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Integral-Direct Hartree-Fock and Møller-Plesset Perturbation Theory for Periodic Systems with Density Fitting: Application to the Benzene Crystal
Abstract
We present an algorithm and implementation of integral-direct, density-fitted Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) for periodic systems. The new code eliminates the formerly prohibitive storage requirements and allows us to study systems 1 order of magnitude larger than before at the periodic MP2 level. We demonstrate the significance of the development by studying the benzene crystal in both the thermodynamic limit and the complete basis set limit, for which we predict an MP2 cohesive energy of -72.8 kJ/mol, which is about 10-15 kJ/mol larger in magnitude than all previously reported MP2 calculations. Compared to the best theoretical estimate from literature, several modified MP2 models approach chemical accuracy in the predicted cohesive energy of the benzene crystal and hence may be promising cost-effective choices for future applications on molecular crystals.
... In recent years, there has been a growing interest in employing quantum chemistry wave function methods, such as Møller-Plesset perturbation theory and coupled cluster (CC) theory [9], to compute ground-state and excited-state properties for periodic systems [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Originally developed to study nuclear physics [26,27], CC theory has become one of the most popular methods in quantum chemistry that involves electronic correlation [28]. ...
... One important driver of this trend is the potential of these methods to provide systematically improvable results. Among the simplest post-HF methods, MP2 has been applied to increasingly large periodic systems along with the development of efficient implementations utilizing density fitting and parallelization techniques [16,23]. Notably, the next-order method, MP3, has also been applied to periodic systems recently for the first time [24]. ...
Coupled cluster theory is one of the most popular post-Hartree-Fock methods for ab initio molecular quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the length of the system. In this study, we review the current understanding of finite-size error in quantum chemistry methods for periodic systems. We introduce new tools that elucidate the mechanisms behind this phenomenon in the context of coupled cluster doubles calculations. This reconciles some seemingly paradoxical statements related to finite-size scaling. Our findings also show that singularity subtraction can be a powerful method to effectively reduce finite-size errors in practical quantum chemistry calculations for periodic systems.
... Recently, our group reported a periodic MP2 study of the benzene crystal, 31 showing promising results and addressing potential basis set incompleteness and finite-size errors in literature MP2 values. We found that MP2 overestimated the magnitude of the cohesive energy by about 18 kJ/mol and, again, that spin scaling could reduce the error to 3-5 kJ/mol. ...
... Moreover, for all calculations requiring more than 2000 GB of storage, we utilized the integral-direct algorithm implemented in PySCF. 31,38 In the Supporting Information, we provide further details on our TDL and CBS extrapolations, including the composite correction, which allow us to conclude that our final numbers are converged to about 2 kJ/mol. Finally, we report the counterpoise corrected cohesive en- Table I. ...
Achieving kJ/mol accuracy in the cohesive energy of molecular crystals, as necessary for crystal structure prediction and the resolution of polymorphism, is an ongoing challenge in computational materials science. Here, we evaluate the performance of second-order M{\o}ller-Plesset perturbation theory (MP2), including its spin-component scaled models, by calculating the cohesive energies of the 23 molecular crystals contained in the X23 dataset. Our calculations are performed with periodic boundary conditions and Brillouin zone sampling, and we converge results to the thermodynamic limit and the complete basis set limit to an accuracy of about 1 kJ/mol (0.25 kcal/mol), which is rarely achieved in previous MP2 calculations of molecular crystals. Comparing to experimental cohesive energies, we find that MP2 has a mean absolute error of 12.9 kJ/mol, which is comparable to that of DFT using the PBE functional and TS dispersion correction. Separate scaling of the opposite-spin and same-spin components of the correlation energy, with parameters previously determined for molecular interactions, reduces the mean absolute error to 9.5 kJ/mol, and reoptimizing the spin-component scaling parameters for the X23 set further reduces the mean absolute error to 7.5 kJ/mol.
... These post-HF methods, however, scale rapidly with system size, making their application to extended systems without additional (sometimes drastic) approximations difficult. Nonetheless, wavefunctionbased methods have become relatively commonplace for periodic systems in recent decades, whether with actual periodic techniques [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] or the so-called method of increments [21][22][23][24][25][26] and other approaches combining periodic and finite-cluster calculations [27][28][29][30][31][32][33][34][35][36][37]. Such techniques, however, have been nearly exclusively applied for closed-shell systems and their generalization to capture multireference character (strong electron correlation) is difficult. ...
To date, computational methods for modeling defects (vacancies, adsorbates, etc.) rely on periodic supercells in which the defect is far enough from its repeated image such that they can be assumed non-interacting. Defects in real solids, however, can be spaced microns or more apart, whereas affordable supercells for density functional theory calculations are no wider than a few nanometers. The relative proximity and periodic repetition of the defect's images may lead to unphysical artifacts, especially if the defect is charged and/or open-shell. Furthermore, to avoid divergence of the periodic electrostatics, a compensating background charge must be introduced if the defect is charged. Even if post-hoc corrections are used, this is a source of unquantifiable error and, in some cases, renders energies useless. In this communication, we introduce an embedding formalism such that a pristine, primitive unit cell may be used for the periodic mean field, after which atoms may be moved or charged within an embedded fragment. This fragment can then be treated with a post-Hartree Fock method to capture important electron correlations pertaining to the defect. By eliminating the need for compensating background charges and periodicity of the defect, we circumvent all associated unphysicalities and numerical issues. Furthermore, the primitive cell calculations drastically reduce computational expense compared to supercell approaches. This method is size-intensive with respect to energy differences and can be routinely applied even to multireference defects, localized excited states, etc. using a variety of fragment solvers. In examining with this approach bond-breaking in a fluorine-substituted graphane monolayer, a difficult testing ground for condensed-phase electronic structure methods, we observe key aspects of the dissociation pathway, specifically a covalent-to-ionic avoided crossing.
... In a preprint version of the present manuscript, 31 we applied our recently developed periodic local CCSD(T) method 32,33 to calculate an adsorption energy in good agreement with the above values. Due to the use of local correlation theory and recent developments in periodic integral evaluation [34][35][36] and correlation-consistent Gaussian basis sets, 37 these calculations were relatively affordable: they were generated from scratch in a few days, requiring about 18k CPU hours. Because of this low cost, we now extend our work to calculate a smooth, two-dimensional PES for CO on MgO, from which we calculate the vibrational frequency shift of CO stretching upon adsorption to be +14.7 cm −1 , which is in nearly quantitative agreement with the infrared experimental result 38 Convergence of the HF interaction energy with surface and basis set sizes. ...
The adsorption of CO on the surface of MgO has long been a model problem in surface chemistry. Here, we report periodic Gaussian-based calculations for this problem using second-order perturbation...
... The theoretical calculation models of cluster energy can be mainly divided into two categories. One category is the ab initio method, rooted in the first principles of quantum mechanics [30,31], which includes Density Functional Theory (DFT) [32,33], Hartree-Focktheory (MF) [34], Second-order Moller-Plesset Perturbation theory [35,36], Complete Active Space Perturbation theory (CASPT2) [37], Multi-Reference Configuration Interaction (MRCI) [38] and others. These methods predict the energy of clusters by describing the electronic structure and interactions among electrons. ...
