Incrementally Corrected Periodic Local MP2 Calculations: I. The Cohesive Energy of Molecular Crystals

Capturing Weak Interactions in Surface Adsorbate Systems at Coupled Cluster Accuracy: A Graph-Theoretic Molecular Fragmentation Approach Improved through Machine Learning

The accurate and efficient study of the interactions of organic matter with the surface of water is critical to a wide range of applications. For example, environmental studies have found that acidic polyfluorinated alkyl substances, especially perfluorooctanoic acid (PFOA), have spread throughout the environment and bioaccumulate into human populations residing near contaminated watersheds, leading to many systemic maladies. Thus, the study of the interactions of PFOA with water surfaces became important for the mitigation of their activity as pollutants and threats to public health. However, theoretical study of the interactions of such organic adsorbates on the surface of water, and their bulk concerted properties, often necessitates the use of ab initio methods to properly incorporate the long-range electronic properties that govern these extended systems. Notable theoretical treatments of "on-water" reactions thus far have employed hybrid DFT and semilocal DFT, but the interactions involved are weak interactions that may be best described using post-Hartree-Fock theory. Here, we aim to demonstrate the utility of a graph-theoretic approach to molecular fragmentation that accurately captures the critical "weak" interactions while maintaining an efficient ab initio treatment of the long-range periodic interactions that underpin the physics of extended systems. We apply this graph-theoretical treatment to study PFOA on the surface of water as a model system for the study of weak interactions seen in the wide range of surface interactions and reactions. The approach divides a system into a set of vertices, that are then connected through edges, faces, and higher order graph theoretic objects known as simplexes, to represent a collection of locally interacting subsystems. These subsystems are then used to construct ab initio molecular dynamics simulations and for computing multidimensional potential energy surfaces. To further improve the computational efficiency of our graph theoretic fragmentation method, we use a recently developed transfer learning protocol to construct the full system potential energy from a family of neural networks each designed to accurately model the behavior of individual simplexes. We use a unique multidimensional clustering algorithm, based on the k-means clustering methodology, to define our training space for each separate simplex. These models are used to extrapolate the energies for molecular dynamics trajectories at PFOA water interfaces, at less than one-tenth the cost as compared to a regular molecular fragmentation-based dynamics calculation with excellent agreement with couple cluster level of full system potential energies.

Can Spin-Component Scaled MP2 Achieve kJ/mol Accuracy for Cohesive Energies of Molecular Crystals?

Attaining kJ/mol accuracy in cohesive energy for molecular crystals is a persistent challenge in computational materials science. In this study, we evaluate second-order Møller-Plesset perturbation theory (MP2) and its spin-component scaled models for calculating cohesive energies for 23 molecular crystals (X23 data set). Using periodic boundary conditions and Brillouin zone sampling, we converge results to the thermodynamic and complete basis set limits, achieving an accuracy of about 2 kJ/mol (0.5 kcal/mol), which is rarely achieved in previous MP2 calculations for molecular crystals. When compared to experimental data, our results have a mean absolute error of 12.9 kJ/mol, comparable to Density Functional Theory with the PBE functional and TS dispersion correction. By separately scaling the opposite-spin and same-spin correlation energy components, using predetermined parameters, we reduce the mean absolute error to 9.5 kJ/mol. Further fine-tuning of these scaling parameters specifically for the X23 data set brings the mean absolute error down to 7.5 kJ/mol.

Discriminating and understanding molecular crystal polymorphism

Polymorph discrimination for a molecular crystal has long been a challenging task, which, nonetheless, is a major concern in the pharmaceutical industry. In this work, we have investigated polymorph discrimination on three different molecular crystals, tetrolic acid, oxalic acid, and oxalyl dihydrazide, covering both packing polymorphism and conformational polymorphism. To gain more understanding, we have performed energy decomposition analysis based on many-body expansion, and have compared the results from the XO-PBC method, that is, the eXtended ONIOM method (XO) with the periodic boundary condition (PBC), with those from some commonly used dispersion corrected density functional theory (DFT-D) methods. It is shown here that, with the XYG3 doubly hybrid functional chosen as the target high level to capture the intra- and short-range intermolecular interactions, and the periodic PBE as the basic low level to take long range interactions into account, the XO-PBC(XYG3:PBE) method not only obtains the correct experimental stability orderings, but also predicts reasonable polymorph energy ranges for all three cases. Our results have demonstrated the usefulness of the present theoretical methods, in particular XO-PBC, while highlighted the importance of a better treatment of different kinds of interactions to be beneficial to polymorph control.

