uncertainty from statistical noise, Q-range, Compton scattering, and self-scattering are discussed. The oxygen-hydrogen contribution to the measured x-ray scattering pattern was subtracted using literature data to yield an exptl. detn., with error bars, of the oxygen-oxygen pair-distribution function, gOO(r), which essentially describes the distribution of mol. centers. The extended Q-range and low statistical noise of these measurements has significantly reduced truncation effects and related errors in the gOO(r) functions obtained. From these measurements and error anal., the position and height of the nearest neighbor max. in gOO(r) were found to be 2.80(1) Å and 2.57(5) resp. Numerical data for the coherent differential x-ray scattering cross-section IX(Q), the oxygen-oxygen structure factor SOO(Q), and the derived gOO(r) are provided as benchmarks for calibrating force-fields for water. (c) 2013 American Institute of Physics.
64. 64
  The CP2K developers group. CP2K , 2018; https://www.cp2k.org, (accessed Apr. 20, 2018).
  There is no corresponding record for this reference.
65. 65
  Hutter, J.; Iannuzzi, M.; Schiffmann, F.; VandeVondele, J. CP2K: atomistic simulations of condensed matter systems. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2014, 4, 15– 25, DOI: 10.1002/wcms.1159
  65
  cp2k: atomistic simulations of condensed matter systems
  Hutter, Juerg; Iannuzzi, Marcella; Schiffmann, Florian; VandeVondele, Joost
  Wiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (1), 15-25CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)
  A review. Cp2k has become a versatile open-source tool for the simulation of complex systems on the nanometer scale. It allows for sampling and exploring potential energy surfaces that can be computed using a variety of empirical and first principles models. Excellent performance for electronic structure calcns. is achieved using novel algorithms implemented for modern and massively parallel hardware. This review briefly summarizes the main capabilities and illustrates with recent applications the science cp2k has enabled in the field of atomistic simulation. WIREs Comput Mol Sci 2014, 4:15-25. doi: 10.1002/wcms.1159 The authors have declared no conflicts of interest in relation to this article. For further resources related to this article, please visit the WIREs website.
66. 66
  VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103– 128, DOI: 10.1016/j.cpc.2004.12.014
  66
  QUICKSTEP: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
  VandeVondele, Joost; Krack, Matthias; Mohamed, Fawzi; Parrinello, Michele; Chassaing, Thomas; Hutter, Juerg
  Computer Physics Communications (2005), 167 (2), 103-128CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
  We present the Gaussian and plane waves (GPW) method and its implementation in which is part of the freely available program package CP2K. The GPW method allows for accurate d. functional calcns. in gas and condensed phases and can be effectively used for mol. dynamics simulations. We show how derivs. of the GPW energy functional, namely ionic forces and the Kohn-Sham matrix, can be computed in a consistent way. The computational cost of computing the total energy and the Kohn-Sham matrix is scaling linearly with the system size, even for condensed phase systems of just a few tens of atoms. The efficiency of the method allows for the use of large Gaussian basis sets for systems up to 3000 atoms, and we illustrate the accuracy of the method for various basis sets in gas and condensed phases. Agreement with basis set free calcns. for single mols. and plane wave based calcns. in the condensed phase is excellent. Wave function optimization with the orbital transformation technique leads to good parallel performance, and outperforms traditional diagonalisation methods. Energy conserving Born-Oppenheimer dynamics can be performed, and a highly efficient scheme is obtained using an extrapolation of the d. matrix. We illustrate these findings with calcns. using commodity PCs as well as supercomputers.
67. 67
  Lippert, G.; Hutter, J.; Parrinello, M. A hybrid Gaussian and plane wave density functional scheme. Mol. Phys. 1997, 92, 477– 488, DOI: 10.1080/00268979709482119
  67
  A hybrid Gaussian and plane wave density functional scheme
  Lippert, Gerald; Hutter, Juerg; Parrinello, Michele
  Molecular Physics (1997), 92 (3), 477-487CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)
  A d.-functional theory-based algorithm for periodic and nonperiodic ab initio calcns. is presented. This scheme uses pseudopotentials in order to integrate out the core electrons from the problem. The valence pseudo wave functions are expanded in Gaussian-type orbitals and the d. is represented in a plane wave auxiliary basis. The Gaussian basis functions make it possible to use the efficient anal. integration schemes and screening algorithms of quantum chem. Novel recursion relations are developed for the calcn. of the matrix elements of the d.-dependent Kohn-Sham self-consistent potential. At the same time the use of a plane wave basis for the electron d. permits efficient calcn. of the Hartree energy using fast Fourier transforms, thus circumventing one of the major bottlenecks of std. Gaussian based calcns. Furthermore, this algorithm avoids the fitting procedures that go along with intermediate basis sets for the charge d. The performance and accuracy of this new scheme are discussed and selected examples are given.
68. 68
  Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868, DOI: 10.1103/PhysRevLett.77.3865
  68
  Generalized gradient approximation made simple
  Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
  Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
69. 69
  Goedecker, S.; Teter, M.; Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 1703– 1710, DOI: 10.1103/PhysRevB.54.1703
  69
  Separable dual-space Gaussian pseudopotentials
  Goedecker, S.; Teter, M.; Hutter, J.
  Physical Review B: Condensed Matter (1996), 54 (3), 1703-1710CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  We present pseudopotential coeffs. for the first two rows of the Periodic Table. The pseudopotential is of an analytic form that gives optimal efficiency in numerical calculations using plane waves as a basis set. At most, even coeffs. are necessary to specify its analytic form. It is separable and has optimal decay properties in both real and Fourier space. Because of this property, the application of the nonlocal part of the pseudopotential to a wave function can be done efficiently on a grid in real space. Real space integration is much faster for large systems than ordinary multiplication in Fourier space, since it shows only quadratic scaling with respect to the size of the system. We systematically verify the high accuracy of these pseudopotentials by extensive at. and mol. test calcns.
70. 70
  Niklasson, A. M.; Tymczak, C.; Challacombe, M. Trace resetting density matrix purification in self-consistent-field theory. J. Chem. Phys. 2003, 118, 8611– 8620, DOI: 10.1063/1.1559913
  There is no corresponding record for this reference.
71. 71
  Schütt, O. Enabling Large Scale DFT Simulation with GPU Acceleration and Machine Learning. Ph.D. Thesis, ETH Zürich, 2017.
  There is no corresponding record for this reference.
Supporting Information
Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.8b00378.
- Representative input files for most simulations (ZIP)
- ct8b00378_si_001.zip (1.84 MB)
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Machine Learning Adaptive Basis Sets for Efficient Large Scale Density Functional Theory SimulationClick to copy article linkArticle link copied!

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Abstract

1. Introduction

2. Methods

2.1. Polarized Atomic Orbitals

2.2. Potential Parametrization

Explicit Form of the Potential Terms

2.3. Machine Learning

Figure 1

2.4. Analytic Forces

2.5. Training Data Acquisition

Regularization

3. Results

3.1. Learning Curve

Figure 2

3.2. Consistency of Energy and Forces

Figure 3

3.3. PAO-ML Molecular Dynamics of Liquid Water

Figure 4

3.4. Check for Unphysical Minima

3.5. PAO-ML Speedup

3.6. Computational Setup

4. Discussion and Conclusions

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Abstract

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Figure 2

Figure 3

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Supporting Information

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Machine Learning Adaptive Basis Sets for Efficient Large Scale Density Functional Theory Simulation
Click to copy article linkArticle link copied!