Background
Clusters, a novel hierarchical material structure that emerges from atoms or molecules, possess unique reactivity and catalytic properties, crucial in catalysis, biomedicine, and optoelectronics. Predicting cluster energy provides insights into electronic structure, magnetism, and stability. However, the structure of clusters and their potential energy surface is exceptionally intricate. Searching for the global optimal structure (the lowest energy) among these isomers poses a significant challenge. Currently, modelling cluster energy predictions with traditional machine learning methods has several issues, including reliance on manual expertise, slow computation, heavy computational resource demands, and less efficient parameter tuning.
Results
This paper introduces a predictive model for the energy of a gold cluster comprising twenty atoms (referred to as Au20 cluster). The model integrates the Multiple Strategy Fusion Whale Optimization Algorithm (MSFWOA) with the Light Gradient Boosting Machine (LightGBM), resulting in the MSFWOA-LightGBM model. This model employs the Coulomb matrix representation and eigenvalue solution methods for feature extraction. Additionally, it incorporates the Tent chaotic mapping, cosine convergence factor, and inertia weight updating strategy to optimize the Whale Optimization Algorithm (WOA), leading to the development of MSFWOA. Subsequently, MSFWOA is employed to optimize the parameters of LightGBM for supporting the energy prediction of Au20 cluster.
Conclusions
The experimental results show that the most stable Au20 cluster structure is a regular tetrahedron with the lowest energy, displaying tight and uniform atom distribution, high geometric symmetry. Compared to other models, the MSFWOA-LightGBM model excels in accuracy and correlation, with MSE, RMSE, and R² values of 0.897, 0.947, and 0.879, respectively. Additionally, the MSFWOA-LightGBM model possesses outstanding scalability, offering valuable insights for material design, energy storage, sensing technology, and biomedical imaging, with the potential to drive research and development in these areas.
Graphical Abstract
... GDF algorithms specialized in all-electron methods were developed to solve the energy cutoff issue. 55,[60][61][62] Despite that GPW and GDF methods provide the possibility to compute HFX with a PBC Gaussian basis, their computational requirements for HFX still make them impractical when targeting the thermodynamic limit (TDL) in periodic systems. The GDF method requires the storage of (N 3 AO N 2 k ) for the intermediate tensors and (N 4 AO N 2 k ) FLOPs for tensor contraction in the HFX matrix. ...
The expensive cost of computing exact exchange in periodic systems limits the application range of density functional theory with hybrid functionals. To reduce the computational cost of exact change, we present a range-separated algorithm to compute electron repulsion integrals for Gaussian-type crystal basis. The algorithm splits the full-range Coulomb interactions into short-range and long-range parts, which are, respectively, computed in real and reciprocal space. This approach significantly reduces the overall computational cost, as integrals can be efficiently computed in both regions. The algorithm can efficiently handle large numbers of k points with limited central processing unit (CPU) and memory resources. As a demonstration, we performed an all-electron k-point Hartree–Fock calculation for LiH crystal with one million Gaussian basis functions, which was completed on a desktop computer in 1400 CPU hours.
... Local density functionals have been shown to yield reasonable results when dispersion corrections are utilized, 49 and recent studies using periodic MP2 have been performed. 50,51 However, these periodic calculations are generally limited to small k-point meshes or small basis sets, or both due to computational constraints. Only a few hybrid functionals have been tested for benzene 49,[52][53][54][55] due to the computational difficulty of obtaining exact exchange energies in the thermodynamic limit. ...
Hybrid density functional theory (DFT) remains intractable for large periodic systems due to the demanding computational cost of exact exchange. We apply the tensor hypercontraction (THC) (or interpolative separable density fitting) approximation to periodic hybrid DFT calculations with Gaussian-type orbitals. This is done to lower the computational scaling with respect to the number of basis functions ($N$), and $\mathbf k$-points ($N_k$). Additionally, we propose an algorithm to fit only occupied orbital products via THC (i.e. a set of points, $N_\text{ISDF}$) to further reduce computation time and memory usage. This algorithm has linear scaling cost with $\mathbf k$-points, no explicit dependence of $N_\text{ISDF}$ on basis set size, and overall cubic scaling with unit cell size. Significant speedups and reduced memory usage may be obtained for moderately sized systems, with additional gains for large systems. Adequate accuracy can be obtained using THC-oo-K for self-consistent calculations. We perform illustrative hybrid density function theory calculations on the benzene crystal in the basis set and thermodynamic limits to highlight the utility of this algorithm.
... Integral screening for crystalline integrals is more complex than for molecular integrals due to the presence of periodicity. Crystalline integrals can be evaluated with different integral algorithms 12,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] . Various parameters, such as the range of truncated Coulomb operator, the multipole expansion order, the energy cutoff, the real space grids, etc. have to be used to efficiently compute integrals. ...
In this document, we briefly review the two-electron integral algorithms based on the range-separated algorithms and Fourier transform integral algorithms that are implement ed in the PySCF package. For each integral algorithm, we estimate the upper bound of relevant integrals and derive the necessary conditions for the screening parameters, including distance cutoff and energy cutoff, to reach the desired accuracy. Our numerical tests show that the proposed integral estimators and screening parameters can effectively address the required accuracy while computational efforts are not wasted on unintended accuracy.
... 54 The conclusions of this work are limited to the class of materials studied, i.e. three-dimensional covalent and ionic insulators, and future work must assess the extension to surfaces, layered materials, and other weakly-bound solids, such as molecular crystals. An early periodic MP2 study by Del Ben et al. 19 observed good SCS-MP2 and double hybrid performance for molecular crystals, and a recent report from our group 23 found that SCS-MP2 yields a very accurate prediction of the cohesive energy of the benzene crystal, while SOS-MP2 yields a nonnegligible underestimation. The present work has demonstrated good performance over a relatively wide range of spin scaling parameters, indicating that additional systems or properties can be incorporated in the identification of optimal parameters for condensed phase materials. ...
We study the performance of spin-component-scaled second-order M{\o}ller-Plesset perturbation theory (SCS-MP2) for the prediction of the lattice constant, bulk modulus, and cohesive energy of 12 simple, three-dimensional, covalent and ionic semiconductors and insulators. We find that SCS-MP2 and the simpler scaled opposite-spin MP2 (SOS-MP2) yield predictions that are significantly improved over the already good performance of MP2. Specifically, when compared to experimental values with zero-point vibrational corrections, SCS-MP2 (SOS-MP2) yields mean absolute errors of 0.015 (0.017) {\AA} for the lattice constant, 3.8 (3.7) GPa for the bulk modulus, and 0.06 (0.08) eV for the cohesive energy, which are smaller than those of leading density functionals by about a factor of two or more. We consider a reparameterization of the spin scaling parameters and find that the optimal parameters for these solids are very similar to those already in common use in molecular quantum chemistry, suggesting good transferability and reliable future applications to surface chemistry on insulators.
The Random-Phase approximation (RPA) provides an appealing framework for semi-local density functional theory. In its Resolution-of-the-Identity (RI) approach, it is a very accurate and more cost-effective method than most other wavefunction-based correlation methods. For widespread applications, efficient implementations of nuclear gradients for structure optimizations and data sampling of machine learning approaches are required. We report a well scaling implementation of RI-RPA nuclear gradients on massively parallel computers. The approach is applied to two polymorphs of the benzene crystal obtaining very good cohesive and relative energies. Different correction and extrapolation schemes are investigated for further improvement of the results and estimations of error bars.