Full Configuration Interaction Quantum Monte Carlo treatment of fragments embedded in a periodic mean field

We present an embedded fragment approach for high-level quantum chemical calculations on local features in periodic systems. The fragment is defined as a set of localized orbitals (occupied and virtual) corresponding to a converged periodic Hartree-Fock solution. These orbitals serve as the basis for the in-fragment post-Hartree Fock treatment. The embedding field for the fragment, consisting of the Coulomb and exchange potential from the rest of the crystal, is included in the fragment's one-electron Hamiltonian. As an application of the embedded fragment approach we investigate the performanceof full configuration interaction quantum Monte Carlo (FCIQMC) with the adaptive shift. As the orbital choice we use the natural orbitals from the distinguishable cluster method with singles and doubles. FCIQMC is a stochastic approximation to the full CI method and can be routinely applied to much larger active spaces than the latter. This makes this method especially attractive in the context of open shell defects in crystals, where fragments of adequate size can be ratherlarge. As a test case we consider dissociation of a fluorine atom from a fluorographane surface. This process poses a challenge for high-level electronic structure models as both the static and dynamic correlations are essential here. Furthermore the active space for an adequate fragment (32 electrons in 173 orbitals) is already quite large even for FCIQMC. Despite this, FCIQMC delivers accurate dissociation and total energies.

Cluster many-body expansion: A many-body expansion of the electron correlation energy about a cluster mean field reference

The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. In the past, applications of MBE to correlation energy have been unfeasible for large systems, but recent improvements to computing resources have sparked renewed interest in capturing the correlation energy using the generalized nth order Bethe-Goldstone equation. In this work, we extend this approach, originally proposed for a Slater determinant, to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean field reference TPS configuration and includes the correlation contribution of the excited TPSs using the MBE. This method, named cluster MBE (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from a RHF reference. We present numerical results for strongly correlated systems, such as the one- and two-dimensional Hubbard models and the chromium dimer. The performance of the cMBE method is also tested by partitioning the extended π space of several large π-conjugated systems, including a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 10⁶⁶ determinants for the exact FCI solution.

Data-efficient machine learning for molecular crystal structure prediction

The combination of modern machine learning (ML) approaches with high-quality data from quantum mechanical (QM) calculations can yield models with an unrivalled accuracy/cost ratio. However, such methods are ultimately limited by the computational effort required to produce the reference data. In particular, reference calculations for periodic systems with many atoms can become prohibitively expensive for higher levels of theory. This trade-off is critical in the context of organic crystal structure prediction (CSP). Here, a data-efficient ML approach would be highly desirable, since screening a huge space of possible polymorphs in a narrow energy range requires the assessment of a large number of trial structures with high accuracy. In this contribution, we present tailored Δ-ML models that allow screening a wide range of crystal candidates while adequately describing the subtle interplay between intermolecular interactions such as H-bonding and many-body dispersion effects. This is achieved by enhancing a physics-based description of long-range interactions at the density functional tight binding (DFTB) level-for which an efficient implementation is available-with a short-range ML model trained on high-quality first-principles reference data. The presented workflow is broadly applicable to different molecular materials, without the need for a single periodic calculation at the reference level of theory. We show that this even allows the use of wavefunction methods in CSP.

XO-PBC: An Accurate and Efficient Method for Molecular Crystals

In this work, we propose the XO-PBC method, which combines the eXtended ONIOM method (XO) with the periodic boundary condition (PBC) for the description of molecular crystals. XO-PBC tries to embed a finite cluster cut out from the solid into the periodic environment, making it feasible to employ advanced molecular quantum chemistry methods, which are usually prohibitively expensive for direct PBC calculations. In particular, XO-PBC utilizes the results from force calculations to design the scheme to fragment the molecule when crystals are made of large molecules and to select cluster model systems automatically consisting of dimer up to tetramer interactions for embedding. By applying an appropriate theory to each model, a satisfactory accuracy for the system under study is ensured, while a high efficiency is achieved with massively parallel computing by distributing model systems onto different processors. A comparison of the XO-PBC calculations with the conventional direct PBC calculations at the B3LYP level demonstrates its accuracy at substantially low cost for the description of molecular crystals. The usefulness of the XO-PBC method is further exemplified, showing that XO-PBC is able to predict the lattice energies of various types of molecular crystals within chemical accuracy (<4 kJ/mol) when the doubly hybrid density functional XYG3 is used as the target high level and the periodic PBE as the basic low level. The XO-PBC method provides a general protocol that brings the great predictive power of advanced electronic structure methods from molecular systems to the extended solids.