Hybrid density functional theory (DFT) remains intractable for large periodic systems due to the demanding computational cost of exact exchange. We apply the tensor hypercontraction (THC) (or interpolative separable density fitting) approximation to periodic hybrid DFT calculations with Gaussian-type orbitals using the Gaussian plane wave approach. This is done to lower the computational scaling with respect to the number of basis functions (N) and k-points (Nk) at a fixed system size. Additionally, we propose an algorithm to fit only occupied orbital products via THC (i.e., a set of points, NISDF) to further reduce computation time and memory usage. This algorithm has linear scaling cost with k-points, no explicit dependence of NISDF on basis set size, and overall cubic scaling with unit cell size. Significant speedups and reduced memory usage may be obtained for moderately sized k-point meshes, with additional gains for large k-point meshes. Adequate accuracy can be obtained using THC-oo-K for self-consistent calculations. We perform illustrative hybrid density function theory calculations on the benzene crystal in the basis set and thermodynamic limits to highlight the utility of this algorithm.
In this work, we developed and showcased the occ-RI-K algorithm to compute the exact exchange contribution in density functional calculations of solids near the basis set limit. Within the Gaussian planewave (GPW) density fitting, our algorithm achieves a 1-2 orders of magnitude speedup compared to conventional GPW algorithms. Since our algorithm is well suited for simulations with large basis sets, we applied it to 12 hybrid density functionals with pseudopotentials and a large uncontracted basis set to assess their performance on band gaps of 25 simple solids near the basis set limit. The largest calculation performed in this work involves 16 electrons and 350 basis functions in the unit cell utilizing a 6 × 6 × 6 k-mesh. With 20-27% exact exchange, global hybrid functionals (B3LYP, PBE0, revPBE0, B97-3, SCAN0) perform similarly with a root-mean-square deviation (RMSD) of 0.61-0.77 eV, while other global hybrid functionals such as M06-2X (2.02 eV) and MN15 (1.05 eV) show higher RMSD due to their increased fraction of exact exchange. A short-range hybrid functional, HSE achieves a similar RMSD (0.76 eV) but shows a notable underestimation of band gaps due to the complete lack of long-range exchange. We found that two combinatorially optimized range-separated hybrid functionals, ωB97X-rV (3.94 eV) and ωB97M-rV (3.40 eV), and the two other range-separated hybrid functionals, CAM-B3LYP (2.41 eV) and CAM-QTP01 (4.16 eV), significantly overestimate the band gap because of their high fraction of long-range exact exchange. Given the failure of ωB97X-rV and ωB97M-rV, we have yet to find a density functional that offers consistent performance for both molecules and solids. Our algorithm development and density functional assessment will serve as a stepping stone toward developing more accurate hybrid functionals and applying them to practical applications.
We study the performance of spin-component-scaled second-order Møller-Plesset perturbation theory (SCS-MP2) for the prediction of the lattice constant, bulk modulus, and cohesive energy of 12 simple, three-dimensional, covalent and ionic semiconductors and insulators. We find that SCS-MP2 and the simpler scaled opposite-spin MP2 (SOS- MP2) yield predictions that are significantly improved over the already good performance of MP2. Specifically, when compared to experimental values with zero-point vibrational corrections, SCS-MP2 (SOS-MP2) yields mean absolute errors of 0.015 (0.017) Å for the lattice constant, 3.8 (3.7) GPa for the bulk modulus, and 0.06 (0.08) eV for the cohesive energy, which are smaller than those of leading density functionals by about a factor of two or more. We consider a reparameterization of the spin scaling parameters and find that the optimal parameters for these solids are very similar to those already in common use in molecular quantum chemistry, suggesting good transferability and reliable future applications to surface chemistry on insulators.
Computationally efficient and accurate quantum mechanical approximations to solve the many-electron Schrödinger equation are crucial for computational materials science. Methods such as coupled cluster theory show potential for widespread adoption if computational cost bottlenecks can be removed. For example, extremely dense k -point grids are required to model long-range electronic correlation effects, particularly for metals. Although these grids can be made more effective by averaging calculations over an offset (or twist angle), the resultant cost in time for coupled cluster theory is prohibitive. We show here that a single special twist angle can be found using the transition structure factor, which provides the same benefit as twist averaging with one or two orders of magnitude reduction in computational time. We demonstrate that this not only works for metal systems but also is applicable to a broader range of materials, including insulators and semiconductors.
While being widely used to understand the chemical reactions in heterogeneous catalysis or other multidisciplinary systems, a great challenge that semilocal and hybrid density functional approximations (DFAs) are facing is to deliver a uniformly accurate description for both finite and extended systems. Herein, we perform reliable and well-converged periodic calculations of two doubly hybrid approximations (DHAs), XYG3 and XYGJ-OS, and demonstrate that the good accuracy of DHAs achieved for molecules is transferable to the semiconductors and insulators. Such an accuracy is not only for energetic properties but also for the first- and second-order response properties, which is general for different kinds of chemical environments, including simple cubic bulks, perovskite-type transition metal oxides like TiO2, and heterogeneous systems like CO adsorption on the NaCl(100) surface. The present finding has strengthened the predictive power of DFT, which not only will inspire the future development of the top-rung DFAs but also will boost their applications in multidisciplinary studies with high accuracy and efficiency.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
Intermolecular interactions play an important role for the understanding of catalysis, biochemistry and pharmacy. Double-hybrid density functionals (DHDFs) combine the proper treatment of short-range interactions of common density functionals with the correct description of long-range interactions of wave-function correlation methods. Up to now, there are only a few benchmark studies available examining the performance of DHDFs in condensed phase. We studied the performance of a small but diverse selection of DHDFs implemented within Gaussian and plane waves formalism on cohesive energies of four representative dispersion interaction dominated crystal structures. We found that the PWRB95 and ωB97X-2 functionals provide an excellent description of long-ranged interactions in solids. In addition, we identified numerical issues due to the extreme grid dependence of the underlying density functional for PWRB95. The basis set superposition error (BSSE) and convergence with respect to the super cell size are discussed for two different large basis sets.
The requirement that the linear density fitting error in the integral exactly vanishes introduces unphysical long range contributions to the approximate density when the auxiliary basis is incomplete. A quasi-robust density fitting formulation is presented where spatial locality is recovered at the expense of permitting a linear error that is made small by the fitting procedure, which involves optimising the Coulomb potential of the approximate charge density. The method is shown to be stable and almost as accurate as standard robust density fitting without local approximations in practical calculations using standard density fitting basis sets.
Python‐based simulations of chemistry framework (P y SCF) is a general‐purpose electronic structure platform designed from the ground up to emphasize code simplicity, so as to facilitate new method development and enable flexible computational workflows. The package provides a wide range of tools to support simulations of finite‐size systems, extended systems with periodic boundary conditions, low‐dimensional periodic systems, and custom Hamiltonians, using mean‐field and post‐mean‐field methods with standard Gaussian basis functions. To ensure ease of extensibility, P y SCF uses the Python language to implement almost all of its features, while computationally critical paths are implemented with heavily optimized C routines. Using this combined Python/C implementation, the package is as efficient as the best existing C or Fortran‐based quantum chemistry programs. In this paper, we document the capabilities and design philosophy of the current version of the P y SCF package. WIREs Comput Mol Sci 2018, 8:e1340. doi: 10.1002/wcms.1340
This article is categorized under: Structure and Mechanism > Computational Materials Science
Electronic Structure Theory > Ab Initio Electronic Structure Methods
Software > Quantum Chemistry
We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order M{\o}ller-Plesset perturbation theory (MP2). In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, $\mathcal O(N^4$), with respect to the system size $N$ and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is solved by an internal basis set extrapolation. The key concept to reduce the scaling of the algorithm is to eliminate all summations over virtual bands which can be elegantly achieved in the Laplace transformed MP2 (LTMP2) formulation using plane-wave basis sets. Analogously, this approach could allow to calculate second order screened exchange (SOSEX) as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.
The Cambridge Structural Database (CSD) contains a complete record of all published organic and metal–organic small-molecule crystal structures. The database has been in operation for over 50 years and continues to be the primary means of sharing structural chemistry data and knowledge across disciplines. As well as structures that are made public to support scientific articles, it includes many structures published directly as CSD Communications. All structures are processed both computationally and by expert structural chemistry editors prior to entering the database. A key component of this processing is the reliable association of the chemical identity of the structure studied with the experimental data. This important step helps ensure that data is widely discoverable and readily reusable. Content is further enriched through selective inclusion of additional experimental data. Entries are available to anyone through free CSD community web services. Linking services developed and maintained by the CCDC, combined with the use of standard identifiers, facilitate discovery from other resources. Data can also be accessed through CCDC and third party software applications and through an application programming interface.
Computation of the Fock matrix is currently the limiting factor in the application of Hartree-Fock and hybrid Hartree-Fock/density functional theories to larger systems. Computation of the Fock matrix is dominated by calculation of the Coulomb and exchange matrices. With conventional Gaussian-based methods, computation of the Fock matrix typically scales as ∼N2.7, where N is the number of basis functions. A hierarchical multipole method is developed for fast computation of the Coulomb matrix. This method, together with a recently described approach to computing the Hartree-Fock exchange matrix of insulators [J. Chem. Phys. 105, 2726 (1900)], leads to a linear scaling algorithm for calculation of the Fock matrix. Linear scaling computation the Fock matrix is demonstrated for a sequence of water clusters at the restricted Hartree-Fock/3-21G level of theory, and corresponding accuracies in converged total energies are shown to be comparable with those obtained from standard quantum chemistry programs. Restricted Hartree-Fock/3-21G calculations on several proteins of current interest are documented, including endothelin, charybdotoxin, and the tetramerization monomer of P53. The P53 calculation, involving 698 atoms and 3836 basis functions, may be the largest Hartree-Fock calculation to date. The electrostatic potentials of charybdotoxin and the tetramerization monomer of P53 are visualized and the results are related to molecular function.
Hybrid density functionals show great promise for chemically-accurate first
principles calculations, but their high computational cost limits their
application in non-trivial studies, such as exploration of reaction pathways of
adsorbents on periodic surfaces. One factor responsible for their increased
cost is the dense Brillouin-zone sampling necessary to accurately resolve an
integrable singularity in the exact exchange energy. We analyze this
singularity within an intuitive formalism based on Wannier-function
localization and analytically prove Wigner-Seitz truncation to be the ideal
method for regularizing the Coulomb potential in the exchange kernel. We show
that this method is limited only by Brillouin-zone discretization errors in the
Kohn-Sham orbitals, and hence converges the exchange energy exponentially with
the number of k-points used to sample the Brillouin zone for all but
zero-temperature metallic systems. To facilitate the implementation of this
method, we develop a general construction for the plane-wave Coulomb kernel
truncated on the Wigner-Seitz cell in one, two or three lattice directions. We
compare several regularization methods for the exchange kernel in a variety of
real systems including low-symmetry crystals and low-dimensional materials. We
find that our Wigner-Seitz truncation systematically yields the best k-point
convergence for the exchange energy of all these systems and delivers an
accuracy to hybrid functionals comparable to semi-local and screened-exchange
functionals at identical k-point sets.
Using the finite simulation-cell homogeneous electron gas (HEG) as a model, we investigate the convergence of the correlation energy to the complete-basis-set (CBS) limit in methods utilizing plane-wave wave-function expansions. Simple analytic and numerical results from second-order Møller-Plesset theory (MP2) suggest a 1/M decay of the basis-set incompleteness error where M is the number of plane waves used in the calculation, allowing for straightforward extrapolation to the CBS limit. As we shall show, the choice of basis-set truncation when constructing many-electron wave functions is far from obvious, and here we propose several alternatives based on the momentum transfer vector, which greatly improve the rate of convergence. This is demonstrated for a variety of wave-function methods, from MP2 to coupled-cluster doubles theory and the random-phase approximation plus second-order screened exchange. Finite basis-set energies are presented for these methods and compared with exact benchmarks. A transformation can map the orbitals of a general solid state system onto the HEG plane-wave basis and thereby allow application of these methods to more realistic physical problems. We demonstrate this explicitly for solid and molecular lithium hydride.
Theorems are derived which establish a method of approximating two‐particle Coulombic potential energy integrals, [ϕa(1) r12−1 ∣ϕb(2)], in terms of approximate charge densities ϕa′ and ϕb′. Rigorous error bounds, ∣[ϕa(1)∣r12−1∣ϕb(2)]−[ϕa′(1)∣r12−1∣ϕb′(2)]∣ ≤ slantδ, are simply expressed in terms of information calculated separately for the pair of densities ϕa and ϕb′ and the pair ϕb and ϕb′. From the structure of the bound, a simple method of optimizing charge density approximations such that δ is minimized is derived. The framework of the theory appears to be well suited for application to the approximation of electron repulsion integrals which occur in molecular structure theory, and applications to the approximation of integrals over Slater orbitals or grouped Gaussian functions are discussed.
We study the performance of spin-component-scaled second-order M{\o}ller-Plesset perturbation theory (SCS-MP2) for the prediction of the lattice constant, bulk modulus, and cohesive energy of 12 simple, three-dimensional, covalent and ionic semiconductors and insulators. We find that SCS-MP2 and the simpler scaled opposite-spin MP2 (SOS-MP2) yield predictions that are significantly improved over the already good performance of MP2. Specifically, when compared to experimental values with zero-point vibrational corrections, SCS-MP2 (SOS-MP2) yields mean absolute errors of 0.015 (0.017) {\AA} for the lattice constant, 3.8 (3.7) GPa for the bulk modulus, and 0.06 (0.08) eV for the cohesive energy, which are smaller than those of leading density functionals by about a factor of two or more. We consider a reparameterization of the spin scaling parameters and find that the optimal parameters for these solids are very similar to those already in common use in molecular quantum chemistry, suggesting good transferability and reliable future applications to surface chemistry on insulators.
Metallic solids are an enormously important class of materials, but they are a challenging target for accurate wave function-based electronic structure theories and have not been studied in great detail by such methods. Here, we use coupled-cluster theory with single and double excitations (CCSD) to study the structure of solid lithium and aluminum using optimized Gaussian basis sets. We calculate the equilibrium lattice constant, bulk modulus, and cohesive energy and compare them to experimental values, finding accuracy comparable to common density functionals. Because the quantum chemical "gold standard" CCSD(T) (CCSD with perturbative triple excitations) is inapplicable to metals in the thermodynamic limit, we test two approximate improvements to CCSD, which are found to improve the results.