Sublimation Properties of α,ω-Diamines Revisited from First-Principles Calculations

Sublimation enthalpies of alkane-α,ω-diamines exhibit an odd-even pattern within their homologous series. First-principles calculations coupled with the quasi-harmonic approximation for crystals and with the conformation mixing model for the ideal gas are used to explain this phenomenon from the theoretical point of view. Crystals of the odd and even alkane-α,ω-diamines distinctly differ in their packing motifs. However, first-principles calculations indicate that it is a delicate interplay of the cohesive forces, phonons, molecular vibrations and conformational equilibrium which governs the odd-even pattern of the sublimation enthalpies within the homologous series. High molecular flexibility of the alkane-α,ω-diamines predetermines higher sensitivity of the computational model to the quality of the optimized geometries and relative conformational energies. Performance of high-throughput computational methods, such as the density functional tight binding (DFTB, GFN2-xTB) and the explicitly correlated dispersion-corrected Møller-Plesset perturbative method (MP2C-F12), are benchmarked against the consistent state-of-the-art calculations of conformational energies and interaction energies, respectively.

Fantasy versus reality in fragment-based quantum chemistry

Since the introduction of the fragment molecular orbital method 20 years ago, fragment-based approaches have occupied a small but growing niche in quantum chemistry. These methods decompose a large molecular system into subsystems small enough to be amenable to electronic structure calculations, following which the subsystem information is reassembled in order to approximate an otherwise intractable supersystem calculation. Fragmentation sidesteps the steep rise (with respect to system size) in the cost of ab initio calculations, replacing it with a distributed cost across numerous computer processors. Such methods are attractive, in part, because they are easily parallelizable and therefore readily amenable to exascale computing. As such, there has been hope that distributed computing might offer the proverbial "free lunch" in quantum chemistry, with the entrée being high-level calculations on very large systems. While fragment-based quantum chemistry can count many success stories, there also exists a seedy underbelly of rarely acknowledged problems. As these methods begin to mature, it is time to have a serious conversation about what they can and cannot be expected to accomplish in the near future. Both successes and challenges are highlighted in this Perspective.

Stability of Hydrogen Hydrates from Second-Order Møller-Plesset Perturbation Theory

The formation of gas hydrates and clathrates critically depends on the interaction between the host water network and the guest gas species. Density functional calculations can struggle to quantitatively capture these dispersion-type interactions. Here, we report wave function-based calculations on hydrogen hydrates that combine periodic Hartree-Fock with a localized treatment of electronic correlation. We show that local second-order Møller-Plesset perturbation theory (LMP2) reproduces the stability of the different filled-ice-like hydrates in excellent agreement with experimental data. In contrast to various dispersion-corrected density functional theory implementations, LMP2 correctly identifies the pressures needed to stabilize the C₀, C₁, and C₂ hydrates and does not find a spurious region of stability for an ice-I_h-based dihydrate. Our results suggest that LMP2 or similar approaches can provide quantitative insights into the mechanisms of formation and eventual decomposition of molecular host-guest compounds.

Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis

We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Møller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N⁴), 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 eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange 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.

Laplace transformed MP2 for three dimensional periodic materials using stochastic orbitals in the plane wave basis and correlated sampling

We present an implementation and analysis of a stochastic high performance algorithm to calculate the correlation energy of three-dimensional periodic systems in second-order Møller-Plesset perturbation theory (MP2). In particular we measure the scaling behavior of the sample variance and probe whether this stochastic approach is competitive if accuracies well below 1 meV per valence orbital are required, as it is necessary for calculations of adsorption, binding, or surface energies. The algorithm is based on the Laplace transformed MP2 (LTMP2) formulation in the plane wave basis. The time-dependent Hartree-Fock orbitals, appearing in the LTMP2 formulation, are stochastically rotated in the occupied and unoccupied Hilbert space. This avoids a full summation over all combinations of occupied and unoccupied orbitals, as inspired by the work of Neuhauser, Rabani, and Baer [J. Chem. Theory Comput. 9, 24 (2013)]. Additionally, correlated sampling is introduced, accelerating the statistical convergence significantly.