Simulating solids with quantum chemistry methods and Gaussian-type orbitals (GTOs) has been gaining popularity. Nonetheless, there are few systematic studies that assess the basis set incompleteness error (BSIE) in these GTO-based simulations over a variety of solids. In this work, we report a GTO-based implementation for solids, and apply it to address the basis set convergence issue. We employ a simple strategy to generate large uncontracted (unc) GTO basis sets, that we call the unc-def2-GTH sets. These basis sets exhibit systematic improvement towards the basis set limit as well as good transferability based on application to a total of 43 simple semiconductors. Most notably, we found the BSIE of unc-def2-QZVP-GTH to be smaller than 0.7 mEh per atom in total energies and 20 meV in band gaps for all systems considered here. Using unc-def2-QZVP- GTH, we report band gap benchmarks of a combinatorially designed meta generalized gradient functional (mGGA), B97M-rV, and show that B97M-rV performs similarly (a root-mean-square- deviation (RMSD) of 1.18 eV) to other modern mGGA functionals, M06-L (1.26 eV), MN15-L (1.29 eV), and SCAN (1.20 eV). This represents a clear improvement over older pure functionals such as LDA (1.71 eV) and PBE (1.49 eV) though all these mGGAs are still far from being quantitatively accurate. We also provide several cautionary notes on the use of our uncontracted bases and on future research on GTO basis set development for solids.
We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, erfc(ωr12)/r12. These estimators are much tighter than the ones based on the Schwarz inequality and can be viewed as a complement to the distance-dependent estimators for four-center short-range Coulomb integrals and for two-center and three-center full Coulomb integrals previously reported. Because the short-range Coulomb potential is commonly used in solid-state calculations, including those with the Heyd–Scuseria–Ernzerhof functional and with our recently introduced range-separated periodic Gaussian density fitting, we test our estimators on a diverse set of periodic systems using a wide range of the range-separation parameter ω. These tests demonstrate the robust tightness of our estimators, which are then used with integral screening to calculate periodic three-center short-range Coulomb integrals with linear scaling in system size.
Despite its reasonable accuracy for ground-state properties of semiconductors and insulators, second-order Møller–Plesset perturbation theory (MP2) significantly underestimates bandgaps. In this work, we evaluate the bandgap predictions of partitioned equation-of-motion MP2 (P-EOM-MP2), which is a second-order approximation to EOM coupled-cluster theory with single and double excitations. On a test set of elemental and binary semiconductors and insulators, we find that P-EOM-MP2 overestimates bandgaps by 0.3 eV on average, which can be compared to the underestimation by 0.6 eV on average exhibited by the G0W0 approximation with a Perdew–Burke–Ernzerhof reference. We show that P-EOM-MP2, when interpreted as a Green’s function-based theory, has a self-energy that includes all first- and second-order diagrams and a few third-order diagrams. We find that the GW approximation performs better for materials with small gaps and P-EOM-MP2 performs better for materials with large gaps, which we attribute to their superior treatment of screening and exchange, respectively.
Coupled-cluster theory with single and double excitations (CCSD) is a promising ab initio method for the electronic structure of three-dimensional metals, for which second-order perturbation theory (MP2) diverges in the thermodynamic limit. However, due to the high cost and poor convergence of CCSD with respect to basis size, applying CCSD to periodic systems often leads to large basis set errors. In a common “composite” method, MP2 is used to recover the missing dynamical correlation energy through a focal-point correction, but the inadequacy of finite-order perturbation theory for metals raises questions about this approach. Here, we describe how high-energy excitations treated by MP2 can be “downfolded” into a low-energy active space to be treated by CCSD. Comparing how the composite and downfolding approaches perform for the uniform electron gas, we find that the latter converges more quickly with respect to the basis set size. Nonetheless, the composite approach is surprisingly accurate because it removes the problematic MP2 treatment of double excitations near the Fermi surface. Using this method to estimate the CCSD correlation energy in the combined complete basis set and thermodynamic limits, we find that CCSD recovers 85%–90% of the exact correlation energy at rs = 4. We also test the composite approach with the direct random-phase approximation used in place of MP2, yielding a method that is typically (but not always) more cost effective due to the smaller number of orbitals that need to be included in the more expensive CCSD calculation.
We present an efficient implementation of periodic Gaussian density fitting (GDF) using the Coulomb metric. The three-center integrals are divided into two parts by range-separating the Coulomb kernel, with the short-range part evaluated in real space and the long-range part in reciprocal space. With a few algorithmic optimizations, we show that this new method—which we call range-separated GDF (RSGDF)—scales sublinearly to linearly with the number of k-points for small to medium-sized k-point meshes that are commonly used in periodic calculations with electron correlation. Numerical results on a few three-dimensional solids show about ten-fold speedups over the previously developed GDF with little precision loss. The error introduced by RSGDF is about 10⁻⁵ Eh in the converged Hartree–Fock energy with default auxiliary basis sets and can be systematically reduced by increasing the size of the auxiliary basis with little extra work.
Compared to common density functionals, ab initio wave function methods can provide greater reliability and accuracy, which could prove useful when modeling adsorbates or defects of otherwise periodic systems. However, the breaking of translational symmetry necessitates large supercells that are often prohibitive for correlated wave function methods. As an alternative, this paper introduces the regional embedding approach, which enables correlated wave function treatments of only a target fragment of interest through small, fragment-localized orbital spaces constructed using a simple overlap criterion. Applications to the adsorption of water on lithium hydride, hexagonal boron nitride, and graphene substrates show that regional embedding combined with focal-point corrections can provide converged CCSD(T) (coupled-cluster) adsorption energies with very small fragment sizes.
The evaluation of the exact [Hartree-Fock (HF)] exchange operator is a crucial ingredient for the accurate description of the electronic structure in periodic systems through ab initio and hybrid density functional approaches. An efficient formulation of periodic HF exchange in a linear combination of atomic orbitals representation presented here is based on the concentric atomic density fitting approximation, a domain-free local density fitting approach in which the product of two atomic orbitals is approximated using a linear combination of fitting basis functions centered at the same nuclei as the AOs in that product. A significant reduction in the computational cost of exact exchange is demonstrated relative to the conventional approach due to avoiding the need to evaluate four-center two-electron integrals, with sub-millihartree/atom errors in absolute HF energies and good cancellation of fitting errors in relative energies. The novel aspects of the evaluation of the Coulomb contribution to the Fock operator, such as the use of real two-center multipole expansions and spheropole-compensated unit cell densities, are also described.
We evaluate the performance of approaches that combine coupled-cluster and perturbation theory based on a predefined active space of orbitals. Coupled-cluster theory is used to treat excitations that are internal to the active space and perturbation theory is used for all other excitations, which are at least partially external to the active space. We consider a variety of schemes that differ in how the internal and external excitations are coupled. Such approaches are presented for ground states and excited states within the equation-of-motion formalism. Results are given for the ionisation potentials and electron affinities of a test set of small molecules and for the correlation energy and band gap of a few periodic solids.
We describe an all-electron G₀W₀ implementation for periodic systems with k-point sampling implemented in a crystalline Gaussian basis. Our full-frequency G₀W₀ method relies on efficient Gaussian density fitting integrals and includes both analytic continuation and contour deformation schemes. Due to the compactness of Gaussian bases, no virtual state truncation is required as is seen in many plane-wave formulations. Finite size corrections are included by taking the q→0 limit of the Coulomb divergence. Using our implementation, we study quasiparticle energies and band structures across a range of systems including molecules, semiconductors, rare gas solids, and metals. We find that the G₀W₀ band gaps of traditional semiconductors converge rapidly with respect to the basis size, even for the conventionally challenging case of ZnO. Using correlation-consistent bases of polarized triple-zeta quality, we find the mean absolute relative error of the extrapolated G₀W₀@PBE band gaps to be only 5.2% when compared to experimental values. For core excitation binding energies (CEBEs), we find that G₀W₀ predictions improve significantly over those from DFT if the G₀W₀ calculations are started from hybrid functionals with a high percentage of exact exchange.
PySCF is a Python-based general-purpose electronic structure platform that supports first-principles simulations of molecules and solids as well as accelerates the development of new methodology and complex computational workflows. This paper explains the design and philosophy behind PySCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PySCF as a development environment. We then summarize the capabilities of PySCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PySCF across the domains of quantum chemistry, materials science, machine learning, and quantum information science.
We present an ab initio study of electronically excited states of three-dimensional solids using Gaussian-based periodic equation-of-motion coupled-cluster theory with single and double excitations (EOM-CCSD). The explicit use of translational symmetry, as implemented via Brillouin zone sampling and momentum conservation, is responsible for a large reduction in cost. Our largest system studied, which samples the Brillouin zone using 64 k-points (a 4×4×4 mesh) corresponds to a canonical EOM-CCSD calculation of 768 electrons in 640 orbitals. We study eight simple main-group semiconductors and insulators, with direct singlet excitation energies in the range of 3 to 15 eV. Our predicted excitation energies exhibit a mean signed error of 0.24 eV and a mean absolute error of 0.27 eV when compared to experiment. Although this error is similar to that found for EOM-CCSD applied to molecules, it may also reflect the role of vibrational effects, which are neglected in the calculations. Our results support recently proposed revisions of experimental optical gaps for AlP and cubic BN. We furthermore calculate the energy of excitons with nonzero momentum and compare the exciton dispersion of LiF with experimental data from inelastic X-ray scattering. By calculating excitation energies under strain, we extract hydrostatic deformation potentials in order to quantify the strength of interactions between excitons and acoustic phonons. Our results indicate that coupled-cluster theory is a promising method for the accurate study of a variety of exciton phenomena in solids.
Recent optimization efforts and extensive benchmark applications are presented illustrating the accuracy and efficiency of the linear-scaling local natural orbital (LNO) coupled-cluster with single-, double,- and perturbative triple excitations [CCSD(T)] method. A composite threshold combination hierarchy (Loose, Normal, Tight, etc.) is introduced, which enables black box convergence tests and is useful to estimate the accuracy of the LNO-CCSD(T) energies with respect to CCSD(T). We also demonstrate that the complete basis set limit (CBS) of LNO-CCSD(T) energies can be reliably approached via basis set extrapolation using large basis sets including diffuse functions. Where reference CCSD(T) results are available, the mean (maximum) absolute errors of LNO-CCSD(T) reaction and intermolecular interaction energies with the default Normal threshold combination are below 0.2-0.3 (0.6-1.0) kcal/mol, while the same measures with the Tight setting are 0.1 (0.2-0.5) kcal/mol for all the tested systems including highly-complicated cases. The performance of LNO-CCSD(T) is also compared with that of other popular local CCSD(T) schemes. The exceptionally low hardware requirements of the present scheme enables the routine calculation of benchmark-quality energy differences within chemical accuracy of CCSD(T)/CBS for systems including a few hundred atoms. LNO-CCSD(T)/CBS calculations can also be performed for more than 1000 atoms with 45000 atomic orbitals using a single, 6-core CPU, about 100 GB memory, and comparable disk space.
Local correlation theories have been developed in two main flavors: (1) “direct” local correlation methods apply local approximation to the canonical equations and (2) fragment based methods reconstruct the correlation energy from a series of smaller calculations on subsystems. The present work serves two purposes. First, we investigate the relative efficiencies of the two approaches using the domain-based local pair natural orbital (DLPNO) approach as the “direct” method and the cluster in molecule (CIM) approach as the fragment based approach. Both approaches are applied in conjunction with second-order many-body perturbation theory (MP2) as well as coupled-cluster theory with single-, double- and perturbative triple excitations [CCSD(T)]. Second, we have investigated the possible merits of combining the two approaches by performing CIM calculations with DLPNO methods serving as the method of choice for performing the subsystem calculations. Our cluster-in-molecule approach is closely related to but slightly deviates from approaches in the literature since we have avoided real space cutoffs. Moreover, the neglected distant pair correlations in the previous CIM approach are considered approximately. Six very large molecules (503-2380 atoms) were studied. At both MP2 and CCSD(T) levels of theory, the CIM and DLPNO methods show similar efficiency. However, DLPNO methods are more accurate for 3-dimensional systems. While we have found only little incentive for the combination of CIM with DLPNO-MP2, the situation is different for CIM-DLPNO-CCSD(T). This combination is attractive because (1) the better parallelization opportunities offered by CIM; (2) the methodology is less memory intensive than the genuine DLPNO-CCSD(T) method and, hence, allows for large calculations on more modest hardware; and (3) the methodology is applicable and efficient in the frequently met cases, where the largest subsystem calculation is too large for the canonical CCSD(T) method.
We introduce a mixed density fitting scheme that uses both a Gaussian and a plane-wave fitting basis to accurately evaluate electron repulsion integrals in crystalline systems. We use this scheme to enable efficient all-electron Gaussian based periodic density functional and Hartree-Fock calculations.
The accurate calculation of intermolecular interactions is important to our understanding of properties in large molecular systems. The high computational cost of the current “gold standard” method, coupled cluster with singles and doubles and perturbative triples (CCSD(T), limits its application to small- to medium-sized systems. Second-order Møller–Plesset perturbation (MP2) theory is a cheaper alternative for larger systems, although at the expense of its decreased accuracy, especially when treating van der Waals complexes. In this study, a new modification of the spin-component scaled MP2 method was proposed for a wide range of intermolecular complexes including two well-known datasets, S22 and S66, and a large dataset of ionic liquids consisting of 174 single ion pairs, IL174. It was found that the spin ratio, ϵΔs=EINTOSEINTSS, calculated as the ratio of the opposite-spin component to the same-spin component of the interactioncorrelation energy fell in the range of 0.1 and 1.6, in contrast to the range of 3–4 usually observed for the ratio of absolute correlation energy, ϵs=EOSESS, in individual molecules. Scaled coefficients were found to become negative when the spin ratio fell in close proximity to 1.0, and therefore, the studied intermolecular complexes were divided into two groups: (1) complexes with ϵΔs< 1 and (2) complexes with ϵΔs≥ 1. A separate set of coefficients was obtained for both groups. Exclusion of counterpoise correction during scaling was found to produce superior results due to decreased error. Among a series of Dunning’s basis sets, cc-pVTZ and cc-pVQZ were found to be the best performing ones, with a mean absolute error of 1.4 kJ mol⁻¹ and maximum errors below 6.2 kJ mol⁻¹. The new modification, spin-ratio scaled second-order Møller–Plesset perturbation, treats both dispersion-driven and hydrogen-bonded complexes equally well, thus validating its robustness with respect to the interaction type ranging from ionic to neutral species at minimal computational cost.