Combining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural Orbitals

In this work, we present a new pair natural orbitals (PNO)-based incremental scheme to calculate CCSD(T) and CCSD(T0) reaction, interaction, and binding energies. We perform an extensive analysis, which shows small incremental errors similar to previous non-PNO calculations. Furthermore, slight PNO errors are obtained by using T_PNO = T_TNO with appropriate values of 10^-7 to 10^-8 for reactions and 10^-8 for interaction or binding energies. The combination with the efficient MP2 focal-point approach yields chemical accuracy relative to the complete basis-set (CBS) limit. In this method, small basis sets (cc-pVDZ, def2-TZVP) for the CCSD(T) part are sufficient in case of reactions or interactions, while some larger ones (e.g., (aug)-cc-pVTZ) are necessary for molecular clusters. For these larger basis sets, we show the very high efficiency of our scheme. We obtain not only tremendous decreases of the wall times (i.e., factors >10²) due to the parallelization of the increment calculations as well as of the total times due to the application of PNOs (i.e., compared to the normal incremental scheme) but also smaller total times with respect to the standard PNO method. That way, our new method features a perfect applicability by combining an excellent accuracy with a very high efficiency as well as the accessibility to larger systems due to the separation of the full computation into several small increments.

A comparison between quantum chemistry and quantum Monte Carlo techniques for the adsorption of water on the (001) LiH surface

We present a comprehensive benchmark study of the adsorption energy of a single water molecule on the (001) LiH surface using periodic coupled cluster and quantum Monte Carlo theories. We benchmark and compare different implementations of quantum chemical wave function based theories in order to verify the reliability of the predicted adsorption energies and the employed approximations. Furthermore we compare the predicted adsorption energies to those obtained employing widely used van der Waals density-functionals. Our findings show that quantum chemical approaches are becoming a robust and reliable tool for condensed phase electronic structure calculations, providing an additional tool that can also help in potentially improving currently available van der Waals density-functionals.

State-of-the-Art Calculations of Sublimation Enthalpies for Selected Molecular Crystals and Their Computational Uncertainty

A computational methodology for calculation of sublimation enthalpies of molecular crystals from first principles is developed and validated by comparison to critically evaluated literature experimental data. Temperature-dependent sublimation enthalpies for a set of selected 22 molecular crystals in their low-temperature phases are calculated. The computational methodology consists of several building blocks based on high-level electronic structure methods of quantum chemistry and statistical thermodynamics. Ab initio methods up to the coupled clusters with iterative treatment of single and double excitations and perturbative triples correction with an estimated complete basis set description [CCSD(T)/CBS] are used to calculate the cohesive energies of crystalline phases within a fragment-based additive scheme. Density functional theory (DFT) calculations with periodic boundary conditions (PBC) coupled with the quasi-harmonic approximation are used to evaluate the thermal contributions to the enthalpy of the solid phase. The properties of the vapor phase are calculated within the ideal-gas model using the rigid-rotor harmonic-oscillator model with correction for internal rotation using a one-dimensional hindered rotor approximation and a proper treatment of the molecular rotational degrees of freedom in the vicinity of 0 K. All individual terms contributing to the sublimation enthalpy as a function of temperature are discussed and their uncertainties estimated by comparison to critically evaluated experimental data.

Molecular Dipole Moments within the Incremental Scheme Using the Domain-Specific Basis-Set Approach

We present the first implementation of the fully automated incremental scheme for CCSD unrelaxed dipole moments using the domain-specific basis-set approach. Truncation parameters are varied, and the accuracy of the method is statistically analyzed for a test set of 20 molecules. The local approximations introduce small errors at second order and negligible ones at third order. For a third-order incremental CCSD expansion with a CC2 error correction, a cc-pVDZ/SV domain-specific basis set (tmain = 3.5 Bohr), and the truncation parameter f = 30 Bohr, we obtain a mean error of 0.00 mau (-0.20 mau) and a standard deviation of 1.95 mau (2.17 mau) for the total dipole moments (Cartesian components of the dipole vectors). By analyzing incremental CCSD energies, we demonstrate that the MP2 and CC2 error correction schemes are an exclusive correction for the domain-specific basis-set error. Our implementation of the incremental scheme provides fully automated computations of highly accurate dipole moments at reduced computational cost and is fully parallelized in terms of the calculation of the increments. Therefore, one can utilize the incremental scheme, on the same hardware, to extend the basis set in comparison to standard CCSD and thus obtain a better total accuracy.

CCSD(T)/CBS fragment-based calculations of lattice energy of molecular crystals

A comparative study of the lattice energy calculations for a data set of 25 molecular crystals is performed using an additive scheme based on the individual energies of up to four-body interactions calculated using the coupled clusters with iterative treatment of single and double excitations and perturbative triples correction (CCSD(T)) with an estimated complete basis set (CBS) description. The CCSD(T)/CBS values on lattice energies are used to estimate sublimation enthalpies which are compared with critically assessed and thermodynamically consistent experimental values. The average absolute percentage deviation of calculated sublimation enthalpies from experimental values amounts to 13% (corresponding to 4.8 kJ mol(-1) on absolute scale) with unbiased distribution of positive to negative deviations. As pair interaction energies present a dominant contribution to the lattice energy and CCSD(T)/CBS calculations still remain computationally costly, benchmark calculations of pair interaction energies defined by crystal parameters involving 17 levels of theory, including recently developed methods with local and explicit treatment of electronic correlation, such as LCC and LCC-F12, are also presented. Locally and explicitly correlated methods are found to be computationally effective and reliable methods enabling the application of fragment-based methods for larger systems.

Modeling Polymorphic Molecular Crystals with Electronic Structure Theory

Interest in molecular crystals has grown thanks to their relevance to pharmaceuticals, organic semiconductor materials, foods, and many other applications. Electronic structure methods have become an increasingly important tool for modeling molecular crystals and polymorphism. This article reviews electronic structure techniques used to model molecular crystals, including periodic density functional theory, periodic second-order Møller-Plesset perturbation theory, fragment-based electronic structure methods, and diffusion Monte Carlo. It also discusses the use of these models for predicting a variety of crystal properties that are relevant to the study of polymorphism, including lattice energies, structures, crystal structure prediction, polymorphism, phase diagrams, vibrational spectroscopies, and nuclear magnetic resonance spectroscopy. Finally, tools for analyzing crystal structures and intermolecular interactions are briefly discussed.

Toward an Accurate Estimate of the Exfoliation Energy of Black Phosphorus: A Periodic Quantum Chemical Approach

The black phosphorus (black-P) crystal is formed of covalently bound layers of phosphorene stacked together by weak van der Waals interactions. An experimental measurement of the exfoliation energy of black-P is not available presently, making theoretical studies the most important source of information for the optimization of phosphorene production. Here, we provide an accurate estimate of the exfoliation energy of black-P on the basis of multilevel quantum chemical calculations, which include the periodic local Møller-Plesset perturbation theory of second order, augmented by higher-order corrections, which are evaluated with finite clusters mimicking the crystal. Very similar results are also obtained by density functional theory with the D3-version of Grimme's empirical dispersion correction. Our estimate of the exfoliation energy for black-P of -151 meV/atom is substantially larger than that of graphite, suggesting the need for different strategies to generate isolated layers for these two systems.

Range-separated double-hybrid density-functional theory applied to periodic systems

Quantum chemistry methods exploiting density-functional approximations for short-range electron-electron interactions and second-order Møller-Plesset (MP2) perturbation theory for long-range electron-electron interactions have been implemented for periodic systems using Gaussian-type basis functions and the local correlation framework. The performance of these range-separated double hybrids has been benchmarked on a significant set of systems including rare-gas, molecular, ionic, and covalent crystals. The use of spin-component-scaled MP2 for the long-range part has been tested as well. The results show that the value of μ = 0.5 bohr(-1) for the range-separation parameter usually used for molecular systems is also a reasonable choice for solids. Overall, these range-separated double hybrids provide a good accuracy for binding energies using basis sets of moderate sizes such as cc-pVDZ and aug-cc-pVDZ.

Method of increments for the halogen molecular crystals: Cl, Br, and I

Method of increments (MI) calculations reveal the n-body correlation contributions to binding in solid chlorine, bromine, and iodine. Secondary binding contributions as well as d-correlation energies are estimated and compared between each solid halogen. We illustrate that binding is entirely determined by two-body correlation effects, which account for >80% of the total correlation energy. One-body, three-body, and exchange contributions are repulsive. Using density-fitting (DF) local coupled-cluster singles, doubles, and perturbative triples for incremental calculations, we obtain excellent agreement with the experimental cohesive energies. MI results from DF local second-order Møller-Plesset perturbation (LMP2) yield considerably over-bound cohesive energies. Comparative calculations with density functional theory and periodic LMP2 method are also shown to be less accurate for the solid halogens.

Ab initio determination of the crystalline benzene lattice energy to sub-kilojoule/mole accuracy

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.

Diffraction of helium on MgO(100) surface calculated from first-principles

In this work we simulate the diffraction peak intensities of He beams scattered on the MgO(100) surface using hierarchical protocol, based on periodic and finite-cluster quantum-chemical calculations.