We present the results of Gaussian-based ground-state and excited-state equation-of-motion coupled-cluster theory with single and double excitations for three-dimensional solids. We focus on diamond and silicon, which are paradigmatic covalent semiconductors. In addition to ground-state properties (the lattice constant, bulk modulus, and cohesive energy), we compute the quasiparticle band structure and band gap. We sample the Brillouin zone with up to 64 k-points using norm-conserving pseudopotentials and polarized double- and triple-zeta basis sets, leading to canonical coupled-cluster calculations with as many as 256 electrons in 2,176 orbitals.
The frozen-virtual natural-orbital (NO) approach, whereby the unoccupied-orbital space is constructed using a correlated density such as that from many-body perturbation theory, has proven to yield compact wave functions for determining ground-state correlation energies and associated properties, with corresponding occupation numbers providing a guide to the truncation of the virtual space. In this work this approach is tested for the first time for the calculation of higher-order response properties, particularly frequency-dependent dipole polarizabilities using coupled-cluster theory. We find that such properties are much more sensitive to the truncation of virtual space in the natural orbital (NO) basis than in the original canonical molecular orbital (CMO) basis, with truncation errors increasing linearly with respect to the number of frozen virtual NOs. The reasons behind this poor performance include the more diffuse nature of NOs with low occupation numbers as well as the reduction in sparsity of the perturbed singles amplitudes in the NO basis and the neglect of orbital response. We tested a number of approaches to improve the performance of the NO space, including the use of a field-perturbed density to define the virtual orbitals and various external-space corrections. The truncation of the CMO space, on the other hand, yields errors in coupled-cluster dipole polarizabilities of less than 2% even after removing as much as 50% of the full virtual space. We find that this positive performance of the CMO space results from a cancellation of errors due to the truncation of the unperturbed and perturbed amplitudes, as well as sparsity of the singles amplitudes. We introduce a simple criterion called a dipole amplitude to use as a threshold for truncating the CMO basis for such property calculations.
Modern linear scaling electron correlation methods are often much faster than the necessary reference Hartree-Fock calculations. We report a newly implemented Hartree-Fock (HF) program that speeds up the most time consuming step, namely the evaluation of the exchange contributions to the Fock matrix. Using localized orbitals and their sparsity, local density fitting, and atomic orbital domains, we demonstrate that the calculation of the exchange matrix scales asymptotically linearly with molecular size. The remaining parts of the HF calculation scale cubically, but become dominant only for very large molecular sizes or with many processing cores. The method is well parallelized and the speedup scales well with up to about 100 CPU cores on multiple compute nodes. The effect of the local approximations on the accuracy of computed HF and local second-order M{\o}ller-Plesset perturbation theory (LMP2) energies is systematically investigated and default values are established for the parameters that determine the domain sizes. Using these values, calculations for molecules with hundreds of atoms in combination with triple-$\zeta$ basis sets can be carried out in less than one hour, using just a few compute nodes. The method can also be used to speed up density functional (DFT) calculations with hybrid functionals that contain Hartree-Fock exchange.
We present a simple, robust and black-box approach to the implementation and use of local, periodic, atom-centered Gaussian basis functions within a plane wave code, in a computationally efficient manner. The procedure outlined is based on the representation of the Gaussians within a finite bandwidth by their underlying plane wave coefficients. The core region is handled within the projected augment wave framework, by pseudizing the Gaussian functions within a cut-off radius around each nucleus, smoothing the functions so that they are faithfully represented by a plane wave basis with only moderate kinetic energy cutoff. To mitigate the effects of the basis set superposition error and incompleteness at the mean-field level introduced by the Gaussian basis, we also propose a hybrid approach, whereby the complete occupied space is first converged within a large plane wave basis, and the Gaussian basis used to construct a complementary virtual space for the application of correlated methods. We demonstrate that these pseudized Gaussians yield compact and systematically improvable spaces with an accuracy comparable to their non-pseudized Gaussian counterparts. A key advantage of the described method is its ability to efficiently capture and describe electronic correlation effects of weakly bound and low-dimensional systems, where plane waves are not sufficiently compact or able to be truncated without unphysical artefacts. We investigate the accuracy of the pseudized Gaussians for the water dimer interaction, neon solid and water adsorption on a LiH surface, at the level of second-order M\{o}ller--Plesset perturbation theory.
In this work, a systematic infrastructure is described that formalizes concepts implicit in previous work and greatly simplifies computer implementation of reduced-scaling electronic structure methods. The key concept is sparse representation of tensors using chains of sparse maps between two index sets. Sparse map representation can be viewed as a generalization of compressed sparse row, a common representation of a sparse matrix, to tensor data. By combining few elementary operations on sparse maps (inversion, chaining, intersection, etc.), complex algorithms can be developed, illustrated here by a linear-scaling transformation of three-center Coulomb integrals based on our compact code library that implements sparse maps and operations on them. The sparsity of the three-center integrals arises from spatial locality of the basis functions and domain density fitting approximation. A novel feature of our approach is the use of differential overlap integrals computed in linear-scaling fashion for screening products of basis functions. Finally, a robust linear scaling domain based local pair natural orbital second-order Möller-Plesset (DLPNO-MP2) method is described based on the sparse map infrastructure that only depends on a minimal number of cutoff parameters that can be systematically tightened to approach 100% of the canonical MP2 correlation energy. With default truncation thresholds, DLPNO-MP2 recovers more than 99.9% of the canonical resolution of the identity MP2 (RI-MP2) energy while still showing a very early crossover with respect to the computational effort. Based on extensive benchmark calculations, relative energies are reproduced with an error of typically <0.2 kcal/mol. The efficiency of the local MP2 (LMP2) method can be drastically improved by carrying out the LMP2 iterations in a basis of pair natural orbitals. While the present work focuses on local electron correlation, it is of much broader applicability to computation with sparse tensors in quantum chemistry and beyond.
A novel algorithm, based on a hybrid Gaussian and plane waves (GPW) approach, is developed for the canonical second-order Møller-Plesset perturbation energy (MP2) of finite and extended systems. The key aspect of the method is that the electron repulsion integrals (ia|λσ) are computed by direct integration between the products of Gaussian basis functions λσ and the electrostatic potential arising from a given occupied-virtual pair density ia. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. In particular, for condensed phase systems, this scheme is highly efficient. Furthermore, our implementation has low memory requirements and displays excellent parallel scalability up to 100 000 processes. In this way, canonical MP2 calculations for condensed phase systems containing hundreds of atoms or more than 5000 basis functions can be performed within minutes, while systems up to 1000 atoms and 10 000 basis functions remain feasible. Solid LiH has been employed as a benchmark to study basis set and system size convergence. Lattice constants and cohesive energies of various molecular crystals have been studied with MP2 and double-hybrid functionals.
An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian-type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one-electron overlap and nuclear attraction, two-electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4-component relativistic Hartree-Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth-order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi-CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20-core node, we are able to achieve an average output 8.3 GB/s for C60 molecule with cc-pVTZ basis. © 2015 Wiley Periodicals, Inc.
© 2015 Wiley Periodicals, Inc.
Computation of lattice energies to an accuracy sufficient to distinguish polymorphs is a fundamental bottleneck in crystal
structure prediction. For the lattice energy of the prototypical benzene crystal, we combined the quantum chemical advances
of the last decade to attain sub-kilojoule per mole accuracy, an order-of-magnitude improvement in certainty over prior calculations
that necessitates revision of the experimental extrapolation to 0 kelvin. Our computations reveal the nature of binding by
improving on previously inaccessible or inaccurate multibody and many-electron contributions and provide revised estimates
of the effects of temperature, vibrations, and relaxation. Our demonstration raises prospects for definitive first-principles
resolution of competing polymorphs in molecular crystal structure prediction.
The second-order Møller–Plesset perturbation energy (MP2) and the Random Phase Approximation (RPA) correlation energy are increasingly popular post-Kohn–Sham correlation methods. Here, a novel algorithm based on a hybrid Gaussian and Plane Waves (GPW) approach with the resolution-of-identity (RI) approximation is developed for MP2, scaled opposite-spin MP2 (SOS-MP2), and direct-RPA (dRPA) correlation energies of finite and extended system. The key feature of the method is that the three center electron repulsion integrals (μν|P) necessary for the RI approximation are computed by direct integration between the products of Gaussian basis functions μν and the electrostatic potential arising from the RI fitting densities P. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. This scheme is highly efficient for condensed phase systems and offers a particularly easy way for parallel implementation. The RI approximation allows to speed up the MP2 energy calculations by a factor 10 to 15 compared to the canonical implementation but still requires O(N5) operations. On the other hand, the combination of RI with a Laplace approach in SOS-MP2 and an imaginary frequency integration in dRPA reduces the computational effort to O(N4) in both cases. In addition to that, our implementations have low memory requirements and display excellent parallel scalability up to tens of thousands of processes. Furthermore, exploiting graphics processing units (GPU), a further speedup by a factor 2 is observed compared to the standard only CPU implementations. In this way, RI-MP2, RI-SOS-MP2, and RI-dRPA calculations for condensed phase systems containing hundreds of atoms and thousands of basis functions can be performed within minutes employing a few hundred hybrid nodes. In order to validate the presented methods, various molecular crystals have been employed as benchmark systems to assess the performance, while solid LiH has been used to study the convergence with respect to the basis set and system size in the case of RI-MP2 and RI-dRPA.
Thresholding criteria are introduced that enforce locality of exchange interactions in Cartesian Gaussian‐based Hartree–Fock calculations. These criteria are obtained from an asymptotic form of the density matrix valid for insulating systems, and lead to a linear scaling algorithm for computation of the Hartree–Fock exchange matrix. Restricted Hartree–Fock/3‐21G calculations on a series of water clusters and polyglycine α‐helices are used to demonstrate the O(N) complexity of the algorithm, its competitiveness with standard direct self‐consistent field methods, and a systematic control of error in converged total energies.
This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.
DOI:https://doi.org/10.1103/RevModPhys.23.69
The crystal structure of deuterated benzene has been refined by single-crystal neutron diffraction analysis at 15 and 123 K. The unit-cell dimensions were also measured at 52.6 and 80 K, and the thermal-expansion coefficients at all four temperatures were calculated. The molecules have C3v symmetry with a small chair-distortion from D6h, which is possibly significant for the carbon atoms and significant for the deuterium atoms. The mean observed bond lengths at 15 K [123 K] are C-C = 1.3972 (5) A [1.3940 (9) A] (1 A = 10.-10 m = 10-1 nm); C-D = 1.0864 (7) A [1.0838 (10) A]. When corrected for molecular libration, the corresponding rest values are 1.3980 A [1.3985 A]; 1.088 A [1.088 A]. Ab initio molecular orbital calculations at the MP-2/6-31G* level gave energy-optimized bond lengths of 1.395 and 1.087 A for the isolated molecule at rest, in agreement with the corrected values from the crystal structure within the experimental errors.
A direct HF algorithm using the resolution of identity for Coulomb and exchange integrals (RI-HF) was implemented within the program system TURBOMOLE. A variational procedure for the optimisation of auxiliary functions is presented as well as optimised auxiliary basis sets for large basis sets up to Br. The accuracy of RI-HF energies and of MP2 energies based on RI-HF wave functions is demonstrated for a large test set of molecules. Accuracy of first order properties is documented for selected cases. The size dependency of the RI errors and the efficiency of the method are investigated for closo-boranes [BnHn]2−
(n=4–12).
We present a scheme for calculating the exact exchange energy in periodic solids within a Kohn–Sham or Hartree–Fock approach, which removes the need to treat the integrable singularities via an auxiliary function. In the exchange integrals, we use a modified Coulomb potential, which tends to the exact potential as the number of k points increases yet has no singularities, and which is also very simple to implement. We compare this approach to the auxiliary function scheme for diamond, graphite, and two allotropes of silicon carbide and show that it converges more rapidly with the number of wave vectors.
A previously developed method for self-consistent-field density-functional calculations involving a variational fit to the charge density is generalized to the case in which the total (electronic plus nuclear) Coulomb potential is fit. Previously the total energy E(ρ,ρ̃) was viewed as a functional of the exact ρ and fitted ρ̃ charge density. The energy expression was modified to be correct when ρ̃=ρ while at the same time allowing ρ̃ to be obtained variationally through ∂E(ρ,ρ̃)/∂ρ̃=0. Herein an expression for E(ρ,Ṽ), where Ṽ is the fit to the Coulomb potential, is derived with similar properties. In particular, Ṽ can be determined variationally through ∂E(ρ,Ṽ)/∂Ṽ=0. In various linear-combination-of-atomic-orbitals calculations on atomic neon, the superiority of variational over conventional least-squares-fitting methods is demonstrated.
We present a new method (LinK) to form the exact exchange matrix, as needed in Hartree–Fock and hybrid density functional theory calculations, with an effort capable of scaling only linearly with molecular size. It preserves the highly optimized structure of conventional direct self-consistent field (SCF) methods with only negligible prescreening overhead and does not impose predefined decay properties. Our LinK method leads to very early advantages as compared to conventional methods for systems with larger band gaps. Due to negligible screening overhead it is also competitive with conventional SCF schemes both for small molecules and systems with small band gaps. For the formation of an exchange-type matrix in coupled perturbed SCF theory our LinK method can exhibit sublinear scaling, or more precisely, independence of the computational effort from molecular size. © 1998 American Institute of Physics.
It is shown that the magnitude of the natural orbital (NO) occupation numbers of second‐order Møller–Plesset (MP2) perturbation theory can be used to select physically reasonable configuration spaces for ground state MCSCF calculations. When the MP2 NO’s are used as an initial guess for the orbitals, a second‐order Newton–Raphson MCSCF calculation is in the local region from the first iteration. Fast convergence is therefore ensured to a stationary point with orbitals of similar structure as the MP2 NO’s, thereby reducing significantly the risk of converging to undesired stationary points.